John E. Gray, Margaret M. Francis, and Allen D. Parks. Naval Surface Warfare Center Dahlgren Division. Systems Research and Technology Department.
Proceedings of the 2005 IEEE International Conference on Information Acquisition June 27 - July 3, 2005, Hong Kong and Macau, China
A Theoretical Framework for Synthesizing Meaning Based Upon Sensor Derived Information John E. Gray, Margaret M. Francis, and Allen D. Parks Naval Surface Warfare Center Dahlgren Division Systems Research and Technology Department Code B10 Dahlgren, VA 22448-5150
Abstract— To take advantage of recent and future advances in communications and information processing technology, Military systems of the 21st century will create systems that enable the elements to operate as nodes in a geographically distributed information collection and processing network. Each network node is immersed in an information environment. An abstract means of characterizing information environments synthesized from sensor information is formulated in this paper. The limitations of such a formulation is discussed as well as directions to complete this concept is mentioned as well.
quality of information drawn from the sensor network to the perceptions drawn from correlation graphs associated with the paths of information flow to a decision processor. The graphtheoretic properties which determine the quality for a sensor include: 1) edges which can be used to count relations between observations; 2) cliques (i.e. complete subgraphs) which count sets of completely cross-correlated observations; 3) isolated vertices which counts completely uncorrelated observations; 4) connected components which measures the coherence of the observations.
Index Terms— sensor networks, interval graphs, phase space
I. I NTRODUCTION st
21 century systems are likely be able to greatly enhance their performance by taking advantage of recent and future advances in communications and information processing technology. These technologies enable the elements of such systems to operate as nodes in a geographically distributed information collection and processing network. Each network node is immersed in its local information environment, hence it has a local perception. If a node is isolated from a more global system, then its perception, as well as its response to the information environment, is strictly local and based only upon data provided by its local sensor observations. However, completely autonomous responses to such locally determined perceptions can be undesirable from a more global systemlevel perspective. When local sensor systems have fields of view that overlap, then actions may require a degree of coordination between system elements so that control functions may be properly exercised. Associated with the sensor measurements, is a representative perception that is synthesized from the pertinent aspects of the system information environment. The decision functions that are derived from the sensor environment have stringent requirements accuracy and timing requirements because they are used to create control commands. These commands and decisions reflect perceptions synthesized from measurements obtained from system sensors. Graph-theoretic methods when applied to sensor networks suggest that there are properties which serve as measures of certain aspects information quality and thus can clarify observations by helping spin the straw of observations into the gold of usable information. Certain graph-theoretic properties are inherent in random graphs can be used as measures of the
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The combination of an abstraction of the sensor concept with a model of perceptions is suggestive of an underlying model of an information space. The technical concept of information has roots that date back to the ancient Greeks and Chinese, but the modern ideas can be traced through two distinct pathways: artistic and statistical mechanics. Perception of space and color changed significantly ([7] and [1]) from the late Renaissance to Da Vinci and culminated with Goethe’s theory of color and color perception. One could argue that the formal aspects of the conception of information space started with Gibb’s book on statistical mechanics with his definition of phase space, though Maxwell should not be neglected, and continued with Boltzman, Ehrenfest, Szilard, Weiner, von Neuman, Shannon, and ended with Jaynes derivation of statistical mechanics from an information theory perspective. Szu has argued that a fully usable theory has not yet been born. An integrated world needs such a theory to successfully integrate networks with sensors to achieve a scalable union of information sources and sinks. To define such as space, we need to go beyond the formulation of information theory that is steeped in the problems of communications, communication information theory (CIT), which is due to Shannon [17]. The essence of the theory of communication as well as the dual problem of synthesizing information from a sensor is captured in the standard figure. The success of CIT was due to the Shannon proposal: "The semantic aspects of communication are irrelevant to the engineering problem." which enabled him to solve the technical aspects of the communication problem. With this fruitful "fiction" he was able to cut the Gordian knot and
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solve the problem of the engineering aspect of communication, while leaving the remaining problems of information, e.g. the communication problem of communications, to future generations to deal with. The other two problems of a more general theory of information: remain to be dealt with as has been pointed out by Weaver in his overview of Shannon’s papers [18]: Level B: How precisely do the transmitted symbols convey the desired meaning? (The semantic problem.) Level C: How effectively does the received meaning affect conduct in the desired way? (The effectiveness problem.) In particular the semantics problem is dual to the reformulation of fuzzy logic [24]-[27] proposed by Zadeh. Additionally it has importance to the formulation of a theory of information space based on sensor information. Szu has suggested that the natural means for discussing information is to formulate in analogy to phase space that is used in statistical mechanics [19]. The reason phase space is such a useful concept in statistical mechanics relates to the goal of statistical mechanics (SM) itself. SM makes no attempt to solve a complicated dynamics problem that consists of many particles. Instead its aim is to predict average properties and their deviations by examining large number of identical systems. Values of desired quantities are computed by performing averages over large ensembles. Behavior is characterized relative to the swarm rather than the individual. Based on the suggestibility of such a concept of dynamic phase space, Szu suggested that a natural information phase space exists as well. He suggested visualizing the space as having axes that consist of the reporter’s standard questions used to formulate a newspaper story, namely: who, what, when, where, why, and how. Such a space we term W5 H-space. Such a space has conserved quantities (volume in traditional phase space) while information flow must be conservative in W 5 H-space. The identity questions are analogous to mass in phase space, while other quantities are related to ideas of individual positions and potentials that give rise to changes in direction. This concept of W5 H-space could be completely developed in analogy to phase space, but we will not discuss it further. Instead we concentrate on establishing the narrative features of the story. This is done with sensors that gather information about the actors on our stage. An abstract theory of sensors as instruments that gather the who, what, where, and when, which are used to synthesize the why and how of a narrative is discussed in this paper. Essentially we discuss how to formulate information starting with multiple sensors and ending with the perceptions that could be drawn from this information. One could think of this is as the formulation of the W 4 -subspace of the W 5 H-space. Besides the formulation of this space, we discuss a number of important issues that arise when one tries to formulate this space. The other aspects of this space are dealt with separately. II. F ROM S ENSOR M ODELS TO P ERCEPTIONS A. Sensor Models When one abstracts the definition of a sensor, it can be a particularly useful means of thinking about the information
environment synthesis. The work of Prasad, [12] and [13], defines an abstract sensor in terms of the parameter it measures and the interval of the reading in which it is contained. Sensor outputs can then be viewed as connected intervals on the real line rather than points. The following definitions capture this abstract viewpoint: Definition 1: An abstract sensor is a sensor which reads a physical parameter and gives out an abstract interval estimate IS , which is a bounded and connected subset of the real line . Definition 2: A correct sensor is an abstract sensor whose interval estimate contains the actual value of the parameter being measured. Definition 3: An incorrect sensor is an abstract sensor whose interval estimate does not contain the actual value of the parameter being measured. Definition 4: Let sensors s1 , s2 , ..., sn feed into a processor P . Let the abstract interval estimate of sj be Ij 1 ≤ j ≤ n, where Ij is the closed interval [aj , bj ] with endpoints aj and bj . Define the characteristic function χj of the j th sensor sj 1 ≤ j ≤ n as follows: Given χj ∈ {0, 1} , then ∀ 1 ≤ j ≤ n 1 ∀x ∈ Ij . (1) χj (x) = 0 ∀x ∈ / Ij In addition to a sensor model, a method is needed for combining sensor measurements that incorporates the temporal aspects associated with the information connectivity particularly in a multi-sensor multiple object environment. One such method draws on elements in random graph theory [10] and in particular the theory of random interval graphs discussed in [15]. The usefulness of such a methodology becomes apparent when they realize that any measurement that is useful has to be applied to a real time system. That system has an interval in which the measurement is useful (the length of the interval can also be interpreted as a fuzzy set.) This can be cast in graph theory language by noting: a graph G with vertex set V and (undirected) edge set E is an interval graph if it is the intersection graph of some family of real closed intervals on the line. That is, if I is the set of all such intervals, then G is an interval graph if and only if there is a mapping ϕ : V → I such that for u, v ∈ V, {u, v} ∈ E, ϕ (u) ∩ ϕ (v) = ∅. Clearly, if I represents time intervals, then G is an abstract representation of overlapping time intervals. A random interval graph with n vertices is generated by independently selecting in the same manner n intervals at random and forming the associated interval graph using the Scheinerman model: each random interval is obtained by first uniformly choosing an interval center x ∈ [0, 1] and then selecting uniformly a halflength ρ ∈ [0, r] for a fixed r > 0.The associated j th random interval is Ij = xj − ρj , xj + ρj and all such intervals are contained within [−r, 1 + r]. The maximum possible length λ of any such random interval is obviously 2r and that of the ”containment” interval is = 1 + 2r = 1 + λ. The associated random interval graph has V = {1, 2, · · · , n} with j → xj − ρj , xj + ρj , j ∈ V, and {j, k} ∈ E when Ij ∩ Ik = ∅. Scheinerman’s work gives a specific probability distribution that is not relevant to further discussion except to
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note that a non-uniform distribution such as a Possion might be a tighter fit to applications we are interested in. Note that both task and resource scheduling fall under the same aegis using this formalism. In summary, random interval graphs characterize simultaneous occurring variable length events. This allows one to develop simple parametric relationships for task processing rate. Processing and decision making consist of finite bounded time intervals so they are inherently interval-based or fuzzy. Any sort of real time scheduling component is an example of this. Applications of interval graphs include: • • • • •
decision control, communication rates for control, estimate conditions which induce temporal decoupling, provide insight into control methodologies to avoid decoupling, characterize resource competition from overlapping intervals
B. Symbolic Sensor Measurements When one is observing an object with a sensor it is reasonable to assume the object has an underlying dynamic law or other type of dynamic characteristic that it obeys. Given a dynamic model, it is always possible to arrive at a symbolic model that is entirely equivalent to the dynamic mode [9], [2] using a sensor as the (symbolic) transformation device that translates measurements into a finite alphabet that is sufficient to span the space of possible descriptions. Note this observation is implicit in all of Shannon’s work on information theory [17]. One can then define a ‘symbol space’ based on the dynamics model that consists of a finite number of symbols N where N is thenumber of symbols. For our purposes, both the elements of N are finite strings of symbols. (Note in other applications the strings are infinite as well.) Measurement leads to speciation of the continuum of possible measurements into a finite symbol alphabet. Example: The simplest alphabet is that used in classical logic gates which is based on voltage signals which are reduced to the symbolic level by a measurement/conversion process to the symbols 1 and 0. The simplest example of a universal symbolization is electrical, specifically the analogto-digital (A-to-D) converter. Formally, an A-to-D converter takes a continuous real time signal t −→ x(t) (t ≥ 0) and generates an output sequence { xk : k ∈ Z + } from a finite alphabet J. The A-to-D converter produces a symbol τ , where τ is a prescribed sampling interval x k at a time k for τ > 0. Note, this process is inherently nonlinear as are most symbolic conversion processes, though this is not widely acknowledged. Thus a digital symbol set 2 consists of the of the two symbols 0, 1 so N = {1, 0}. When dealing with communication or digital encoding, words or messages consist of long strings of ones and zeros. Any measurement or observation can be translated from the measurement process M (n) at discrete time intervals n of a finite number of symbols drawn from a set {s1 , s2 , ..., sm } into an observation sequence vector |O(n) which can be
represented as
⎡ ⎢ ⎢ ⎢ |O(n) = ⎢ ⎢ ⎢ ⎣
s1 (1) s1 (2) s5 (3) si (k) ... sm (N )
⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦
(2)
Real measurement is always disturbed by noise, thus one must consider the role of noise relative to the mapping process. Any symbolic representation must be connected by understanding of the physical process from which it is abstracted. This leads to the question of how symbolic noise should be characterized. Voss has discussed [20] this problem with respect to the type of symbolic dynamics that occurs in classical physics, our perspective is somewhat different. What Voss did was to assume a finite time series that could be matched to a finite time series developed from a Bernoulli process. The two finite symbol sets were matched and if there is sufficient agreement, then the models are said to match. Reduction of signals to symbols exhibit a variety of symmetries. Because the alphabet of symbols is finite, they represent classes that have a degree of invariance because of the inability to distinguish between them (an example of this is translational invariance). Invariances discussions can often be deferred until one knows the specifics of a physical signal. At least one form of invariance that is required for any type of symbolization is the identity. Also, one can define an equivalence relationship among signals if S is the mapping from signals v onto symbols denoted by [22] s (3) v → S(v) and u and v are arbitrary signals, then one can write u ≈ v, then it is clear that we have the three properties of an equivalence relation (transitivity, symmetry, and reflexivity) for ≈. With an alphabet, there is an inherent uncertainty in the assignment of signal to symbol which can be symbolized as P (s/v), which is the probability that the symbol s is the correct representation of some part of the signal space spanned by the signal set given that the corresponding signal is v. This allows a symbolic basis for measurement to be derived. It is possible to implement this type of algebra either optically or with a radar with the symbols being coded waveforms. There is a symbolic algebra associated with the physical signal and the reduction of signal to a finite alphabet of symbols. Other forms of physical symbolizations are realizable as well [21]. There is also a separate aspect of the symbolization process. Given the symbols are generated by a dynamic system, one has a separate problem associated with the symbol strings that are generated by the symbolization process. String spaces have their own underlying algebras, which when properly understood, can affect how they can be manipulated and hence detected. While understanding the algebra associated with the symbolization process is important for single sensors, it is even more important in the multi-user environment. Introduction of an algebra with the appropriate string symbols allows one to try and characterize which symbols are permissible in adjacent positions and which aren’t. One
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is seeking an algebra that allows one to associate permissible and non-permissible symbols that follow each other–a string algebra so to speak. It seems that the natural algebra to be associated with such a problem is the permutation group [14]. The underlying algebra of a permutation group gives us a basis to discuss both transitions between states (model switching) and noise. The permutation or symmetry group is a finite set of symbols, say three for example {a, b, c}. A mapping of
symbols onto themselves is denoted by the symbol a bthese d b d a which is interpreted as the transformations a maps to b, b maps to d, and d maps to a. Thus the set of all permissible transitions between symbols could be represented this way. An identity transform for five symbols is denoted this way a b c abc
.
(4)
For an alphabet of three symbols, we have a group that consists of 3! or 6 symbols like this that represent potential mappings or transitions between one symbol set and another. Rules can be represented in terms of a sub-algebra of the permutation group. One can use these algebras to recognize where some string groupings are not possible, so one can develop pruning rules that concatenated the strings because the string is inconsistent with the algebra for a particular threat type. When measurement data is viewed as a result of a symbolic dynamics process, noise causes a transcription error relative to a symbol algebra. If one has the algebra already worked out, then string matching tells us that the transcription error has occurred. One can then replace the incorrect symbol with one that is consistent with the algebra. The symbolic viewpoint has an other useful feature when expressed in fuzzy language. The symbols are not ambiguous, but the algebra may be. If the transcription error rate is high enough in the symbolization process, we cannot infer mappings from observations into specific symbols. At best, the symbols can determine the mappings into the subgroups of the permutation group rather than into specific symbols within the subgroup. Noise prevents mappings from being one-to-one, instead element mappings are one-to-many. Thus "fuzziness" is the breakdown of the identity transform when we are trying to preserve this communication paradigm viewpoint. The lack of exactness is the breakdown of the concept of identity transform definiteness to fuzziness. Any transform into the symbolic is necessarily inexact. The language of exactness is complete only when it is closed and not subject to the external world. C. General Properties of Sensor Perceptions Any method for quantifying the quality perceptions has several important system level problems that need clarification. In order to do this, we first define the universal information environment for a system as the set U of all possible measurements that could be made at any time by the system’s sensors. These measurements may include such traditional quantities as target position, velocity, and polarization state, as well as environmental data (e.g., electromagnetic propagation conditions). Definition 5: A perceptqualm is the pair (O, R) where
1) O ⊂ U is a finite set of abstract sensor measurements called observations which represent some aspect of a system’s information environment, and 2) R ⊂ O × O is a non-reflexive (i.e. ∀x ∈ O, (x, x) ∈ / R) and symmetric (i.e. (x, y) ∈ R ⇐⇒ (y, x) ∈ R) binary relation of observation pairs called correlations. A perception has a mathematical representation as a simple undirected graph, which is termed the correlation graph for the perception. Randomness in a correlation graph parameterizes the quality of the system’s knowledge about the relationships between its observations in terms of probability. The numerical determination of the parameters associated with the graph can be defined as the system’s correlation quality. The expected value of certain quality related graph-theoretic properties leads to properties that are inherent in these random correlation graphs. They can be used as measures of the quality of perceptions associated with the correlation graph for various domains. The graph-theoretic properties which determine the quality of a perception are the same as those for a sensor, namely: 1) edges which count the pair wise correlations in the perception; 2) cliques (i.e. complete subgraphs) which count sets of completely cross-correlated observations; 3) isolated vertices which counts completely uncorrelated observations; 4) connected components which measure the coherence of the perception. The application of a specific perception quality problem requires sufficiently consistent and context dependent definitions for O, correlation, and correlation quality. No explicit relationship exists between correlation quality and knowledge of specific related system factors such as observation accuracy and attribute classificatorily. An information environment can be synthesized from a finite set of system observations that provide an overall correlation quality for the system which can be adequately defined for observations in terms of relevant probabilities. One can define the correlation graph C(O) for the relation R to be the simple graph with vertex set O and edge set E such that there is an undirected edge connecting vertices x and y when observations x and y can be correlated, i.e. when (x, y) ∈ R and (y, x) ∈ R. The graph Cp (O) is a random correlation graph if each edge of Cp (O) exists with some probability p. Each edge in the graph represents the fact that observations x and y can be correlated by a system which has its correlation quality parameterized by the probability value p. Example 1: A possible application is to characterize the quality of synthesized perceptions gathered from a kinematic information environment (e.g. a correlated multi-target track file synthesized by a multi-radar system from many distributed geographically local radar observations.) An appropriate meaning for correlation quality in this context is a number between 0 and 1. This number reflects how well observations can be correlated by the system based solely upon its ability to identify and classify common dynamic attributes shared by the track observations. The closer this number is to one indicates
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the more capable the system. Example 2: To characterize the overall coherence associated with a synthesized perception of a system-wide status, availability, and planned actions; information must be synthesized from that environment using a system of sensors. An integrated representation of the time intervals associated with the system wide operational status is needed that includes sensors and controlled agents, remaining expendable agents, positions and velocities of all friendly and competitive controllers, and current agent controller assignment schedules. The observations for this example constitute multiple attribute classes that are intended to address the aspects of the systems total information environment. The meaning for correlation quality, as well as its numerical value, reflects the system’s overall knowledge of how all the observations relate and antirelate to each other as well as the quality of the commands to agents that are synthesized from them. The method for construction of Cp (O) is to independently introduce a linear order by adding one edge at a time that has a probability p into each of the (n2 ) available edge slots in C(O). At the end of this procedure, the random correlation graph Cp (O) has been generated that provides a topological representation of a system’s perceptqualm. Universal information U is synthesized from a finite set O of observations drawn from a collection of abstract sensors observing its environment. The probability p parameterizes the overall quality of the system’s ability to correlate observations in O. These observations account for the existence or non-existence of common correlation attributes between the observations. System dependent knowledge relates factors such as sensor observation quality, attribute discrimination, temporal concurrency of the collection and correlation processes, etc. which are discernible from the perceptqualm. Zero probability quantifies the worst correlation quality since the associated system possesses no (useful) knowledge about relationships between observations because no two observations in O can be correlated. Thus, correlation is the impossible event. For this case, C0 (O) is the empty graph on n vertices (i.e. |E| = 0) and the system is unable to synthesize for its use even a perceptqualm. This represents the worst case that can be synthesized from a set of n observations; it is called a null perceptqualm. On the other hand, p = 1 quantifies the best possible case for correlation quality from a set O of observations. All observations are completely inter-related, so the system has sufficiently knowledge to complete association of all relationships. All observations in O can be correlated with every other observation in O; thus complete correlations are certain for a given event. The C1 (O) is the complete graph on n vertices (i.e. |E| = (n2 )). The complete graph represents the fact that the system can synthesize the best possible perceptqualm possible using the available set O of n inter-related observations. This case is called the complete perceptqualm of the environment and it is used to form a single component of n-correlations. An m-correlation, 2 ≤ m ≤ n, is a subset M ⊆ O of |M | = m observations which can be completely inter-related. Not only can correlation graphs be associated with the null and complete perceptions of a system’s information environment, they also reflect topological models
which represent synthesized perceptions of the worst and best quality synthesized perceptions. Any Cp (O) with |O| = n and 0 < p < 1 so that 0 < |E| < (n2 ) can be used to model system observation sets to form perceptqualms which are of intermediate quality. The coherence of a perception (when it makes sense to do so) is the connectivity of the associated correlation graph. If Cp (O) is connected, the associated perception synthesized from O is a coherent perception (e.g. C1 (O) is a single component and represents a completely coherent perception of n-correlations). If Cp (O) represents the perception of an information environment which is not connected, it comprises several components each of which is connected. The associated perception is an incoherent perception. Observations of each component of Cp (O) are coherent, thus they can serve as adequately correlated local perceptions. The overall quality of the perception of the information environment contained within O depends upon the context of the application. Thus, such disconnectedness in Cp (O) indicates a degradation of the coherence of the associated synthesized perception. We have provided a mathematical characterization that is essence of “a system’s awareness and understanding of its information environment”. The definition of a perceptqualm can be represented as a simple undirected graph termed the correlation graph for the perceptqualm. Randomness associated with the set of observations O provides an important feature of correlation graphs which enables one to parameterize the quality of the system’s knowledge about the relationships between its observations. This parameter is called the system’s correlation quality. The framework discussed above is a component of a general theory that is explored in more detail in [4],[11], and [5] The general theory assumes no explicit relationship between the correlation quality and such factors as observation accuracy and attribute classification. Thus, this random correlation graph analysis technique can be used to study the quality associated with a system’s perceptqualm for any aspect of its information environment that is synthesized from a finite set of system observations. It provides a means for associating overall correlation quality for any system that can be adequately defined from observations in terms of relevant probabilities. III. C ONCLUSIONS The construction of "information space" that we have discussed has three principal aspects associated directly with the sensor: picture, symbolization, and the perception picture synthesized from sensors that allows one to abstract these concepts in the manner we have discussed. There are two additional problems associated with sensor based information space that need significant further development. Szu’s W5 H space needs to be formulated in a mathematical manner that space has prevented us from doing. Once formulated in a manner analogous to traditional phase space, it is possible to formulate "information operators" that represent the dynamics associated with incompressible flows of information in these spaces. One can use these to formulate the concept of a "selfreference expansion" data series (Szu and Gray in preparation)
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that are formulated so that the series adoptively determines the law that generates it without reference to a fixed sampling rate. The other aspect of information space that has not been dealt with adequately is the human interface in the perception of the information drawn from the sensor picture, which is distinct from the sensor perception problem. At some level of the sensor space problem, humans are always inserted into the loop. The map of sensor space does not encompass all aspects of the territory, one recognizes that Human System Integration (HSI) must necessarily encompass the functionality of human cognition as part of the portrait. One therefore must recognize both the strengths and weaknesses of human intelligence in the loop.. Additionally, in any operational environment one recognizes that underlying environment has many types of complex interactions occurring over all scales of human perceptions. The information environment has both natural and artificial components some of which have been naturally adapted to by human cultural background and some artificial components which haven’t any such history. So, a biological approach is useful in determining integration factors. The underlying operational sensor environment occurs at the level of human, both individual and statistical, which enforces a perception perspective. These perceptions are functioning within a synthesized artificial sensory environment. Care must be taken in reliance on their judgements without careful understanding how the "human sensor" performs in different environments and under a variety of stresses. The capabilities of humans must always be factored in design with particular care taken to understand their noise characteristics. One final point is worth mentioning related to the measurement process associated with sensors. Measurements or perceptions associated with "human measurement" often produce numbers that lack the clarity to capture an attribute, a less crisp concept such as a "word descriptions" might work better. Any symbolic process that reduces an observation to a number amounts to imposing the algebraic structure of a ring upon the observations. By saying that an observation does not lead to the crisp concept of a number, we are saying that the algebra associated with the measurement process has less structure than a ring. It might have the algebra of a semi-group for example. Structures are interpreted as words, which allow one to treat the symbolization process like that of a transformational process like that resembling that of an automata [8]. Translations of one word into another rather than the computation of one number (an estimate) starting from another (representing a measurement) is symbolic computing in disguise which allows one to view all computations in a uniform manner [3].With words, one doesn’t even know the algebra initially, so one is interested in the possible word translations so one can obtain an inference algebra that can be used to characterize the words. One lets Nature impose the grammar on the words rather than imposing it. We choose to interpret it as freedom. Thus symbolization allows additional degrees of freedom (in physics parlance) which means that one can deconstruct into models rather than a single deconstruction that is governed by numbers.
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