[] []
INTERNATIONAL SOCIETY FOR THE INTERDISCIPLINARY STUDY OF SYMMETRY (ISIS-SYMMETRY) Prestdent D~nes Nagy, Instttute of Applied Physics, University of Tsukuba, Tsukuba Scmnce City 305, Japan;
[email protected] [Geometry. Crystallography, History of Science and Technology, Lmguishcs] Chaimtun of the Advisory Board Arthur L Loeb, Carpenter Center for tile Visual Arts, Harvard University, Cambridge, MA 02138, U S.A ;
[email protected] edu [Visual Arts, Choreography, Music, Crystallography] Honorary Presidents Konslantin V. Frolov, Russian Academy of Sciences, Mechanical Engineering Research Institute, ul Griboedova 4, 101830 Moscow, Russia [Mecbamcnl Engineering] and Yuval Ne’eman, R. and B. Suckler Faculty of Exact Sciences, TeI-Aviv University, TeI-Aviv, Israel 69978;
[email protected]. and Center for Particle Physics, Umverstty of Texas, Austin, TX 78712, U.S.A. [Pamcle Physics]
Secretary General Gy6rgy Darvas, Symmetrion - The lnsmute for Advanced Symmetry Studies, and Instttute for Research Organisation of the Hungarian Academy of Scmnces, N,"idor u. 18, P O. Box 994, Budapest, I-1-1245 Hungary;
[email protected] [Philosophy of Science, Theoret,c.al Physics] Secretaries Marlha Pardavi-llorvalh, Department of Electrical Engineering and Computer Science, The George Washington University, Washington, D.C. 20052, U.S.A.;
[email protected] [Materials Science] and Janusz Rebielak, Department of Architecture, Faculty of Building Structures, Tec.hnical University of Wrodaw, ul. B. Prusa 53/55, PL-50-317 Wroc~w, Poland;
[email protected] pl [Architecture, Building Engineering]
Mcmbcr~ of the Board Fred L Bookslein, Center for Human Growth and Development, Umversity of Michigan, 300 North lngalls Building, Ann Arbor, Mich*gan 48109-0406, U.S.A.;
[email protected] [Biology, Statistics] Siglind Bruhn, Department of Musmology and Inst,tutc for the Humanities, University of M~chigan; 1308 Broadway, Ann Arbor, 48105, D S.A.;
[email protected] [Music, Semiot,cs, Literature] Belly Collings, 1991 Hillside Drive, Columbus, OH 43221, U.S.A.;
[email protected] [Sculpture} Liz Edward% Visual Arts Program, Academy of the Arts, Queensland Umv.ersity of Teclmology, P.O. Box 362, Redhill, Brisbane, QLD 4059, Australia;
[email protected] [Fme Art, }-hstory of Art] Ruslan I. Koslov, Faculty of Geology and Prospect rag, University of Mining and Geology "St. Ivan Rilsk,", Sofia 1100, Bulgaria;
[email protected] [Geology, M|nearology, History of Science] Sergei V. Pelukhov, Mechanical Engineering Research Institute, Russian Academy of Sciences, ul. G rlboedova 4, 10 t830 Moscow, Russia,
[email protected] [Mechanical Engineering, Biomechanics] hi formation Network Advisor Mihfily Szoboszlai, Faculty of Architecture, Techmcal Univers,ty of Budapest, Mfiegyetem rkp. 3, Budapest, H-I 111 Hungary;
[email protected] [Architecture, Geometry, Computer Aided Architectural Design] Aasociate Editor (of Symmetry: Culture attd Science) John llosack, Department of Mathematics and Computing Science, University of the South Pacific, P.O. Box 1168, Suva, Fij~; j.hosack@usp ac.fj [Mathematical Analysis, Phdosophy] See theAd~isoo, goa,,d inside back cover
Symmetry: The Quarterly of the International Society for the Interdisciplinary Study of Symmetry (iSIS-Symmetry) Temporarily edited by the curatory of the International Symmetry Foundation ’ Published by the International Symmetry Foundation Volume 8, Number 2, 113-224, 1997
SPECIAL ISSUE: SYMMETRY AND INFORMATION Guest Editors: Koichiro Matsuno and Pedro C. Marijufin
CONTENTS PREFACE
115
EDITORIAL
117
SYMMETRY: CULTURE & SCIENCE ¯ The arrow of mind: Symmetry-breaking, information, and biological complexity, James Barham ¯ Generation of fractals from incursive automata, digital diffusion and wave equation systems, DanielM. Dubois ¯ Information processing and symmetry-breaking in memory evolutive systems, Andr4e C. Ehresmann, Jean-Paul Vanbremeersch ¯ Con.stancy, uniformity and symmetry of living systems: the computational functions of morphological invariance, Avshalom C. Elitzur ¯ Information processing in biosystems: Quantum mechanical background and relation to symmetry-breaking, Abir U. lgamberdiev ¯ Is symmetry informative.9 John E. Gray, Andrew Vogt ¯ Symmetry and symmetry-breaking between suppliers and consumers in natural ecosystems, Yoshinori Takahara, Nobual# Ono
119 133 157 177 193 207 215
SYMMETRY: CULTURE AND SCIENCE is edited by the Board of the International Society for the Interdisciplinary Study of Symmetry (ISIS-Symmetry), and published by the International Symmetry Foundation. The views expressed are those of individual authors, and not necessarily shared by the Society or the editors.
Curatory of the International Symmetry Foundation: Szaniszl6 Brrczi (head), J~nos F. Brhm, Gyrrgy Darvas, S~ndor Kabai
Any correspondence should be addressed to the International Symmetry Foundation c/o Symmetrion P.O. Box 994, Budapest, H-1245 Hungary Phone: 36-1-331-8326 Fax: 36-1-331-3161 E-mail: sym@,freemail.hu, bercziszani@,ludens.elte.hu,
[email protected] http://isis-symmetry.org
Subscription: Annual membership fee of the Society: Benefactors, US$780.00; Ordinary Members, US$78.00 (includes the subscription to the quarterly), Student Members, US$63.00 (includes the subscription to the quarterly); Institutional Members, please contact the Secretary General. Annual subscription rate for non-members: US$100.00. Make checks payable to 1SIS-Symmetry and mail to G. Darvas, Secretary General, or transfer to the following account number: ISIS-Symmetry, International Symmetry Foundation, 504-00004-2100-4015, Hungarian Foreign Trade Bank, Budapest, Szt. I stvfin trr 11, H-1821 Hungary (Telex: Hungary 22-6941 extr-h; Swift MKKB HU HB).
ISIS-Symmetry. No part of tiffs publication may be reproduced without written permission from the Society, ISSN 0865-4824
Cover layout: Gunter Schmitz Images on the front and back cover: Student works from the Basic Design Studio by William S. Huff (Department of Architecture, SUNY at Buffalo, N.Y.) Ambigram on the back cover: Douglas R. Hofstadter
Symmetry: Culture and Sc:ence Vol. 8, No.2, 115, 1997
PREFACE This issue continues to publish thematic papers on SYMMETRY AND INFORMATION started in the Vol. 7 (1996), No. 3. Thanks should be expressed to the two guest editors, Koichiro Matsuno and Pedro C. Marijuan, who not only edited the issue, but also produced it technically full prepared and printed in a camera ready form for reproduction. Due to the delay in the production, mentioned in the previous issues, a part of the papers were published in the periodical BioSystems in the meantime. Now these four papers are reprinted here with the kind permission of the publisher, Elsevier. The original source of the concerned papers: Dubois, D. M. (1997) Generation of fractals from incursive automata, digital diffusion and wave equation systems, BioSystems, 43, 97-114. Ehresmann, A. C., Vanbremeersch, J. P. (1997) Information processing and symmetrybreaking in memory evolutive systems, BioSystems, 43, 25-40. Elitzur, A. C. (1997) Constancy, uniformity and sym .metry of living systems: The computational functions of morphological invanance, BioSystems, 43, 41-53. Gray, .1. E., Vogt, A. (1997) Is symmetry informative? BioSystems, 43, 55-61.
Symmetry" and Science Vol.Culture 8, No. 2, 117, 1997
EDITORIAL
SYMMETRY AND INFORMATION: AN UNDERLYING THEME This is the second special issue on Symmetry and Information in series. The underlying theme is our persistent endeavor for elucidating the connection between symmetry and information. Symmetry has been a time-honored concept that enables us to perceive the external object in a most respectful and trustworthy manner. It is quite Platonic in its tradition. In contrast, information presumes a contrast between the knower and the known or between being and becoming. It is Aristotelian. The dichotomy between symmetry and information in fact reminds us of a resurrection of the old and historically repeated confrontation between Platonic idealism and Aristotelian empiricism. Of course, there is no easy answer. Nonetheless, this has been serving as an inexhaustible source of our tireless intellectual activities. One attempt for overcoming the Platonic-Aristotelian dichotomy has been Kantian transcendentalism. Although it provides a profound and reliable philosophy on the matter of symmetry and information, the Kantian perspective does not squarely face the issues of science being practiced at the turn of the twenty first century. At issue is how one can ground our serious discussion on the relationship between symmetry and information on a concrete scientific basis in the contemporary cultural tradition. A likely candidate for groping in the dark could be a notion of time. Newtonian absolute time is still quite influential but metaphysical at best, because Newton himself declared it to be absolute as being relative to nothing. Kantian time is specific to the transcendental ego and anthropomorphic at best. This observation encourages us to take a new perspective towards the issue of time. The authors invited to the present special issue touch upon various aspects of the issue of time. Although each paper discusses the dichotomy of symmetry and information in its own light, a common thread is to make the notion of time empirical beyond simply being metaphysical or philosophical. These efforts may open a challenging opportunity for making a newer and more fruitful synthesis of symmetry and information. May we be on a right track!
Koichiro Matsuno (Nagaoka University of Technology) Pedro C. Marijuan (CPS, University of Zaragoza)
Symmetry: Culture and Science Vol. 8, No 2, 119-132, 1997
SYMMETRY." CULTURE AND SCIENCE
THE ARROW OF MIND: SYMMETRY-BREAKING, INFORMATION, AND BIOLOGICAL COMPLEXITY James Barham Address: 404 St Joseph St, Apt. 2, Lancaster, PA 17603, U S.A.
Abstract: There are two senses in which symmetry-breaking produces information. In the first sense, we human beings interpret spatial symmetry-breaking in a physical system, whether inorganic or organic, as an increase in its information content (where the information is meaningful for us). This provides an information-theoretic measure of complexity, but one which is extrinsic or subjective, because relative to the human observer. In the second sense, temporal symmetry-breaking in a biological system may be interpreted as an increase in its information capacity (where the information is meaningful for the system itselJ). A dynamical model of the meaning of information is sketched which would provide an intrinsic or objective measure of biological complexity.
1. INTRODUCTION Information is one of the most vexed concepts on the contemporary intellectual scene. On the one hand, if we restrict it to its original, syntactic, use as a measure of the carrying capacity of a communications channel (Shannon & Weaver, 1963), then it is mathematically rigorous but hardly relevant to theoretical biology, cognitive science, or epistemology. On the other hand, if we relax our usage so as to encompass its semantic content, or meaning, then the concept of information becomes relevant to wider philosophical and scientific concerns, but only at the price of remaining ill-defined and largely mysterious. It is due to equivocation between these two usages that information has come to assume its aura of a fundamental physical principle on a par with matter and energy. However, as Ho (1993; p. 96) has remarked, "’information’ is not something separate from energy and organization"; rather, it is a patterned matter or energy structure which acquires meaning by virtue of the role it plays in the organization of functional action (Barham, in press). The concept of complexity is scarcely less contentious than that of information. On the one hand, information theory has seemed to provide us with various ways of quantifying the complexity of physical objects, including "algorithmic complexity" (def’med as the length in bits of the shortest computer program capable of specifying an object) (Chaitin, 1990), "logical depth" (the number of operations, or logical steps, actually executed by such a program) (Bennett, 1988), and other similar measures (Wackerbauer et al., 1994). On the other hand, with respect to the class of physical objects which would appear to be the most complex of all -- namely, biological organisms--it has been claimed that the notion of complexity is an anthropomorphic bias lacking objective validity (Gould,
120
J. BARHAM
1989; McShea, 1991). The notion of symmetry-breaking provides a link between the concepts of information and complexity. This is fortunate in two ways. First, symmetry and symmetry-breaking are mathematically rigorous concepts which cannot help but clarify the relatively vague and confused notions of semantic information and complexity. Second, the essence of spatial symmetry-breaking is the creation of discernible differences, or structure, in the world where previously none existed, thus defining a "before" and an "after," and hence throwing light on the time irreversible nature of most real processes (aka, the "arrow of time"). This is useful because the very idea of complexity would seem to contain a directional component, as well; that is, we must assume that the different sorts of things in the world which we call "simple" and "complex" have come into being at different times, with the former preceding the latter. A world in which prokaryotic cells led to eukaryotic cells, on to plasmodial colonies, and ultimately to metazoans, makes intuitive sense to us, whereas a world in which this sequence ran backwards would appear absurd. However, the currently dominant school of thought in evolutionary biology, neo-Darwinism, either dismisses this directional component as subjective (as mentioned above), or else, if it recognizes the phenomenon as real, has only ad-hoc explanations to offer (Bonner, 1988). The notion of symmetry-breaking potentially offers a deeper and more unified explanation of this fundamental aspect of the evolutionary process. In this paper, I will discuss some of the ways in which symmetry-breaking, information, and biological complexity are related to each another. In addition, using conceptual tools borrowed from nonlinear dynamics, I will construct a naturalistic model of semantic information. Finally, I will show how this way of understanding information suggests an objective metric of biological complexity.
2. SYMMETRY-BREAKING AND INFORMATION On the surface, the link between symmetry-breaking and information would appear to be straightforward and unproblematic. The essence of the notion of symmetry-breaking is the creation of differences, or distinctions, where there were none before. Thus, if a featureless sphere is rotated about an arbitrary axis any number of degrees you like, it is indistinguishable from the unrotated object. As soon as we specify a great circle on the sphere as a benchmark, however, the symmetry is broken; now only a certain subset of rotations (namely, those about the diameter perpendicular to the plane of the great circle) will leave the object invariant. As a concrete example, consider an unfertilized ovum, on the one hand, and a newly-formed zygote, on the other. The former cell is spherically-symmetrical insofar as gross anatomical features are concerned.1 This means it would be impossible to tell it had been rotated, unless it were marked in some way. At fertilization, however, the diameter formed by the entry point of the sperm cell and the point opposite defines the dorsal-ventral axis around which the process of gastrulation will largely unfold (Gilbert, 1988; pp. 124-25). Following this event, rotations can be discerned by measuring the displacement of this axis. Prior to symmetry-breaking, the ~Needless to say, the unfertilized ovum Is itself highly structured. However, although the future germ layers of the zygote can in fact be predicted from the orientation of the so-called "animal-vegetal" polarity of the ovum, the eventual dorsal-ventral axis of the zygote is not determined untd the moment of fertilization (Gilbert, 1988; pp. 124-125). Therefore, the account in the text is an accurate, albeit ~dealized, description of the gross anatomical symmetry of the embryo
THE ARROW OF MIND
121
ovum had no discernible gross-anatomical features: the concepts "front" and "back" simply did not apply to it. After symmetry-breaking, the zygote has acquired parts which can then be distinguished by these labels. Now, information in the Shannonian sense consists precisely in the specification of a particular set of elements out of a probability space consisting of all the permutations of such sets. On this view, then, the information content of an object is just the number of distinguishable parts it contains. Thus, the close connection between syntactic information and symmetry-breaking is readily apparent. On the one hand, spatial symmetrybreaking is the physical process which produces heterogeneity and differentiation in the world. On the other hand, information is a measure of the number of ways in which a region of the world can be subdivided into parts. In short, symmetry-breaking creates information in the Shannonian sense, and the Shannonian information content of an object is a measure of the reduction in its degrees of freedom. It would appear that here we have penetrated to a deep truth about the nature of things. Thanks to this seemingly intimate link between symmetry and information, it appears possible to proceed directly from cosmogenesis to cognition, thus overleaping the chasm between matter and mind at a single bound. Indeed, it is quite common nowadays to encounter in scholarly writings the notion that information is a fundamental physical concept (e.g., Wicken, 1987), that organic evolution is a computational process (Dennett, 1995), and even that the universe as a whole is a gigantic cosmic computer (Wheeler, 1990). However, despite the conventional wisdom, these ideas are in fact antinaturalistic; their superficial appeal is due to equivocation between the syntactic and semantic senses of the word "information." As we have seen, the notion of information implies an act (the nature of the agent usually being glossed over) of distinguishing one part from another, of specifying one pattern as opposed to other possible patterns, of preferring one thing over another. All of this is quite foreign to the traditional mechanistic view of nature. In short, the world cannot be reduced to pure syntax, because the very idea of syntax already presupposes semantics -- i.e., a cognitive agent for whom the result of the syntactic operation is meaningful. Therefore, the notion of the cosmic computer implies that mind precedes matter -- hence, it comes to imply the existence of a supernatural Hacker who interprets the results of the computations (Rosen, 1991). But if the notion of the cosmic computer is anti-naturalistic, then how can the seemingly close connection between symmetrybreaking and information be understood in a naturalistic way? Shannonian information theory is no help, because it simply ignores the problem of meaning, as Shannon himself was the first to admit (Shannon & Weaver, 1963; p. 31). What is needed, rather, is an objective, naturalistic understanding of the meaning of information on the basis of fundamental physical theory.
3. SYMMETRY-BREAKING AND BIOLOGICAL FUNCTION The first step toward this goal is to clarify the reason why information cannot be simply equated with structuration due to symmetry-breaking: it is not because (as some socalled "postmodern" thinkers pretend) there is no objective structure to the world absent the human knower, or that the human mind somehow imposes its own structures on the external world (Barham, 1995). Rather, it is because the existence of structure is independent of any cognitive agent, whereas the existence of information about that structure depends on the prior existence of a cognitive agent for which the information is mean-
122
J BARHAM
ingful. Information is simply not the sort of thing that can exist in the absence of living things. True, one might speak of a particular structure as having an informational potential for a certain kind of organism even if such an organism did not actually exist. But even in that case, the potential would inhere in the structure by virtue of the ability of a possible organism to actualize it, not as a fundamental physical property of the structure per se. However you look at it, information is a feature of the world which derives from, and exists only in relation to, living things. With this distinction in mind, then, it becomes clear that our goal is to explain the relationship between the structure of the world and the knowledge of it possessed by organisms, for that is what a physical theory of the meaning of information would amount to. In the first place, it may be noted that the tendency of matter to arrange itself into one configuration, or structure, rather than another, is a matter of degree. There is clearly a sense in which it is correct to say that inanimate objects -- from crystals to planets -have axes of symmetry and thus also have "preferred" orientations in space. The axes of symmetry of the orbiting earth or of a diamond are perfectly real and objective -- they owe nothing to human categorization and would be exactly the same if human beings had never existed. In comparison with organisms, of course, this sort of "preference" is very attenuated. The question is, What is it that chiefly distinguishes the higher degree of preference exhibited by living things in comparison with inanimate objects? The answer would appear to be that it is the nature of the symmetry-breaking involved. In the case of inanimate objects, structuration occurs primarily with respect to the three spatial dimensions. In the case of living things, it occurs above all with respect to time. The dynamical evolution, or motion, of most inanimate objects is remarkably uniform, which is to say that structuration due to symmetry-breaking along the time dimension is of a very low order for this class of objects. For example, the motions of the planets in their orbits may be modeled to a very high degree of approximation by linear equations producing one-to-one, time-symmetric mappings. This fact of celestial mechanics is, of course, what allows us to produce ephemerides that are valid for a period of time on the order of millennia. It is true that the apparent temporal symmetry of the planetary motions is an idealization which breaks down over longer periods of time; nevertheless, in comparison with the dynamics of the living state, temporal symmetry-breaking in the dynamics of most inanimate objects is of a very low order of magnitude. Thus, the "preferences" exhibited by inorganic systems are, for the most part, a matter of spatial rather than temporal structuration. Matters stand very differently with living things. Here, it is dynamical structure -- that is, temporal symmetry-breaking- that is of the essence. As Yates has observed (1993, p. 190-191): [B]iological order is unlike order in physics or mathematics. Biological order is remarkable not for its degree, but for its specialness. It is a functional order that serves to correlate relevant biochemical and physiological events; but it is difficult to formulate mathematically the condition of invariance that must be fulfilled, which can be stated broadly as the need to keep the characteristics of one species constant during all the transformations that give rise to biochemical events during development. In contrast, in the case of a crystal lattice the spatial order is best expressed by the presence of correlations among the positions of equal atoms, and this order is further characterized by a condition of invariance toward the space transformations allowed by the symmetry class of the lattice in question. In functional order the correlations must be formed among the times at which different
THE ARROW OF MIND:
123
events occur." (emphasis in original) While inanimate objects may seem to have preferred configurations in space, they are indifferent for the most part to the passage of time; one moment, for them, is much like any other. It seems a matter of indifference to them, so to speak, whether they continue to exist as a whole with a collective identity, or whether their coherence is destroyed and their parts are scattered. Living things, on the other hand, show a distinct preference for continued existence as organized wholes. Indeed, the chief characteristic of the living state is its striving to avoid disintegration. All living things actively resist the inexorable trend towards thermodynamic equilibrium in accordance with the Second Law through the cunning exploitation of physics and chemistry. Perhaps the most fundamental physical principle which life employs to win its temporary victory over the Second Law is symmetry-breaking in time. In thermodynamically-isolated systems, local energy potentials will be smoothed out as the system relaxes to equilibrium in accordance with the Second Law. In systems thermodynamically open to energy and material flows in which the flux rate is greater than the thermal relaxation time, a steady state will be established away from equilibrium so long as the flow continues. In such nonequilibrium systems, it has been shown (Matsuno, 1989; Morowitz, 1979; Prigogine, 1980; Swenson, 1992) that global, coherent cycling is the expected result, since under these conditions the spontaneous creation of macroscopic structures dissipates energy more rapidly than thermal relaxation can do (hence the name "dissipative structures"). This is the fundamental physical reason why cycles, or oscillations, are discernible in almost all functional activity (Glass & Mackey, 1988; Lloyd & Rossi, 1992; Winfree, 1990). Although all such functional cycles are embedded within densely-nested hierarchical networks of other functions -- from individual enzymes to metabolic networks to organ systems -- nevertheless, at any given level, each one enjoys a limited degree of coherence and autonomy. This suggests that we might model fimctional activity in general by means of the notion of a nonlinear oscillator and its associated phase-space attractor. Instead of a one-to-one mapping of initial states onto final states, we have a many-to-one mapping of a large ensemble of possible initial states of the system onto a single final state (the "goal state"). This set of equifinal phase-space trajectories is referred to as the system’s "basin of attraction"; the goal state is its "attractor." A nonlinear attractor is a mathematical object whose property of equifinality nicely captures the goal-directedness, or teleonomy, that is the essential feature of the dynamical evolution of living things (Delattre, 1986). Once we have taken this step, then the notion of the "success" of a functional action may be identified with the preservation of the dynamical stability of its associated nonlinear oscillator (and hence its continued cycling). From such a viewpoint, it is clear that, with respect to the dynamical evolution of biofunctions, all times are no longer equal. The activity of each of an organism’s subsystems, or biofunctions, may be viewed as a diachronic structure created by symmetrybreaking along the time dimension in the same way that an ordinary object is a synchronic structure created by symmetry-breaking in three-dimensional space. The temporal symmetry is broken in the sense that the dynamical evolution of a biofunction is no longer invariant under time reversal, since its final state cannot be traced back to a unique initial state. How is it possible for mere molecules, through "signalling" and "recognition," to "regulate" and "coordinate" chemical events in time in such a way as to keep the Second Law temporarily at bay in order to make life possible? In spite of the staggering ad-
124
J. BARHAM
vances in molecular biology over the past half century, many aspects of this question remain unanswered. However, it does appear that the strong preferences, or goaldirectedness, exhibited by living things must be accompanied by a power of discrimination of external conditions in order to be successful. In Section 4, I will sketch a dynamical model of the connection between biological function and cognition. From this model may be derived a physical interpretation of information that is intrinsically meaningful to organisms themselves (Section 5). Finally, in Section 6 I will attempt to show how this interpretation of semantic information can be used as the basis for an objective metric of biological complexity.
4. A DYNAMICAL MODEL OF THE MEANING OF INFORMATION The resistance to the Second Law which is the chief mark of the living state translates, in physical terms, into a partial autonomy with respect to local energy potentials. More precisely, biological systems are able to vary their rate of energy consumption independently of variations in local gradients thanks to their "on-board" energy supply in the form of ATP and related compounds. This ability allows living things to avoid slavish dependence on their surround and to resist disintegration when external conditions deteriorate by actively seeking out more favorable conditions, either in time (by slowing down their metabolism in order to conserve on-board supplies) or in space (by speeding it up in order to move about). This limited independence of living things from local energy potentials has a crucial corollary: a biofunction must be able to distinguish between those conditions external to itself which will support its continued oscillation, and those which will not. In other words, energy autonomy -- and hence life itself-- implies a degree of cognition. How is it possible for a biofunction, conceived of as a nonlinear oscillator, to acquire this ability to distinguish between those external conditions which will support its continued oscillation and those which will not, and to coordinate its functional action accordingly so that its dynamical stability will be preserved? The answer seems to be that living systems achieve limited energy autonomy from local, high-energy potentials by becoming sensitive to nonlocal, low-energy fluxes. As Swenson has put it (1992; pp. 140-141): ...whereas in nonliving systems the dynamics are governed by local field potentials (with dimensions of mass, length, and time, viz, "mass-based" fields), the dynamics of the living are governed by nonlocal potentials linked together through observables with dimensions of length and time (kinematic or information fields). Thus, the chief difference between an organism and an inorganic nonlinear oscillator (like a hurricane) is the organism’s ability to use low-energy fluxes fxom a distal source in order to detect high-energy potentials before it becomes thermodynamically coupled with them. In order to explain this seemingly mysterious ability, I have proposed (Barham, 1990) that we postulate a fundamental differentiation within all biofunctions between: (1) a high-energy interaction of the oscillator with a set of constraints in its surround (the functional action as a whole); and (2) a low-energy interaction of a subsystem of the oscillator with a second set of constraints which are highly correlated with the first set. Note that this postulated differentiation implies the existence of a component within every biofunction capable of undergoing the low-energy interaction; I have pro-
THE ARROW OF MIND.
125
posed that this subsystem be called the "epistemon." (In essence, the epistemon is a generalization of the notion of a sense organ.) This conjecture leads naturally to a tetradic model of perception and action, as follows. (See Figure I)
SURROUND High-Energy Enviromental Constraints
Low-Energy Constraints (Information)
Epistemon
SYSTEM
Figure 1: A biological function modeled as a nonlinear oscillator with a low-energy trigger
First, we posit an existing biofunction (a) coupled to its surround in such a way that its functional action is ordinarily successful (i.e., a dynamically-stable nonlinear oscillator). Next, we pick out those high-energy constraints (b) in the surround with which (a) ordinarily interacts. Then, we identify a second set of environmental constraints (c) which
126
J. BARHAM
are highly correlated with the first set, but which are lower in energy. Finally, we have the postulated universal subsystem (d) (the epistemon) which is capable of interacting with these low-energy constraints. As a result of the interaction between the low-energy constraints and the epistemon, the latter undergoes a state transition which acts as a trigger for the functional action (a), thus completing one perception-action cycle. Now, provided only that the two sets of exogenous constraints -- high-energy (b) and lowenergy (c) -- stand in some causal relation to each other (the precise nature of which will vary from case to case), then the correlation (d)-(a) between the epistemon state transition and the oscillation of the biofunction becomes, in effect, an internal projection of the correlation (b)-(c) between the high- and low-energy constraints. That is, the lowenergy constraints act as semantic information with respect to the functional action, indicating the presence of those external conditions which successfully support its action. Another way of putting it is to say that the interaction between the low-energy (or informational) constraints and the epistemon (call it the "epistemic interaction") predicts that the overall functional action, if undertaken now, will be successful (meaning that the dynamical stability of the oscillator will be preserved). In short, the meaning of information is the prediction of the success of functional action.
5. INFORMATION: SUBJECTIVE AND OBJECTIVE At first glance, this conception of information would appear to be inconsistent with the usual Shannonian notion. However, it is easy to see that in fact they are complementary. On the one hand, the low-energy inputs carrying semantic information can be analyzed syntactically (i.e., broken down into "bits") in the usual way. Of course, even here, syntax remains dependent on semantics, since how the bits are counted will depend on the purpose of the counter, but then that is always true. On the other hand, we can now give a naturalistic account of this dependency of syntax on semantics, as opposed to merely noting it as a brute fact. According to this model, to ascribe a syntactic information content to an object is tantamount to listing the number of different epistemic interactions we can have with that object. For example, ifI say that a sheet of paper with some marks on it contains 100 bits of information, what this means is that there are I00 distinct states of the sheet, each of which is capable of interacting with a correlated epistemon in my brain. Thus, we can see the reason why the traditional information-theoretic measures of complexity are inherently subjective. As Grassberger has noted (1989, p. 496): "We really cannot speak of the complexity of a pattern without reference to the observer." This is because all that the syntactic notion of information content is really measuring is the capacity of the object in question to produce distinct states in us. In applying the theory sketched above reflexively to ourselves in this way, we are stepping outside of the traditional circle of ideas in artificial intelligence, and cognitive science generally, which attempted to explain human cognition in purely syntactic terms on the model of the digital computer. In so doing, we may rely on a growing body of work-which, if it has not yet won widespread mainstream recognition, nevertheless by now constitutes a substantial and respectable rival school of thought -- that views brains as nonlinear dynamical systems (Brooks, 1995; Freeman, 1995; Kelso, 1995; Port & van Gelder, 1995; Pribram, 1994; Thelen & Smith, 1994). Freeman’s work, in particular, is congenial to the viewpoint adopted in this paper. For example, he has demonstrated that identifiable chaotic attractors in EEG data, which are generated by coherent oscillations of nerve cell assemblies, can be predictably correlated with individual perceptual cate-
THE ARROW OF MIND:
127
gories. These nerve cell assemblies -- or rather, their collective modes of action -- are natural candidates for epistemons according to the model of perception and action sketched above. But even if the syntactic view of information based on Shannonian information theory and the semantic view sketched above based on nonlinear dynamics can be reconciled with one another, nevertheless they remain very different approaches which are useful for different purposes. The syntactic information content of an object is extrinsic and subjective in the sense that the information is only meaningful for us, not for the object itself; on the other hand, the semantic information capacity of an organism (roughly, the number of epistemic interactions it is capable of entering into) is intrinsic and objective in the sense that the information is meaningful for the organism itself.2 One must be careful about the terms "subjective" and "objective," here. There is an epistemological sense in which the Shannonian information content of an object is objective: namely, it can be formalized and publicly agreed upon. However, I am using the terms in an ontological sense in order to draw attention to where, in the physical world, the meaning of the information is actually located. In the syntactic, Shannonian case, the meaning is located in the human being, and is extrinsic to the object to which the information content is attributed; therefore, one may say that it is ontologically subjective in the sense that it has no existence independent of the human observer. In the semantic, dynamical case, on the other hand, the meaning of the information is located in the organism itself; it is an intrinsic, objective fact about the world which does not depend in any way on the existence of a human observer. (The fact that the organism in question may also be a human being is irrelevant, since science has long since grown accustomed to viewing the human being in objective terms, ontologically speaking.) This distinction between the extrinsic, or subjective, and the intrinsic, or objective, senses in which information may exist is of fundamental importance for clarifying a number of difficult scientific and philosophical problems, including that of defining an objective metric of biological complexity. It is to this question that I turn in the next section.
2Some authors (e.g., Brooks & Wiley, 1988) use the term informatton "capacity" to refer to the probability space against which the actual information "content" of a system is supposed to be measured. For example, the information "content" of an orgamsm might be the number of base pairs present in ~ts genome, whde the information "capacity" of the same orgamsm would be the factorial of th~s number! There are many objections to this way of looking at things. In the first place, this notion of information capacity ~s subjective and arbitrary, because it is tied directly to the subjective and arbitrary notion of information content (what should count as a "bit" of information? base pa~rs, or genes? ~f genes, how do we identify them? do we count proteins, or actual biofunctlons? if we try to count biofunctions, how are we going to work backwards to the genes again? and what about introns? etc.) Furthermore, even if an objectwe measure of reformation content were possible, what good would it be? Only infinitesimally few of the alternative states in probability space would be functionally v,able, and we would stdl have no way of knowing which were whtch, therefore, as a theoretical construct, information "capacity" in this sense would be useless. Of course, this just points up the fundamental fallacy of applymg extrinsic and subjective information-theoretic measures to organtsms as though they were intrinsically and objectively meaningful (see, also, Lewontin, 1993) Unlike the information-theoretic use of the term, my use of information "capactty" describes an objective property of the organism itself I hope that the other meaning of the term is not yet so entrenched as to foreclose the possibility of diverting tt to a more appropriate use.
128
J BARHAM
6. BIOLOGICAL COMPLEXITY AS EPISTEMIC DEPTH The idea that there exists a natural hierarchical continuum in which all living things have their place -- sometimes referred to as the scala naturae ("ladder of nature") -- is a very old one (Lovejoy, 1936). Although many specifically European ideological features have been grafted onto this notion over the centuries, particularly during the period of the domination of Christianity, the fundamental insight cannot be attributed to mere cultural conditioning, since similar conceptions have arisen independently in other cultures (e.g., China -- see, Tu, 1984). Certainly, when one compares a prokaryotic with a eukaryotic cell, an amoeba with an ant, or an oyster with an octopus, a compelling case can be made for the existence of a vector of increasing complexity over the course of evolutionary history. If that is so, then the scala naturae is a striking and important phenomenon which demands scientific explanation. Nevertheless, as was noted above, many view this intuition as little more than an anthropocentric bias, and certainly it has never been successfully translated into quantitative or operational terms. Therefore, until recently, the whole subject has been scientifically disreputable. At present, it is undergoing renewed scrutiny (Bonner, 1988; Cowan et al., 1994; Nitecki, 1988; Weber et al., 1988; Zurek, 1990). However, most recent authors who have studied the fossil record with a view to quantifying the intuitive notion of an increase in morphological complexity have come to a pessimistic conclusion. For exampie, Boyajian and Lutz (1992) and McShea (1992, 1993) have shown that the history of repetitive skeletal structures does not support the hypothesis of an overall trend toward increasing complexity. It is true that one occasionally meets with a grudging acknowledgement of the existence of some trend or other in the fossil record -- such as Gould et al.’s (1987) study of species diversity (coupled, however, with the earnest denial that the trend in question constitutes evidence of genuine complexification). There is even a minority viewpoint which allows that complexity increase may be real -- see, e.g., Valentine et al.’s (1994) study of the increase in the number of cell types in metazoans over time. On the whole, though, the majority opinion within evolutionary biology remains highly skeptical. Most practitioners still feel it is preferable to explain the impression of a natural hierarchy of living things as an artifact of perspective which causes us to attach greater value to organisms which are similar to ourselves.
THE ARROW OF MIND:
y = functional complexity (see Legend)
129
IV
HighlyEncephalized Metaphytes Metazoans
III
Metazoans
& Protoct~sts
II Prokaryotes
I 4 LEGEND I=
3
2
1
0
X = time (Bilhon Years Before Present)
Active transport across membrane (charge separation, osmoregulation), protein synthests, ATP-based metabolism, photoreceptton, chemoreception, chemotaxis. II= Phagocytosis & predation, oxygen-dependent motility, pheromones, eye spot, habituation & sensit~zatlon. Nerve nets & proto-encephalization, neurotransmit-ters, hormones, pigment-cup III = eye, associative learning IV= Advanced encephalization, complex eye. insight learning Sources: Broda, 1975; Brusca & Brusca, 1990, Jerison, 1973: Loom~s, 1988; Margulis, 1984; McMenamin &, 1990; Schopf, 1992. Figure 2: Main stages in the evolution of functional complexity
A glance at Figure 2 ought to be sufficient to convince an unbiased observer that the intuitive idea of the scala naturae- that is to say, a time-irreversible vector of increasing biological complexity -- is, in fact, objectively valid. The functional repertoire of a highly-encephalized animal (say, the octopus) is built up out of the general metazoan repertoire (e.g., that of the flatworm), which in turn is constructed from the general eukaryotic repertoire (amoeba), which is derived from the prokaryotic repertoire (bacterium) that is shared by all living things. It is tolerably clear that we are dealing here with a natural phenomenon on a par with any other, whose structure is not understood and which therefore deserves a scientific explanation. How to explain this structure is, of course, another matter. It is clear that none of the various information-theoretic metrics of complexity that have been proposed (e.g., Bennett, 1988; Chaitin, 1990) can help us here, because they measure the extrinsic information content of objects; what we want is a way to quantify the intrinsic information capacity of organisms. At first glance, Lloyd and Pageis’ (1988) "thermodynamic depth" would appear to fill the bill. Whereas Bennett’s "logical depth" is the number of steps required to execute the genetic "program" that supposedly produced an organism, the thermodynamic depth of the same organism would be the number of evolutionary steps it took to construct it. This is a step forward because, unlike the information-theoretic notion of a genetic "program," the evolutionary history of an organism in the fossil record gives us an objective quantity that can be estimated independently of human choices or purposes. Furthermore, the concept of thermodynamic depth points to an important aspect of the problem of complexity: namely, the fact that the more intuitively simple creatures must precede the more intuitively complex ones in evolutionary history. Unfortunately, though, what thermodynamic depth is really measuring is not the complexity of
130
J. BARHAM
an organism as such, but merely its evolutionary "age." It is an explanation of sorts of how a complex organism got to be the way it is, but it still does not give us an independent way of measuring complexity, apart from the fossil record. Rather, by tacitly assuming the correlation between length of evolutionary history and degree of complexity, the notion of thermodynamic depth begs the crucial question: How do we know that one organism is more or less complex than another, in the first place? What we really need is a thermodynamic (i.e., non-syntactic) approach which focuses, not on phylogeny, but on the individual organism. The dynamical model of semantic information outlined in Section 4 above provides a way of doing this. If we think of each of the myriad, hierarchically-nested biofunctions within a given organism -- enzyme species, metabolic networks, organ systems, nerve cell assemblies, and so forth -- as a coherent nonlinear oscillator, then a natural measure of intrinsic biological complexity suggests itself. Namely, the complexity of an organism, on this model, may be measured by the number of different types of epistemic interactions it is capable of undergoing (that is, by the number of different kinds of epistemons it contains). Let us call this value the epistemic depth of the organism. Loosely speaking, then, one might say that the epistemic depth of an organism is the sum of the number of distinct biological functions it contains. Since one of the chief means nature employs for creating new biofunctions is the integration of old functions into new emergent wholes, the notion of epistemic depth accords well with the intuitive conception of the scala naturae illustrated by Figure 2. Obviously, it would be out of the question to attempt to assign an actual empirical value to this quantity for a particular organism, because the notion of an epistemon is as yet too abstract to be of much operational value. However, in this respect, the notion of epistemic depth fares no worse than other proposed complexity metrics. Its advantage is that, unlike its information-theoretic rivals, it tackles directly the problem of what we intuitively mean by biological complexity. The basic mistake that most investigators have made is to assume that what mattered was the number of different parts (broken symmetries in space) an organism contained. But surely our intuitive idea of the scala naturae is captured better by counting the number of different functions (broken symmetries in time) an organism is capable of performing. Counting biological functions comes much closer to capturing what we intuitively mean by biological complexity because, unlike the counting of parts, it takes into account the information capacity (i.e., intelligence) of an organism. An octopus seems to us more complex than an oyster because it is capable of doing more things, which is equivalent to saying that it knows more things. This explains why whose who have focused on cell types -- which differ largely according to their function -- have come to a different conclusion from those who have studied repetitive structural parts: namely, that a vector of increasing complexity over the course of the evolutionary history of the metazoans really does exist. The notion of epistemic depth is potentially an even better metric than cell type number, since it may eventually allow us to measure complexity increase from single cells to animals with sophisticated brains on a single scale. By pointing to the fundamentally epistemic aspect of all functional activity, it allows us to characterize the apparent increase in complexity over the course of organic evolution as an increase in information capacity, and to explain this irreversible process as a result of temporal symmetrybreaking analogous to the spatial symmetry-breaking that has occurred over the course of cosmic evolution. In this way, organic evolution may be viewed as a stage in the overall structuration of the universe that is ultimately traceable to the nonequilibrium conditions created by the Big Bang (Frautschi, 1982; Layzer, 1990). One might even go
THE ARROW OF MIND.
131
so far as to say that, with the emergence of life, the cosmological arrow of time assumed the form of a biological "arrow of mind."
REFERENCES Barham, J. (1990) A poincarn approach to evolutionary epistemology, Journal of Soctal and Btological Structures, 13, 193-258. Barham, J. (I 995) Ni Darwin m Derrida, Journal of Soctal and Evolutionary Systems, 18, 277-308. Barham, J. (m press) A dynamical model of the meaning of information, BtoSystems. Bennett, C. H. (1988) Dissipation, information, computational complexity and the definition of organization, In: Pines, D., ed., Emerging Syntheses m Sctence, Redwood City CA’ Addison-Wesley, pp. 215-233. Bonnet, J. T. (1988) The Evolution of Complextty by Means of Natural Selection, Princeton, NJ: Princeton University Press. Boyajian, G. and Lutz, T. (1992) Evolution of btological complexity and its relation to taxonomic longevity in the ammono~dea, Geology, 20, 983-986 Broda, E. (1975) The Evolution of the Bioenergetic Processes, Oxford: Pergamon. Brooks, D. R. and Wiley, E. O. (1988) Evolutton as Entropy: Towards Untried Theory ofBwlogy, 2nd ed, Chicago: University of Chicago Press. Brooks, R. A. (1995) Intelligence without reason, In: Steels, L. and Brooks, R, eds., The Artifictal Ltfe Route to Artificial Intelligence, Hillsdale N J" Erlbaum, pp. 25-81 Brusca, R. C. and Brusca, G. J. (1990) Invertebrates, Sunderland MA: Sinauer. Chaitin, G. J. (1990) Algorithmic information theory, In: Chaitin, G. J., Informatton. Randomness and Incompleteness. Papers on Algortthmtc Information Theory, 2nd ed, Singapore: World Scientific, pp. 44-58. Cowan, G.A., Pines, D., and Meltzer, D., eds. (1994) Complextty: Metaphors. Models, and Reahty, Reading, MA: Addison-Wesley. Delattre, P. (I 986) An approach to the notion of finalityaccording to the concepts of qualitative dynamics, In: Diner, S., Fargue, D., and Lochak G., eds., Dynamtcal Systems: A Renewal of Mechanism, Singapore World Scientific, pp. 149-154. Dennett, D. C (1995) Darwm ’s Dangerous ldea. Evolutton and the Meanings of Ltfe, New York Simon & Schuster. Frautschi, S. (1982) Entropy in an expanding umverse, Sctence, 217, 593-599. Freeman, W. J. (1995) Soctettes of Brains, Hdlsdale NJ. Erlbaum. Gilbert, S. F. (1988) Developmental Btology, 2nd ed., Sunderland MA: S~nauer. Glass, L. and Mackey, M. C. (1988) From Clocks to Chaos. The Rhythms of Ltfe, Princeton NJ: Pnnceton University Press. Gould, S. J. (1989) Wonderful Ltfe, New York: Norton. Gould, S. J., Gihnsky, N. L, and German, R. Z. (1987) Asymmetry of lineages and the direction of evolutionary time, Sctence, 236,1437-144 I. Grassberger, P. (1989) Problems in quantifying self-generated complexity, Helvetica Phystca Acta, 62, 489511 Ho, M.-W. (1993) The Rambow and the Worm. The Physics of Orgamsms, Singapore: World Scientific. Jenson, H. J. (1973) Evolution of the Brain andlntelhgence, NewYork: Academic Press Kelso, J. A. S. (1995) Dynamtc Patterns: The Self-Orgamzatton of Bram and Behavior, Cambridge MA’ Bradford Books/MIT Press Layzer, D. (1990) Cosmogenests: The Growth of Order m theUmverse, New York. Oxford University Press. Lewontin, R. C. (1993) Biology as Ideology: The Doctrine of DNA, New York: Harper Perennial/HarperCollins. Lloyd, D. and Rossi, E. L., eds (1992) Ultradian Rhythms m Ltfe Processes, London" Spnnger-Verlag. Lloyd, S. and Pagels, H (I 988) Complexity as thermodynamic depth, Annals of Physics, 188, 186-213, Loomis, W. F (1988) Four Bilhon Years An Essay on the Evoluttonof Genes and Organtsms, Sunderland
132
J. BARHAM
MA: Sinauer. Love.loy, A. O (1936) The Great Chain of Being, Cambridge, MA: Harvard University Press. Margulis, L (1984) Early Life, Boston: Jones and Bartlett. Matsuno, K. (1989) Protobiology: Physical Basis of Biology, Boca Raton FL: CRC Press McMenamin, M. A. S. and McMenamin, D. L. S. (1990) The Emergence of Ammals: The Cambrian Breakthrough, New York: Columbia University Press McShea, D. W. (1991) Complexity and evolution: What everybody knows, Biology and Philosophy, 6, 303324. McShea, D. W. (1992) A Metric for the study &evolutionary trends in the complexity of serial structures, Biological Journal of the Lmnean Society. 45, 39-55. McShea, D. W. (1993) Evolutionary change in the morphological complexity of the mammalian vertebral column, Evolution, 47, 730-740. Morowitz, H. J. (1979) Energy Flow in Biology, Woodbridge CT" OxBow Press. (Originally published by Academic Press, New York, tn 1968.) Nitecki, M. H., eds. (1988) Evolutionary Progress~, Chicago’ University of Chicago Press. Port, R. F. and van Gelder, T., eds. (1995) Mind as Motion. Explorations in the Dynamics of Cogmtion, Cambridge MA: Bradford Books/MIT Press. Pnbram, K. H., ed. (1994) Origins: Brain and SelfOrgamzation, Hfllsdale N J" Erlbaum. Prigogine, I. (1980) From Being to Becoming, San Francisco. W. H. Freeman and Company. Rosen, R. (1991) Life ltself. A Comprehensive lnqutry into the Nature, Origin, and Fabrlcation of Life, New York: Columbia Umversity Press Schopf, J. W., ed. (1992) Major Events m the History of Life, Boston" Jones and Bartlett. Shannon, C. E. and Weaver, W. (1963) The Mathematical Theory of Communication, Urbana IL: University of lllinois Press (Originally publisl~ed in 1949.7 Swenson, R. (1992) Order, evolution, and natural law’ Fundamental relations in complex system theory, In: Negolta, C. V., ed., Cybernetics and Apphed Systems, New York. Marcel Dekker, pp. 125-148. Thelen, E. and Smith, L. B. (1994) A Dynamic Systems Approach tothe Development of Cognition and Action, Cambridge MA: Bradford Books/MIT Press Tu W-M (1984) The Continuity of being: Chinese v~stons of nature, In: Rouner, L.S., ed, On Nature, Notre Dame IN: University of Notre Dame Press, pp. 113-129. (Reprinted in. Callicott, J.B. and Ames, R T., eds., Nature in Asian Traditions of Thought, Albany, NY: State University of New York Press, 1989, pp 67-78, 298-301 .) Valentine, J. W., Collins, A. G., and Meyer, C P (1994) Morphological complexity increase in metazoans, Paleoblology. 20, 131-142. Wackerbauer, R., Witt, A., Atmanspacher, H, Kurths, J., and Scheingraber, H. (1994) A comparative classification of complexity measures, Chaos. Solitons and Fractals, 4, 133-173. Weber, W. H., Depew, D. J., and Smith, D. J., eds. (1988) Entropy, Information, and Evolution: New Perspectives on Physical and Biological Evolution, Cambridge MA: Bradford Books/MIT Press Wheeler, J. A. (1990) Information, physics, quantum: The searchfor links, In: Zurek, W. H., ed, Complexity, Entropy and the Physics oflnformatlon, Redwood City CA: Addison-Wesley, pp. 3-28 Wicken, J. S. (1987) Evolution, Thermodynamics. and Information: Extending the Darwinian Paradigm, New York: Oxford University Press. Winfree, A. T. (1990) The Geometry of Biological Time, corrected reprint, Berlin: Springer-Verlag. (Originally published in 1980.) Yates, F. E. (1993) Self-organizing systems, In: Boyd, C. A. R.and Noble, D., eds, The Logic of Life The Challenge of Integrative Physiology, Oxford: Oxford Umversity Press, pp. 189-218 Zurek, W. H, ed. (1990) Complexity, Entropy and the Physics oflnformation, Redwood City, CA: AddisonWesley.
Symmetry" Culture and Sctence Vol. 8, No. 2, 133-156, 1997
SYMMETRY. C UL TURE AND SCIENCE
GENERATION OF FRACTALS FROM INCURSIVE AUTOMATA, DIGITAL DIFFUSION AND WAVE EQUATION SYSTEMS Daniel M. Dubois Address. Umversity of LIEGE, Instltute of Mathematics, Avenue des Tdleuls 15, B-4000 LIEGE, Belgium, Fax */32/41/669489, E-marl’
[email protected] be
Abstract: This paper provides modelling tools for formal systems design in the field of information and physical systems. The concept and method of incursion and hyperincursion are.firstly applied to the Fractal Machine, an hyperincursive cellular automata with sequential computations with exclusive OR where time plays a central role. Simulations will show the generation of fractal patterns. The computation is incursive, for inclusive recursion, in the sense that an automaton is computed at the future time t+ 1 in function of its neighbour automata at the present and/or past time steps but also at the future time t+ 1. The hyperincursion is an incursion when several values can be generated at each time step. External incursive inputs cannot be transformed to recursion. This is really a practical example of the Final Cause of Aristotle. But internal incursive inputs defined at the future time can be transformed to recursive inputs by self-reference defining then a self-referential system. A particular case of self-reference with the Fractal Machine shows a non deterministic hyperincursive field The concepts of incursion and hyperincursion can be related to the theory of hypersets where a set includes itself Secondly, the incursion is applied to generate fractals with different scaling symmetries. This is used to generate the same fractal at different scales like the box counting method for computing a fractal dimension. The simulation of fractals with an initial condition given by pictures is shown to be a process similar to a hologram. Interference of pictures with some symmetry gives rise to complex patterns. This method is also used to generate fractal interlacing. Thirdly, it is shown that fractals can also be generated from the digital equations of diffusion and wave, that is to say from the modulo N of their.finite difference equations with integer coefficients.
1. INTRODUCTION The recursio~ consists of the computation of the future value of the variable vector X(t+l) at time t+l from the values of these variables at present and/or past times, t, t-l, t-2 ....by a recursive function : p) X (t+ 1) =f(X(t), X(t-1) .....
where p is a command parameter vector. So, the past always determines the future, the present being the separation line between the past and the future.
134
D. M. DUBOIS
Starting from cellular automata, the concept of Fractal Machines was proposed in which composition rules were propagated along paths in the machine frame. The computation is based on what I called "INclusive reCURSION", i.e. INCURSION (Dubois, 1992ab). An incursive relation is defined by: X(t+l) =f(..., X (t+l), X(t), X(t-1) ..... p)
which consists in the co~nputation of the values of the vector X(t+l) at time t+l from the values X(t-i) at time t-i, i=1, 2 .... , the value X(t) at time t and the value X(t+j) at time t+j, j=l, 2, .... in function of a command vector p. This incursive relation is not trivial because future values of the variable vector at time steps t+l, t+2 .... must be known to compute them at the time step t+ 1. In a similar way to that in which we define hyper recursion when each recursive step generates multiple solutions, I define HYPERINCURSION. Recursive computational transformations of such incursive relations are given in Dubois and Resconi (1992, 1993a-
b).
I have decided to do this for three reasons. First, in relativity theory space and time are considered as a four-vector where time plays a role similar to space. If time t is replaced by space s in the above definition of incursion, we obtain X(s+ l) =f( ..., X(s+ 1), X(s), X (s-l) ..... p)
and nobody is astonished: a Laplacean operator looks like this. Second, in control theory, the engineers control engineering systems by defining goals in the future to compute their present state, similarly to our haman anticipative behaviour (Dubois, 1996a-b). Third, I wanted to try to do a generalisation of the recursive and sequential Turing Machine in looking at space-time cellular automata where the order in which the computations are made is taken into account with an inclusive recursion. We have already proposed some methods to realise the design of any discrete systems with an extension of the recursion by the concept of incursion and hyperincursion based on the Fractal Machine, a new type of Cellular Automata, where time plays a central role. In this framework, the design of the model of any discrete system is based on incursion relations where past, present and future states variables are mixed in such a way that they define an indivisible wholeness invariant. Most incursive relations can be transformed in different sets of recursive algorithms for computation. In the same way, the hyperincursion is an extension of the hyper recursion in which several different solutions can be generated at each time step. By the hyperincursion, the Fractal Machine could compute beyond the theoretical limits of the Turing Machine (Dubois and Resconi, 1993a-b). Holistic properties of the hyperincursion are related to the Golden Ratio with the Fibonacci Series and the Fractal Golden Matrix (Dubois and Resconi, 1992). An incursive method was developed for the inverse problem, the NewtonRaphson method and an application in robotics (Dubois and Resconi, 1995). Control by incursion was applied to feedback systems (Dubois and Resconi, 1994). Chaotic recursions can be synchronised by incursion (1993b). An incursive control of linear, nonlinear and chaotic systems was proposed (Dubois, 1995a, Dubois and Resconi, 1994, 1995). The hyperincursive discrete Lotka-Voiterra equations have orbital stability and show the emergence of chaos (Dubois, 1992). By linearisation of this non-linear system, hyperincursive discrete harmonic oscillator equations give stable oscillations and discrete solutions (Dubois, 1995). A general theory of stability by incursion of discrete
DIGITAL DIFFUSION AND WAVE EQUATION SYSTEMS
equations systems was developed with applications to the control of the numerical instabilities of the difference equations of the Lotka-Volterra differential equations as well as the control of the fractal chaos in the Pearl-Verhulst equation (Dubois and Resconi, 1995). The incursion harmonic oscillator shows eigenvalues and wave packet like in quantum mechanics. Backward and forward velocities are defined in this incursion harmonic oscillator. A connection is made between incursion and relativity as well as the electromagnetic field. The foundation of a hyperincursive discrete mechanics was proposed in relation to the quantum mechanics (Dubois and Resconi, 1993b, 1995). This paper will present new developments and will show that the incursion and hyperincursion could be a new tool of research and development for describing systems where the present state of such systems is also a function of their future states. The anticipatory property of incursion is an incremental final cause which could be related to the Aristotelian Final Cause.
2. INCURSION AND ARISTOTLE’S FINAL CAUSE Aristotle identified four explicit categories of causation: 1. Material cause; 2. Formal cause; 3. Efficient cause; 4. Final cause. Classically, it is considered that modem physics and mechanics only deal with efficient cause and biology with material cause. Robert Rosen (1986) gives another interpretation and asks why a certain Newtonian mechanical system is in the state (phase) Ix(t) (position), v(t) (velocity)]: 1. Aristotle’s "material cause" corresponds to the initial conditions of the system [x(0), v(0)] at time t=0. 2. The current cause at the present time is the set of constraints which convey to the system an "identity", allowing it to go by recursion from the given initial phase to the latter phase, which corresponds to what Aristotle called formal cause. 3. What we call inputs or boundary conditions are the impressed forces by the environment, called efficient cause by Aristotle. As pointed out by Robert Rosen, the first three of Aristotle’s causal categories are tacit in the Newtonian formalism: "the introduction of a notion of final cause into the Newtonian picture would amount to allowing a future state or future environment to affect change of state in the present, and this would be incompatible with the whole Newtonian picture. This is one of the main reasons that the concept of Aristotelian finality is considered incompatible with modern science. In modern physics, Aristotelian ideas of causality are confused with determinism, which is quite different.... That is, determinism is merely a mathematical statement of functional dependence or linkage. As Russell points out, such mathematical relations, in themselves, carry no hint as to which of their variables are dependent and which are independent." The final cause could impress the present state of evolving systems, which seems a key phenomenon in biological systems so that the classical mathematical models are unable to explain many of these biological systems. An interesting analysis of the Final Causation was made by Emst von Glasersfeld (1990). The self-referential fractal machine shows that the hyperincursive field dealing with the final cause could be also very important in physical and computational systems. The concepts of incursion and hyperincursion deal with an extension of the recursive processes for which future states can determine present states of evolving systems. Incursion is defined as invariant functional relations from which several recursive models with interacting variables can be
136
D M. DUBO1S
constructed in terms of diverse physical structures (Dubois & Resconi, 1992, 1993b). Anticipation, viewed as an Aristotelian final cause, is of great importance to explain the dynamics of systems and the semantic information (Dubois, 1996a-b). Information is related to the meaning of data. It is important to note that what is usually called Information Theory is only a communication theory dealing with the communication of coded data in channels between a sender and a receptor without any reference to the semantic aspect of the messages. The meaning of the message can only be understood by the receiver if he has the same cultural reference as the sender of the message and even in this case, nobody can be sure that the receiver understands the message exactly as the sender. Because the message is only a sequential explanation of a non-communicable meaning of an idea in the mind of the sender which can be communicated to the receiver so that a certain meaning emerges in his mind. The meaning is relative or subjective in the sense that it depends on the experiential life or imagination of each of us. It is wellknown that the semantic information of signs (like the coding of the signals for traffic) are the same for everybody (like having to stop at the red light at a cross roads) due to a collective agreement of their meaning in relation to actions. But the semantic information of an idea, for example, is more difficult to codify. This is perhaps the origin of creativity for which a meaning of something new emerges from a trial to find a meaning for something which has no a priori meaning or a void meaning. Mind dynamics seems to be a parallel process and the way we express ideas by language is sequential. Is the sequential information the same as the parallel information? Let us explain this by considering the atoms or molecules in a liquid. We can calculate the average velocity of the particles from in two ways. The first way is to consider one particular particle and to measure its velocity during a certain time. One obtains its mean velocity which corresponds to the mean velocity of any particle of the liquid. The second way is to consider a certain number of particles at a given time and to measure the velocity of each of them. This mean velocity is equal to the first mean velocity. So there are two ways to obtain the same information. One by looking at one particular element along the time dimension and the other by looking at many elements at the same time. For me, explanation corresponds to the sequential measure and understanding to the parallel measure. Notice that ergodicity is only available with simple physical systems, so in general we can say that there are distortions between the sequential and the parallel view of any phenomenon. Perhaps the brain processes are based on ergodicity: the left hemisphere works in a sequential mode while the right hemisphere works in a parallel mode. The left brain explains while the right brain understands. The two brains are complementary and necessary. Today computer science deals with the "left computer". Fortunately, the informaticians have invented parallel computers which are based on complex multiplication of Turing Machines. It is now the time to reconsider the problem of looking at the "right computer". Perhaps it will be an extension of the Fractal Machine (Dubois & Resconi, 1993a). I think that the sequential way deals with the causality principle while the parallel way deals with a finality principle. There is a paradox: causality is related to the successive events in time while finality is related to a collection of events at a simultaneous time, i.e. out of time. Causality is related to recursive computations which give rise to the local generation of patterns in a synchronic way. Finality is related to incursive or hyperincursive symmetry invariance which gives rise to an indivisible wholeness, a holistic property in a dia-
DIGITAL DIFFUSION AND WAVE EQUATION SYSTEMS
137
chronic way. Recursion (and Hyper recursion) is defined in the Sets Theory and Incursion (and Hyperincursion) could be defined in the new framework of the Hypersets Theory (Aczel, 1987; Barwise, Moss, 1991). If the causality principle is rather well acknowledged, a finality principle is still controversial. It would be interesting to re-define these principles. Causality is defined for sequential events. If x(t) represents a variable at time t, a causal rule x(t+l) = f(x(t)) gives the successive states of the variable x at the successive time steps t, t+l, t+2, ... from the recursive functionf(x(t)), starting with an initial state x(0) at time t=0. Defined like this, the system has no degrees of freedom: it is completely determined by the function and the initial condition. No new things can happen for such a system: the whole future is completely determined by its past. It is not an evolutionary system but a developmental system. If the system tends to a stable point, x(t+l) = x(t) and it remains in this state for ever. The variable x can represent a vector of states as a generalisation. In the same way, I think that determinism is confused with predictability, in modern physics. The recent fractal and deterministic chaos theory (Mandeibrot, 1982; Peitgen, Ji~rgens, Saupe, 1992) is a step beyond classical concepts in physics. If the function is non-linear, chaotic behaviour can appear, what is called (deterministic) chaos. In this case, determinism does not give an accurate prediction of the future of the system from its initial conditions, what is called sensitivity to initial conditions. A chaotic system loses the memory of its past by finite computation. But it is important to point out that an average value, or bounds within which the variable can take its values, can be known; it is only the precise values at the successive steps which are not predictable. The local information is unpredictable while the global symmetry is predictable. Chaos can presents a fractai geometry which shows a self-similarity of patterns at any scale. A well-known fractal is the Sierpinski napkin. The self-similarity of pattems at any scale can be viewed as a symmetry invariance at any scale. An interesting property of such fractals is the fact that the final global pattern symmetry can be completely independent of the local pattern symmetry given as the initial condition of the process from which the fractal is built. The symmetry of the fractal structure, a final cause, can be independent of the initial conditions, a material cause. The formal cause is the local symmetry of the generator of the fractal, independently of its material elements and the efficient cause can be related to the recursive process to generate the fractal. In this particular fractal geometry, the final cause is identical to the final cause. The efficient cause is the making of the fractal and the material cause is just a substrate from which the fractal emerges but this substrate doesn’t play a role in the making.
3. THE HYPERINCURSIVE FRACTAL MACHINE A one-dimensional network of cellular automata (Feynman, 1982; Gardner, 1971; Schroeder, 1991; Weisbuch, 1989; Wolfram, 1994, Zuse, 1969) is represented by a vector of automata states, each automaton state having an integer numerical value at the initial time t=0. A set of rules defines how the states change at every clock time. A simple rule consists of replacing the value of each automaton by the sum of itself and its left neighbour at each clock time. Figure 1 shows a one-dimensional network of cellular automata giving rise to the Pascal triangle. The recursive model of the Pascal triangle network is (1)
X(n, t+l) =X(n, t) + X(n-l, t)
138
D M. DUBOIS
with t=0, 1, 2 .... and n=l, 2, ..., starting with initial conditions X(n, 0), n=l, 2 .... at time t=0 and boundary conditions X(0, t) at each time step t= 1, 2 ....
t=O t=l t=2 t=3 t=4 t=5 t=6 t=7 t=8
n=O
1
2 3 4
5
0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8
0 0 0 0 1 5 15 35 70
0 0 1 3 6 I0 15 21 28
0 0 0 1 4 i0 20 35 56
6
7 8
0 0 0 0 0 0 0 0 0 0 1 0 6 1 21 7 56 28
0 0 0 0 0 0 0 1 8
Figure h The Pascal triangle generated by the recursive eq. 1
The recursive equation 1 can be reversed in replacing t+l by t-I (la) X(n,t-1)=X(n,t)+X(n-l,t) In making a time translation, replacing t by t+l, one obtains (lb) X(n, t+l) =X(n, t))-X(n-1, t+l)
This eq. lb can be computed in an incursive way, that is to say in a sequential order, in giving initial conditions X(n, 0) and boundary conditions X(0, t+l) at the future time t+l, for each time t-=0, 1, 2 .... It is absolutely impossible to build a real physical systems governed by such an equation because" How to give to the system the boundary conditions with inputs defined at the future time step t+l ? ". It will be shown below in this paper that such incursive system can work in practice if the boundary conditions are zero (no inputs) or the system defined itself its boundary conditions in a self-referential way (for example, in defining periodical boundary conditions). But some problems can appear: uncertainty or indecidability. In such a case, a solution would be to define a purpose to the system so that the future inputs can be replaced by inputs at the present time with a feed-back process as made in cybernetics and control theory (Rosenblueth, Wiener, Bigelow, 1943; Van de Vijver, 1992). With modulo N, the recursive eq. (1) becomes (2) X(n, t+l) = (X(n, t) +X(n-1, t)) modN
with t=0, 1, 2 .... and n=l, 2 .....
with initial conditions X(n, 0) and boundary conditions X(0, t). With N=2, the pattern is given by the fractal Sierpinski napkin given in Figure 2a. In this recursion the present time step always determines the next future time step, even for the boundary conditions X(0, t). In the Fractal Machine (Dubois, 1992), the following incursive digital equation is defined (3) X(n, t+l) = (X(n, t) +X(n-1, t+l)) mod N with n=l, 2, ..., 8, and t=0, 1, 2 .....7
DIGITAL DIFFUSION AND WAVE EQUATION SYSTEMS
139
where X(n, t) is the automaton state at position n and time t. The modulo with N=2 is exclusive OR., XOR. The computation of eq. (3) with N=2, given at Figure 2b, gives rise to a time reverse Sierpinski napkin (Dubois, 1990,1991). Let us remark that with the modulo 2, the negative term in eq. lb is without importance. n=O 1 2 3 4 5 6 7 8 t=O t=l t=2 t=3 t=4 t=5 t=6 t=7 t=8
0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 i 1 i
0 1 0 1 0 i 0 1 0
0 0 1 1 0 0 1 1 0
0 0 0 1 0 0 0 1 0
0 0 0 0 1 I 1 i 0
0 0 0 0 0 1 0 1 0
0 0 0 0 0 0 1 1 0
0 0 0 0 0 0 0 i 0
n=O 1 2 3 4 5 6 7 8 0 0 0 0 0 0 0 0 0
I 1 i i i 1 1 i i
0 i 0 1 0 i 0 1 0
0 1 1 0 0 1 i 0 0
0 1 0 0 0 1 0 0 0
0 1 1 1 1 0 0 0 0
0 1 0 1 0 0 0 0 0
0 1 1 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
Figure 2a-b: Recursive Sierpinsk, napkin and incursive S~erpxnsh napkin
3.1. Self-Referential Fractal Machine In the Fractal Machine, if it is natural to consider the successive time steps t in the increasing order, it is also necessary to consider the successive computations in the increasing order of the number n of automata which can be considered as an internal time. Explicitly it is possible to define two times: an external time and an internal time. The duration of the external time is the sum of the sequential computational internal times. For n= 1, the future inputs X(0, t+l) must be defined at each time step in view of computing the automata X(n, t+l) as a final cause which controls the dynamics of the system. In transforming eq. (3) in a quasi recursive equation system (Dubois, 1996a) (3a)
X(0, t+l) = external inputs =final cause X(1, t+l) = (X(1, t) + X(0, t+l)) mod N X(2, t+l) = (X(2, t) +X(1, t) +X(0, t+l)) rood N X(3, t+l) = (X(3, t) +X(2, t) +X(1, t) +X(0, t+l)) modN /;(8, t+l)= (X(7, t)+ X(6, t)+ ... +X(i, t) +X(0, t+l))mod N
it is explicitly seen that the external inputs must be defined in the future time like a final causation which controls completely all the automata at the same time step in a holistic way. Indeed the inputs X(0, t+l) are present in each automata at the same external time. It is impossible to transform external inputs defined in the future time t+l to inputs defined in the present time t. In this, we can say that we are dealing with a strict incursive system. Thus the final causation is really the 4th causation which must be taken into account in systems modelling as Aristotle had proposed. It seems also impossible to construct a real working engineering system where real working external future inputs would control its current present state. But it is possible to define internal future inputs in considering self-referential systems.
D M DUBOIS
140
For example, in taking the following initial conditions X(n, 0)=0, n=0 .... ,8 and boundary conditions X(0, t+l)=X(8, t+l), t=0, 1, 3, it is shown in Figure 2c that there are two solutions at each time step.
t=O t=l t=2
n=O 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1
Figure 2e Uncertainty in the self-referential fractal machine.
Indeed ifX(0, 1)=1 then X(8, 1)=1 and ifX(0,1)=0 then X(8,1)=0. Thus, this is an hyperincursive system because we have two different solutions at each time step. Moreover in some cases, contradiction can appears. For example, starting with the following different initial conditions at time t=0 given in Figure 2d in taking X(0, 1)=1 then X(8, 1)=0 and if X(0, 1)=0 then X(8, 1)=1. This case could be resolved in deciding that X(0, t+l)=l-X(8, t+l), then the first example will give a contradiction. The Fractal Machine can become non deterministic or non algorithmic, what I suggest to call an HYPERINCURSIVE FIELD where uncertainty (indecidability) or contradiction (exclusion principle) occur (Dubois, 1996a).
t=O t=l
n=O 1 2 3 4 5 6 7 8
0 1 2 3 4 5 6 7 8
0 1 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1
Figure 2d: Contradiction in the self-referential fractal machine.
4. GENERATION OF FRACTALS FROM INCURSIVE AUTOMATA Lets us consider numerical simulations on computer of a few incursive automata. Figures 3a-b give the simulation of the incursive automata given by eq. (3) for N=2 and N=3, the initial conditions and boundary conditions are the same as in Figure 2-b.
Figure 3a: Simulation of hyperineursive equation (3) with N=2.
Figure 3b: Simulation ofhyperincursive equation O) with N=3.
DIGITAL DIFFUSION AND WAVE EQUATION SYSTEMS
141
Figures 4a-b show an other Sierpinski napkin and a fractal pattern from the incursive relation (4)
X(n, t+l) = (X(n-1, t-l) + X(n-2, t+l)) mod 2
where n=2, 3, .. and t=l, 2 .... The boundary conditions are O’s. Two different initial conditions at time t=l are considered: (4a) X(n, 1)=0 for n=0,1,2 .... (4b) X(1 l.p,1)=l forp=0,1,2 ....
Figure 4a: Simulation of hyperincursive equation 4 with initial condition 4a.
Figure 4b: Simulation of hyperincursive equation 4 with initial condition 4b.
The following incursive relation depending on three automata (5)
X(n, t+l) = (X(n, t) + X(n-1, t+l) + X(n-1, 0) mod 3
with n=l, 2, ... and t=0, 1, 2, ... generates a square fractal given in Figure 5a with the initial condition X(1, 0)=1 and the boundary conditions X(0, t)=0 for t=0, 1, 2 ....
Figure 5a: Simulation ofhyperincursive equation (5) giving a square fractal.
142
D M. DUBOIS
Similarly with the Pascal Triangle, an incursive square with modulo 3 can be generated as follows: n=O 1 2 3 4 5 6 7 8 9 t=O t=l t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9
0 0 0 0 0 0 0 0 0 0
0 1 1 1 1 1 1 1 1 1
0 1 0 2 1 0 2 1 0 2
0 1 2 1 1 2 1 1 2 1
0 1 1 1 0 0 0 2 2 2
0 1 0 2 0 0 0 2 0 1
0 1 2 1 0 0 0 2 1 2
0 1 1 1 2 2 2 1 1 1
0 1 0 2 2 0 1 1 0 2
0 1 2 1 2 1 2 1 2 1
Figure 5b: Incursive square fractal generated by eq. 5, starting with X(1,1)= 1. This fractal pattern ~s similar to the Sierpmski carpet in considering the O’s and non O’s pixels.
The time reverse of the eq. 5 is given by (5a) X(n, t+l) = (X(n, t) -X(n-1, t+l)-X(n-1, t)) mod 3 which is again an incursive equation with the same terms with different signs. Contrary to the Sierpinski gasket, the time reverse of this square fractal is not a recursive equation, but another incursive one (see Figure 5c). Let us recall that a definition of modulo 3 for negative values is: (-1) mod 3=2; (-2) mod 3=1. n=O 1 2 3 4 5 6 7 8 9 t=O t=l t=2 t=3 t=4 t=5 t=6 t=7 t=8 t=9
0 0 0 0 0 0 0 0 0 0
0 1 1 1 1 1 1 1 1 1
0 2 0 1 2 0 1 2 0 1
0 1 2 1 1 2 1 1 2 1
0 2 2 2 0 0 0 1 1 1
0 1 0 2 0 0 0 2 0 1
0 2 1 2 0 0 0 1 2 1
0 1 1 1 2 2 2 1 1 1
0 2 0 1 1 0 2 2 0 1
0 1 2 1 2 1 2 1 2 1
Figure 5e: lncursive Sierpinski carpet starting from X(I,I)=I. The O’s are exactly at the same posmon in the direct and reverse time incursive process.
The direct and reverse times patterns show a reflection symmetry for each space-time (n, t) value. The symmetry of the whole pattern is the symmetry of the elementary part (the generator), that is to say a self-similarity as shown in Figure 5d. Indeed, the symmetry is space-time scale invariant in considering units in 3", m= 1, 2, 3 ....
DIGITAL DIFFUSION AND WAVE EQUATION SYSTEMS
n=l 2 3 t=l t=2 t=3
1 2 1 1 0 1 1
n=l 2 1
t=l t=4 t=7
143
4
7
1 2 1 1 0 2 1 1 1
Figure 5d: Elementary square fractal as the generator and self-symmetry at space-time scale Dn=3 and Dr=3, giving a self-similar pattern characterizing a fractal.
The two diagonals are identical and represent the Cantor set in one dimension: 1 0 1 0 0 0 1 0 1 Figure 5e: Cantor set in one dimension given by the diagonals of the square fractal.
Let us remark that the square fractal is similar to the Sierpinski carpet in the sense that the zeros are at the same places. In the classical Sierpinski carpet all non zero values are given by the same value 1. In this incursive square fractai, it is necessary to consider 1 and 2, that is to say the modulo 3. With modulo 2, equation 5 gives rise to a uniform Euclidean pattern of dimension D=-2. Figure 6 shows a Pentagon fi’actal generated by the incursive equation with four automata
Figures 6: Simulation of hyperincursive equation (6) showing a pentagon fractal.
(6)
X(n, t+l) = (X(n, t) +X(n-1, t+l) +X(n-1, t-l) + X(n-2, t)) mod 2
with n=l, 2 .... and t=0, I, 2 .... with the initial condition X(1, 0)=1 and the boundary conditions X(0, t)=0 for t=0, 1, 2 ....
144
D. M DUBOIS
5. HOLOGRAPHIC GENERATION OF FRACTALS AT DIFFERENT SCALES Different patterns at different scales can be generated in the following way. Let us take the following equation which will gives the Sierpinski gasket (N=2) in different space scales: (7)
X (n, m, t+zlt) = (X (n, m, t) + X (n-An, m, t+At) + X (n, m-Am, t+dt)) mod N
with the first space parameter n=l, 2 .... , for each successive second space parameter m=l, 2, ... in function of the time t=0, 1, 2, ... The space steps An and Am define the scale at which the fractal is generated, starting with an initial condition given by a picture. If the picture is a black square (7a)
X(n, rn, 0) = 1 for n=l to An and m=l to Am
different Sierpinski gaskets are generated in one time step, t=0, with space steps given, by example, by An--2p, p=0, 1,2, 3 and ZIm=2q, q=0, 1, 2, 3, as shown in Figures 7a-b-c-d.
d
Figures 7a-b-c-d
With this incursive process 7, the fractai dimension is easy to compute by "Box counting" (see for example Peitgen, Jtirgens, Saupe, 1992). When p=q, with the same initial condition 7a, the same fractals can be generated from
DIGITAL DIFFUSION AND WAVE EQUATION SYSTEMS
(7b)
145
X(n, m, t+At) = (X(n, m, t) + X(n-l, m, t+At) + X(n, m-l, t+At)) mod N
in several time steps given respectively by t=0 to 2p, p=0, I, 2, 3. This is possible by the symmetry property of this Sierpinski fractal, that is to say the self-similarity at scales 2p. Starting with an initial condition given a particular picture, instead of the black square, this three-dimensional equation 7b was already simulated (see Dubois, 1992, 1995). The basic evolution of the system is the multiplication of the initial picture given as initial condition through the whole frame with order/chaos transitions (the chaos transitions are obtained for odd time steps) and then their fusion by interference. The process is similar to a hologram but with the interference of the multiple identical pictures. Lets us show that the same holographic effect can be obtained directly in one time step from eq. 7 in taking, for example, the space steps At=l, At=15 and At=64 as shown in Figures 8b-c-d, the Figure 8- giving the initial picture.
C
Figures 8a-b-e-d: Holographic generation of fractals with chaos-order transitions
D. M. DUBOIS
146
The same method can be used for any other fractals. For example, from the following equation (8) X(n, m, t+At) = (X(n, m, t) + X(n-dn, m, t+At) + X(n, m-Am, t+At) + X(n-An, m-Am, t+At)) rood N
the square fractal is obtained for N=3. For N=2, with a black square as initial condition, the pattern is a uniform pattern of dimension D=-2. With an initial condition given by a black and white pattern with a certain symmetry, the Figures 9a-b-c-d give several simulations.
a
Figures 9a-b-e-d: Different patterns from eq. 8 with symmetrical initial conditions.
DIGITAL DIFFUSION AND WAVE EQUATION SYSTEMS
147
6. GENERATION OF FRACTAL INTERLACING Let us consider again eq. 7 (9)
X (n, m, t+At) = (X (n, m, t) + X (n-An, m, t+At) + X (n, m-Am, t+At)) mod N
with the following initial condition (gb)
X(n, 1, 0)=1 for n=An+l, An+2 ....and X(I, m, 0)=1 for m=Am+l, Am+2 ....
a Sierpinski interlacing can be generated as shown in Figure 10a, for p=q=3.
Figure 10a: Generation ofa Sierpinski interlacing.
This is really an interesting result because a Sierpinski gasket can be generated with a unique line (this line is given by successive 1 ’s and has sometimes one value 0 in changing of direction) travelling in the space (n,m). The fractal dimension of this interlacing is D=I for space scales (n, m) < (2p ,2q), and D=log3/log2 for larger scales. With the same method, many other fractal interlacing can be generated. Figure 10b was generated from the equation (10) X(n,m,t+l)=(X(n,m,t)+X(n,m-4, t+l)+X(n-4, m-4, t+l)+X(n-8, m, t+l)) mod 2 (10a) X(n, 1, 0)=1 for n=16, 17, 18 .... andX(1, m, 0)=1 for m=16, 17, 18 ... and Figure 10c, from the equation (11) X(n,m,t+l)=(X(n,m,t)+X(n,m-16, t+l)+X(n-12, m-16, t+l)+X(n-16, m-12, t+l) + X(n-16, rn, t+l)) mod 2 (lla) X(n, 1, 0))=1 forn--16, 17, 18 .... andX(l, m, 0)=1 for m=16, 17, 18 ....
148
D. M DUBOIS
Figure 10b: Fractal interlacing generated from equation 10 with initial condition 10a
Figure 10c: Pentagon fractal ~nterlacing generated from equation 11 wtth initial condition 1 la
The conclusion of this section on interlacing is that the local rule given by incursive equations give rise to a global pattern exhibiting only one simple line travelling in a very complex way through the whole space.
7. FRACTALS FROM DIGITAL DIFFUSION EQUATION Many physical, chemical or biological systems deal with diffusive reactions systems. Let us show that fractal can be generated from a digital diffusion equation and that the incursive diffusion equation can be algorithmically deterministic with a self-reference for the inputs def’med at future time. Indeed, let us consider the one dimension diffusion equation (12)
~x(s, t)/~t = a.x(s, t) + D.82x(s, t)/~s2
With a discrete forward time derivative, (13a)
Df(x(s, t))=((x(s, t+At)-x(s, t))/At
the discrete diffusion equation (12a) x(s, t+At)-(s, O =-a.dt.x(s, t) + D.dt.[x(s+As, t)-2.x(s, O + x(s-As, t)]/As2
gives unstable solution for integer parameters. In defining a diffusion difference equation modulo N, with a=0, D=I and At=As=l, the following "digital" diffusion equation gives a fractal pattern shown in figure 11 a (Dubois, 1996a) (12b) x(s, t+ l )=[-x(s, t) + x(s+ l, t) + x(s-1, t)]modN The term "digital" was proposed by Konrad Zuse (1982). With a=l, D=I and At=As=l, the following digital equation gives the Sierpinski fractal pattern given in Fig. lib (Dubois, 1996a) (12c) x(s, t+l) = [x(s+l, t) + x(s-1, t)] mod N
DIGITAL DIFFUSION AND WAVE EQUATION SYSTEMS S~
t=O t=l
01234567...
01234567...
00000000100000000 00000001110000000 00000010101000000 00000110101100000 00001000100010000 00011101110111000 00101000100010100 01101101110110110 i0000000100000001
00000000100000000 00000001010000000 00000010001000000 00000101010100000 00001000000010000 00010100000101000 00100010001000100 01010101010101010 i0000000000000001
149
Figure 11 a-b: Generation of fractal patterns by eqs. (10b-c) with N=2.
The numerical simulations of eq. 12b with N--2 and N=3 are given in Figures 12a-b. They give very special symmetries.
Figure 12~: Fractal generated from digital diffusion eq. 12b with N=2.
DIGITAL DIFFUSION AND WAVE EQUATION SYSTEMS
151
inclusive or implicit recursion" I proposed. Indeed, it is the inclusion of each iterate in the others which defines them in a self-reference way. In taking periodic boundary conditions x(0, t+l)=x(S, t+l) and x(S+l, t+l)=x(1, t+l), the system defines itself the values of the boundaries. For example, with a constant diffusion D=I and S=-3, we obtain, in inverting the matrix A, the equation system 14d. The simulation of this system in Figure 13 shows that the convergence is very rapid, a few time steps. I think that this example is a good one to explain "holistic" properties of self-referential system. (14d)
x(1, t+ 1) = x(1, 0/2 + x(2, t)/4 + x(3, t)/4 x(2, t+l) = x(2, 0/2 + x(3, t)/4 + x(l, t)/4 x(3, t+l) = x(3, 0/2 + x(1, 0/4 + x(2, 0/4 s=
t=O t=l t=2 t=3
1
1 ~ 6/16 22/64
0 ¼ 5/16 21/64
0 ~ 5/16 21/64
Figure 13. Simulation of self-referential diffusion equatton system 14d
Each automaton at time t+l is related to itself at the preceding time t and at the future time t+l and to its direct space adjacent neighbours at the future time t+l. Due to the self-reference of each automaton with its neighbours, it is possible to compute a new transformed recursive system where now each automaton is computed in function of itself only at the preceding time step but in function of all the automata of the system at the preceding time. This is really an important result which shows that an incursive holistic non-local property comes from local interaction dealing with a recursive system depending on future states. With boundary conditions as external inputs x(0, t+l) and x(4, t÷l ), the system becomes (Dubois, 1996a) (11 e)
x(l, t+l) = [8.[x(1, t) + x(0, t+l)] + 3.x(2, t) + [x(3, t) + x(4, t+l)]]/21 x(2, t+ I) = [3.x(2, t) + 9. [x(3, t)+x(4, t+ 1 )] + 3. Ix(l, t) + x(0, t+ 1)] ]/21 x(3, t+l) = [[x(3, t) +x(4, t+l)] + 3.[x(l, t)+x(0, t+l)]+8.x(2, t)]/21
We remark that the inputs are defined at the future time t+ 1 and they are present in the three equations. It means that the inputs at the boundaries are transmitted instantaneously, that is to say during the time step 1, to each automata, and that their effect are immediate because it is the same time step that the equations are computed. Why this phenomenon? As the movement equations are defined in the future time by the backward derivative, the inversion of the matrix A has the effect of mixing all the automata together at the present time t (and the inputs at the future time t+l) to compute their future values at time t+l. The inversion of the matrix A transforms a local incursive system to a non-local recursive system, that is to say a folding of each automaton to the other ones from the future time t+l to the present time t. It must be pointed out that classical mathematical analysis deals with derivatives defined for a unit time interval At tending to zero, so the backward and the forward derivatives
152
D. M. DUBOIS
are identical for derivable continuous systems. But the majority of physical systems seem discrete and not continuous, for example, the molecules of all actual systems are discrete entities and not a continuous medium. Newtonian mechanics is called the rational mechanics of the point, i.e. the dynamics of a Newtonian system are described by differential equations where all the objects are represented by sets of points without spatial dimension. It is well-known that Physicists have many mathematical problems with classical mathematical analysis due to the appearance of infinities. Maurice Jessel (1993) believed in the Non-Standard Analysis of Abraham Robinson as a first attempt to avoid this problem. He thought also that a Finality Principle should be established after the proposition of the French Nobel Prize winner Alfred Kastler. So any evolving system can be defined at any time step t by its backward derivative Ab(X(t)) or its forward derivative A~(X(t)): the forward derivative is related to the formal causation of Aristotle and the backward derivative, to the final causation, because the resulted equation takes into account the future time step t+At (Dubois, 1995). Final causation is a potential causation because it is not yet realised. Let us remark that we defined here an incremental final causation, changing at each time step. The principle of finality or teleonomy defined classically deals with the final value of a variable when the system reaches its stationary state, i.e. when it is no longer evolving but only developing. A mathematical model theory of evolving systems would be helpful. If evolving systems reach a stationary state (which can show a complex behaviour like a living system), the mathematical model can be simplified to deal only with its stationary state: in this case, no more information exists to explain how the system evolved to this state. It will be a developing system rather than an evolving system. The development of organisms from their birth to their death is a sub-class of the evolution of these organisms viewed at the level of species evolution. The interaction between a system and its environment can be viewed as a whole system (Bohm, 1987). The environment views the system and the system views the (external) environment. Anticipation seems to be the rule for physical structures in living systems: how can we justify the fact that mathematical models are recursive processes based only on past events? The anticipative nature of evolving systems is difficult to observe and to model because it is included in the recursive model of the system (some incursive control can be transformed to a recursive control, although not always), i.e. in the formal cause. It was shown with the hyperincursive field that it is not always possible to change, mathematically, a final cause to a formal or efficient cause (Dubois, 1996a).
8. FRACTALS FROM DIGITAL WAVE EQUATION Let us consider the one dimension wave equation
(15) ,~2x(s, t)/c~t~ = -w~.x(s, t) + c~O~x(s, t)/Os~ where x(s, t) is the value of the wave at position s at time t, c the velocity and w is the pulsation of oscillators. This differential equation can be replaced by the finite difference equation: (15a) x(s, t+At) - 2.x(s, t) + x(s, t-At) = -w~.Ata.x(s, t) + c~.At~.[x(s+As, t) - 2.x(s, t) +
x(s-As, t)]/As~
DIGITAL DIFFUSION AND WAVE EQUATION SYSTEMS
153
In taking As=l, At=l, c=l and w=l, the following digital wave equation is obtained [a similar equation was studied with w=0 in Dubois, 1996b]: (15b) x(s, t+l) + x(s, t-l) =-x(s, t) + x(s+l, t) + x(s-1, t) In taking the modulo N of this equation, the following digital wave equation is obtained: (15c) x(s, t+l) = [-x(s, t) -x(s, t-l) + x(s+l, t) + x(s-1, 0] mod N The Figures 14a-b show the numerical simulations of this equation for N=2 and N=3, with the initial condition (15d) x(255, 0) = +1 and x(257, 0)= -1 where s=l to 512 and t=l, 2 .... The first one gives a particular new ffactal and the second one is similar to a square fractal as shown previously in this paper.
Figure 14a: Fractal generated from the digital wave equation 15c with initial condition 15d for N=2 A new type of fractal is generated.
Figure 14b: Fraetal generated from the digital wave equation 15c with initial condition 15d for N=3. The fractal is similar to a square fractal, that ts to say a Sierpinski carpet.
154
D M. DUBO1S
9. CONCLUSION This paper deals with the concept and method of incursion and hyperincursion to model discrete systems. These are an extension of recursive processes where the computation of future time steps only depends on present and past steps. With incursion and hyperincursion, future time steps can be introduced to compute these future steps. Hyperincursion is an incursion when several future values can be generated at each time step. These can be interpreted as an incremental anticipation similar to the Aristotelian Final Causation. They are related to the definition of backward and forward time derivatives. The method of incursion is a powerful tool for modelling discrete systems. With the Fractal Machine, it is explicitly seen that the external inputs must be defined in the future time like a final causation which controls completely all the automata at the same time step in a holistic way. Indeed the inputs are present in each automata at the same external time. It is impossible to transform external inputs defined in the future time t+l to inputs defined in the present time t. In this, we can say that we are dealing with a strict incursive system. Thus the final causation is really the 4th causation which must be taken into account in systems modelling as Aristotle had proposed. It seems also impossible to construct a real working engineering system where real working external future inputs would control its current present state. But it is possible to de/me internal future inputs in considering self-reference systems. The Fractal Machine can become non deterministic or non algorithmic, what I suggest to call an HYPERINCURSIVE FIELD where uncertainty (indecidability) or contradiction (exclusion principle) occur. It was shown that the incursive diffusion equation can be algorithmically deterministic with a self-reference for the inputs defined at future time by space periodic conditions. As the movement equations are defined in the future time by the backward derivative, the recursive transformation has the effect of mixing all the automata together at the present time t in view of computing their future values at time t+l. The transformation of a nonlocal incursive system to a local recursive system leads to a folding of each automaton to the other ones from the future time to the present time. Several fractals were simulated from incursive automata giving rise to square and pentagon symmetries. An interesting case was given by the generation of fractal interlacing represented by only one travelling line in the space. It was shown that fractals can also be generated from the digital diffusion and wave equations in using the modulo N of the finite difference equations. Finally, the concepts of incursion and hyperincursion can be related to the theory of hypersets which are defined as sets containing themselves. This theory ofhypersets is an alternative theory to the classical set theory which presents some problems as the incompleteness of G6del: a formal system cannot explain all about itself and some propositions cannot be demonstrated as true or false (undecidability). Fundamental entities of systems which are considered as ontological could be explain in a non-ontological way by self-referential systems.
REFERENCES Aczel, P. (1987) Lectures in nonwellfounded sets, CLS1 Lecture Notes n° 9 Barwise, J., Moss, L. (1991) Hypersets, in the mathematical intelligencer, 13, 4, pp. 31-41. Bohm, D. (1987) Wholeness and the Implicate Order, La plenitude de l’univers (French translation), Ed du Rocher, I’Espnt et la Matin:re
DIGITAL DIFFUSION AND WAVE EQUATION SYSTEMS
155
Dubois, D. (1990a) Self-organisation of fractal objects in XOR rule-based multilayer networks, In’ Neural networks &thetr apphcattons, Neuro-Nfmes ’90. Proceedings of the 3ra lnternattonal Workshop, edited by EC2, pp. 555-557. Dubois, D. (1990b) Le labyrmthe de l’tntelhgence: de l’tntelhgence naturelle ~ l’tntelhgence fractale, 2ride ~:dition, lnterEditions/Paris, Academia/Louvain-la-Neuve, 331 p. Dubois, D. (1991 a) Mathematical fundamentals of the fractal theory of artifictal intelhgence, Communlcatzon and Cogmtton -Artifictal lntelhgence, 8, 1, pp. 5-48 Dubois, D. (1991b) Fractal algorithms for holographic memory of ~nvers~ble neural networks, Commumcatton and Cogmtton - A rtifictal 1ntelhgence, 8, 2, pp. 137-189. Dubo~s, D. (1992a) The Fractal Machine, Presses Umvers~taires de Lt~ge, 375 p. Dubois, D. M. (1992b) The hypermcurs~ve fractal machine as a quantum holographic bra~n, desigmng new intelligent machines (COMETI" programme), Communtcatton& Cognttton -Arttfictal lntelhgence, 9 (4), pp. 335-372. Dubois, D. (1993a) Hyperincursivity: reclusive recursivity w~thout time arrow, In: Proceedings of the 13th International Congress on Cybernettcs, Namur, 1992, ~dit6 par I’Associat~on Internationale de Cybem~t~que, pp 152-156. Dubois, D. (1993b) The fractal machine" The wholeness of the memory chaos, In: Proceedings of the 131h lnternattonal Congress on Cybernetics. Namur 1992, ~d~t~ par I’Associat~on lnternationale de Cybem~t~que, pp. 147-151. Dubois, D. M. (1993c) Les idles, toujours d’actualit~, de l’inventeur du premier ordinateur, le Dr lr K ZUSE, in: Actes du Colloque International, Htstoire de l’lnformattque, Sophia Anttpolis 13-15 octobre 1993, France, ~dit~ par I’Institut de Recherche en Informatique et en Automat~que, 20 p. Dubo~s, D. (1995) Total incursive control of hnear, non-linear and chaotic systems, In: Proceedmgs of the 7th lnt. Conf on Systems Research. lnformattcs and Cybernettcs, Baden-Baden, August 15-21. 1994. In G. Lasker (ed.): Advances in Computer Cybernetics. Int. Inst. for Advanced Studies in Syst. Res and Cybernetics, vol. II, 167-171 (1995) Dubois, D. M (1996a) Introduction of the Aristotle’s final causation in CAST: Concept and method of incursion and hyperincursion, In: Pichler, F., Diaz, R M., Albrecht, R., eds., Computer Aided Systems Theory - EUROCAST’95. Lecture Notes in Computer Science, 1030, Springer-Verlag Berlin Heidelberg, pp. 477-493. Dubois, D. M (1996b) A semantic logic for CAST re~ated to Zuse, Deutsch and McCulloch and Pitts computing pnnciples, In: Pichler, F., Diaz, R M., Albrecht, R., eds., Computer Aided Systems Theory EUROCAST’95. Lecture Notes in Computer Science, 1030, Springer-Verlag Berlin Heidelberg, pp 494-510 Dubois, D. and Resconi, G. (1992), HYPERINCURSIVITY. A New Mathemattcal Theory, Presses Universitaires de Liege, 260 p. Dubois, D M. and Resconi, G. (1993a) The hyperincurs~ve fractal machine beyond the turing machine, In. Proceedmgs of the 4th lnternattonal Sympostum on Systems Research and Cybernettcs, Baden-Baden, edited by G. Lasker, 5 p. Dubo~s, D. and Resconi, G. (1993b) Introduction to hyperincursion w~th applications to computer science, quantum mechanics and fractal processes, designing new intelligent machines (European COMETI~ programme), Commumcation & Cognition - Artificial lntelhgence, 10 1-2, pp 109-148 Dubo~s, D. and Resconi, G. (1994) Hohstic control by incursion of feedback systems, fractal chaos and numerical instabilities, In: CYBERNETICS AND SYSTEMS’94, edited by R. Trappl, World Scientific, pp.71-78 Dubois, D. and Resconi, G. (I 995) Advanced Research tn lncurston Theory Applied to Ecology, Physics and Engineertng, COMETT European Lecture Notes tn lncurston, ~d~t~ par I’A.I.Lg,, Ed. by A.I.Lg, Association des Ing~nieurs de I’Umversit~ de Liege, D/1995/3603/01, 105 p. Dubo~s, D and Resconi, G. (1994) Hyperincursive fractal machine beyond the tunng machine In Lasker, ed., Advances in Cognitive Engineenng and Knowledge-based Systems. Int. Inst. for Adv. Studtes in Syst. Res. and Cybernetics, 212-216.
156
D. M. DUBOIS
Feynman, R. (1982) Simulating physics with computers, lnternattonal Journal of Theorettcal Phystcs, 21, 67, pp. 467-488. Gardner, M. (1971) Mathematical games on cellular automata, self-reproduction, the garden of Eden and the game life, Scientific American, February 1971, pp. 112-117. Hasslacher, B. (1988) Beyond the tunng machine, In: Herken, R., ed., The Universal Turing Machine. A Half-Century Survey, Oxford: Oxford University Press, pp. 417-43 I. Jessel, M. (1973) ,4cousttque thdortque, propagatton et holophonte, Masson et Cie. Jessel, M. (1993) Une math~matique non standard au service de la cybern~tique et des autres sciences de la nature, in: Proceedings of the 13th International congress on cybernetics, Namur 1992, ~dit~ par I’Assoctation Intemationale de Cybem6tlque, p. 165 Levy, H., Lessman, F (1959) Ftmte Difference Equations. Pitman, London. Mandelbrot, B. (1982) The Fractal Geometry of Nature, Freeman. Marcer, P. and Dubois, D. (1993) An outline model of cosmological evolution, In: Proceedings of the 13th International Congress on Cybernetics, Namur 1992, ~dit6 par I’Association Intemationale de Cybem6ttque, pp. 157-160. Peltgen, H.-O., Jtlrgens H., Saupe, D. (1992) Chaos ans Fractals" New Frontiers of Sctence, SpringerVerlag. Pithier, F, Scwartzel, H., eds. (1992) CAST Methods in Modelhng: Computer A tded Systems Theory for the Destgn of Intelligent Machines Springer-Verlag. Pribram, K. (1971) Languages of the Brain, Prentice Hall. Rosen, R. (1986) Causal structures in brains and machines, International Journal of General Systems. 12, pp. 107-126. Rosenblueth, A., Wiener, N., Bigelow, J (1943) Behawor, purpose and teleology, Phdosophy of Sctence, 10, 18-24. Scheid, F. (1986) Theory and Problems of Numerical Analysts, McGraw-Hall Inc. Schroeder, M. (1991) Fractals, Chaos, Power Laws " Minutes from an Infinite Paradise, New York: W. H. Freeman and Company. Van de Vijver Gertrudis (1992) ed., New Perspecttves on Cybernettcs, Synthese Library, Vol. 220, Kluwer Academic Publishers. Von Glasersfeld E. (1990) Teleology and the concepts of causation, In: G. Van de Vijver, ed., Self-organizing and Complex Systems, Ph~losophica 1990, 46, pp. 17-43 Weisbuch, G. (1989) Dynamtque des systkmes complexes: une mtroduction aux rdseaux d’automates, InterEditions/Editions du CNRS Wolfram, S. (1994) Cellular Automata and Complextty- Collected Papers, Addison-Wesley Publishing Company. Zuse, K. (1969) Rechneneder Raum, Friedr Vieweg + Sohn-Braunschweig, 70 p. Zuse, K. (1970) Der Computer mein Lebenswerk, Verlag Modeme Industrie, 221 p. Zuse, K. (1982)The computing universe, lnternattonalJournal ofTheorettcal Phystcs, 21, 6/7, pp. 589-600. Zuse, K. (1993) The Computer - My Life, Spnnger-Verlag, 245 p Zuse, K. (1994) Discrete mathematics and rechnender raum (computing space) - Part 1 - cellular structured space (rechnender raum) and physical phenomena - Part 2 -, Konrad-Zuse-Zentrm far lnformattonstechntk, Berlin. Technical Report TR 94-10.
Symmetry: Culture and Sctence Vol. 8, No. 2, 157-175, 1997
SYMMETRY." CULTURE AND SCIENCE
INFORMATION PROCESSING AND SYMMETRYBREAKING IN MEMORY EVOLUTIVE SYSTEMS Andr6e C. Ehresmann and Jean-Paul Vanbremeersch Address: Facult~ de Math~mat~ques et d’Informat~que, 33 rue Saint-Leu, F-80039 Amiens, France; E-mail:
[email protected] fr
Abstract: The aim of this paper is to evaluate the role of symmetry and symmetrybreaking processes on the complex information processing developed by hierarchical evolutionary natural systems, such as biological, neural, social or cultural systems. The study is conducted in the frame of the Memory Evolutive Systems, which give a mathematical model of these systems. The dynamics of a MES is modulated by the competition between a net of internal regulation centers, which act apart but encode overlapping strategies which have to be equilibrated The main characteristics of these systems, at the root of their complexity and adaptability, is a symmetry-breaking in the passage from a higher (or macro) level to a lower (or micro) level: several disparate subsystems with different comportments at the micro level can be undistinguishable at the higher macro level because of a similar macro behavior (Multiplicity Principle). It is responsible for the development of a dialectics between heterogeneous regulation centers, and for the emergence in time of more and more complex objects. An application to neural systems vindicates an emergentist dynamical reduction of mental states to physical states.
1. INTRODUCTION For the Webster, symmetry is defined as "Proper proportion in the arrangement of parts; correct balance between the two halves of an outline; graceful proportioning; the beauty of harmonious arrangement of parts". The mathematical notion of symmetry is more precise: It supposes given a set of objects on which acts a group G of transformations, and two objects of the set are said to be symmetrical (for G) if there exists a transformation in G transforming one into the other. In particular spatial symmetries around a point, or a line, or a plane, are ubiquitous in Art, and are also taken into account for taxonomic classifications. In complex evolutionary natural systems, such as biological, neural, social or cultural systems, symmetry and symmetry-breaking occur in a multitude of different ways, at a structural, functional and temporal level. They modulate the evolution of the system and allow for a complex information processing, not relying on a convention between an emittor and a receptor, but distributed over a variety of overlapping and possibly conflicting internal regulations. Classical physical models, tailored for a specific process in a well-delimited environment, are not flexible enough to deal with these processes, though chaos theory allows for some symmetry-breaking.
158
A. C. EHRESMANN, AND J. P. VANBREMEERSCH
The aim of this paper is to show how these processes can be studied in the frame of the Memory Evolutive Systems (or MES, introduced in preceding papers, e.g., Ehresmann and Vanbremeersch 1987, 1991, 1996), which give a mathematical model for selforganized hierarchical evolutionary systems in which the global dynamics follows from a balance between competing regulations. Each regulation depends on a particular internal organ, modeled as a sub-system of the system called a Center of Regulation (CR). The components, or actors, of a CR have the same complexity level, and cooperatively develop a stepwise more or less cyclic process with its own time-scale. In this process, the CR acts first as a filter and decoder of informations to form its landscape, which represents its internal model of the system; then it selects a strategy on this landscape to give the best answer at its level, and finally it encodes commands to effectors to realize this strategy; its choice is helped by a differential recourse to a central memory in which past experiences are stored. We prove that one of the main characteristics of such complex systems, which give them their flexibility and adaptability, is the existence of ’multifold’ objects, that is an object of a higher level which has several possible lower level decompositions in sub-systems with disparate comportment at this level, though they cannot be distinguished at the higher level (Multiplicity, or Degeneracy, Principle). It entails a symmetry-breaking in the passage from a higher to a lower level, and is responsible for the emergence in time of objects with increasing complexity order. It has also a strong influence on the dynamics of the MES which is modulated by the cooperation and/or competition between the various CRs, specially on the equilibration process between their strategies to attain a global coherence, possibly with fractures for some CRs. It leads to a dialectics between CRs with disparate com.plexity level and time-scales, and may cause evolutionary symmetry-breakings m the form of de/resynchronisations of the CRs. Though developed in a different frame, these results have strong convergences with the "Physics of becoming" proposed by Matsuno (1989, 1994). They can also be compared with the features by which Farre (1994) characterizes evolutionary systems, or with the mechanisms singled out in the C8 taxonomy of Chandler (1991; cf. also Chandler, Ehresmann and Vanbremeersch 1995).
2. GLOBAL DESCRIPTION OF AN EVOLUTIVE SYSTEM We first give an ’external’ description of an evolutionary system. 2.1. Systemist Models To model a system consists in organizing the informations that an observer (the modeler) can obtain on the system and its dynamics. In Systems theory, the informations taken into account are related to the class of its components (individuals or parts of the system) considered as more or less stable, and to the interactions between them and their temporal variation, generally measured thanks to quantitative observables (e.g., distance, or attraction forces, or energy constraint...). Geometrically, these data can be modelled by a graph: the nodes N of the graph represent the components of the system and the arrows represent interactions between them;
SYMMETRY AND INFORMATION 1N MES
159
we may have none, one or several arrows from N to N’, and also loops from a node to itself. Though it is often supposed that the interactions between components are symmetrical, here they are oriented, and graph is always taken in the sense of such an oriented polygraph. The elements of this graph can be labelled by some observables. In most models, it is assumed that the organization represented by the graph remains the same during all the period on which the system is observed, and the dynamics is entirely described by the variation of the labelling observables, which can often be computed by solving some differential equations. If there is only a finite number of nodes N, and at most one arrow between two nodes, the labelled graph can be replaced by the matrix of transitions with indices i corresponding to the nodes and in which the (i,j)-term is the label s,j(t) of the arrow from N, to N~ if it exists, and is 0 otherwise. For instance in the Ising model inspired from Thermodynamics, the nodes are assimilated to spins localized on the sites of a lattice, with s,,(t) taking two possible values 0 or 1 (’up’ or ’down’), and the interactions between two spins are measured by the energy necessary for correlating their behavior. This model has been applied in the most disparate systems, from ferromagnets to cormexionist neural networks or economy problems. 2.2. Evolutive Systems: The State-Categories However in the complex autonomous evolutionary systems which we consider, we cannot restrain the study to transformations on a well-fixed (high-dimensional) space. Indeed, these systems are ’open’ in the sense that they have external exchanges, and their organization varies in time, with possible destruction or rejection of some components, and emergence of new elements, either taken from the environment or internally generated. The Evolutive Systems (or ES) introduced in (Ehresmann and Vanbremeersch 1987) give a model for such systems, based on Category Theory (cf. Mac Lane 1971), a recent domain of Mathematics. An ES describes the successive states of the system at each date of a given time-scale, and the transformations between them; the time-scale can be a sequence of reals, or an interval of the real line. (Thus it describes the closure and the conformation of the system in the sense of Chandler 1991 .) The state of the system at a date t of the time-scale is modeled by a category, called the state-category at t. A category is a graph on which there is given some ’transitivity’ of interactions, generated by a law associating to two consecutive arrows, say f from N to N’ and g from the same N’ to N" a well-determined arrowf.g from N to N", called their composite. This composition law must satisfy two axioms: the first to ensure that a path of consecutive arrows gives by composition the same unique arrow in whatever way it is 2-2 decomposed (associativity); the second to ensure that each element has an ’identity’ arrow.
The state-category embodies all the informations an observer with a complete view of the system could get on it at the date t. The nodes (called objects) of this category represent all the components of the system as they appear at t, and the arrows (called links) from N to N° represent the various relations between them at this date. Some of these links are more or less invariant ’structural’ links such as causal or topological relations (e.g., desmosomes between 2 contiguous cells, or canals through which informations can be transmitted from N to N’, as a synapse between two neurons), whatever be the nature of these informations: spatial, temporal or energy constraints, communication exchanges or commands... Other links are more labile connections, representing a temporary interaction between two objects. The composition law, extended to paths, determines which paths represent equivalent information canals between the two extreme objects, so that it
160
A. C. EHRESMANN, AND J P. VANBREMEERSCH
delimits the constraints on information processing. 2.3. ES: The Transition Functors The above category models the state of the system at t. The change of states from t to a later date t’ will be modeled by the transition functor from this category to the statecategory at t’. (A functor is a map which is compatible with the graph structure and the composition law.) The transition functors must satisfy the transitivity of changes: if t < t’ < t", the transition functor from t to t" is the composite of the transition functor from tto t’with that from t’to t". For an observer, the transition functor gives informations on the becoming of the components between the two dates. Let us explain this, because it is different from usual models. When we speak, say, of a cell in an organism, we consider that it remains ’the same’ cell at different times. Here this cell will be represented not by one object, but by the sequence of all the objects in the different state-categories which represent its successive states; if we denote C, (the object representing) its state at a time t of its life, these objects are connected by the fact that C,, is the image of C, by the transition functor from t to t’. In other terms, the cell as a unit is represented not by one of its states, say C,, but by the trajectory of C~ formed by its successive states. Naturally this trajectory supposes that the observer can ’transcend’ the present and compare successive states to recognize symmetrical occurences of an object in different state-categories (which might not be possible internally). Depending on the context and when no confusion might arise. the terms component or object may refer either to a particular state or to the whole trajectory. Now the cell is destructed at some date s. To represent the possible disappearance of elements or links in time, we suppose that the state-categories possess an objet 0 on which are mapped all the components having been suppressed (destructed or rejected out of the system). Thus the death of the cell at s will be represented by the fact that its state C.~ becomes 0 while its preceding states were not 0. The emergence of new components (coming from the environment, or internally generated) is also revealed by the transition functors. Indeed, there may exist objects in the state-category at t’ which are not the images of any object by the transition functor from t to t’; such objects have emerged at some date between t and t’. The data of the time-scale, of the state-categories at its different dates, and of the transition functors between them is called an Evolutive system. 2.4. Ponderations on a Category The above representation of a system is geometrical but not quantitative. In concrete applications, quantifications are introduced by labelling the links of the state-categories by real numbers (or possibly by vectors), called their weights (e.g. strength of the link, its capacity as an information canal, its propagation delay...). Then we say that the category is pondered if there is given a law to deduce the weight of a path from the weights of its factors (most often, the weight of the path is the sum, or the product, of those of its factors). The weights may be invariant of the structure, or may be function of time. By analogy with the case of a canal activated when an information is transmitted, we say that a link is activated at a time t if its weight increases at t, and quiescent if it does not vary.
SYMMETRY AND INFORMATION 1N MES
161
A pondered category is often constructed from an initial labelled graph G as follows. We first extend the graph by adding all the paths of G, such a path being weighted by the sum of the labels of its factors. We so obtain the category of paths if we define the composite of two successive paths by their concatenation. In this category there may exist several arrows from N to N’ with the same weight, i.e., which transmit an activation from N to N’ in exactly the same way. This is redundant on a functional perspective. So we will identify two paths with the same extremities and the same weight. After this identification, we obtain a new category in which the objects are still the nodes of G, but in which an arrow from N to N’ is a class of paths with the same label. Many structural (or ’geometrical’) properties of the system will be deduced from a study of this category, forgetting how it has been constructed and the weights of the arrows, these properties depending only on which paths have the same composite, hence play symmetrical roles to transfer informations. Let us remark that this identification of paths is an essential step for it completely modifies the internal organization of the category. (In particular it allows for the existence of colimits for non trivial patterns, which is very important in the sequel, but is not the case in a category of paths as proved in Ehresmann, forthcoming.) For instance the category of neurons is obtained by such a construction from the graph with nodes the neurons and arrows the synapses (from the pre-synaptic neuron to the post-synaptic one), the label representing the force of the synapse, related to its manner to transmit the influx from one neuron to the other.
3. DIFFERENT MEANINGS OF INFORMATION IN AN ES In an ES, informations will be processed along the links of the state-categories. However the word ’information’ must be given a precise definition in this context. 3.1. Classical Information Theory The usual meaning of information is "communicated knowledge or news", and so it supposes some intention in the process. In mathematical theories of information, of which the theory of Shannon is the best known, we have an emittor which encodes a message, a canal which transmits the coded message, and finally a receptor decoding the resulting signal. The message can correspond to the transfer of some material object (as a string of letters, or a quantity of motion), or to a modification of the state of the receptor, e.g., by a wave or a vibration. Its information content depends on a statistical evaluation of the probability of each signal used. The aim of the theory is to study the number of bits of information which will be faithfully transmitted in the process, without any regard for the meaning of this information. This pre-supposes some convention between emittor and receptor determining the class of possible messages. Without such a convention, any message would appear as pure hazard, and it is proved (Benzecri 1995) that the practical establishment of an efficient code between actors through action is a complex process. This theory can be interpreted in a subjective perspective: what is the information gain for the receptor, measured by the difference between its incertitude with respect to the message before the reception, and its incertitude after reception. More objectively, i.e., for an external observer knowing the codes, the information content of the message is its specificity, that is the difference between the variety of all possible messages and the sub-variety of those messages which can lead to the received signal.
162
A. C EHRESMANN, AND J. P. VANBREMEERSCH
3.2. Information through Action But the notion of information processing which is at the root of the dynamics of complex ES such as biological, social or cultural systems is much more general, in that it resorts of an ’action language’, in which the mirror symmetry of receptor and emittor through their common convention is destructed. While usual theories are concerned with the correct material transmission of a message between two actors, here the importance of messages, be they endogeneous (commands, constraints, spontaneous oscillatory processes) or caused by external perturbations, comes from the interactions they generate or in which they participate and the feedbacks they trigger. The same message can be decoded by several actors in a different way; for instance, a color center and a shape center in the brain ’decode’ different informations from the same visual target. Conversely, several messages can be decoded as the same signal, e.g., different blue objects are not separated by a color center. In our categorical model, information transfers are effected through the links between objects. At a date t, an object N can receive informations only through the links to N which are activated at t ; these links form what we call the reception field ofN. It may send informations to other objects by activating its links toward them, which form its operation field. (These two fields are exchanged by the mirror-symmetry which replaces a category by its opposite category obtained by inverting the direction of each link; but in natural systems this formal operation has no meaning.) In any case the transmission of a message depends on the structure of the link, on its weight (to measure the amplitude or the frequency of the signals), and on its propagation delay which determines the transmission delay. But the activation of a link is not sufficient to speak of information. 3.3. Information Transfer in an ES
The activation of a link from N to N’ represents an information transfer only in the following cases: -- If N’ receives the signal (a letter can be lost, or written in an unknown language!), decodes it and takes it into account in its subsequent action. The response can be immediate, or delayed, or opposed to the message, or even a non-response (the differentiation of a cell consists in omitting to express some genes). The signal may be non-intentional, such as the traces left by a prey which reveal its presence to a predator, or some instability in a cell, as a change in an oscillatory process, which is transmitted to contiguous cells. For a cell, the signal can also result from the diffusion of a product (say, an hormone) secreted in a far off cell and which diffuses through the circulation or the conjonctive tissue, after a more or less long delay. -- If the emittor N sends a message (constraint, command...) with the specific objective to modify the action of N’, and later on receives some feed-back of this action. It is specially the case in human societies, where the sending and reception of the message can be intentional. But natural selection has led to the development of organisms able to send instinctive signals which are decoded as messages by other organisms, possibly after modification during their transfer from the source to the receiver (e.g., emission of pheromones by a female insect to attract the male).
SYMMETRY AND INFORMATION IN MES
163
-- If there has been established (intentionally or not) some convention between N and N’ for a real transfer of informations between them. In this case, we can speak of a communication exchange. But this ’communication’ can be an artefact caused by an external observer, who detects that a particular signal determines a correlation between some state of the emittor and some state of the receptor.
4. HIERARCHICAL SYSTEMS In the definition of a category, all the objects seem to play a symmetrical role. But in a complex natural system, an observer can distinguish all a hierarchy of types: individuals, groups of cooperating individuals able to perform a task that they could not do separately, groups of groups, and so on. For instance, in an organism: atoms, molecules, cells, tissues. How can we distinguish such a hierarchical structure in the purely categorical framework? 4.1. Patterns of Linked Objects Any cooperation between components of a system requires a possibility of exchange of informations between them. We have said that such exchanges are done through the links of the system. Thus a group of cooperating objects will be represented by a pattern in the state-category. A pattern consists of a family of objects N, , and some distinguished links between these objects. A collective link from the pattern to an object N’ is a family of individual links f from N, to N’, correlated by the distinguished links of the pattern. Such a collective link models a common action (emission of a message, constraint, command,...) performed by the pattern as a whole, and which could not be realized by the objects of the pattern acting separately. The cooperation can be temporary, as in a group of people who decide to unite to realize a particular task. But if it lasts for a long period, their association can take an identity of its own, and become institutionalized in the system by the formation of a higher order object N (their concatenation in the sense of Chandler 1991), which ’binds together’ the pattern and such that the links of N to any N’ correspond to the collective links of the pattern. 4.2. Binding of a Pattern into its Colimit Categorically, this binding object will be modeled by the colimit (or cohesive binding) of the pattern. The colimit (cf. Mac Lane 1971) of a pattem is an object N such that there is a canonical collective link (c,) from the pattern to N, and each collective link (f) from the pattern to anyN’ binds into a unique linkffrom N to N’ (in formula: fi= c,.f for each i ). The colimit does not necessarily exist, but if it exists it is unique (up to an isomorphism). For instance, a molecule is the colimit of the pattern formed by its atoms with the distinguished chemical bonds which determine its spatial configuration. Roughly, in the colimit the degrees of liberty of the objects are freezed to ensure a better cooperation along their distinguished links. The situation can also be seen ’upside-down’: an object N which is the colimit of a pattern can be considered as a complex object admitting the pattern as an internal organization into more elementary components, and the pattern will be called a decomposition of N. But while the pattern univocally determines its colimit (if it exists), the inverse is not true because the same object may have several (non-equivalent) decompositions, so that
164
A. C. EHRESMANN, AND J. P. VANBREMEERSCH
the map associating to a complex object its decompositions is one-to-many. We’ll come back on this important property later on. For instance, the same amino acid can be obtained from different codons (degeneracy of the genetic code). 4.3. Limit of a Pattern Symmetrically, the pattern can act collectively to decode some messages which could not be decoded by its objects separately. We define a common message from an object M to a pattern as a family of individual links from M to the different objects of the pattern, which are correlated by the distinguished links of the pattern. And an object representing the pattern in its capacity of acting as a common decoder will be represented by the limit L of the pattern (defined in the same way as the colimit, but by inverting the arrows).
Though the same pattern can operate collectively as an emittor of messages, or as a receptor, there is a symmetry-breaking in the way its distinguished links operate in both cases, and it is revealed by the fact that the higher objects which represent it in these two roles, namely its colimit and its limit, are different. It follows that several patterns which play symmetrical roles as emittors (they have the same colimit) have different (nonisomorphic) limits, thus play non-symmetrical roles as receptors. 4.4. Hierarchical Systems An ES is called a Hierarchical ES (or HES, cf. Ehresmann and Vanbremeersch 1987) if each state-category is hierarchical in the following sense: the objects are divided into a finite number of complexity levels, numbered 0, 1, ..., rn, so that each object N of a level n+ 1 is the colimit of at least one pattern P of linked objects of the lower level n. Then N has also a complex organization as an iterated colimit of lower levels. Indeed, each object of P is itself the colimit of a pattern of linked objects of the level n-l, so that the links from N to any object can be deduced by an iterative process from the collective links of these lower level patterns; we say N is a 2-iterated colimit, or admits a 2ramification based on the level n-1. And so on down the ladder, to the 0-1evel. However the hierarchy is not purely ascending. The different levels are intertwined, with interlevel and intralevel interactions from lower to higher levels, and vice-versa. An object can receive informations simultaneously from objects of any level, and in response send messages to any level, as long as the energy constraints are satisfied. For instance, an enterprise has such a hierarchical organization with several levels: The objects of level 0 represent its employees. More complex objects represent departments, from small producing units affected to a specific task, to higher directorial levels. The links between the members of a department represent the canals by which they exchange informations and collaborate to achieve a common goal. The hierarchy is intertwined, because higher levels can send orders to lower levels, but at the same time they can depend on some work done by the lower levels. The construction of a machine is stopped if some material necessary for its construction is not produced in sufficient quantity by a lower-level unit.
SYMMETRY AND INFORMATION IN MES
165
5. SYMMETRY-BREAKING BY PASSAGE TO A LOWER LEVEL We have said that a complex object has several decompositions in more elementary objects. We are going to precise this situation and indicate some of its important consequences. 5.I. Simple Links Binding a Cluster In a HES, there are some links between complex objects which are deducible from a lower level. Indeed, let N be the colimit of a pattern P and N’ the colimit of a pattern P’. A cluster from P to P’ is a family of individual links between the components of P and P’, correlated by the distinguished links of the two patterns, so that this cluster binds into a link from N to N’, called a (P, P3-simple link. These simple links only transmit informations mediated through individual members of the patterns. A composite of simple links binding adjacent clusters is still a simple link. In particular, two decompositions P and Q of N are said to be equivalent if the identity of N is a (P, Q)-simple link, so that P and Q are connected by a cluster. If moreover Q is a sub-pattern of P, we say that Q is a representative sub-pattern of P. Roughly it means that all the collective actions of P are entirely determined by those of Q, the elements of P not in Q being ’constrained’ by the comportment of those in Q. An example is given by the Representatives of a nation (whence the terminology). The notion of a representative sub-pattern is used to define the stability span of an object (cf. Ehresmann and Vanbremeersch 1987). 5.2. Stability Span of a Complex Object We have seen (cf. 2.3.) how, in an ES, we can keep track in time of a component by its trajectory. Now if the component is a complex object N of level n+l, its internal organization of level n at t may be different from its later organization; for instance the ’same’ cell perdures though its different components are renewed in time. If we know a decomposition P of N at t, how can we pass from P to a decomposition P’ of(the state of) N at a later date t’ so that we may recognize directly at the level n that P’ is still a decomposition of the ’same’ object? For this, the change in P must be smooth enough, and this ’smoothness’ is measured by defining the span of the decomposition at t : It is the longest period during which there exists a representative sub-pattern of P whose successive states remain a decomposition of the successive states of N. The stability span of N at t is then defined as the lowest upper bound of the spans of its different decompositions at t. This stability span is related to the renewal and degradation rates of the components of N, and N preserves its complex identity as long as this stability span does not become too small. For an organism, the decreasing of the stability span denotes its aging. The fact that the stability span of N is less than its lifetime emphasizes that, on the long term, there is an evolutionary symmetry-breaking in the passage from a level to a lower level, since the stability of an object during its evolution at its level is not reflected in a stability of its organization of the lower level.
166
A. C. EHRESMANN, AND J. P. VANBREMEERSCH
5.3. Multifold Objects Complexity can also cause another type of symmetry-breaking in the passage to a lower level, more ’structural’ since it is observable at a given date as follows. A complex object generally admits several equivalent decompositions. However there may exist objects N of level n+l at t admitting also non-equivalent (i.e., non connected by a cluster) decompositions of level n in the state-category at t. Such an object is called a multifold object (Ehresmann and Vanbremeersch 1995). If two patterns represent equivalent decompositions of N, this can be recognized directly at the level n of these decompositions by the existence of a cluster of links between their components. But for non-equivalent decompositions there is no possibility to recognize at their level n that they act as symmetrical emittors; this symmetry emerges at the higher level where only their collective comportment is taken into account. Thus there is a symmetry-breaking in the passage from the ’macrolevel’ n+l (where the patterns are binded into the same object) to the ’microlevel’ n where the patterns are not connected (in particular they do not act symmetrically as receptors). For instance 2 different codons of the same amino acid seem unrelated at the atomic level, though they give rise to the same molecule at the molecular level. Or the two possible images in an ambiguous figure gain their ’symmetry’ only when they are interpreted in relation to the complete figure, not when apprehended separately. By analogy with the entropy of a gaze in Thermodynamics, we define the n-entropy of a multifold object of level n+l as the number of(classes of) its non-equivalent decompositions of level n, thought of as its different possible ’microscopic’ states. It is a measure of its structural (or ’horizontal’) complexity.
6. MULTIPLICITY PRINCIPLE AND COMPLEXIFICATION 6.1. Multiplicity Principle We say that a HES satisfies the Multiplicity Principle (or MP) if there exist multifold objects of level n+l, and if, conversely, an object of level n can be a component of several objects of level n+ 1 . As shown above, the first property entails a symmetry-breaking in the passage to a lower level, that might be thought of as a degeneracy (as we say that the genetic code is degenerated). In fact the MP has initially been called the degeneracy principle, following a terminology introduced by Edelman (1989) in the case of neural systems. The MP allows for much flexibility in the comportment of the system, and that makes it one of the main characteristics of complexity for evolutionary systems. For instance, it is related to the well-known though not easily understood scaling and universality properties, which entail that disparate complex systems behave similarly near their critical points though their microscopic comportment is very different. Indeed, near a critical point, all these systems are represented at the macrolevel by a same multifold object.
SYMMETRY AND INFORMATION 1N MES
167
6.2. Complex Links In a HES with no multifold objects (so that the MP is not satisfied), a composite of simple links is still simple. But this result does not extend to the HES which satisfy the Multiplicity Principle. Indeed, in such a system, there may exist non-simple links obtained by composing a path of two (or more) simple links which bind non-adjacent clusters; they are called complex links (Ehresmann and Vanbremeersch 1995). More precisely, the composite of a simple linkffrom N to a multifold object M of level n+l, with a simple link g from M to N’ must exist. However, since M is multifold, it admits non-equivalent decompositions of level n, so that f may bind a cluster from a decomposition P of N to a decomposition Q of M, while g binds a cluster from a nonequivalent decomposition Q’ ofMto a decomposition P’ of N’. In this case, the composite off and g is generally a complex link. For instance, the link from the group of authors of a Journal to the group of its subscribers is a complex link, mediated by the Journal as such, considered as a multifold object representing both its editorial staff and its administration. Though a complex link of level n+l is not simple, it can partially been handled at the lower level n, through the links of the clusters that its factors, sayfand g, bind, except for the ’switch’ between the two non-equivalent decompositions of the intermediate object M; this switch can be recognized as such only at the level n+l, where a symmetry between the two non-equivalent decompositions is generated. A composite of complex links is generally a complex link (though it might exceptionally be simple). 6.3. Complexification The change of states in biological systems consists in exchanges with the environment, formation of new objects by association of components, and decomposition of some complex components. For instance for a cell: endocytosis and exocytosis, synthesis of new proteins, decomposition of some products. To model these operations in an ES, we define the complexification of a category with respect to a strategy. In the ES we consider in the sequel, we suppose that the transition functors are obtained by iteration of this complexification process with respect to adequate strategies on the state-categories. A strategy on a category consists in the data of a set of new elements ’to absorb’, a set of components ’to destruct’, a set of patterns without a colimit ’to bind’ and a set of colimits ’to decompose’. Then we construct a functor embedding the initial category into a new category, called its complexification, in which the objectives of the strategy are fulfilled in the most economical way, both from an algorithmic point of view and with respect to the energy cost. The complexification can be constructed explicitly (cf. Ehresmann and Vanbremeersch 1987). In particular in this new category the components to destruct become the 0 object, and there is added a new complex object for each pattern to be binded; this object, which becomes the colimit of the pattern in the complexification, can be thought of as the pattern in itself, taken as an integrated unit. In concrete examples, its formation (e.g., synthesis of a protein) entails the strengthening of the distinguished links of the pattern. The links between these new objects are both simple links binding together a cluster of links between the patterns they bind, and complex links obtained by composing simple
168
A C. EHRESMANN, AND J. P. VANBREMEERSCH
links. 6.4. Emergentist Reductionism In a hierarchical evolutive system, an object N of level n+l is (by definition) the colimit of at least one pattern P of level n, but also a 2-iterated colimit based on the level n-l, and so on down the scales. In some cases, we can ’skip’ the level n, and directly present N as the colimit of a ’large enough’ pattern of level n. An important result (Ehresmarm and Vanbremeersch 1995) is that such a presentation is possible if all the links of P are simple, but not if some of the distinguished links of P are complex. Thus the level of an object is not a faithful mirror of its dynamic (or ’vertical’) complexity. To measure this complexity, we define the order of N as the smallest k such that N is the (non-iterated) colimit of a pattern of level k-1 . From the results recalled above, it follows that the existence of complex links (and a fortiori the MP) is a requisite for the existence of objects of order more than 2. In particular, the objects which emerge in a complexification can be of the same order as the pattern they bind (horizontal complexification) or of a higher order (vertical complexification). For instance, the synthesis of a protein remains at the molecular level, but Evolution has led to the emergence of more and more complex organisms. An iteration of horizontal complexifications can be replaced by a unique complexification with respect to a suitable ’large’ strategy which subsumes all the successive strategies. But in HES satisfying the MP, we may have an iteration of vertical complexifications which cannot be replaced by a unique complexification, hence which necessitates a ’dynamical’ construction in several steps. And it leads to the emergence of objects of strictly increasing orders. This result (Ehresmann and Vanbremeersch 1995) vindicates an emergentist reductionism in the sense of Bunge (1979).
7. MES AND THEIR CRS 7.1. Memory Evolutive Systems We are going to study how the dynamics of self-organized autonomous systems, such as living systems, is modulated by internal overlapping regulations. In the global description of a HES given up to now, we have seen that there are symmetry-breakings between the levels, so that an internal actor cannot transcend them for having a complete view of the system. It explains that the regulation must be distributed between several cooperative and competitive intemal centers of various levels; each one operates at its own timescale, with some access to a Memory, in which informations on the preceding experiences met by the system are organized for a better adaptation when the same circumstances occur anew. This situation is modeled by the notion of a Memory Evolutive System (or MES, Ehresmann and Vanbremeersch 1991). It is a hierarchical evolutive system, generally satisfying the Multiplicity Principle, in which the evolution is partially regulated by a net of internal organs of regulation, called Centers of Regulation, or CR. Each CR is a subsystem of the system whose objects, called actors, have a definite complexity level, and cooperatively direct a stepwise dynamic process at a specific discrete time-scale; the CR acts by itself, in the best possible way for it, depending on the informations it can inter-
SYMMETRY AND INFORMATION 1N MES
169
nally decode and encode. But the realization of its strategies is subjected to coherence constraints coming from the confrontation with the strategies of other competitive CRs and of external perturbations. We also suppose that the system contains and develops a sub-system which represents a central hierarchical Memory to which the various CRs have a differental access, both for storing and retrieval. Let us first describe one step of a CR. Its length is determined by the time-scale of the CR, and constitutes the actual present of the CR. The step is divided in several more or less overlapping phases. 7.2. The Decoding Phase In the first phase, the CR acts as an internal observer to form its actual landscape, which represents the informations, coming from receptors of external processes or from various internal components, that the CR can decode during its actual present. It is constructed as follows. The informations received by a specific actor A are modelled by the links (or aspects) in its reception field which are activated during the actual present of the CR and propagate some signal (e.g., transfer of energy) to A. For instance in a neural system, it could be the arrival of a photon on a cone A of the retine, or the ear vibration generated by a spoken word. In any case, A can only extract some characteristics of an object, because of structural constraints (a photon interacts only with specific molecules), or energy constraints (a neuron must receive a sufficient input to fire), or temporal constraints (during the latency delay of a neuron it cannot receive any information). But the actors of the CR must respond as a group and not individually. So there is an exchange of informations between them through their distinguished links in the CR; for instance if the CR is a tissue, through direct contact between its cells. Two aspects which are correlated by a zig-zag of distinguished links between the actors give the ’same’ information to the CR; we say they are in the same perspective. The landscape of the CR is the category in which the objects are the perspectives for the CR, and the links come from links in the system compatible with two perspectives. The landscape is an internal model of the system for the CR; the possible error in this representation can be measured only externally, by the distortionfunctor from the landscape to the system. 7.3. The Decoding Phase In the second phase of the step, the actors analyze the informations extracted from this landscape and select a common strategy to adapt to the situation. Each actor corresponds to a degree of liberty, but the fact that the actors must act cooperatively introduces constraints transmitted through their communication links. The choice of the strategy is facilitated by a recourse to that part of the Memory which the actors can attain, and in which preceding strategies and their results may have been memorized. The third phase consists in encoding commands to effectors to realize the selected strategy. The anticipated landscape for the next step should be the complexification of the actual landscape with respect to the chosen strategy. Its adequation will be internally evaluated at the next step, by comparing it with the next effective landscape (through the comparison functor). If some errors are so detected, one of the objectives of the next strategy will be to correct them. Another objective of this strategy will be to memorize the preceding strategy and its result for the CR (possibly under the form of iterated
170
A. C. EHRESMANN, AND J P. VANBREMEERSCH
colimits) to allow for better choices in similar situations later on, 7.4. Information Processing by the CR To sum up: the CR acts first as a collective decoder to form its landscape. But the signals it so collects from various parts become informations for it only in so far as it uses them to select a strategy. After that, the CR emits commands to realize this strategy, and evaluates the result not directly on the fidelity of the transmission to the effectors, but indirectly in its next landscape, by comparison with its anticipated feedback (whence a loop between the commands sent by the CR and their later effects for it). In some way, we could say that the CR handles the informations in a solipsist manner, to obtain the best return of its internal measurement process (in the sense of Matsuno, 1989). Naturally it does not always succeed, for its actions may be counteracted by the other CRs, up to a premature interruption of its step, if a strategy cannot be found, or if, once selected, it cannot be realized. Then we say that there is a fracture for the CR. On a quantitative (energy) level, the reception of informations in the landscape corresponds to an increase of the weight of a perspective (computed as the mean of the weights of the aspects which compose it), and thus transfers a gain in energy to the actors. This energy is dissipated through the choice and realization of a strategy, so that the energy of the CR is at its lowest at the end of the step. For lower CRs, this mechanism has been described by Schneider (1991) under the name of a "molecular machine", of which an example is the union of EcoRI and DNA leading to a specific cutting of the DNA.
8. EQUILIBRATION BETWEEN STRATEGIES AND MEMORY 8.1. The Interplay among Strategies Each CR operates its decoding/encoding process on its own landscape, which is only its formal and more or less partial and deformed model of the system. But the effective realization of its strategy involves the system itself. Thus, at a given date, the actual strategies of the different CRs in their respective landscapes are repercuted to the system via the distortion functors. All their commands enter in competition, with more or less conflict, so that an equilibration process is necessary to produce a coherent global strategy which will be effectively realized by the system. This equilibration process is called the interplay among strategies. As a result, some commands of the CRs may be rejected or opposed, possibly leading to fractures for the corresponding CRs or even to a blocage of their action. The interplay among strategies is not directed. It is a free competition, in which each command (link to an effector) of the different strategies intervenes with a specific weight. It also profits from the symmetry-breaking in the passage from a higher to a lower level (entailed by the MP, cf. 6.1.). Indeed, if a command concerns a multifold object, its activation can be done through any of its non-equivalent decompositions, which may have different weights. This allows for a flexibility in the response, depending on the context. For instance if we have to recognize the classical ambiguous figure with a young and an old woman, and if this occurs in the course of a story which speaks of fairies, it will be interpreted as the young woman. Another example is the recurrence of a same word when we write quickly or are tired.
SYMMETRY AND INFORMATION 1N MES
171
8.2. Development of the Memory It develops from an initial innate kernel, to retain the results of past strategies for a better adaptation when the same inputs occur anew. This Memory acts. as a detector of symmetry between consecutive inputs or strategies. Among the commands of the strategies of each CR figures the storage of the preceding informations decoded in its landscape, of the commands of the strategy it has chosen and of its result for the CR. If it is accepted after the interplay among strategies, the subsequent complexification adds these items to the existing Memory, under the form of (iterated) colimits and complex links which become images of the corresponding items, and are accessible for a CR under some perspective. The recognition of an item is induced by the symmetry between this item and its image memorized as (iterated) colimit. Indeed, if the ’same’ situation recurs later on in the CR landscape, its image in the Memory is activated, thus allowing its recognition by the CR, and the retrieval from the Memory of a suitable strategy. The recognition and the retrieval of an item can proceed through any of the different (non-equivalent) decompositions or ramifications of its image, thus affording some flexibility to the process. A symmetry-breaking occurs when the strategy who had succeeded before does no more succeed, and then the CR will encode a modification of the Memory at the following step. The memorized strategies and their results form a sub-system of the Memory which is called the Procedural Memory. 8.3. Measure of the Information Content
In systems with only a small number of well-adapted comportments and in a stable environment, there is generally few conflicts, the strategy interplay does not cause fractures, and the CR strategies are realized, if not immediately at least with only a small delay. In this case, the Memory remains stable and it allows to quantify the information content decoded by a CR. Indeed, the informations that a CR could recognize during one of its steps come from the perspectives issued from the Memory in its landscape, and their probability is function of the weight of the perspective. The information content effectively decoded by the CR is measured by a comparison between the number of these perspectives, pondered by their probability, and the number of those perspectives which are activated during the step. The information content encoded by the CR strategy is defined symmetrically, if we consider the perspectives coming from available strategies in the Procedural Memory. But in changing environments, and for systems with a great number of CRs and a large choice of strategies represented by multifold objects, the strategy interplay can take a great importance and thoroughly modify the comportment of the CRs, thus leading to new strategies and necessitating changes in the Memory. For instance, in an Ising model, the correlations between the comportment of two spins should decrease exponentially with their distance. However it has been theorized long ago (Gaunt and Domb 1970, and recently verified experimentally by Back et al 1995) that, for a specific ’critical’ temperature, the correlations extend to spins far apart, and the rapid exponential decay is replaced by a long-range power-law decay, with a critical exponent which is the same for all an invariance class of systems. In our model it is explained by the fact that, when the number of spins increase, there are more complex
172
A. C. EHRESMANN, AND J P. VANBREMEERSCH
links along which they can be correlated. Among the applications of this result: correlations between DNA base-pairs, or lung inflation, even city growth or economics.
9. DIALECTICS BETWEEN HETEROGENEOUS CRS Here we will distinguish different causes of fractures, and study their implications for the evolution of the system. 9.1. Structural Temporal Constraints ofa CR Each step of a CR may have a different length, in particular this length might be very short if there is a fracture. A step of a CR which is completed without a fracture is called a normal step. We define the period of the CR at a given date as the mean length of its preceding normal steps since a fracture. Generally the period of a CR increases with its complexity level. Lower CRs with few available strategies (e.g., the promotor of a gene in a cell), have a more or less cyclic comportment, with periodical recurrence of the same situations and strategies; and the period remains almost constant. But for more complex CRs, the period may vary, as long as it remains subjected to some structural temporal constraints which must be respected for a step to be completed in due time, and avoid fractures. These constraints are formulated in the form of inequalities (and not equalities): the period of the CR at a given date must be less than the mean time-lags of its actual decoding and encoding processes (measured through the propagation delays of the corresponding links), and more than the stability spans of the objects which send perspectives in its landscape. They leave some flexibility in the operation of the CR and, as long as they are satisfied, its period can be maintained. Though they concern the temporal dimension, they have implications on the energy requirements, because propagation delays and stability spans ultimately depend on the availability of energy resources. If these constraints cannot be realized during a long enough time, there will be a lasting fracture, or dyschrony. Its repair may impose some temporal symmetry-breaking in the form of a change of period, leading to a de-resynchronisation of the CR with respect to the other CRs. In particular, we have proposed (Ehresmann and Vanbremeersch 1993) a Theory of Aging for an organism, based on such a cascade of de-resynchronisations for CRs at increasing levels; this theory seems to unify the various physiological theories. 9.2. Dialectics between Heterogeneous CRs Generally there is no direct concertation between two CRs, though higher associative CRs supervise some lower level CRs and may impose strategies on them, possibly to help repair a fracture. But in any case, the comportment of a CR may affect the comportment of another one, through the interplay among strategies and the fractures which might issue from it. Let us consider this situation in the case of two heterogeneous CRs, say a lower level (or micro) CR with a short period, and a higher level (or macro) CR with a long period. Let us consider one step of the macro CR, and its corresponding unique actual landscape.
This macrostep covers many steps of the micro CR, during which it can accumulate changes which are not individually perceived in real time by the macro CR because of
SYMMETRY AND INFORMATION IN MES
173
the propagation delays, or because each one does not affect the stability of a higher level multifold object due to the symmetry-breaking in the passage from the macrolevel to the microlevel. However their long term accumulation makes the unchanging landscape of the macro CR more and more unreliable, and may ultimately cause a fracture to this CR, for instance if they have progressively destructed all the components of some object which plays a role in its strategy. To repair its fracture, the macro CR will have to initiate a new step with a new landscape, and a new strategy. In some cases this change of strategy will retroact sooner or later on the micro CR and impose on it a change of strategy, as a consequence of an interplay among strategies in which the new macrostrategy takes precedence over the microstrategy. This back-firing of fractures between the two CRs generates a dialectics between heterogeneous CRs, which modulates the dynamics of a MES. This process explains why long-term prediction is impossible for a complex system (for instance in meteorology). Because of the symmetty-breaking between the levels, the analytic models usually proposed for physical systems can only describe the situation at a specific level, say at the level of a macro CR. In other terms such a model represents not the system as such, but a macro landscape. As we have just seen, the approximation afforded by this landscape, hence by the model, becomes more and more unreliable in time, up to a fracture, at which time the model must be completely modified to accommodate the change of landscape (cf. Rosen, ! 985). The dialectics between CRs has also some bearing on the problem of the irreversibility of time: is the time-arrow an artefact? We cannot discuss this question here. Let us just say that, even if we could reverse the time at the microlevel, the reverting process itself would be transmitted to the macro CR with a latency, as above, so that there can be no common reversibility for the two CRs. And here we only consider two CRs, while a complex system has all a net of CRs which interfere in the interplay among strategies. 9.3. Application to Neural Systems A neural system can be modelled by a MES based on the category of neurons (def’med at the end of 2.4.). A pattern of neurons generally has no colimit in the category of neurons, except for some patterns corresponding to important features of the environment or to innate comportments, for instance the grasping reflex of the child. However it can acquire a colimit by a complexification process. The formation of this colimit corresponds to the synchronization of the pattern, which becomes an assembly of synchronous neurons (in the sense of Hebb 1949, or a neuronal group for Edelman 1989); medical imagery has shown that mental processes activate such assemblies. The colimit, which can be considered as representing the assembly itself, but taken as an integrated unit, has been called a category-neuron (Ehresmann and Vanbremeersch 1991). It takes its own identity in time, and can become activated by other equivalent or non-equivalent (in the sense of 5.3.) assemblies of neurons. As the Multiplicity Principle is satisfied, successive complexifications can lead to the emergence of a hierarchy of category-neurons of increasing order, integrating assemblies of assemblies of neurons, assemblies of assemblies of assemblies, and so on (cf. 6.4.). In particular in the development of the Memory, there emerge such category-neurons, which represent complex comportments or higher cognitive processes. Their activation consists in the ’dynamical’ unfolding of one of their ramifications, down to the neuron
174
A. C EHRESMANN, AND J P. VANBREMEERSCH
level. For a category-neuron of order 1, representing a simple assembly of neurons, it reduces to the physical state in which this assembly is excited. But for category-neurons of order more than 2, the unfolding necessitates at least 2 steps and cannot be effected in a unique step by skipping the intermediate levels (cf. 6.4.); so it does not reduce to a pure physical state, though it may be described as a physical dynamical event, in which all the operations are well characterized. Thus we could be justified to say that mental states ’dynamically’ emerge from physical states of brain but are not identical to them (emergentist monism in the sense ofBunge 1979). 9.4. Semantics and Consciousness
In the neural system, the net of CRs contains lower CRs corresponding to different areas of the brain decoding some specific informations, for instance a color CR, or a shape CR, supervised by higher associative CRs of increasing complexity. The actors of a CR can be neurons or, in higher CRs, category-neurons. The CRs can act as detectors of symmetries between items in the Memory. We cannot describe here how this operation is done in a 2-step process, first leading to an ’acted’ classification, which (in the case of higher animals) is internally reflected so that each class of symmetrical items becomes represented by a new category-neuron, called its concept (it is the limit of the pattern of actors activated by the items of the class; cf. Ehresmann and Vanbremeersch 1992). The concepts form a sub-system of the Memory, called the Semantical Memory. The activation of a concept relies on a double indeterminacy: firstly choice of a particular instance of the concept, and then choice of a particular decomposition of this instance, by-passing twice the symmetry-breaking in the passage from higher to lower levels. Thus the development of this semantical memory affords a greater flexibility, e.g., in an interplay among strategies chosen under the form of concepts. This flexibility, which makes the animal more adaptable, could be at the root of the emergence of consciousness. We have proposed to say that a CR is conscious if it is able, by an increase of attention after a fracture: (i) to extend its actual landscape retrospectively to past lower levels; (ii) to operate an abduction process in this extended landscape to find the probable cause of the fracture; (iii) and finally to planify a strategy for several steps ahead, through the formation of internal ’virtual’ landscapes in which strategies (as concepts) can be tried without energy costs to realize them. In this sense, consciousness would amount to an internalization of Semantics and Time which allows to punctually transcend both the stream of time and the complexity symmetry-breakings, thus giving a selective advantage.
REFERENCES Back, C. H., Wursch, C. H., Vaterlaus, A., Ramsperger, V., Maler, U. and Pesire, D. (1995) Experimental confirmation of universality for a phase transition in two dimensions, Nature, 378, 597-600. Benzecri, J.-P. (1995) Convergence des processus..., Cahters de 1 ’Analyse des donn~es, 20 (4), 473-482 Bunge, M. (1979) Treatise on Bastc Philosophy, Vol. 4, Dordrecht: Reldel, Chandler, J. L. R. (1991) Complexity" A phenomenological and semantic analys~s of dynamical classes of natural systems, WESS communicattons, 1(2), 34-42. Chandler, J. L. R., Ehresmann, A. C and Vanbremeersch L-P (1995) Contrasting two representations of emergence of cellular dynamics, In. Farre, eds., Proceedings Sympostum on Emergence. InterSymp ’95.
SYMMETRY AND INFORMATION 1N MES
175
Edelman, G. M. (1989) The Remembered Past, New York. Basic Books Ehresmann, A C. and Vanbremeersch, J.-P. (1987) Hierarchical evolutive systems, Bulletin of Mathemattcal Btology. 49 (1), 13-50. Ehresmann, A. C. and Vanbremeersch, J.-P. (1991) Un mod~:le pour des syst~mes t3volutifs, Revue lnternattonale de Systdmtque, $ (1), 5-25. Ehresmann, A. C. and Vanbremeersch, J.-P (1992) Outils math~matiques pour mod61iser les syst~mes complexes, Cahters de Topologte et Gdomdtrte Dtff~renttelle Cat~goriques, XXXIII-3,225-236. Ehresmann, A C. and Vanbremeersch, J -P. (1993) R61e des contraintes structurales; In: AFCET 1993, Vol. 8, Versailles, 103-112. (It also contains a list of all our preceding papers, with a guide through their maze.) Ehresmann, A. C and Vanbremeersch, J.-P. (1995) Multiplicity pnnciple and emergence in MES, Journal of Systems Analysts. Modelhng, Simulatton; to appear. (Condensed translation of Emergence et T~l~ologte. Amiens 1993.) Farre, G L. (1994) Reflections on the question of emergence, In Advances tn Synergetics, Volume I: Systems research on emergence, Windsor The International Institute for advanced studies m Systems research and Cybernetics. Gaunt, D S. and Domb, C. (1970) Journal of Physics C, 3, 1442-1461 Hebb, D. O (1949) The Orgamzatton of Behavtour, New York" Wdey. Mac Lane, S. (1971) Categortesfor the Working Mathematictan, Berlin: Springer. Matsuno, K. (1989) Protobtology: Phystcal basts of Btology, Boca Raton: CRC Press. Matsuno, K. (1994) Hidden symmetries, symmetry-breaking and emergence of complexity, In: Gruber, B., ed., Symmetrtes m Sctence VIII, New York Plenum. Rosen, R. (1985) Anttcipatory Systems, New York: Pergamon. Schneider, T. D. (1991) Theory of molecular channels (I and II), Journal of Theorettcal Btology, 148, 83137.
Symmetry: Culture and Sc,ence Vol 8, No 2, 177-192, 1997
SYMMETRY." CULTURE AND SCIENCE
CONSTANCY, UNIFORMITY AND SYMMETRY OF LIVING SYSTEMS: THE COMPUTATIONAL FUNCTIONS OF MORPHOLOGICAL INVARIANCE Avshalom C. Elitzur Address: School of Physics and Astronomy, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel, and The Seagram Center for Soil and Water Sciences, The Hebrew University, 76100 Rehovot, Israel; E-mail. cfeh@weizmann welzmann.ac.d
1. INTRODUCTION Living systems, whether organs or entire organisms, display various forms of morphological invariance. Why? These phenomena are studied here from the Proto-Cognitive perspective, according to which evolution proceeds by processing information about the environment. The evolution of constancy, uniformity and symmetry is studied in detail. The study later focuses on the invariances of the plant’s leaves, and it is proposed that these invariances play a crucial role in the plant’s development of individual form. Experiments are proposed to test these hypotheses. Morphological invariance is further examined in the light of thermodynamics and information theory. New thermodynamic restrictions are imposed on the processes of measurement and information recording. Uniformity and symmetry are shown to meet these restrictions. Science-fiction movies depicting extraterrestrial life often disappoint the biologist with their dull imagination. Nearly always, the alien in these movies appears highly anthropomorphic or resembling some other terrestrial organism. From the viewpoint of evolutionary theory, where chance plays such a cardinal role, the probability for such coincidences is much smaller than, say, the chance that two unrelated cultures, undergoing their independent histories, would raise two musicians who would write identical symphonies. One bet, however, seems to be safe when one tries to imagine how extraterrestrial beings, if we ever encounter any, would look like: Very probably, their bodies will demonstrate some form of symmetry. Whether their entire body or only parts of it would be symmetric, whether the symmetry would be continuous (e.g., spherical or radial) or discrete (e.g., bilateral or threefold); dextrosinisteral (right-left), ventrodorsal (bottomtop) or anteroposterior (front-rear), is too risky to predict. But, based on all the known forms of living organisms on Earth, the general expectation of symmetry seems to be quite sound. Why, then, do living organisms exhibit various forms of symmetry? Numerous discussions have addressed the biological asymmetries, while symmetry has been often taken
178
A. C ELITZUR
for granted. However, the former only highlights the latter. Take, for example, the famous biomolecular asymmetry: All living forms possess proteins composed of "left handed" amino acids while their nucleic acids, DNA or RNA, are made of "righthanded" sugars. Various hypotheses have been proposed for the origins of this asymmetry, but nearly all of them agree that life could just as well exhibit the opposite handedness. Once primordial life has assumed left- or right-handedness, for whatever reason, the next generations simply had no choice but to adhere to it. But the initial preference is probably due to mere chance. Symmetry, in contrast, has been dictated by evolution time and again, in independent lineages, even on the basis of previous asymmetries. It therefore reflects necessity rather than chance (B+rczi et al., 1993). In animals, for example, evolution has imposed somatic (external) symmetry upon the visceral (internal) asymmetry. Even in cases where topological constraints require asymmetry of the external structure, the initial symmetry is restored in another way. The left-right asymmetry of the conches of several snails is well-known, stemming fi-om the unique difficulties posed by the task of magnifying a solid structure. Still, in those species who demonstrate a very high conch asymmetry, we witness a counterbalancing tendency of the conch to turn backwards rather than to the side. For example, in the Cerithium, whose conch is an elongated cone, the tip points straight backwards, thereby almost completely restoring the right-left symmetry. A peculiar example of symmetry (Neville, 1976) is seen in the eel Leptocephalus diptychus: During its post-larval stage, it has four spots on one side and three on the other. Asymmetry? No, because the body is transparent, thus displaying seven spots to both sides! Why, then, does symmetry dominate the form of most organisms? And why, coming to think about that, should there be a common form to all members of the same species? And why, moreover, should organisms have constant forms in the first place? The search for explanations will carry us to domains as diverse as theoretical physics, morphology and epistemology. The issue is profound, prompting novel questions about the very nature of life.
2. THE EVOLUTION OF MORPHOLOGICAL INVARIANCE Symmetry is only one example of morphological invariance, so let us first review the broader issue. I would like to show that, although the data looks inconsistent at first, overall morphological invariance increase with evolution. We first notice that a few unicellular organisms, like Amoeba, do not have a constant form but ceaselessly change their form according to the varying circumstances. This plasticity enables the organism to respond to any change of the environment by an immediate change of its own body. This luxury, however, had to be given up by the multicellular organisms, who developed tissue specialization and constant form. (In fact, even many unicellular organisms, such as the Class Sarcodina to which the Amoeba belongs, have constant form.) In multicellular organisms, form constancy has developed gradually. The bodies of Molusca and Annelida exhibit a greater plasticity than Arthropoda and Vertabrata, who have a rigid skeleton. Also, Vertabrata are more rigid in that their growth does not involve metamorphosis but retains nearly the same form throughout the animal’s life. Constancy of form is therefore the first step in the evolution of morphological invariance.
CONSTANCE UNIFORMITY AND SYMMETRY OF LIVING SYSTEMS:
179
Once the constancy of form has been attained, the next stage was the development of uniformity. Two main trends have developed. Most sessile organisms, like plants and sponges, have a relatively constant form, yet it varies from one individual to another. In contrast, most mobile and locomotive organisms are uniform, with a typical bodily form shared by all members of the species. The distinction between mobile and locomotive organisms is important in this context. "Mobile" denotes primitive organisms that are passively carried by water or air, whereas "locomotive" refers to organisms that move voluntarily. We shall see that sessility, mobility and locomotion pose different morphological challenges. The differences are more notable in respect to the next level of invariance, namely, symmetry. Whereas form constancy exhibits the organism’s resemblance to itself over time, and uniformity exhibits resemblance of various individuals to one another, symmetry is the resemblance of various parts of the same organism to one another. All uniform organisms exhibit also some symmetry, but its degree varies.~ Symmetry is maximal in some of the most primitive mobile organisms such as the above Sarcodina, some of which have a nearly perfect spherical symmetry. Upper on the evolutionary tree we encounter locomotive organisms, such as Nematoda, that have a biradial symmetry. Echinodermata basically have a fivefold symmetry. Symmetry declines further in animals like Arthropoda and Vertabrata, where it is usually only bilateral. Symmetry too, then, is related to evolution, although apparently showing a trend opposite to that of form constancy and uniformity. To summarize, morphological invariances appear to be fundamental characteristics of the organism’s structure, increasing or decreasing with evolution. Could they be telling us something important about the nature of life?
3. THE PROTO-COGNITIVE MODEL: LIFE IS A COMPUTATIONAL PROCESS Evolution presents a paradoxical form of "wisdom," namely, ingenuity of solutions to the environment’s challenges without there being a sentient mind devising these solutions. This riddle lies at the basis of the Proto-Cognitive Model (Elitzur 1994, 1995, 1996a,b), which I would like to apply to the present discussion. A brief introduction to the model follows. Central to the model is the assertion that evolution is mainly a computational process. This assertion follows from the trivial fact that the living state is an optimum state. Change, for example, the glucose percentage in an animal’s blood; interfere with the length of a tree’s branches; or slightly alter the position of a bee-hive -- organisms invariably react with manifest distress, trying to restore the upset optimum, or, when the change exceeds some limit, they just die. Now, these biological optima strictly depend on the environmental conditions to which the organism adapts. Hence, in order to achieve these optima -- and, moreover, to achieve many optima that together create a higher optimum -- the genome needs precise quantitative information about the envi~ Significantly, when domestication of animals and birds creates patters of fur and feathers that deviate from the uniform pattern, the change is asymmetric as well.
180
A. C. ELITZUR
ronment. The gravitational force, average temperature, pressure, chemical composition, etc. -- only such quantitative data, encoded in the genome, allows the evolution of phenotypes optimally adapted to a particular environment. Like the cognitive mechanisms used by the individual organism during its ontogenetic development, the species as a whole employs proto-cognitive processes during its philogeny. The species, as it were, "perceives" the environment, "measures" its relevant variables, and "computes" the data for the purpose of increasing adaptation. But how can such quantitative tasks as measurement and computation be carried out and recorded as genetic information? Evolution does it with a very efficient mechanism, namely, differential reproduction. Each genome grants the organism a certain life-time, determined by the genome’s fitness to the environment. This life-time is eventually converted into the number of the organism’s offsprings. Consequently, in each generation there are different percentages of genes in the population’s gene-pool. In populations of rapidly-breeding organisms, even the slightest advantage can lead to considerable differences in the number of the mutant’s offsprings, giving the mutation a boost for further development. When sexual reproduction also allows the assembly of new combinations between the surviving genomes, then the population as a whole is engaged, in effect, in a very efficient computational process, capable of gradually reaching optimal solutions to conflicting environmental challenges. This proto-cognitive process is performed mainly by exploiting statistical rules. Consider, for example, the way by which a population "detects" very subtle environmental factors. The fate of each individual organism is determined both by t) conditions that are inherent to that environment and prevail in space and time ("signal"), and ii) local, random fluctuations ("noise"). The individual’s fate is therefore subject to the whims of chance. However, the overall population is governed by more deterministic laws, due to the famous ",/-; rule" (Schr6dinger, 1945). This rule states that the random fluctuations that inflict a measurement have the magnitude of the square root of the number of atoms involved in the measured process. Evolution takes advantage of this rule by producing multitudes of individuals of the same species. When natural selection interferes with the population growth, the fraction of the population that eventually survives reflects mainly those environmental factors that repeatedly occur at various sites and times. The random factors "cancel out," as it were, at the statistical level, enabling the environmental signal to outweigh noise (Elitzur, 1995). Every organism, to summarize, constitutes a cognitive organ, whose fate adds to the information accumulated in the species’ genome. At the same time, the organism is also a carrier of evolutionary information. This is similar to our sense organs that, while receiving information, already embed enormous information in their very structure. The structure of the eye, for example, is based on a very rich evolutionary knowledge about the nature of light. The same holds for the entire organism: It contains information that solicits further information. To use the illuminating expression of Lorenz (1973), every organism constitutes a hypothesis about the world, put to test by the struggle for survival. Life thus gains information about the world by the active strategy of constantly putting forward hypotheses, to be later proved or disproved by natural selection.
With these formulations in mind, we return to our initial question. Why are is the form of so many living systems constant, uniform and symmetric?
CONSTANCY, UNIFORMITY AND SYMMETRY OF LIVING SYSTEMS"
181
4. MORPHOLOGICAL INVARIANCE AND THE INVARIANCE OF PHYSICAL LAW One basic clue comes from theoretical physics, where invariance plays an important role. Three elementary invariances of physical law are 0 invariance under translation in time, ii) invariance under translation in space, and iii) invariance under reflection. Now (i) seems to be related to the living systems’ above constancy of form, (iO to their uniformity and (iiO to their symmetry." Is this affinity between the physical and the biological invariances only coincidental er does it reflect a more profound relation? The ProtoCognitive Model suggests an affirmative answer to the second option. Perhaps the organism’s invariant form reflects cognizance of the environmental invariances. Let us systematically follow this affinity. If the organism’s form ought to adapt to those features of the environment that are not random, then sessile organisms on the one hand, and mobile and locomotive organisms on the other hand, should indeed differ in this very respect. The sessile organism adapts its form to the local environmental conditions that sustain temporary fluctuations. An example of such an adaptation is given by the plant, whose individual form accords with the local slope of the ground, direction of light, etc. Hence the great individual variety in most plants’ forms. Notice, however, that the plant’s form is nevertheless constant: Unlike few, short lived unicellular organisms, the longevity of multicellular organism’s necessitates a form that adapts to the persisting environmental conditions. For the mobile and locomotive organisms, the task is much harder: They have to adapt to conditions that wildly fluctuate not only from one moment to another, but also from one site to another. Hence, their body is adapted to those features of the ground, water or air that prevail everywhere. The hydrodynamic or aerodynamic forms of fish and birds are only two examples for a structure that relies on the invariant characteristics of the medium in which the animal moves. For this reason, rabbits should resemble one another more than potatoes: The rabbits are adapted not only for eating a particular clover or escaping a particular fox, but rather for eating all kinds of grass and escaping all predators everywhere. Uniformity therefore represents a higher level of constancy, extending the constancy over time to constancy over space as well. But then, if uniformity increases with the evolution of motility, why does symmetry not increase too? Well, as a matter of fact it does. Like uniformity, overall symmetry increases in spacetime. Let us follow its development in some detail. Mobile organisms, i.e., those that lack independent modes of locomotion, often possess perfect symmetry. The reason is simple: They equally interact with the environment on all their sides, hence their form is spherically symmetric. Such a perfect spherical symmetry is found in many Bacteria and Protista, that passively flutter in water. Breaking of this perfect symmetry appears when a preferred posture and/or a preferred direction for locomotion develops. Schypozoa, for example, such as the medusas, have developed a specialized direction for swimming, giving rise to anteroposterior asymmetry while retaining radial symmetry in all perpendicular directions. Echinodermata, on the other hand, such as the sea urchin, have developed a preferred posture but not a preferred direction for locomotion. Here too, asymmetry has developed in the ventrodorsal axis 2 The basic laws of physics are also invariant under time reversal. This invariance, however, holds only at the microscopic level, while macroscopic processes exhibit the famous time asymmetry of entropy increase. Significantly, this exception too is reflected in the clear time-asymmetry of biological processes.
182
A. C. ELITZUR
while fivefold or even radial symmetry has been preserved in the other directions. If the echinoderm is approached from the side by a predator or food, it will not rotate; it has the same defensive measures and eating organs in all its sides. If, however, it is turned upside down, it will rotate along this axis, in which it is not symmetric, in order to resume its specialized up-down position. Higher on the evolutionary scale we find animals like those belonging to Phylum Arthropoda (e.g., insects) and Chordata (e.g., vertebrates), who have developed both a preferred posture and a special direction for movement. They therefore present symmetry breaking both in the ventrodorsal and the anteroposterior axes. Yet, in spite of this evolutionary reduction of morphological symmetry, the constant motion and rotation of these animals gives rise to increasing invariance at a more complex level. The locomotive animal remains far away from the predator and close to the prey; it always turns to the food its face while its predator gets its back. A new invariance thereby appears: The animal faces the environmental challenge from similar distances and angles. To better comprehend this invariance it is instructive to consider a mammal that is capable of giving up its asymmetry and returning to an almost spherical symmetry, namely, the hedgehog when assuming the defensive posture. Significantly, it remains motionless while in this posture, since it does not have to rotate or move: It appears the same from all its sides (A sphere’s projection from all sides is the same circle). Compare this with the behavior of the squirrel when approached by a predator: It keeps moving and turning, such that the predator never sees its back. Thus both animals, so different in their reaction to danger, nevertheless demonstrate the same basic invariance: Their form remains the same to predators from every side. Other vertebrates that has regressed to the earlier, higher symmetry are the snakes. Here too, loss of the ventrodorsal asymmetry (and partly of the anteroposterior asymmetry) in favor of biradial symmetry parallels the loss of the specialized mode of walking on legs in favor of unspecialized crawling in which all sides of the elongated body are equally used. 3 A surprising support for this explanation of uniformity and symmetry comes from plants. The lack of uniformity in plants has an exception in the form of some trees that possess rigid, constant forms, such as the fir or the cypress. Now, like primitive mobile animals, all such trees also possess radial symmetry in addition to their uniformity. Significantly, all these trees also assume a precisely straight upright posture. The explanation for their symmetry is therefore simple: These trees, like mobile animals, have adapted their form to the one environmental condition that does not vary from one spot to another, namely, the gravitational fiel!! Morphological invariance is thus related to the organism’s mode of interaction with the environment. Organisms that maintain an invariant position in respect to other objects have low morphological invariance, whereas organisms that cannot maintain a positional invariance have a high morphological invariance instead. In this respect, plants and locomotive animals are similar while mobile organisms are distinct. Both sessility and locomotion represent attempts to maintain constant position in respect to important agents in the environment, whereas passive mobility must be compensated by high symmetry, enabling equal adaptation of the all the organism’s sides to all locations. 3 Hydrodynamic and aerodynamic forms are, by definition, symmetric for this very reason: Their interaction with the medium in which they move is equal in the s~des perpendicular to that of thelr motion.
CONSTANCY, UNIFORMITY AND SYMMETRY OF LIVING SYSTEMS.
183
We can therefore give a general formulation of the relation between the formal invariance of physical law and the morphological invariance of living systems. The macroscopic laws of physics are invariant under translation (in space and in time) and under rotation (in space). Statistically, this means that that, in any given environment, most events are equally likely to occur to the organism at all sites, at all instances, and fi’om all angles. The organism’s form, adapted to the invariant features of its environment, must therefore manifest similar invariances. And indeed, all organisms are invariant in one or more of the following senses: 0 Organisms that ceaselessly change their form assume the same form in response to similar events. ii) Organisms with a constant but individual form assume the same position at the same site throughout their lives. iii) Organisms with a constant and uniform form that have no control on their position are highly symmetric, exhibiting the same projection to all sides. iv) Organisms that are uniform but asymmetric along some axis assume the same relative position and relative angle to similar events.
5. MORPHOLOGICAL INVARIANCE OF SUB-UNITS Morphological invariances are often more pronounced at the microscopic than at the macroscopic level. Even in non-uniform organisms, most cells of the same tissue are uniform. Many cells possess symmetry of their own. In fact, even the entire asymmetric organisms is cytolocgially symmetric in that the membranes in all its sides are composed of similar cells. Here too, it is not surprising to learn that neurons, whose function depends on their unique synaptic associations with the neighboring tissues, possess a very low uniformity and symmetry, whereas leucocytes, carried by the blood all over the body, are uniform and spherical. Invariance of sub-units appears at the macroscopic level too. Such, for example, are plants, who usually have non-uniform and asymmetric forms, yet their leaves are fairly uniform and symmetric. In some plants, the number of leaves on each branch is constant, thereby creating appealing symmetry of the branches. In other plants, the branches themselves are arranged symmetrically. Many flowers possess threefold, fivefold, sixfold or even perfect radial symmetry. Many fruits have spherical symmetry. In is in plant formation that I believe morphological invariance conceals some of the most profound biological insights. The plant’s individual growth provides a fascinating subject for the student ofmorphogenesis because the plants’ growing parts are, in effect, still embryos, continually growing new organs and yet accessible to direct observation. Leaves’ uniformity and symmetry might prove instructive in the study of these processes.
6. COMPUTATIONAL FUNCTIONS OF LEAF FORMATION Turning to the issue of plants’ form, I hope I will be forgiven for beginning with a personal confession. Perhaps it does not merely reflect idiosyncratic taste; an interesting scientific problem seems to be involved too.
184
A C, EL1TZUR
Of all civilized man’s ways of interfering with Nature in the name of aesthetics, nothing strikes me as arrogant as the common practice of trimming trees. But why should such an innocent action offend me so much? It has to do with the feeling of awe and admiration that I, a dilettante biologist, feel when looking at an old wild tree. Erecting such a complex and stable structure only by molecular processes looks almost miraculous. Overcoming the gravitational force that threatens to break the tree branches and trunk, creating appropriate balance, avoiding branches’ intersection with one another, the delicate distribution the leaves for an optimal exploitation of sunlight -- all these difficult and even conflicting tasks, as well as many others, underlie the profound harmony and beauty of a mature tree’s form. So, apart from its beauty, the tree’s natural form presents a very elaborate solution to the environment’s challenges. Bearing in mind the central biological role of computation discussed above, surely the formation of a tree is a typical example of such a computation-based process. I would therefore like to call attention to the role of leaves in this computation. Our initial question is therefore posed in a new form: Why are the plant’s leaves uniform and symmetric? In most plants the branches grow more intensively towards the more illuminated side of the plant, thus being non-uniform and asymmetric. Why then not the leaf itself?. One possible answer is that symmetric and uniform leaves require shorter genetic programs for their production. However, bearing in mind the high complexity at the microscopic level that the genome has to create, leaf symmetry seems to make only little difference. An intriguing explanation comes from the study of plant-parasite relations. Leaves are the main food resource of many insects and larvae. Luckily for the trees, many birds feed on these pests. It is thus desirable for the tree to inform the birds which of its leaves is harassed by the parasites. Could leaves’ uniformity and symmetry enable the tree to do so? Heinrich (1979, 1992; Heinrich and Collins 1983) has provided three observations that support this hypothesis: /) When the shapes of tree leaves were artificially damaged, the insect-eating birds frequented these trees or leaves more than undamaged leaves or trees, ii) Some caterpillars carefully nibble the leaf margins in such a way that its symmetric form is maintained, thus making it harder for the birds to locate the damaged leaves, iii) Many oak species, whose leaves are neither uniform nor symmetric, seem to be immune to leaf-eating pests. Shmida (1992) has suggested a complementary hypothesis concerning flower symmetry, namely, that symmetry is positively correlated with high amounts of sugar in the flower. For this reason, so goes the hypothesis, pollinating insects prefer symmetric flowers. Symmetry, then, might enable the plantbeneficial animal to evaluate the plant’s well-being, while asymmetry discloses the plant’s illness. Leaf uniformity and symmetry thus seem to provide efficient means of information transmission from the plant to other organisms. Could they also enhance the reception of information by the plant? The Proto-Cognitive Model seems to indicate an affirmative answer, supported by the following facts. The leaves of many plants change their angle throughout the day so as to get maximum sunlight. In order to best function as an "antenna," for detecting the direction from which the light comes from, the leaf must be symmetric in relation to the direction that it faces. It is the comparison between the light absorbed by the leaf’s right and left halves that enables precise location of the source’s direction. Clearly, identical halves would perform this task most accurately.4 4 The comparison between the hght impingtng on the leaf’s two halves can be carried out in two ways. In many plants the leaf’s two halves maintain an angle that is smaller than 180°. In these leaves the compari-
CONSTANCY, UNIFORMITY AND SYMMETRY OF LIVING SYSTEMS:
185
The explanation for the leaves’ uniformity is similar. The formation of branches poses many demands to the plant: overcoming gravitational forces, optimal exploitation of light, avoidance of intersection between the branches, etc. In other words, reaching optimal form requires a great deal of computation to be carried by the plant. This task of branch formation consume much more resources than the formation of the small, shortlived leaves. Now leaves, besides fulfilling their known photosynthetic goal, are also capable of affecting the plant so as to grow towards those directions where light is abundant. For the plant to properly detect those directions in which light is abundant, it should rely on the overall sum of signals coming from a multitude of leaves. It is only with such a statistical averaging that random fluctuations "cancel out." Our question thus gets a new form: Would this statistical task be better performed with uniform leaves or with non-uniform ones? Clearly, have the leaves had different sizes and forms, the task would be much harder. Only with uniform and symmetric leaves can the plant compare and average all the environmental signals received by all of them. There is a zoological analogue to this presumed mechanism. Increasing evidence suggests that flocking behavior among social animals enables the herd to reach optimal decisions by averaging all the individual movements (Deneunbourg et al., 1991; Kerlinger, 1989). Here too, the entire herd might reach an optimal decision by a "referendum" based on the sum of the individual decisions.
7. EMPIRICAL PREDICTIONS AND FURTHER QUESTIONS The above hypotheses concerning the proto-cognitive functions of morphological invariance yield a plenitude of predictions that are worth being tested by experiment or observation. They also prompt a host of new questions. When a rule is conjectured, an important test of the rule’s soundness is this: Exceptions from the rule should prove to disclose exceptions in other aspects as well. We have seen this when discussing the exceptions displayed by radially symmetric trees and the spherically and biradially symmetric vertebrates: Each case discloses non-trivial deviations in the organism’s general biology. It is therefore instructive to discuss a few other organisms that deviate from the morphological rules we have formulated above; they should reveal other biological exceptions as well. In animals and birds, deviations from uniformity and symmetry in the fur or feathers pattern are nearly always the result of domestication. Still, there are a few exceptions in the wild. The males of the ruff (Philomachus pugnax) display highly non-uniform patterns during the breeding season. In the African wild dog (Lycaon pictus) the fur pattern is not only individual but asymmetric as well. Significantly, both these species are exceptional among their fellow species of birds and mammals, respectively, in their social life and breeding habits. Greater exceptions are those displayed by locomotive animals that are asymmetric not only in their pattern but in their very body. Snails are the most notable example, but then they are also notable in their slowness, which makes them exempt from the need for the symmetry imposed on hydrodynamic or aerodynamic forms. The asymmetry of chelea (claws) in Crustacea presents other interesting examson can be made between the angle in which the hght ~mpinges on the leaves. In other plants, who have completely fiat leaves, the comparison can be made between the mere amounts of light absorbed by the two halves
186
A C ELITZUR
pies. The hermit crab (Coenobita) is asymmetric, but for a good reason: It is adapted to the empty snail shell in which it lives. The heteroechely of other Crustacea, on the other hand, does not lend to this explanation as they do not dwell in shells, It is therefore worth studying these cases to see whether these deviations from the common symmetry are similarly related to special biological specializations. Interestingly, one of evolution’s latest products, namely, Homo sapiens and the hominid apes, display an inhom bilateral asymmetry in terms of the functioning of the hands. This breaking of the bilateral symmetry, the last that has remained intact in most locomotive animals, seems easy to explain once we recall that when these animals perform manual work, the object is kept in an invariant position in relation to the two hands, and therefore the left and right hand can differentiate and specialize in their roles. From these few examples we can derive a general prediction for all organisms that deviate from the morphological invariances characterizing their taxonomic groups: they should all disclose biological adaptations unique to those species. An interesting phenomenon seems to occur in descendants of domesticated animals that have become wild again. The domestic pigeon, descendent of the rock dove (Columba livia), has lost its original wild pattern of feathers and acquired a wealth of patterns and colors. Large populations of such pigeons became wild again, inhabiting cities all over the world. Having observed many such flocks, it seems to me that their average pattern gradually converges again, not to the original pattern but to a darker, less elaborate and much less beautiful form, but nonetheless considerably uniform and symmetric. A systematic study is therefore warranted to see whether selection pressure enforces again uniformity and syrmnetry on the new patterns of previously domesticated animals. The hypothesis concerning the computational role of leaf formation can be tested in a number of ways. Similarly to Heinrich’s experiment mentioned above, leaves could be artificially cut so as to have asymmetric forms. If the hypothesis is correct, this interference should affect the leaves’ ability to follow the movement of the sun during the day. Similarly, it should affect the normal formation of the branch on which the leaves grow, in comparison to control branches. Plants that exhibit symmetry in the arrangement of leafs on the branch provide further opportunities for such experiments. Another test is to look for proportions between leaf uniformity and symmetry on the one hand and the form of the entire tree on the other hand. As noted above, trees that have a uniform shape exhibit also radial symmetry, thereby disregarding local conditions in favor of environmental invarianceso From our model it follows that such trees should have less uniform, less symmetric and/or less elaborate leaves. Indeed, a few symmetric trees that I have occasionally inspected do exhibit such a relation, but only a systematic study could yield a reliable correlation. An important step in the quantitative measurement of symmetry is the method developed by Avnir et al. (Zabrodsky & Avnir, 1995, and references therein). The method, originally devised to measure molecular chirality, has been later extended to other domains. Rather than distinguishing between chiral and achiral objects, Avnir et al. have provided an exact quantitative measure that allows one to distinguish chiral objects from one another by their degree of shape chirality. The application of this method to our subject in order to measure the degrees of leaf symmetry in various plants would provide further tests for the hypotheses proposed above.
CONSTANCY, UNIFORMITY AND SYMMETRY OF L1VING SYSTEMS."
187
Of course, the explanation proposed here for biological uniformity and symmetry as serving proto-cognitive purposes does not exclude the prevailing explanations (embryological, functional, etc.) but complement them. Indeed, in is by now commonly accepted that many biological systems, besides carrying out their main tasks, also constitute information-processing systems. The most notable example is the vertebrates’ respiratory system, that has developed the additional function of vocal communication. To take another example, the blood system in the brain serves not only to transport nutrients and oxygen to the brain, but also enables the brain to measure the body’s temperature, chemical balance, etc. Information abounds everywhere, especially in living systems. It is only natural to assume that evolution has learned how to use such sources of available information. More particularly, plant development has been shown to be directed by auxins, i.e., hormones secreted by growing branches and affecting other parts of the plant (Kagan et al., 1992; Sachs et al., 1993). It has even been shown that leaves can distinguish between ordinary light and light that has passed through other leaves, thereby avoiding the branches intersecting one another (Novoplansky et al., 1990). Further research is very likely to reveal other "sensory" functions of leaves and, consequently, other cognitive functions of the plant. We can therefore conclude that the introduction of information-theory concepts such as cognition and computation into biology is likely to provide many insights into the physical uniqueness of the living state. The fertility of this approach is demonstrated by the two rival hypotheses concerning the biological role of body symmetry: 0 MiSller (1992) argued that females’ preference for symmetrical males, observed in swallows, reflects preference for healthier males, while i0 Enquist & Arak (1994) and Johnstone (1994) argue that this preference stems from the fact that symmetrical forms, being invariant under rotation, are easier to recognize from various angles. In the Proto-Cognitive framework, the two hypotheses are complementary: By the same reasoning that asserts that an organism’s form has to be rotationinvariant in relation to observers, this form should be similarly invariant in relation to the environment. Adaptation poses equal demands to both right and left sides of the mobile organism. A symmetric form, therefore, both conveys information about the organism’s well-being and is easier to be recognized by other animals. The above studies of leaf-symmetry (Heinrich 1979, 1992; Heirtrich and Collins 1983) lend some support for (i), in that they show that asymmetry is an indication of sickness, yet they do not exclude (iO (Notice, by the way, that the latter hypothesis provides a very reasonable explanation for the high symmetry of most flowers and fruits: They should appear rotation invariant to the animals they have to attract!) Indeed, once the two hypotheses have yielded quantitative predictions, observations gave varying support for both of them in a few species (Ridley 1992).
8. SYMMETRY, ASYMMETRY AND THE PHYSICS OF MEASUREMENT To reiterate, the Proto-Cognitive model extends the notion of cognition from the ontogenetic to the phylogenetic level. Like the individual organism that perceives the environment through sense organs, so does the species "perceive" the environment through the overall population of organisms. In both cases, quantitative data about the environment (e.g., distances, concentrations, temperatures, etc.) is recorded, whether in the nervous system or in the genome. Such recording of quantitative data constitutes, in fact, a fundamental action of information processing, namely, measurement. Now, a closer look into the physical principles underlying information recording and measurement
188
A, C ELITZUR
shows that these processes are intimately related to symmetry. A discussion of this affinity might shed further light on the issue of biological symmetry. Rosen (1995), in a surprising theorem against the prevailing cosmological theories of symmetry breaking, has proven that asymmetry is a more basic concept than symmetry, the latter being devoid of meaning in the absence of the former. In other words, asymmetry could not have originated in a perfectly symmetric universe, because there is no meaning to symmetry without there being something asymmetric to serve as a frame of reference for the invariance under rotation or reflection. This theorem’s significance goes beyond theoretical physics. In the present context, it has an important bearing on the very notion of measurement. Take, for example, a ruler. It is asymmetric in that its figures regularly differ from one another. Otherwise, have the figures been identical, the ruler would be invariant under translation and therefore incapable of measuring an object’s position. However, the concept of information requires a somewhat surprising twist of Rosen’s conclusion: For the measuring instrument to precisely measure a specific variable, other variables must not interfere with this measurement; they are "noise" in relation to the specific variable to be measured. We can therefore extend this conclusion to a general requirement for the operation of measurement: The measuring instrument’s structure must be symmetric in relation to all irrelevant variables. This, for example, is the reason why the ruler must be straight: While assigning different values to the points along its edge, it assigns the same value to all points in the dimensions perpendicular to its edge. Has it not been so, that is, has the ruler not been straight, it would be measuring a mixture of two or more dimensions, losing its precision. When the measuring device can also record the result, then, similarly to the above spatial combination of symmetry and asymmetry in the device’s structure, a syrnmetryasymmetry combination is needed in the time dimension as well. Prior to measurement there is a symmetry of probabilities, which, upon measurement, gives its place to the asymmetry of the actual result. In other words, prior to measurement the measuring device must be in an initial state So that can evolve into any one out of the possible consequent states $1, St, S~ .... All these subsequent states are, prior to the interaction with the measured object, equally likely to occur. Once, however, measurement has taken place, the consequent state is irreversible: The initial state is not likely to be restored. Consider, for example, an unexposed film: All points must be equally capable of blackening, whereas after exposure the blackened points cannot become white again. Let us generalize this rule into the second requirement for the operation of measurement: Any measuring instrument must begin with a dynamically-symmetric initial state, so as to end up in the appropriate asymmetric final state. Biological perception meets both the above requirements, the structural and the dynamic one, for carrying out efficient measurements. Most perceptions constitute, in essence, measurements of asymmetry: Perceiving an object’s location gives information whether it is on the organism’s right or left side, above or behind it, etc. Now, in order for the perceptual mechanism to be unbiased when performing such an asymmetry measurement, the perceptual mechanism itself must be symmetric, lest its own asymmetry interferes with the result. Hence the parity of eyes and ears, the symmetry of antennas, and, as proposed above, the symmetry of leaves.
CONSTANCY, UNIFORMITY AND SYMMETRY OF LIVING SYSTEMS:
189
This structural symmetry meets also the second requirement, namely, dynamical symmetry. Consider again the antenna, the pair of eyes and the symmetric leaf. In all these cases, the radiation impinges on two nearly identical halves of the apparatus. This is So in the above notation. However, the two halves absorb the radiation differently, depending on the relative position of the light’s source (S~ or $2). The structural symmetry ensures an unbiased operation of the asymmetric response, yielding accurate information about the source’s relative position. Biological symmetry thus seems to play a basic role of information recording: It provides the organism an unbiased background (symmetry of probabilities) upon which environmental information (asymmetry of the actual result) can be efficiently recorded and processed.
9. SYMMETRY, ORDER AND INFORMATION: A THERMODYNAMIC PERSPECTIVE If symmetry serves information processing, its function should accord with thermodynamic laws. A brief discussion of this aspect will complement our discussion. As stated above, any recording of information, such as measurement, requires a system to be in a state where the system has equal chance to end up in any one out of a number of possible outcomes. Let this state be called "pre-set state." Thermodynamically, this state is characterized by two seemingly-opposite conditions: low entropy and low information. Let us examine the reasons for this requirement. First, the system has to be in low entropy, i.e., high order. An ordered state, by Boltzmann’s classic definition, is compatible with only a few microscopic arrangements, while a disordered state can be one out of numerous microscopic states (put in commonsense terms: there are only a few arrangements that make a room ordered, but numerous ways to make it disordered). Put dynamically, the ordered state is far from equilibrium. The Second Law of Thermodynamics makes such states highly unstable. It is this instability of the pre-set state of the recording device that the incoming information upsets. The system then moves to one of the many states of higher entropy, closer to equilibrium. The new equilibrium that has been reached is more stable, giving an irreversible record of the result. Second, in the pre-set state the recording or measuring system must contain no information, lest that information interferes with the incoming information. Previously-recorded information might exist in the system’s memory, but not in the recording part, where it would be nothing but "noise." Such a pre-set state characterizes, for example, the unexposed photographic film or the clean writing-paper. They manifest both low entropy and low information, and record information by irreversible processes that increase both their entropy and information content. The thermodynamic uniqueness of the pre-set state is illustrated by the resetting, or calibration, of a measuring instrument: Resetting erases all previous information and brings the instrument back to the unstable state of low equilibrium. Interestingly, Bennett’s (1987) resolution of the paradox known as "Maxwell’s demon" shows that the main thermodynamic cost of any measurement is paid during the resetting stage. Resetting,
190
A C ELITZUR
whereby previous information is erased yet the system’s order is increased, returns the system to the pre-set state. Following Bennett, I have elsewhere (1994) pointed out constant resetting as one of the most basic characteristics of all living systems. Clearly, symmetry provides an efficient basis for the pre-set state. A symmetric object is an ordered one, as it exhibits improbable correlations between its two or more sides. At the same time, symmetry represents lack of information (’Ne’eman, 1990). A symmetric arrow cannot yield spatial information; only asymmetry provides the necessary signal. This is why measuring instruments, as noted above, possess initial symmetry, to be broken by the interaction with the measured object. Let us now return to the proto-cognitive processes by which the genome records information about the environment. Do these processes employ pre-set states? Indeed, such are the three forms of morphological invariance considered in this paper: /) the constancy of the organism’s form, ii) the resemblance of all individuals of the same species to one another, and i0 the similarity of two or more sides of the organism’s body. Consider constancy first. An organism that lives longer than a few hours exposes its phenotype to the challenges of the environment for a long enough period to survive various fluctuations. The orderliness of this state, namely, the endurance of the same structure over a long period, provides an efficient test of the organism’s fitness to that environment, just like the film exposed for a few hours to get a satisfactory picture of a dim star. Uniformity is another example for a pre-set state, operating on the philogenetic level. When a single genome a-sexually multiplies, it gives rise to a myriad of identical individuals that invade that environment. The initial population is highly ordered, yet it contains no specific information about the environment. Only after natural selection has played its role, destroying many organisms, does the number of surviving organisms reflect the genome’s fitness to that environment. Here, in fact, a reliable experiment is performed, testing the survival value of that genome. The result, i.e., the number of surviving individuals indicating the genome’s fitness, is very reliable due to the numerous replications of the experiment. Uniformity thus plays in evolution a role similar to that of replication in scientific experiments. Now, in a realistic situation, where reproduction is sexual, the population as a whole contains not one genome but many. These genomes share certain genes and differ with other genes. Therefore, they constitute control experiments for each other. Uniformity thus constitutes an ideal pre-set state, enabling a reliable test of each genome’s fitness. Finally, consider symmetry. A pair of sense organs, placed exactly on the opposite locations on the animal’s head, exhibit an ordered state, yet the organs themselves give no information about the location of the object that they sense. It is their unequal response to external objects that gives that information. The def’mition of the pre-set state as a combination of low entropy and low information might seem contradictory to readers who are used to the common definition of information as the opposite of entropy. A deplorable confusion still prevails in the literature concerning the relation between entropy, order, information and complexity, especially in the context of life sciences (Elitzur, 1996a). Equally, I believe, a correct application of these notions provides one of the most fruitful paradigms for modem biology. A thorough analysis of these terms and their applications is carried out elsewhere (1996b).
10. CONCLUDING COMMENTS Three types of morphological invariance, namely, constancy, uniformity and symmetry, turn out to offer penetrating insights into the nature of life. In the framework of the
CONSTANCY, UNIFORMITY AND SYMMETRY OF LIVING SYSTEMS:
191
Proto-Cognitive model employed here, constancy, uniformity and symmetry appear to be simple means by which life records and processes information about the environment, on both ontogenetic and philogenetic levels. Morphological invariances exhibit high degrees of order, creating a blank, unbiased background for the recording of information. They are thus akin to the basic scientific requirement of replicability: For the purpose of ruling out local, random artifacts, the replicated experiment or measurement should use identical, calibrated and unbiased instruments. This fi’amework has been shown to provide a wealth of testable hypotheses that warrant further research, as well as a new way of asking questions. The enormous variety of structures displayed by living systems, especially the highly accessible plants, is very appealing in this respect. The precise relation between order, information, complexity and other relevant terms, especially in the biological context, warrants further study, now in progress (Eiitzur, 1996b). Needless to say that the preceding discussion has dealt only with a few of the fascinating questions posed by morphology and morphogenesis.
REFERENCES Bennett, C. H (1987) Demons, engines, and the second law, SctentificAmertcan. 257(5), 88-96. B~rczi, S., B~la, L, & Moln~, I. (1993) On symmetry and topology of organisms m macroevolution, Symmetry: Culture and Science, 2, 123-137 Deneunbourg, J. L., Gross, S., Becker, S., and Sandtni, G. (1991) Collectively self-solving problems, In. SelfOrgamzation, Emerging Properties, and Learning (A. Babloyantz, Ed ), NATO-ASI series B-260, New York: Plenum Elitzur, A. C. (1994) Let there be life" Thermodynamic reflections on biogenesis and evolution, Journal of Theoretical Biology, 168, 429-459 Elitzur, A. C. (1995) Life and mind, past and future: Schrodinger’s vision fifty years later, Perspectives m Biology and Medicine, 38, 433-458 Ehtzur, A. C. (1996a) L~fe’s emergence ~s not an axmm A reply to Yockey, Journal of Theoretical Biology, in press. Elitzur, A. C. (1996b) Information, complexity, and the physical uniqueness of the living state, Prepnnt WISCP-96bc. Ehtzur, A. C & Shmida, A. (1995) Biological symmetry and information processing, Symmetry 6, 178-181. Enquist, M., and Arak. (1994) Symmetry, beauty and evolution, Nature, 372, 169-172. Heinrich, B. (1979) Foraging strategies in caterpdlars’ Leaf damage and possible predator avoidance strategies, Oecologta, 42, 325-337 Heinrich, B. (1992) Awan predatmn constraints on caterpillar foraging, In’ Caterpdlars" Ecological and Evolutionary Constramts on Foraging (N. E. Stamps and T. M Casey, Eds.). New York: Chapman and Hall Heinnch, B., and S. L. Collins (1983) Caterpillar leaf damage and the game of hide-and-seek with birds, Ecology, 64, 592-602 Johnstone, R A. (1994) Female preference for symmetrical males as a by-product of selection for male recogmtion, Nature, 372, 172-175. Kagan, M., Novoplansky, N., & Sachs, T. (1992) Variable cell lineages during the development of somatal patterns, Annals of Botany, 69, 303-312. Kerlinger, P. (1989) Fhght Strategies of Migratmg Hawks. Chicago’ University of Chicago Press. MOiler, A P. (1992) Female swallow preference for symmetrical male sexual ornaments, Nature, 357, 238240.
192
A. C. EL1TZUR
Ne’eman, Y. (1990) The interplay of symmetry, order and information in physics and the tmpact of gauge symmetry on algebraic topology, Symmetry. Culture and Sctence, 1,229-255 Neville, A C (1976) AmmalAsymmetry, London: Edward Arnold Novoplansky, A, Cohen, D, & Sachs, T (1990) How Portulaca seedlings avmd their neighbors, Oecologia, 82, 490-493 Ridley, M. (1992) Swallows and scorpmnflies Ond symmetry ~s beautiful, Sctence. 257, 327-328. Rosen, J. (1995) Symmetry m Sctence. An lntroductton to the General Theory, New York. Springer. Sachs, T., Novoplansky, A., & Cohen, D. (1993) Plants as competing populations of redundant organs, Plant Cell and Envtronment, 16, 765-770 Sachs, T. (1991) Pattern Formatton tn Plant Ttssues, Cambridge. Cambridge University Press. Shmida, A. (1992) The ecology and evolution of the beauty of flowers. Prepnnt, The Hebrew University of Jerusalem. SchrOdinger, E. (1945) What is Ltfe? Cambridge. Cambridge University Press. Zabrodsky, H, & Avnir, D. (1995) Continuous symmetry measures 4. Chirahty, Journal of the Amertcan Chemical Soctety, 117, 462-473.
Symmetry: Culture and Science Vol. 8, No. 2, 193-205, 1997
SYMMETRY." CULTURE AND SCIENCE
INFORMATION PROCESSING IN BIOSYSTEMS: QUANTUM MECHANICALBACKGROUND AND RELATION TO SYMMETRY-BREAKING Abir U. Igarnberdiev Address: Department of Plant Physiology and Biochemistry, Voronezh State University, Voronezh 394693, Russia
Abstract: Life is characterized by the information processing which distinguishes it from inorganic matter. The background of information generation in living systems is a central problem for understanding of life existence and expansion. The symmetrybreaking phenomenon based on quantum irreversibility is preconditioned to all transformations which lead to the appearance of information transfer in biosystems. It underlies the origin of life and corresponds to the emergence of the genetic code and other semiotic relations essential for biological structures. In the present paper we discuss these fundamental aspects continuing the framework drafted in our previous publications (Igamberdiev 1992, 1993, 1994).
1. RECOGNITION ACTIVITY OF BIOMACROMOLECULES Biological system possesses a hierarchy, according to which the potentialities of constituent elements at lower levels are restricted by a higher level. The subdivision of a non-equilibrium biological system into levels corresponds to ’the vertical picture’ of life processes in which the transduction-amplification cascades link molecular level processes to the macroscopic structure and behavior of organisms. The transductionamplification cascades provide ’vertical flow’ of information and that enables organisms to exploit the nonpicturable quantum dynamics at submolecular level to control their classical macroscopic behavior (Conrad, 1994a). The reliability of information transfer inside the biosystem is determined by the specific features of the operation of enzymes and other biomacromolecules. The specificity of biomacromolecules for the strictly determined interactions (i.e. their recognition activity) can be explained by low dissipation of energy during their operation, which provides registration of signals not distinguished by their energy from the surrounding noise. In analysis of operation of these molecular machines we should take into account their quantum properties. As it was stated by Marijuan (1994), every enzyme is a pattern recognition machine. We cannot recursively formalize its internal dynamics as we can in Turing machines and conventional computers: enzyme is a structurally nonprogrammable device (Conrad, 1992). There appears a substantional difference between the switching function provided by a logic gate in a digital computer and the switching
194
A U. IGAMBERDIEV
provided by enzymes and proteins. It seems that the Turing-Church thesis about the equivalence between computation and dynamics cannot be substantiated in cellular systems (Conrad and Liberman, 1982; Conrad, 1992; Mariju~in, 1994). During enzyme operation a non-picturable quantum dynamics takes place. Enzymes determine the boundary conditions which direct the course of a reaction into a certain route. Under these conditions certain states of particles (electrons) are not allowed, whereas in a coordinate interval defined by the active site, the new wave function is coincident with the one that existed before the action of enzyme. An electron is therefore evolved, being directed into the passage of routes which is determined by the coordinate scale defined by an enzyme. This results in the prohibition of some previously probable trajectories of electron movement in the substrate molecule, whereas other trajectories become more probable leading to the redistribution of electron density and hence to the internal polarization of the molecule determining substrate conversion into a product (Green and Vande Zande, 1981; Igamberdiev, 1993).
According to Conrad (1994b), each input signal at the molecular level triggers the appearance of a specific conformational change or a release of a specifically shaped macromolecule. The signal pattern is thus represented as a set of conformations. Electronicconformational interactions that accompany molecular recognition processes allow the non-picturable superposition of electronic states to percolate to higher levels of function. The self-assembly process converts a symbolic pattern recognition phenomenon into a free energy minimization process. This process is non-integrable, i.e. it cannot be evolved deterministically from the primary structure of macromolecule and from the nature of input signal by the finite number of operations. The situation is similar to the well-known "three-body problem". Conformational relaxation of macromolecular systems is considered to be an elementary action of the bioenergetic process (Blumenfeld, 1983), in which the fast quantum effect (e.g. the capture of an electron by macromolecule) is followed by a slow conformational transition being the mechanical motion of a macromolecule. During this transition, the energy is not dissipated remaining stored for a total lifetime long enough for the work to be performed (Kremen, 1992). Thus, its motion is many times slower than the initial quantum effect, and the rate ofa bioenergetic process is therefore determined by the rate of conformational relaxation. The latter takes place only after the action of a force converting the system into the new conformational state, i.e. after the generation of a nonequilibrium state resulting from the fast initial interaction. From this point of view the specificity of biomacromolecules is connected with the recognition of specific configurations of electron clouds in certain compounds and should therefore be described using quantum mechanical formalism. Braginsky et al. (1980) had analyzed the conditions necessary for the detection of weak forces. It was shown that according to the Heisenberg uncertainty ratio, interactions between a quantum system and a macroscopic measuring device can take place by a path that provides practically non-demolition registration of strictly determined weak signals. These interactions are characterized by high precision and certainty of the result of measurement, as the sensitivity of the detector is determined by its relaxation properties. Quantum measurement is connected with low energy dissipation in the case where the relaxation period of a macroscopic oscillator is many times larger than the time interval of measurement. Under the condition of quantum non-demolition (QND) measurement, internal fluctuations of the oscillator will not unmask the action of detected weak force, and certain motions in a macroscopic oscillator can be transformed into high-frequency
INFORMATION PROCESSING 1N BIOSYSTEMS:
195
vibrations without information loss. This provides electron movements over considerable distances with great speed and specificity through the metabolic network. Low energy dissipation during the recognition of weak forces is provided by mobile proton states appearing in biomacromolecules. According to Witten (1982) the key feature in QND measurement is repeatability. The observable in such operation must be that which can be repeatedly measured, with the result of each measurement, in the absence of classical force, being completely determined by the result of an initial precise measurement (Caves et al., 1980) One way to achieve this capability is to measure an observable that does not become contaminated by uncertainties in other noncommuting observables. The concept of QND arises out of need to monitor the values of single observable over and over again. The existence of QND observables and measurements provides a means of escape from the problem of nonrepeatable experiments (Witten, 1980). Internal QND measurements are inherent for biological organization and determine the essential features of living systems. Low energy dissipation in these measurements provided by slow conformational relaxation of biomacromolecular complexes (regarded as measuring devices) is the main precondition of enzyme operation and information transfer determining the steady non-equilibrium state of biosystems (Igamberdiev, 1993). We show here that quantum mechanical uncertainty that underlies the appearance of bifurcations is the main physical foundation of complication and irreversible transformation of biosystems, i.e. of their information capacity.
2. QUANTUM-MECHANICAL BACKGROUND OF SYMMETRYBREAKING The transition from the set of possible worlds to the description of the real world results from the process of reduction of potentialities. In the conditions of QND measurements, provided by long relaxation times, the possibility of non-predictable alternative result of measurement (i.e. of bifurcation) is minimal. If relaxation periods are shorter, the system is less stable but in can evolve in a different state via generation of the alternative description. This generation is a symmetry-breaking phenomenon which underlines the formation of new structures. Therefore the ratio between the measurement time and the relaxation period determines the state of equilibration between the stability of the system and its ability to change. The generation of an alternative description corresponds to the arising of bifurcation within the system. Bifurcation analysis is developed in the catastrophe theory. It claims that at a certain stage of evolution the parameters of the system attain critical values at which the steady state bifurcates and hence the stability is lost. In addition to the customary catastrophe-theoretic model of bifurcation, which operates with non-linearities, it is important to state that the initial instability arises from the non-absolute character of the internal QND measurements. As was shown by Matsuno (1992), local fluctuations are accompanied by the non-vanishing rate of variation because of the uncertainty relationship, and the endogenous transformations refer to the symmetry breaking of the Hamiltonian, which has its own dynamics. Irreversible symmetry-breaking emerges from indefinite states, and indefiniteness is provided by the quantum measurements. Under this consideration, macroscopic bifurcations seem to be the consequence of the quantum properties of the biosystem, and only the measurement process is responsible for the branching behaviour of bio.dynamics. At the macroscopic level, the alterations in relaxa-
196
A. U. IGAMBERDIEV
tion processes lead to the redistributions between different steady states within the biosystem. Thus, non-linearity arises from high order relaxation processes. This leads to instabilities and to initially unpredictable transformations, resulting in macroscopic bifurcations. Bifurcations being symmetry-breaking phenomena generate ’voids’ which are acknowledged by the whole system and internalized during ontogenesis and evolution in generated higher structures (Aires and Marijuhn, 1995). The bifurcations that arise during the operation of the genome and enzymes and provided by the non-absolute character of non-demolition measurements (i.e. primarily non-internalized bifurcations) are essential in the evolutionary process. Therefore the evolutionary process seems to be a consequence of the quantum uncertainty that appears at the macroscopic level. In general, a change in relaxation time leads to the alteration in specificity of biomacromolecules to certain interactions, and this can result in branching behaviour. Perturbed versions of the original functions can interact in such a way that a property of their commutativity breaks down, and a new global function for the system is generated (Matsuno, 1985). In physics the concept of symmetry is connected with the most generalized characteristics of the Universe. Physical symmetries correspond to conservation laws, and violations of symmetry lead to the formulations of more general, global symmetries. The CPT theorem claims the invariant character of all physical processes in relation to the turn of charge, space and time constituents. In physics violation of symmetry is an initial point for seeking a global symmetry, and the final dream of physics, ’the Generalized Field Theory’ should be based on formulation of invariant characteristics which are constant under all possible transformations. W. Heisenberg named such fundamental theory as ’Urgleichung’, i.e. as the basic generalized equation. For understanding of the symmetries in living systems we should realize that every irreversible process is reflected in space coordinates as a symmetry breakage. Followed aRer this, the generation of a new organization is realized as a building of the symmetry of a higher level.
3. QUANTUM MEASUREMENTS AND INTERNAL COMPLEXITY OF BIOSYSTEMS The transition from the set of possible worlds to the description of the real world results from the irreversible process of reduction of potentialities in which the reflection from the set of complex variables to real numbers takes place (Rosen, 1977). During this process, the system, considered a device, can generate independent descriptions which are alternative constructions without an implicative relation between them. The point of discrimination between these two descriptions is considered to be the bifurcation point (Rosen, 1979). The relation of two biological systems or two states of one biosystem in different moments of time (i.e. of two reflections to real numbers) is realized by the principle of epimorphism, i.e. of many-to-one mapping. These ideas arise to Rashevsky (1961). The complexity of biological system in this paradigm is a result of the interference of restrictions based on the internal formal description, which can act reciprocally with the set of physical laws. These restrictions are called constraints, in contrast to physical laws. Constrains in which many-to-one mapping is realized are called nonholonomic constraints (Pattee, 1970). In different physical models we reveal the reflection from imaginary to real numbers.
INFORMATION PROCESSING 1N BIOSYSTEMS:
197
The idea arises to P. A. Florensky who suggested at the beginning of the 20th century that imaginary points in the time-space serve for gluing together of the single points of the Universe into the whole entity (Florensky, 1993). Time coordinate in the theory of relativity is described by imaginary numbers, whereas spatial coordinates are real. In the model of the Universe suggested by S. Hawking (1988) the imaginary time is introduced in which the Universe is appeared as a closed holistic structure without frames. After the transition to real time the reduction to the model in which "the beginning" of the Universe and its inflation are present. But the most developed the idea of reduction of potentialities described as a reflection from complex to real numbers is present in quantum mechanical formalism. Complex psi-function of Schr0dinger equation during measurement is reduced into the set of real numbers. The generalization of quantum theory was suggested using biquatemions - hypercomplex numbers (Conte, 1994). The reduction to real numbers can be determined as dissymmetrization in the field of potentialities and can be considered as an important precondition of semiotic structures (Igamberdiev, 1992). A specific recognition is characterized by the minimal energy dissipation during interaction between the measuring device and the measured object. The chemical reactions which proceed under enzyme control retain all the equilibrium and other thermodynamic characteristics they possessed before this control. They are only influenced by the enzymatic constraint insofar as the substrate conversion proceeds via a strictly determined route according to enzyme specificity. Under this control the transition to macroscopic time takes place, as the conformational relaxation of slow enzymes is realized in seconds. This can be explained as the effect of a strictly determined electron state on the enzyme molecule. As a result, the system is subdivided into two subsets: one controlling and the other being controlled. Controlling (information) level gives the appearance of placing extra restriction on the system (constraints). The system operates not only according to physical laws, but also according to its own restrictions (constraints), encoded in the internal description which determines the specificity or ’individuality’ of the system. These internal restrictions can be conceived as ’arbitrary’ in relation to physical laws, so we can introduce the idea of ’arbitrariness’ between the signifiant and the thing it signifies (signifie). From a physical point of view, this connection is presented as arbitrary or casual, and its reproduction can be represented as a result of the storage of casual choice. The problem of interference between physical laws and internal constraints can be seen as the problem of interconnection between the physical and the biological in organic life. Biological system operates according to both physical laws and its own internal constraints, which determine its specifity; under these constraints ’the molecule becomes a message’ (Pat-tee, 1970). H. Pattee (1989) emphasized that measurement itself is a non-formal process and cannot be programmed but its results are symbols that can be used in a formal system as information. The act of recognition (based on QND measurement) involves a low-energy interaction between a component of a non-linear system (macromolecular device) and an environmental (epigenetic) input signal that causes the component to undergo a state transition (Barham, 1990). In such a system a low energy recognition stroke and a high energy or work stroke constitute the work cycle. Both phases of the cycle are viewed from a physical perspective in which the low energy (information or recognition) constraints act as signs with respect to high energy (pragmatic) constraints, leading to ’semiotic correlations’ that have predictive values (Yates, 1992). The relative autonomy of the biological system from local energy gradients is provided
198
A U. IGAMBERD1EV
by the ’on board’ energy supply in the form of macroergs. Owing to this, biological system is able to distinguish between the conditions external to itself which will support its continued oscillations, and those which will not support them. The low-energy environmental inputs which happen to be correlated (semiotically, i.e. based on the internal constraints) with those propitious conditions in order to correlate oscillator’s high energy interactions with the environment are recognized. Barham (1990) states that these low energy inputs constitute ’information’ in its semantic sense. In other words, biological system possesses an ability to measure certain environmental signals which are transformed into the actual work. In accordance with this, every biofunction from enzymes on up, contains a subsystem (’epistemon’), a sort of sense organ which acts as a trigger for the functional action of the whole system. Active sites of enzymes, various receptors are considered as such epistemons. The structural stability and reliability of the biological system can be considered in relation to physical grounds, making apparent various semiotic relations (i.e., constraints) within the system. The state of non-equilibrium in which biological system exists is stabilized, and internalized within its cyclic organization. The system itself seeks for non-equilibrium flows which maintain its own existence in accordance with the selection of distinct signals from a chaotic set of environmental factors forming ’a noise field’. Therefore, the system is not only internally organized; it also organizes the surrounding environment. As mentioned by Prigogine (1980), the first mechanism of non-equilibrium evolution is the one that resulted in the system’s movement out into strongly nonequilibrium conditions.
4. NON-FORCE CORRELATIONS IN BIOSYSTEMS Operation of biosystem as an entity is provided not only by the interactions between the informational and energetic levels, but also by the formation of the other level - the level at which the system acts as a whole entity. The physical foundation of such a level is a consequence from the Einstein-Podolsky-Rosen (EPR) paradox. Low energy dissipation during the conformational relaxation of biomacromolecules provides for the possibility of long-distance non-locality transfer for electron and proton flows through the metabolic networks. In such systems a non-locality in the quantum mechanical sense and non-force correlations can arise between subsystems of the biological system in accordance with the EPR-paradox. The EPR correlations can appear within systems which realize QND measurements. Two particles arising from a single system (e.g. two electrons with opposite spin values from the same atomic sublevel) can store ’knowledge’ about a previous state when they are later non-disturbed, i.e. when non-controlled quantum measurements hiding the initial picture have not happened. Otherwise, information about the whole system will be unavoidably lost. Therefore the preservation of knowledge (memory) about the whole system is possible only in the case of non-demolition measurements that are realized on its subsystems, and low energy dissipation during conformational relaxation of biomacromolecules can be considered as a main precondition for providing and maintaining EPR correlations. The verification of Bell’s inequalities determining the existence of non-locality effects which can really prove actual wholeness of biosystems arising to their non-local properties is important for the confirmation of such an approach. The verification procedure could show that certain correlations in biosystem result from non-local interactions aris-
INFORMATION PROCESSING IN BIOSYSTEMS:
199
ing from the EPR effects. Non-force interactions could explain the co-ordination of the parameters of elementary particles during the action of protein molecules and their complexes. In the scheme of slow conformational changes, Bell’s inequalities can be interpreted as the quantum mechanical background of the operation of macromolecular subsystems in a biological system. The hypothesis of non-force quantum field in biosystems was discussed by V. E. Zhvirblis (1993). He proposed that the non-force three-dimensional field, provided by different topological structures in DNA (according to the quantum Aaronov-Bohm effect) may determine structure generation. In this model the basic morphogenetic bioinformation (’Gestalt-information’) arises from the solenoidal non-force field not consisted of real physical interactions but only derived from them. The vectorial movement of particles during morphogenesis may be determined by these quantum properties. According to the presented ideas the holistic properties of life can be rationally exo plained in correspondence with the concepts of modern physics.
5. TIME ASYMMETRY AND BIOLOGICAL SYSTEMS Each temporal event is an actualization, i.e. the irreversible transition from the field of possibilities to real events. It can be represented as a reflection into real numbers, where the previous event is put into correspondence to the appearing event and considered as its possible cause. This shows the resemblance of temporal process with the quantum reduction of potentialities, i.e. with quantum measurement. The temporal process is based on quantum irreversibility (Igamberdiev, 1985). Although the quantum laws are time-reversal invariants, a contradiction appears if two measurements performed by a single observer, and described according to these laws, are performed in two opposite directions of time. This contradiction leads to bringing forth the concept of an observer’s private time, and then to building up a temporal parameter common to several observers from their private times. Time asymmetry turns out to be a consequence of the latter construction (Bitbol, 1986). The concept of time flow as a consequence of quantum measurement is developed by Mensky (1993). The ’energy-time’ uncertainty ratio can be considered as a complementarity ratio between time and alteration. In this ratio, time appears as the time of the observer but not of the quantum system (de Broglie, 1982), which leads to the impossibility of formulation of the Hamiltonian for this ratio. Possibilities of disturbances and bifurcations in the system arise from this ratio. Irreversibility of time in quantum mechanics appears to be the consequence of subsequent measurements at the stage of information gathering on the whole sequence of outcomes (Dicke, 1989). Branched evolutionary processes lead to the actual irreversibility, which contradicts the formal reversibility of SchrOdinger’s equation (Toyozawa, 1989), i.e. irreversibility arises as a symmetry-breaking at the macroscopic level and is connected with bifurcations. The latter can therefore be considered as the precondition for irreversible development in ontogenesis and evolution and the reason for the complication of organization. The original approach to understanding of time was drawn out in "the causal mechanics" ofN. A. Kozyrev (1991). He considers the transition from the cause to its result as a basic event on which the conceptual basis of mechanics can be built. According to Kozyrev, the transition from the cause to the result being an initial irreversible process is realized as a jump through the "empty" (in our consideration - imaginary) point. This
200
A. U. IGAMBERDIEV
can be actually compared with the quantum theory of measurement in which the reflection to real numbers is realized. The speed of transition from cause to result, according to Kozyrev, is described by a pseudoscalar and could be connected with the finite speed of"a jump" through the "empty" point. In our approach it may correspond to the time of relaxation of a measuring device. In the direction of the time flow an absolute distinction between the past and the present is reflected in the similar absolute distinction in the symmetrian properties of space which is realized not in the differences in the spatial directions, but in the absolute distinction between the right and the left. The flow of time is described by the value which determines the linear rate of turning. Kozyrev proposed that life uses the flow of time as an additional source of energy and that the optical dissymmetry of biological molecular structures (of nucleic acids, proteins, etc.) corresponds to the actual irreversible processes in living systems, to high "density of time" in biological systems. "The density" of time in Kozyrev’s conception may correspond to the intensity of irreversible processes, i.e. of the reductions of potentialities, which influences on other processes by the nonforce way (we may correspond this to the influence via EPR-correlations). The possibility of a connection through time in quantum mechanical paradigm we can explain as a connection via EPR-correlations, i.e. as the effects of a wholeness, and EPR-correlations themselves are the result of the reduction of potentialities (i.e. of time in general sense). In the spatial region where irreversible process takes place, "the density of time" is changed, and a non-force action of one process on the other without contact by physical forces is possible. This was demonstrated by the experiments conducted by Kozyrev with rotating balance. Irreversible process absolutely separated spatially from the balance (i.e. evaporation process separated from the balance by a wall) nevertheless turns the balance. Active properties of time (its ’flow’ and ’density’) connect all events in the world into a total wholeness and can realize the interaction of different phenomena without causal (material) connection. In Kozyrev’s theory the irreversible process changes "properties of time" (and vice versa - anisotropy of space causes irreversible process), generates dissymmetry and the effects of wholeness. This is reflected in biological systems in their dissymmetry and chirality. The degree of"the density of time" generates the complexity, hierarchical structuration, i.e. the decrease of entropy. Some fruitful ideas in Kozyrev’s conception are expressed in inadequate way and many of his conceptual innovations are not clear and difficult for interpretation. Translation of these ideas into the language of quantum measurement theory may clear up some his thoughts and develop them in positive direction. Understanding of the temporal process as a basic event for the description of the Universe and non-derivable from something else is the main suggestion which can explain fundamental symmetrian properties of space. From this point of view time represents as an internal process of actualization. Intensive actualization process in living systems corresponds to their complex molecular, metabolic and morphologic spaces.
6. METABOLIC ORGANIZATION AS A CONSEQUENCE OF SYMMETRY-BREAKING What is a background of metabolic organization? The formation of an alternative product in enzymic reaction because of the non-absolute specificity of the enzyme is a direct consequence of the dissymmetrization which basically is derived from the non-absolute character of QND measurements. This possibility of appearance of a void within nonequilibrium biological organization can be an initial point of further informational
INFORMATION PROCESSING IN BIOSYSTEMS
201
growth realized in the complication of hierarchical organization. Therefore the dynamical organization of biosystems is characterized by the possibility of appearance of a functional void within biological cycles. Functional voids are symmetry-breakings which counteract within the system (Mariju~n, 1995; Alves and Mariju~n, 1995). The separation between the direct and reverse routes of biochemical reaction, one being linked (coupled), whereas the other non-coupled with the pool of certain (often energyrich) compounds, results in the symmetry-breaking and in the arising of "the inequality" between the direct and reverse pathways. Consequently, the separation of the direct and reverse reactions leads to the generation of elementary metabolic substrate cycle. Reactions connected with the significant changes of free energy are preferable for the generation of such cycles. The realization of the direct and reverse reactions by two different enzymes leads to the transformation of initially linear pathway into the cyclic one. The initial uncoupled structure of futile cycle seems to be very disadvantageous, but the transition to the hierarchical organization with strongly coupled reaction networks converts it into a powerful mechanism of reciprocal regulation of the direct and reverse metabolic pathways. The allosteric and covalent modification are the most effective means for the reciprocal regulation of two enzymes forming the futile cycle. They result in the effective regulation of the direct and reverse routes in such a way that these reactions become separated in time and the energy loss is minimized (Koshland, 1984). The logic of futile cycle transformation into metabolic cycle was considered in our previous paper (Igamberdiev, 1994). The enzymes catalyzing correspondingly the direct and reverse reactions are distinguished (at least slightly) by their specificity to their substrates. This leads to the generation of bifurcation in one branch and consequently to the appearance of the alternative pathway of metabolic conversion. The latter can result in the formation of a compound identical to the initial substrate. Therefore the futile cycle subsequently unfolds into the complete metabolic cycle. When a certain subset of a substrate set of the catalytic system realizes the function of a matrix which determines the formation and reproduction of this catalytic system, the structure arises defined as hypercycle, It can be considered as a generalization of the selfreproducing metabolic system. Hypercycle corresponds to the appearance of internal invariants within the system which are conserved during its reproduction. Low energy dissipation in the recognition, as well as in the reparation and editing processes, ensures the stability of the hypercyclic structures. The emergence of reflectively autocatalytic sets of peptides and polypeptides is considered by Kauffman (1986) to be an essentially inevitable collective property of any sufficiently complex set of polypeptides. The participation of nucleic acids provides a new means to select for peptides with useful properties. It becomes evident that the selfreplication is an emergent property arising from local interactions in systems that can be much simpler than it generally believed. This property is a result of a spontaneous dissymmetrization arising from internal possibilities of autocatalytic sets.
7. GENERATION OF MORPHOLOGY Morphological structure of biosystems is based on their metabolic organization. This determination is connected with the symmetry breaking processes in metabolic pathways which are considered as the main precondition of generation of morphogenetic informa-
202
A U IGAMBERDIEV
tion. The kinetic parameters of metabolic reactions provide spatial geometric effects, and the structures appear as morphological fixations of kinetic processes. The cyclomeric concept of Petukhov (1987) claims that the biochemical cycle being a system of incomes and outcomes of certain compounds can be defined as an axis of symmetry relatively to which biological structures are organized. Even insignificant changes in the parameters of cycles can lead to the essential morphological reconstructions. Morphology is formed by the stable trajectories of the formation and deposition of compounds formed in biochemical cycles. The formation of new cycles leads to the overbuilding of new metabolic trajectories, and this corresponds to morphological changes. Regulators of growth and development via influence on bifurcations make it possible for the system to transform into a new state. Under their influence the system becomes more dissipative (consequently less stable) and can more easily be transformed. Thus, morphology is a result of the temporal and spatial organization of matter and energy flows in biosystems. Morphological transformations occur via changes of organization of these flows. This results in the transformations of coordinate scales describing biological forms (Thompson d’Arcy, 1917). The degree of curvilinearity of ’the space of biological forms’ is determined by the periods of cycles’ turnover and by the differences in the rates of reactions providing depositions from the cycles. If the deposited compounds can turn into reverse transformations, oscillations are possible, but if they turn into the insoluble form, they participate in the construction of rigid skeleton. The formation of morphological structures is a result of interference between concentration oscillations of compounds participating in the structure formation. The morphology being a reduction of potentialities of the metabolic system is a projection from the multi-dimensional space of kinetic equilibria and processes into the threedimentional space. Alterations of the time intervals of the cycles (i.e. changes of cyclic symmetrian characteristics) can lead to the changes of such projections which results in the modifications of morphology. The morphogenetic field operates in the space of physical fields, and the metabolic cycle as being a primary morphological generator provides the conditions of structure formation. The problem of morphogenetic information cannot be reduced only to the linear information of the genetic code. It was proposed that DNA may be a source of coherent photon storage, and besides the genetic information it can be a carrier of the information for ’pattern recognition’ (Popp, 1989). This may be realized via coherent interactions in DNA-sequence-specific biophoton transfer. The coherence of biophotons may form ’Gestalt’-information, essential for morphogenesis. Ultraweak or even "non-force" (according to the quantum Aaronov-Bohm effect) three-dimentional field, provided by different topological structures in DNA is proposed for explanation of the structure generation (Zhvirblis, 1993). The problem of morphogenetic information was stated for the first time by A. G. Gurvich who considered non-equilibrium macromolecular constellations as a possible source of biological informational field. These constellations may be connected not only with DNA. Electromagnetic bio-information may be generated via extinctions of photons in the enzymatic reactions connected with changes of free energy and generation of free radicals during enzymatic conversions (Gurvich and Gurvich, 1948). It was shown that light-emitting processes are connected with normal reactions in living tissue, and peroxidases and other enzymes utilizing peroxides and active oxygen forms are involved
INFORMATION PROCESSING IN BIOSYSTEMS:
203
in such a process. The extinction is increased before cell division and morphogenetic process, and it is most expressed in highly developed species. Therefore the ultraweak photon emission may be proposed to be a biological phenomenon, which is a carrier of morphogenetic information (Popp, 1989). The separations of the direct and reverse metabolic flows leading to the transition from the simple uncoupled processes to the integrated strongly coupled networks, the depositions from the cycles and the appearance of rhythms of concentrational oscillations are the basical events determining the temporal and spatial organization of biosystems. Topological reconstructions of DNA and enzymatic reactions of oxidative metabolism may provide low energy informational interactions via electromagnetic extinctions. This can result in the coordination of different metabolic fluxes which provides threedimensional pattern of morphology of the whole organism. Low-energy influences on bifurcations provide cyclomeric transformations which alter dissymmetric relations between the direct and reverse routes within the cycles. They interact with high-energy non-equilibrium processes via the transduction-amplification cascades realized in cyclic processes. The low-energy influences themselves are connected in the total informational set directing morphogenetic events, and the minimization of disturbances in this set provides the effects of non-locality which tie system’s parts into the whole entity.
8. CONCLUSION Life is a specific form of existence, internally determining itself in the Universe. It can be defined as the self-organizing and order-generating activity of open non-equilibrium systems based on their semiotic structure. Biological activity corresponds to specific informational organization which provides high energy interactions connected with nonequilibrium states, low energy interactions being a background of the recognition processes and realized as quantum non-demolition measurements, and non-force (no energy) correlations of quantum mechanical nature which determine operation of living system as a whole entity. Biological organization provides the ’spontaneous’ activity of life which corresponds to the specific understanding of time as an internal process of actualization. Intensive actualization process in living systems corresponds to their complex molecular, metabolic and morphologic spaces. The emergence of a new order cannot be recursively calculated: The generation of bifurcations in living systems determined by quantum uncertainty appears to be the physical basis of this symmetry-breaking process. It becomes internalized within the system and results in ’the self-growing Logos’ (Herakleitos) realized in the development of complex informational network ofbiosystems.
REFERENCES Alves, D. and Marijuhn, P. (1995) Information and symmetry in the cellular system. Third lnterdtsctphnary Congress on Symmetry, (Washington 1995) (Symmetrian Institute Budapest) Barham, J. (1990) A poincarean approach to evolutionary epistemology, Journal of Soctal and Btologtcal Structures, 13, 193-258. Bitbol, M. (1986) Time symmetry and quantum measurements, Physics Letters A, 115, 357-362. Blumenfeld, L A. (1983) Phystcs of Btoenergetic Processes, Berlin’ Springer-Verlag
204
A. U IGAMBERDIEV
Braginsky, V. B., Vorontsov, Yu. K and Thome, K. S. (1980) Quantum non-demolition measurements, Science, 209, 547-557. Broghe, L. de (1982) Les incertitudes d’Hetsenberg et I ’mterpretatton probabthste de la mecamque ondulatotre, Paris: Gauthier-Villars. Caves, C. M., Thome, K. S., Drever, W P., Sandberg, V. D. and Zimmerman, M. (1980) On the measurement of a weak classical force coupled to a quantum mechanical oscdlator. I, lssues ofPrmctple Revtews of Modern Phystcs, 52, 341-398. Conrad, M. (1992) Molecular computing: The lock-key paradigm, Computer, 25, i 1-20. Conrad, M. (1994a) From microphys~cal dynamics to macrophysical architecture. The vertical flow of information in b~olog~cal systems. Foundations of Information Science. From Computers and Quantum Physics to Cells, Nervous Systems and Societies Conference held in Madrid 11-15 July 1994. Book of Abstracts. Zaragoza. P.7. Conrad, M. (1994b) Quantum neuromolecular computlng. Foundations of information science. From computers and quantum physics to cells, nervous systems and societies. Conference held in Madrid 11-15 July 1994 Book of Abstracts. Zaragoza, P.8. Conrad, M. and Liberman, E A. (1982) Molecular computing as a link between biological and physical theory, Journal of Theorettcal Btology, 98, 239-252. Conte, E. (1994) Btquaternion Quantum Mechamcs Vol L Theorettcal Foundations, L.I.U.I.: Academic Press. Dicke, R. H. (1989) Quantum measurements, sequential and latent, Foundattons of Physics,, 19, 385-395 Eigen, M and Schuster, P. (1979) The Hypercycle: A Prmciple of Natural Self-Orgamzation, Berlin. Springer-Verlag. Florensky, P. A. (1991)lmaginaries m Geometry [In Russian], Moscow: Lazur. Green, D. E. and Vande Zande H D. (1981) Universal energy principle of biological systems and the unity of bioenergetics, Proceedings of the National Academy of Sclences of the USA, 78, 5344-5347. Gurvich, A, and Gurvich, L. (1948) lntroductton to the Study of Mitogenesis [in Russian], Moscow: lzdatelstvo Akademii Meditsinskikh Nauk SSSR. Hawking, S. (1988) A Br~ef Htstory of Time. From the Big Bang to Black Holes, Toronto: Bantam Books. Igarnberdiev, A. U. (1985) Time in biological systems [in Russian], Zhurnal Obshchei Btologu [Journal of General Biology], 46, 471-482. Igamberdiev, A. U. (1992) Organtzatton of biosystems: A semiotic approach, In. Sebeok, Th. A. and UmtkerSebeok, J, eds., Btosemtottcs. The Semtotic Web 1991, Berlin: Mouton de Gruyter, pp 125-144 lgamberdiev, A U. (1993) Quantum mechanical properties of biosystems" A framework for complextty, structural stability and transformations, BtoSystems, 31, 65-73. lgarnberdiev, A. U. (1994) The role of metabolic transformations in generation of biological order, Rtvtsta dt Btologta (Biology Forum), 87, 19-38. Kauffman, S. A. (I 986) Autocatalyt~c sets of proteins, Journal of Theorettcal Biology, 119, 1-24 Koshland, D. E. Jr (1984) Regulation of enzyme activity and metabolic pathways, Trends m Btocherntcal Sctences. 9, 82-92. Kozyrev, N. A. (1991) Selected works [In Russian], Leningrad" Umversity Publishers Kremen, A. (1993) Btologtcal molecular energy machines as measuring devices, Journal of Theoretwal Biology, 154, 405-413. Mariju~, P. C. (1994) Enzymes, automata and artificial cells In: Ray Paton, ed., Computing with B~ologtcal Metaphors, London: Chapman & Hall Mariju~, P. C. (1995) lnformatton science and symmetry. On the emergence of a new disciplinary/interdisciplinary avenue of enquire. Third Interdisctplinary Congress on Symmetry (Washington 1995) (Symmetrian Institute Budapest). Matsuno, K. (1985) How can quantum mechanics of material evolution be possible?: symmetry and symmetry-breaking in protobiological evolution, BtoSystems, 17, 179-192. Matsuno, K. (1992) The uncertainty principle as an evolutionary engine, BioSystems, 27, 63-76. Mensky, M. B. (1993) Continuous Quantum Measurements andPath Integrals, Bristol: IOP Publishing. Pattee, H. H (1970) The problem of biological hierarchy, In: Towards a Theoretical Biology. V.3, Draets
INFORMATION PROCESSING IN BIOSYSTEMS"
205
(Edinburgh Umversity Press), p.117-136. Pattee, H. H. (1989) The measurement problem m art,fic~al world models, BtoSystems. 23, 281-290. Petukhov, S. V. (1987) The cychc groups of non-linear automorphisms ~n biostructures and the cyclomenc theory [In Russian], In. Presnov, E V, ed, Theorettcal and Mathemattcal Aspects of Morphogenests, Moscow: Nauka), pp. 218-224. Popp, F A (1989) Coherent photon storage of b~ological systems, In’ Popp, F A, ed., Electromagnettc Btolnformatton, Munchen’ Urban & Schwarzenberg, pp. 144-167. Prigogine, I. (1980) From Being to Becoming." Ttme and Complextty tn the Physical Sciences, San Francisco. W. H. Freeman. Rashevsky, N. (1961) Biological epimorphism, adequate design, and the problem of regeneration, Bulletin of Mathemattcal Biology, 23, 109-113. Rosen, R (1977) Observation and b~ological systems, Bulletin of Mathematlcal Btology, 39, 663-678. Rosen, R. (1979) Bifurcations and biological observables, Annals of the New York Academy of Sciences, 316, 178-187. Thompson, d’Arcy W (1917) On Growth and Form, London: Cambridge University Press. Toyozawa, Y. (1989) The irreversibd~ty inherent in quantum mechanics, Journal of the Phystcal Society of Japan, 58, 2215-2218. Witten, M. (1980) A note on the structure of system state spaces and its implications on the existence of nonrepeatable experiments, Bulletin ofMathemattcal Btology, 42, 267-272 Witten, M. (1982) Some thoughts on quantum non-demolition measurements tn biological systems, Bulletin of Mathematical Btology, 44, 689-696. Yates, F. E (1992) On the emergence of chemical languages, In: Sebeok, Th. A and Umiker-Sebeok, J., eds, Btosemtottcs. The Semiotic Web 1991. Berhn" Mouton de Gruyter, pp. 471-486. Zhvirblis, V. E. (1993) Generation of form [m Russian], Khtmiya 1 Zhtzn" [Chemistry and Life], 8, 42-49.
Symmetry. Culture and Sctence Vol. 8, No. 2, 207-214, 1997
SYMMETRY. CULTURE AND SCIENCE
IS SYMMETRY INFORMATIVE? John E. Gray and Andrew Vogt address: Code N92, Department of Mathematics, Naval Surface Warfare Center/DD Georgetown Umverstty, Dahlgren, VA 22448 Washington, DC 20057-0996, U.S.A, E-mail:
[email protected] mil, andy@gumath I math georgetown edu
1. INTRODUCTION Is symmetry informative? The answer is both yes and no. We examine what information and symmetry are and how they are related. Our approach is primarily mathematical -- not because mathematics provides the final word, but because it provides an insightful and relatively precise starting point. Information theory treats transformations that messages undergo from source to destination. Symmetries are transformations that leave some property of interest unchanged. In this respect the studies of information and symmetry can both be regarded as a Quest for the identity transformation.
2. WHAT IS INFORMATION? Shannon and Weaver (the latter wrote an introduction to Shannon’s papers (Shannon and Weaver, 1964)) explicitly called Shannon’s work communication theory. In fact, even more narrowly they called it "a mathematical theory of communication." Despite popular usage, no comprehensive theory of information exists. The common notion of "information" -- i.e., facts, knowledge, data, structure -- is probably too broad for scientific and mathematical characterization. Information theory is as applicable to the communication of a stream of nonsense as it is to the most profound emanations of the human psyche. Shannon (1964), in a fruitful oversimplification, declared that "... the semantic aspects of communication are irrelevant to the engineering aspects." Every form of communication has three components: a message from a source (e.g., words spoken into a telephone), a signal representing this message that travels along a channel (e.g., a series of electrical pulses traveling along a telephone wire), and the message actually received by the destination (e.g., the reconstituted words heard by a listener on another telephone). The message is encoded as a signal, transmitted along the channel, and received and decoded at the destination. Enroute noise (e.g., the static on a telephone line) may alter the signal, so that the message received differs from the message sent.
208
J E. GRAY, ANDA. VOGT
The problem that information theory addresses is how best to encode and decode the message, taking into account the characteristics of source, channel, noise, and destination. "Best" is equated with maximum accuracy and speed subject to cost constraints. On the assumption that one-time-only messages are of no importance for design, the theory concerns itself with average accuracy, average speed, and average cost. A message is a succession of elements (e.g., letters, words, sentences, digits, images, musical notes, etc.). These elements -- call them symbols -- are drawn from a preexisting set of possible symbols (e.g., an alphabet, dictionary, etc.), and over the course of many communications each symbol is assumed to occur with a definite relative frequency. (This is sometimes referred to as the stationarity assumption. It is false in most applications, but as frequencies usually change slowly with time, it is a useful approximation.) Information theory associates with each symbol the quantity Iog2(1/p)
where p is the relative frequency or probability of the symbol. This quantity came to be called the "information’’t of the symbol. It is a measure of infrequency: the more infrequent the symbol, the greater the "information." This quantity has no direct connection with the particular information represented by the symbol. Indeed, since it depends only on the probability of the symbol, not its content or meaning, no such connection is possible. We call this quantity the "infrequency." If a symbol has a fifty percent probability of occurring, its infrequency is 1og2(1/(1/2)) = 1og2(2) = 1 and the symbol is said to consist of one "bit," a term coined by John Tukey. If it has probability 1/(2"), the infrequency is 1og2(2~) = n bits. The infrequency is related to binary coding. Encoding replaces one symbol set by another (e.g., alphanumeric computer keyboard input by the O’s and l’s of machine language). A binary code has two symbols (say 0 and 1) and encodes a message as a f’mite string of successive O’s and l’s. Binary codes are convenient in electronics and computer science since 0 and 1 can be translated into on/off states of electric circuits. If only three source symbols are to be sent- one of them of probability I/2, and the other two of probability 1/4 each, the ftrst symbol might be encoded as 0 and the other two as 1 and 01. Another choice is to use 0, 10, and 11 respectively. The latter code, called a Huffinan code, allows us to interpret the sequence as soon as it is sent rather than waiting to see if more symbols will be sent. A "good" binary code, as the examples suggest, assigns shorter strings (fewer digits) to more frequent symbols and longer strings to less frequent symbols. The infrequency of a symbol is approximately the number of O’s and 1 ’s needed for the symbol in a good binary code. Parenthetically we note that the number of O’s and 1 ’s in a binary signal is called the "bit count" and is itself spoken of as a measure of information. The bit count has nothing directly to do with either frequency or information content. It is just the total raw binary data actually transmitted between sites or stored at a site. Curiously, bit count treats O’s and l’s as if they are equal, but depending on the implementation the two digits may have ~ There is only one funct~onf." p ~fl, p) taking real numbers between 0 and 1 to real numbers and satisfying: (i) 3~1/2) = 1 (binary normalization); (fi) ftP~P2) =tiPs) +tiP2) (additiwty), and (iii) ffp~ >_ p2, thenflp~)
