The Plinius ontology of ceramic materials Paul E. van der Vet
Piet-Hein Speel April 1995
Nicolaas J.I. Mars
Abstract The Plinius ontology of ceramic materials covers the conceptualisation of the chemical composition of materials. The design decisions underlying ontology development at our group are discussed. The ontology of ceramic materials is given as a conceptual construction kit, involving several sets of atomic concepts and construction rules for making complex concepts. One of its implementations, that in Ontolingua, is presented.
c 1994, 1995 by the authors. This paper has appeared originally in Workshop Notes ECAI'94
Workshop Comparison of Implemented Ontologies, Nicolaas J.I. Mars (ed.), Amsterdam, 1994, pp. 187{205. The authors are indebted to Wilco ter Stal for comments on a former version of this paper, and to Tom Gruber for help with the Ontolingua implementation. Corresponding author: Paul E. van der Vet, Knowledge-Based Systems Group, Dept. of Computer Science, University of Twente, P.O. Box 217, 7500 AE Enschede, the Netherlands; phone +31 53 89 36 90, fax +31 53 33 96 05; e-mail
[email protected]
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1 Setting and scope The present paper accompanies an implemented version of the Plinius ontology of materials and serves to explain the main design decisions. More detailed information on the ontology itself (van der Vet & Mars [1993]) and on the Plinius project in general (Mars & Schreiber [1985]; Mars et al. [1994]; van der Vet et al. [1994]) is available elsewhere. The ontology is still under development. The version presented here has been `frozen' for the purpose of a research aiming at the comparison of knowledge representation languages. This has been part of a Ph.D. research carried out by Piet-Hein Speel, currently completed (Speel [1995]). The current version has been implemented in the kl-one systems classic, c-classic, back, and back++, and in Ontolingua version 4 (see the appendix). It will be implemented in two other kl-one systems (kris and loom) in the near future. An older version has also been implemented in Ontolingua version 3, Prolog, life, and two kl-one systems, the taxon part of the colab system and, again, loom; and formalised in Conceptual Graphs. The ontology covers the domain of ceramic materials and is developed speci cally for the Plinius project. Plinius is concerned with semi-automatic knowledge extraction from short texts. The corpus consists of about 300 abstracts of primary publications on mechanical properties of ceramic materials. In the present context, the corpus serves as an operational speci cation of the domain to be covered by the ontology. It helps assess the scope of the ontology. The ontology is required to cover every relevant concept that occurs in the texts (coverage) and to make every relevant distinction made in the texts (granularity). Additional parts of the Plinius ontology (not treated here) cover properties and processes. In this paper, we will rst discuss general design decisions (section 2) and then explain the ontology itself including more speci c design decisions (section 3).
2 Major design decisions 2.1 Function
We approach the subject of ontologies from an engineering perspective. We regard the design of an ontology as an essential step in the design and actual implementation of knowledge bases and knowledge-based systems. The importance of the availability of an ontology is particularly evident when the knowledge base gets large (Mars [1993]). At some point in the building process of a knowledge-based system there will be a representation of the knowledge in a formal language. We require an ontology to supply the meaning of every non-logical constant that occurs in the representation. We treat logical constants and their associated inferential properties as being given.1 This position implies that we represent knowledge always in a formal language with a well-de ned syntax and semantics.
2.2 Independence of particular knowledge representation languages
The representation of an ontology in some particular representation language is a separate problem that can be solved when the ontology at the knowledge level is suciently developed. The idea is that the ontology contains speci cations of the knowledge base to be built. Hence, an ontology has to be independent of any particular choice of knowledge representation language. The knowledge representation language has to be chosen so as to meet the speci cations. The advantages of this approach are: the ontology can serve to unify several co-operating knowledge bases written in di erent representation languages within a single system; and, more generally, the support of sharing and reuse. The Plinius ontology uses the language of sets and tuples, among other reasons because this language is well understood and completely neutral with respect to knowledge representation 1 Mark Willems (Free University of Amsterdam) has pointed out that it might additionally be advantageous to address the issue of an ontology of logical constants. We agree but will not pursue this issue here.
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languages. The latter point is demonstrated by the very di erent implementations made. As far as we are aware no other ontology has been implemented in so many di erent ways.
2.3 The principle of conceptual construction kit
An ontology can be just a at list of concepts with some explanation or display structure. Structure has the advantage of allowing for a parsimonious design since most concepts can be de ned implicitly rather than be enumerated explicitly. This corresponds to what we call the principle of conceptual construction kit. An ontology built according to this principle consists of atomic concepts, serving as primitives, and construction rules that de ne all other concepts. Formally, the ontology is given as a tuple hA; Ci with A the set of atomic concepts and C the set of construction rules. The tuple can be seen as a calculus with A the alphabet and C the transformation rules. The deductive closure of this calculus produces the explicit list of concepts.
2.4 Knowledge as assertions about entities
Scienti c knowledge is the kind of knowledge primarily studied by us. In science, it is vital to be able to assess whether a new proposal is in con ict with what we already know. For a knowledge base with scienti c knowledge, a similar requirement holds. In a rst approximation, the problem is solved in science by distinguishing between de ning and non-de ning properties of entities (whether individuals or classes of individuals). Knowledge consists of assertions about entities. Two assertions can only con ict if they are about the same entity, and this can be found out by inspecting the de ning properties of the entities. This strategy can be adopted for scienti c knowledge bases, so that the detection of con icts can be automated. It requires at least the introduction of identifying labels, but it is better to have many concepts for entities de ned so that the system can automatically inspect the de ning properties.
2.5 Bottom-up approach
There are various ways to realise a conceptual construction kit. One approach, embodied in kl-one-like systems,2 starts with a concept for everything there is and proceeds by successive di erentiation. This is a modern variant of the Aristotelian genus-species approach, where the species are di erentiated from the genus and from each other by means of di erentiae (roles in kl-one). By contrast, our approach is based on atomism. Atomism proceeds bottom-up in that it builds objects out of smaller objects. Chemical elements, kinds of atoms, are among the more important atomic concepts. The construction rules construct concepts for more complex assemblies. Each construction rule is written as a de nition that expresses constraints on the construction of every complex concept of the type in question.
2.6 Anticipating concepts for new entities
An added advantage of this approach is that it allows for the de nition of all relevant concepts of a given type in a single step. This is important for Plinius. Although we can easily assemble a list of all concepts occurring in our corpus, we do not want to re-design the whole ontology when we move to other corpora. Thus, we need a design that within the scope of the Plinius application in principle accommodates every concept of interest. Consider, for an illustration, chemical concepts for pure substances. Pure substances, like table salt (NaCl) or water (H2 O) consist of chemical elements in whole number proportions. When chemical elements and natural numbers are given as atomic concepts, an appropriate construction rule de nes every pure substance there is and many (technically, in nitely many) more besides.3 2 Kl-one refers to a family of representation languages and systems, but is is also a way to organise the universe of discourse. 3 As readers with a chemical background will note, isomerism poses some problems. These are, however, immaterial for the purposes of the present illustration.
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In Fregean terms, all concepts have sense but only some have reference. Since the concepts are de ned implicitly, there is no burden on the system. The construction moreover anticipates new developments to some extent. Thus, 35 years ago chemists generally held that noble gases such as helium and xenon do not form compounds. This belief was shown unfounded by Bartlett's synthesis of XePtF6 in 1962; other noble gas compounds soon followed. An ontology with explicit de nitions of concepts for pure substances would have had to be rewritten in 1962, while a Pliniuslike ontology would already have contained the concept implicitly.
2.7 Implicit de nition of taxonomies
The bottom-up approach utilising sets and tuples not only allows for parsimonious implicit definition of most concepts, but also for implicit de nition of taxonomies. The logical basis is abstraction over sets. We write arbitrary(X ), where X is a set, to mean that for the concept that contains this term it is immaterial which element of X is substituted. For a simple illustration, consider again concepts for pure substances. Starting with a concept for any amino acid, we can derive the concept for amino acid in general by substituting an appropriate arbitrary(X ) for all parts of the speci cation save those that specify the NH2 and COOH-parts of the molecule. The procedure can be repeated. Substitution of COOH by arbitrary(X ) yields a concept for amino compounds, and substitution of the only remaining speci c part, that for amino, nally yields the concept for pure substance in general.
2.8 Engineering perspective
The task to be performed by the knowledge-based system has a great in uence on the design of the ontology. In particular, the decision whether concepts are elaborated or not is governed by the outcome of a cost-bene t analysis. In materials science, for instance, it is important to classify materials according to the role they play in the production process. A typical ceramic starts its career as a powder; it is turned into something called greenling; and the greenling is densi ed to yield the nished product. A detailed conceptualisation of microscopic features at a scale of 10?5{10?8 m allows unambiguous de nition of these concepts. This has been omitted for Plinius, however, because it would serve mainly esthetic goals. Instead, the concepts powder, greenling, and finished have simply been added as primitives to our ontology.
3 The ontology
3.1 Introduction: materials and samples
Ceramic materials come in two variants: glasses and polycrystalline materials. Polycrystalline materials consist of a large number of so-called grains. In between the grains and in the grains themselves there might be pores. The structure of the material at the scale of grains is called the microstructure, see gure 1. A great number of properties of polycrystalline materials are determined not only by the chemical composition in terms of phases and substances, but also by the microstructure. This poses the dicult problem of choosing the right level of abstraction. The best possible description of the microstructure lists properties such as geometry, chemical composition, and aggregation state, of each and every grain in the sample and thus makes every sample unique. However, such a description is seldom, if ever, available. Worse, the level of detail is wrong. Materials science aims to generate assertions that can be used by others by incorporating abstractions. The level of abstraction can be chosen anywhere in a continuum with the grain-for-grain description on one extreme and a bare speci cation of the chemical composition on the other. An ontology ideally provides the whole range, so that the appropriate level can be chosen depending on the application. Since the microstructure is determined by the production process, one also nds materials speci ed by telling how they can be produced. The speci cation of the production process then serves as an indirect speci cation of the microstructure. Mixed speci cations also occur. One 4
Figure 1: The microstructure of a ceramic material. This photograph shows a crosscut of a typical ceramic sample, coarse-grained zirconia (ZrO2) which is stabilized by 17 wt% Y2 O3 and densi ed by heat treatment at 1550 C. The average grain size of the present sample is 8 2 m, while abnormal grain growth has resulted in the presence of smaller grains (about 2 m). The photograph also shows a number of included pores, visible as black spots surrounded by white. Photograph courtesy of the Inorganic Materials Science Group, University of Twente. attractive feature of specifying the production process is that it solves the problem of choosing the appropriate level of generalisation. If the process is suciently controlled, the samples produced will exhibit a number of predictable properties. Some spread in microstructural detail will inevitably remain, but apparently it does not matter. For an ontology, this means that it has to accommodate speci cations of materials involving production processes. To accommodate the many levels of generalisation, the Plinius ontology construes materials as abstractions from samples. Each sample is unique. In the analysis of input texts, Plinius initially interprets every reference to a material as reference to a sample and assigns a unique label to it. The label itself is interpreted as a sample characteristic and serves as shorthand for local production conditions. The connection between sample and material can be made by means of two predicates, said-toconsist-of and consists-of. The said-to-consist-of predicate is used to express that the authors of a certain text claim that their sample consists of a particular material. Such an assertion is true no matter what the sample composition really is. The consists-of predicate expresses the quite di erent assertion that a sample consists of a particular material. Initially, all assertions that specify the composition of a sample are expressed in terms of the said-to-consist-of predicate. In a separate step, a rule can be applied to the e ect that if no con icts arise this way an occurrence of said-to-consist-of can be exchanged for consists-of.
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3.2 Overview
In chemistry, glasses and polycrystalline materials are instances of systems. Systems come in two sorts. When no inhomogeneities are present, the system is called homogeneous, else heterogeneous. Two clari cations are in order. (1) The concept of inhomogeneity is understood at a level above that of atoms and molecules, for otherwise there would be inhomogeneities in any system. (2) The de nition of heterogeneous system given here di ers from the de nition usually given in introductory textbooks, that de nes a heterogeneous system as a system consisting of more than one phase. Our de nition also covers single-phase polycrystalline systems. Glasses are homogeneous systems while systems composed of grains are heterogeneous. The composition of systems is given by a list of constituent phases, where appropriate with a number specifying relative proportions. A homogeneous system by de nition consists of a single phase. A heterogeneous system can consist of one or more phases. A phase, in turn, consists of one or more substances in certain proportions. Substances, nally, consist of chemical elements. The ontology will be built up in the reverse order.
3.3 Atomic concepts
The Plinius ontology for materials uses the following atomic concepts, distributed over di erent sets for clarity: Chemical elements as characterised by atomic number and identi ed by their standardised chemical symbol. Examples include Na, H, O, Ca. The set of chemical elements is called E . Real numbers as de ned in number theory; the set is called R. Aggregation states of phases, identi ed as a set of unique labels: gaseous, liquid, vitreous,4 liquid crystal, and six concepts for solid states: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic. The set of aggregation states is called A. Material tags serving to distinguish materials according to the role they play in the production process. For the time being, the set T of material tags consists of three labels: powder, greenling, and finished. It is allowed to single out speci c subsets of these sets and give them names. Here, we will use N , the set of natural numbers (in the classical sense, i.e., integers > 0), R+, the set of real numbers > 0, the half-open interval [0; 1) R, and the set Sol A that embraces all six concepts for the solid state. We can now write the concept for the solid state in general as arbitrary(Sol).
3.4 Construction rules
Within the domain of ceramics, the so-called gross formula, specifying the chemical elements and their proportions, is unambiguous. Due to the presence of nonstoichiometric compounds, the proportions have to be chosen out of R+ rather than out of the set of natural numbers. Compared to the gross formula, we have added detail for two reasons. (1) The unambiguous nature of the gross formula is a contingency. Adding detail anticipates the advent of ambiguous gross formulae. (2) Adding detail considerably facilitates the conversion of our concepts into systematic names or formulae and vice versa. Detail here takes the form of the introduction of the concept of group to have groups composed of chemical elements and substances composed of groups. This gives rise to two de nitions:
De nition 1 (Group) A group is a non-empty set of tuples t such that for each t t = he; ni with e 2 E and n 2 N 4 That is, the glassy state. Many chemists will argue that glasses are just liquids with an exceptionally high viscosity, but others disagree and draw attention to long-range ordering phenomena. The best way out is to introduce a special state.
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such that no e occurs in more than one tuple. The set of groups is called GR.
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De nition 2 (Chemical substance) A chemical substance is a non-empty set of tuples t such
that for each t t = hg; ni with g 2 GR and n 2 R+ such that no g occurs in more than one tuple. The set of chemical substances is called CS .
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Phases are built up out of chemical substances in relative proportions chosen in the interval [0; 1). A phase is further characterised by its aggregation state. We obtain: De nition 3 (Phase) A phase is a tuple hS; ai with a 2 A S speci es the composition in terms of chemical substances and their proportions. It is a non-empty set of tuples t such that for each t t = hc; ni with c 2 CS and n 2 [0; 1) The sum of the n-values for all tuples 2 S is required to be exactly one. Also, no c can occur in more than one tuple. The set of all phases is called PH .
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Phases form the constituents of homogeneous and heterogeneous systems. The two cannot be told apart by counting the number of phases, because single-phase polycrystalline systems are by de nition heterogeneous in the present ontology. Therefore, separate de nitions are needed. That for homogeneous system is obvious: De nition 4 (Homogeneous system) A homogeneous system is de ned as a tuple consisting of a single member p 2 PH . The set of all homogeneous systems is called HO.
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For heterogeneous systems, a speci cation of constituent phases with relative proportions chosen in the interval [0; 1) is not sucient, because the microstructure has to be speci ed as well. The speci cation of microstructure can take the form of: a set of microstructural quantities such as porosity, average grain size, and the like; a speci cation of the production process; or any combination of the two. Because properties and processes are not treated here, the speci cation of microstructure cannot be elaborated. It will be abbreviated in the de nition as Mic. De nition 5 (Heterogeneous system) A heterogeneous system is a tuple
hS; Mici
S speci es the composition in terms of constituent phases and their proportions. It is a non-empty set of tuples t such that for each t t = hp; ni with p 2 PH and n 2 [0; 1) The sum of the n-values for all tuples 2 S is required to be exactly one. Also, no p can occur in more than one tuple. The set of all heterogeneous systems is called HS .
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We are now able to de ne a material.
De nition 6 (Material) A material is a tuple hh; ti with h 2 HO [ HS and t 2 T It is further required that: If h 2 HO, then the aggregation state of the only phase must be vitreous. If h 2 HS , then: { if there is one phase, its aggregation state must be solid; { if there is more than one phase, the aggregation state of the phase(s) forming the majority of the system must be solid. \Solid" here means either one of the elements of the set Sol or arbitrary(Sol). The set of all materials is called MA.
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3.5 Primitive or de ned
It has been observed by Gruber [1991] that a good ontology consists of two mutually supplementary parts: a natural-language part for explanatory purposes and a formal part for unambiguous de nitions. The two parts together de ne the concepts. The important di erence is that automated reasoning is supported by the formal part only.5 It is therefore important to inquire how much of the de nitions of concepts given above is formal. For the atomic concepts the answer is simple. Atomic concepts are formally unde ned. They acquire their meaning through the natural-language account that anchors them into domain knowledge. For the complex concepts, the answer is less obvious. Within the scope of each de nition given above, all concepts that fall under the de nition are fully de ned. Thus a certain group g is fully de ned in terms of necessary and sucient conditions given that it is a group. However, the six types of concept introduced by the de nitions are not fully de ned with respect to each other. The construction is such that automated procedures can always tell the type of any concept, but automated reasoning about the di erences and relations between the concepts is not possible. The reason is that the di erences between the concepts are not made explicit by the de nitions. For example, the distinction between homogeneous and heterogeneous systems is made by means of a criterion that involves the absence or presence of inhomogeneities in the system. The ontology has no concepts for the inhomogeneities themselves and is a fortiori unable to infer whether a system is homogeneous or heterogeneous on the basis of the criterion. The conclusion is, that complex concepts are de ned to the extent that they can always be assigned to the correct type automatically and are primitive in the sense that there is no way to automatically reason about their relations if they are of di erent types. One way to solve the problem partly is to de ne special relations to hold between complex concepts of di erent types. An obvious candidate is the part-of relation that holds between elements and groups, between groups and chemical substances, between chemical substances and phases, and so on. i
5 That is assuming that the formalisation does not introduce additional axioms. We would regard such an introduction unwanted. If it is possible and desirable to formalise further, then it can and should be done in the ontology itself.
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4 Formalisation and implementation In the appendix, the current implementation of the ontology in Ontolingua version 4 is presented.6 Here, we will brie y discuss the decisions made. The focus is on the Ontolingua implementation, but the classic7 and Prolog8 implementations will be touched upon as well.
4.1 Atomic concepts
In the Ontolingua implementation, the atomic concepts of the sets E (chemical elements), A (aggregation states), and T (material tags) have been de ned by means of extensional speci cation of sets. Symbols for chemical elements, and names for aggregation states and material tags become object constants. KIF (and, hence, Ontolingua) has complex numbers built-in, but for clarity the following classes have been de ned explicitly: positive real numbers, real numbers in the half-open interval [0; 1), and natural numbers (integers > 0). In the classic implementation, all extensionally speci ed atomic concepts are de ned as classic-things by means of the construct cl-define-disjoint-primitive. One of the issues addressed in this implementation (conveniently ignored in the Ontolingua implementation) is that of inferential power. Inferential eciency is greatly enhanced by using numbers exclusively as cardinality restrictions on the AT-LEAST and AT-MOST constructs. Only natural numbers are allowed, so the use of real numbers has to be simulated. This has been done by multiplying all reals by a large integer value. For instance, the fractions to be speci ed in a complex concept for a phase (compare de nition 3 in sec. 3.4 above) are then expressed as integers > 0 and, say, < 106. The value of the upper bound can be chosen such that the error introduced this way is smaller than the random error of the empirically determined values. An alternative would be to use test functions. This alternative, however, carries disadvantages (apart from loss of eciency): test functions can only be applied to individuals, and part of the conceptualisation becomes procedural. In the Prolog implementation, a predicate chemical element :- member(X,[ac,.....,zr]).
(where \[ac,.....,zr]" of course stands for the complete list of element symbols) and similar predicates for A and T are de ned. The built-in predicates integer/1 and float/1 are used for de ning the other atomic concepts.
4.2 Complex concepts
The de nitions given in sec. 3.4 have to be formalised before an implementation can be made. Four issues arise.
1. The `chemical constraint'. The rst four de nitions all de ne complex concepts as sets of
ordered pairs, where the rst member of the pair is always a constituent and the second a number (natural number, positive real, or fraction) expressing a proportion. It is further required that the set be non-empty and that, for every set, no constituent occurs in more than one ordered pair. The latter requirement is called by us the chemical constraint. All requirements can be expressed in
6 For background information on Ontolingua, see Gruber [1993]. There is a hard-copy manual for version 3, Gruber [1992]. The manual for version 4 is a HyperText document that comes with the Ontolingua package and can also be consulted through WWW from URL http://www-ksl.stanford.edu/knowledge-sharing/ontolingua/ontolingua.html. This is also the location for retrieving the package. Ontolingua is written in KIF version 3.0 (Genesereth & Fikes [1992]). 7 For background information see Brachman et al. [1991]. An extensive discussion of the representation of the present ontology in several kl-one languages is provided by Speel [1995]. 8 This is a Quintus Prolog implementation of an older version of the ontology, prepared in 1993 by Dimiter Dobrev of So a University, Bulgaria, when on leave at the University of Twente in the context of the TEMPUS exchange programme of the European Community. The implementation de nes a number of checking predicates that, when given an argument, determine whether it is of the correct type. A newer Prolog version, based on somewhat di erent principles, is currently under development.
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rst-order predicate logic (and hence in KIF). The principle is demonstrated by giving the formal corollary of the de nition of group; the other de nitions follow the same principle.
Corollary de nition 1 (Group) A group is any member of the set GR de ned as GR =Df fg j g E N ^ g 6= ; ^ 8vwxy[(hv; xi 2 g ^ hw; yi 2 g ^ x 6= y) ! v 6= w]g In the present de nition, the chemical constraint is imposed by the last clause. It functions as follows. As before, any concept for a group is a set of tuples. The tuples hv; xi and hw; yi are both members of the same set; v and w are chemical elements and x and y are natural numbers. Two cases can be distinguished. If v 6= w, then the chemical constraint is trivially satis ed. If v = w, then the chemical constraint is violated if also x 6= y. Finally, if both v = w and x = y, then the tuples hv; xi and hw; yi are equal; from a set-theoretical point of view this means that there is actually only a single tuple that has been listed twice. The pattern also occurs in de nitions 2, 3, and 5. The Ontolingua implementation is straightforward, see the appendix. The inclusion of these constructions has an adverse e ect on portability, since it uses KIF functions that (at least for now) cannot be handled by the translators delivered with Ontolingua. The principles followed in the classic and Prolog implementations can also be explained by means of the construction of group concepts. In the classic implementation, there is a parent role de ned for the concept group, with a value restriction to chemical elements. There are exactly 103 child roles, each with a value restriction to a particular chemical element. Since chemical elements are de ned as disjoint primitives, no chemical element can occur as ller of two roles of the same group. The proportion n is speci ed by means of the cardinality restrictions AT-LEAST n and AT-MOST n. In the Prolog implementation, groups are lists of tuples (X,Y) where X is required to be a chemical element and Y a natural number. The checking predicate maintains a second list in which all elements are collected. The check fails if an element occurs more than once.
2. Addition of second elements of tuples in a set. De nitions 3 and 5 involve addition of fractions to yield exactly one. In both cases, the concepts involved are written as sets of ordered pairs hx; yi, where y is a real number in the half-open interval [0; 1). A function pair-range-sum can be de ned that takes as argument a set of ordered pairs where the second member of each pair is a number and returns the sum of all those numbers. The (recursive) de nition is as follows: pair-range-sum(;) = 0 pair-range-sum(fhx; y ig [ Z ) = y + pair-range-sum(Z ) The Ontolingua implementation is again straightforward. As above, it has an adverse e ect on portability. The Prolog implementation is similar. In the classic implementation, the cardinality restrictions can be used when (as indicated above) the fractions are turned into natural numbers by multiplication. 3. Formalising `predominantly solid'. The demand that a material, if a heterogeneous system, is `predominantly solid' also requires addition of fractions, this time of all fractions that occur in ordered pairs of which the other member is a solid phase. The sum must be a number x that is to be speci ed in background knowledge. (The decision what goes into the ontology and what into background knowledge is dicult and seldom clear-cut.) As far as the ontology is concerned, x is a real number in the half-open interval [0; 1). The complication here is that the addition is to be performed over a subset of the relation rather than over all pairs. In the Ontolingua implementation this issue is addressed by adding concepts for solid and non-solid phases. These are then used in turn for de ning heterogeneous systems as consisting of solid and non-solid phases subject to two constraints on the fractions specifying the proportions of constituent phases: the fractions of all phases have to add up to one; and the fractions of all solid phases have to add up to a number greater than or equal to a number x (called threshold 10
in the implementation). For applications in which the proportion of solid phases is unimportant these additions can be left out.
4. Implicit de nition of taxonomies. The arbitrary(X ) function has not yet been implemented in the Ontolingua version. KIF has an operator for -abstraction, so it will be possible to implement arbitrary in Ontolingua. The corollary of arbitrary in classic amounts to leaving out cardinality restrictions on roles. Classic then assumes that roles can be added (subject to constraints imposed by roles attached to the parent concept), which is another way of expressing what arbitrary stands for.
5 Bibliography Ronald J. Brachman, Deborah L. McGuiness, Peter F. Patel-Schneider, Lori Alperin Resnick & Alexander Borgida [1991], \Living with CLASSIC: when and how to use a KL-ONE-like language," in Principles of semantic networks. Explorations in the representation of knowledge, John F. Sowa, ed., Morgan Kaufmann, San Mateo CA, 401{456. Michael J. Genesereth & Richard E. Fikes [1992], \Knowledge Interchange Format, version 3.0. Reference manual," Computer Science Department, Stanford University, Report Logic 92-1, Stanford CA. Thomas R. Gruber [1991], \The role of common ontology in achieving sharable, reusable knowledge bases," in Proceedings of the Second International Conference on Principles of Knowledge Representation and Reasoning (KR'91), Cambridge, MA, April 22-25, 1991, James Allen, Richard Fikes & Erik Sandewall, eds., 601{602. Thomas R. Gruber [1992], \Ontolingua: a mechanism to support portable ontologies," Knowledge Systems Laboratory, Stanford University, Technical Report KSL 91{66, Palo Alto, CA. Thomas R. Gruber [1993], \A translation approach to portable ontology speci cations," Knowledge Acquisition5, 199{220. Nicolaas J.I. Mars [1993], \The role of ontologies in structuring large knowledge bases," in Proceedings of the International Conference on Building and Sharing Very Large-Scale Knowledge Bases '93 (participant's edition), Japan Information Processing Development Center, Tokyo, 235{243. Nicolaas J.I. Mars & A.T. Schreiber [1985], \Direct access to knowledge in bibliographic databases," in Proceedings of the ARTINT Workshop on Arti cial Intelligence and Information Retrieval, Luxembourg, 13 September 1985, Commission of the European Communities, Luxembourg, 83{86. Nicolaas J.I. Mars, Wilco G. ter Stal, Hidde de Jong, Paul E. van der Vet & Piet-Hein Speel [1994], \Semi-automatic knowledge acquisition in Plinius: an engineering approach," in Proceedings of the Eighth Ban Knowledge Acquisition for Knowledge-Based Systems Workshop, Ban , January 30 { February 4, 1994, Brain Gaines & Mark Musen, eds., 4{1 { 4-15. Piet-Hein Speel [1995], Selecting knowledge representation systems, University of Twente, Enschede, the Netherlands, Ph.D. thesis. Paul E. van der Vet, Hidde de Jong, Nicolaas J.I. Mars, Piet-Hein Speel & Wilco G. ter Stal [1994], \Plinius intermediate report," University of Twente, Memoranda Informatica 94-35, Enschede, the Netherlands. Paul E. van der Vet & Nicolaas J.I. Mars [1993], \Structured system of concepts for storing, retrieving, and manipulating chemical information," Journal of Chemical Information and Computer Sciences 33, 564{568.
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A Ontolingua implementation Below we give the complete listing of the Ontolingua code of the ontology. Care has been taken to keep de nitions on a single page.
;;; PLINIUS ONTOLOGY OF CHEMICAL COMPOSITION, version 2.0 ;;;;;;;;;;;;;;;;;; ; ; ; Ontolingua implementation, made July 2-8, 1994 ; ; ; ; Checked by Ontolingua version 4 on July 8, 1994 ; ; No errors reported ; ; ; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ; ; ; This version prepared by (C) Paul E. van der Vet ; ; ; ; Address of the author: ; ; Knowledge-Based Systems Group ; ; Dept. of computer Science, University of Twente ; ; P.O. Box 217, 7500 AE Enschede, the Netherlands ; ; e-mail
[email protected] ; ; ; ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;; PREAMBLE ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; (in-package :ol-user) (define-theory Plinius-chem-comp (frame-ontology) "This part of the Plinius ontology covers the chemical composition of materials") (in-theory 'Plinius-chem-comp)
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;;; ATOMIC CONCEPTS ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;; Classes of atomic concepts with explicit enumeration ;;; (define-class chemical-element (?ce) "Chemical elements are the basic building blocks of materials. They are defined extensionally, as a list of element symbols. Another choice would be to define elements by their atomic number, since numbers are already defined in KIF. However, for the purposes of the present ontology, atomic numbers are not numbers but rather discrete labels. Element symbols avoid confusion. Most of these elements will not occur in ceramics. However, there are only 103 in the list and it seems unwise to make a selection that will have to be changed later. The list is not complete: it contains only the first 103 elements in the periodic system. Elements with higher atomic numbers are so unstable that they will never occur in any material. Isotopes are not distinguished in the domain. For the implementation it has to be noted that Ontolingua wants the instances of an extensionally defined class to be introduced twice: once in the extensional definition of the class itself, and then by means of a define-instance for each instance separately." :iff-def (member ?ce (setof Ac Ag Al Am Ar As At Au B Ba Be Bi Bk Br C Ca Cd Ce Cf Cl Cm Co Cr Cs Cu Dy Er Es Eu F Fe Fm Fr Ga Gd Ge H He Hf Hg Ho I In Ir K Kr La Li Lr Lu Md Mg Mn Mo N Na Nb Nd Ne Ni No Np O Os P Pa Pb Pd Pm Po Pr Pt Pu Ra Rb Re Rh Rn Ru S Sb Sc Se Si Sm Sn Sr Ta Tb Tc Te Th Ti Tl Tm U V W Xe Y Yb Zn Zr))) (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance
Ac (chemical-element)) Ag (chemical-element)) Al (chemical-element)) Am (chemical-element)) Ar (chemical-element)) As (chemical-element)) At (chemical-element)) Au (chemical-element)) B (chemical-element)) Ba (chemical-element)) Be (chemical-element)) Bi (chemical-element)) Bk (chemical-element)) Br (chemical-element)) C (chemical-element)) Ca (chemical-element)) Cd (chemical-element)) Ce (chemical-element)) Cf (chemical-element)) Cl (chemical-element)) Cm (chemical-element))
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(define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance
Co (chemical-element)) Cr (chemical-element)) Cs (chemical-element)) Cu (chemical-element)) Dy (chemical-element)) Er (chemical-element)) Es (chemical-element)) Eu (chemical-element)) F (chemical-element)) Fe (chemical-element)) Fm (chemical-element)) Fr (chemical-element)) Ga (chemical-element)) Gd (chemical-element)) Ge (chemical-element)) H (chemical-element)) He (chemical-element)) Hf (chemical-element)) Hg (chemical-element)) Ho (chemical-element)) I (chemical-element)) In (chemical-element)) Ir (chemical-element)) K (chemical-element)) Kr (chemical-element)) La (chemical-element)) Li (chemical-element)) Lr (chemical-element)) Lu (chemical-element)) Md (chemical-element)) Mg (chemical-element)) Mn (chemical-element)) Mo (chemical-element)) N (chemical-element)) Na (chemical-element)) Nb (chemical-element)) Nd (chemical-element)) Ne (chemical-element)) Ni (chemical-element)) No (chemical-element)) Np (chemical-element)) O (chemical-element)) Os (chemical-element)) P (chemical-element)) Pa (chemical-element)) Pb (chemical-element)) Pd (chemical-element)) Pm (chemical-element)) Po (chemical-element)) Pr (chemical-element)) Pt (chemical-element)) Pu (chemical-element)) Ra (chemical-element)) Rb (chemical-element)) Re (chemical-element)) Rh (chemical-element)) Rn (chemical-element)) Ru (chemical-element))
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(define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance
S (chemical-element)) Sb (chemical-element)) Sc (chemical-element)) Se (chemical-element)) Si (chemical-element)) Sm (chemical-element)) Sn (chemical-element)) Sr (chemical-element)) Ta (chemical-element)) Tb (chemical-element)) Tc (chemical-element)) Te (chemical-element)) Th (chemical-element)) Ti (chemical-element)) Tl (chemical-element)) Tm (chemical-element)) U (chemical-element)) V (chemical-element)) W (chemical-element)) Xe (chemical-element)) Y (chemical-element)) Yb (chemical-element)) Zn (chemical-element)) Zr (chemical-element))
(define-class material-tag (?mt) "In materials science, it is often convenient to distinguish between materials according to the role they play in the production process. This is accomplished here by extensional definition of a class of identifying labels. A more fundamental definition would go far beyond the scope of this ontology. For the implementation it has to be noted that Ontolingua wants the instances of an extensionally defined class to be introduced twice: once in the extensional definition of the class itself, and then by means of a define-instance for each instance separately." :iff-def (member ?mt (setof powder greenling finished))) (define-instance powder (material-tag)) (define-instance greenling (material-tag)) (define-instance finished (material-tag))
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(define-class microstructure-specification (?mi) "Introduced here for completeness only. A heterogeneous system is defined, among others, by its microstructure. The ontology of microstructural specifications has not been elaborated yet. Instead, a single primitive serves for the time being as such. For the implementation it has to be noted that Ontolingua wants the instances of an extensionally defined class to be introduced twice: once in the extensional definition of the class itself, and then by means of a define-instance for each instance separately." :iff-def (member ?mi (setof microstructure))) (define-instance microstructure (microstructure-specification))
; Atomic concepts for aggregation states (define-class solid-aggregation-state (?sas) "Aggregation states are states of matter such as the gaseous and liquid states. They are defined extensionally here, as strings that identify states. A more fundamental treatment requires a detailed analysis of interatomic and intermolecular forces. This is far beyond the scope of the present ontology. Aggregation states are initially partitioned into solid and non-solid aggregation states. Where there is just one liquid state, solid states can be subdivided according to the crystal symmetry. Crystal symmetries are described by group theory. There are 32 distinct space-filling lattices (also called Bravais lattices). It is customary to employ a coarser classification, that distinguishes six kinds of lattices. The latter classification is followed here. The class of solid aggregation states thus comprises six atomic concepts. A seventh atomic concept, called unknown-sas, is added to make it possible to refer to a solid aggregation state without, however, being able to specify crystal symmetry. For the implementation it has to be noted that Ontolingua wants the instances of an extensionally defined class to be introduced twice: once in the extensional definition of the class itself, and then by means of a define-instance for each instance separately." :def (member ?sas (setof triclinic monoclinic orthorhombic tetragonal hexagonal cubic unknown-sas))) (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance (define-instance
triclinic (solid-aggregation-state)) monoclinic (solid-aggregation-state)) orthorhombic (solid-aggregation-state)) tetragonal (solid-aggregation-state)) hexagonal (solid-aggregation-state)) cubic (solid-aggregation-state)) unknown-sas (solid-aggregation-state))
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(define-class non-solid-aggregation-state (?nas) "This class assembles all non-solid aggregation states. The vitreous state (for glasses and other amorphous solids) is included to sidestep discussions about the true nature of this aggregation state: just a liquid with a very high viscosity or something special. Experts differ. For the implementation it has to be noted that Ontolingua wants the instances of an extensionally defined class to be introduced twice: once in the extensional definition of the class itself, and then by means of a define-instance for each instance separately." :def (member ?nas (setof gaseous liquid vitreous liquid-crystal))) (define-instance (define-instance (define-instance (define-instance
gaseous (non-solid-aggregation-state)) liquid (non-solid-aggregation-state)) vitreous (non-solid-aggregation-state)) liquid-crystal (non-solid-aggregation-state))
(define-class aggregation-state (?as) "The class of aggregation states is finally defined as the class of all solid and non-solid aggregation states. The definition includes the requirement that solid and non-solid aggregation states are mutually exclusive." :axiom-def (exhaustive-subclass-partition aggregation-state (setof solid-aggregation-state non-solid-aggregation-state)))
;;; Sets of atomic concepts with intensional specification ;;; ;;; Used for the handling of numbers ;;; (define-class positive-real (?pr) "The class of real numbers greater than zero." :iff-def (and (real-number ?pr) (> 0 ?pr))) (define-class fraction (?fr) "Real numbers in the half-open interval [0,1)." :iff-def (and (real-number ?fr) (< 0 ?fr) (>= 1 ?fr)))
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; A function to define the sum of the elements making up the ; range of a relation. (define-function pair-range-sum (?x) "This function takes a set of ordered pairs (in other words, a binary relation) as input and returns the sum of the second members of of all pairs. In the definition, the first clause says that the pair-range-sum of the empty set is zero. The second clause defines the pair-range-sum of a non-empty set recursively for binary relations (sets of ordered pairs) of which the second member is a real number." :axiom-def (and (forall (?x) ( (empty ?x) (zero (pair-range-sum ?x)))) (forall (?x ?y ?z) (=> (and (binary-relation ?x) (real-number ?z)) (= (pair-range-sum (union ?x (listof ?y ?z))) (+ (pair-range-sum ?x) ?z))))))
;;; COMPLEX CONCEPTS ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;; GROUPS AND CHEMICAL SUBSTANCES ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(define-class group (?gr) "Within the domain, a group is a set of elements bonded by chemical bonds. This implies that the proportions of the elements that make up the group can only be whole numbers. A group is defined as a non-empty set of ordered pairs, where the first member is a chemical element and the second a number expressing proportions. From a chemical point of view, a group is ill-defined if a chemical element occurs in more than one ordered pair." :axiom-def (forall (?x) ( (instance-of group ?x) (and (binary-relation ?x) (subclass-of exact-domain ?x chemical-element) (subclass-of exact-range ?x natural) (not (empty ?x)) (forall (?p ?q ?v ?w) (=> (and (holds ?x ?p ?v) (holds ?x ?q ?w) (/= ?v ?w)) (/= ?p ?q)))))))
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(define-class chemical-substance (?cs) "Within the domain, a chemical substance is a non-empty set of groups together with proportions. The occurrence of so-called nonstoichiometric compounds in the domain entails that the proportions have to be positive reals. A chemical substance is defined as a non-empty set of ordered pairs, where the first member is a group and the second a number expressing proportions. From a chemical point of view, a chemical substance is ill-defined if a group occurs in more than one ordered pair." :axiom-def (forall (?x) ( (instance-of chemical-substance ?x) (and (binary-relation ?x) (subclass-of exact-domain ?x group) (subclass-of exact-range ?x positive-real) (not (empty ?x)) (forall (?p ?q ?v ?w) (=> (and (holds ?x ?p ?v) (holds ?x ?q ?w) (/= ?v ?w)) (/= ?p ?q)))))))
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;;; PHASES ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; (define-class phase-chemical-composition (?pcc) "In chemistry, a phase is defined by its chemical composition and its aggregation state. The present class defines the chemical composition of a phase as a non-empty set of ordered pairs, of which the first member is a chemical substance and the second member a fraction specifying the proportion of the substance in the phase. The requirement that the sum of the fractions for all ordered pairs in the set be one has been formalised using the pair-range-sum function. As was the case for groups and chemical substances, a phase-chemical-composition is ill-defined if a chemical substance occurs in more than one ordered pair." :axiom-def (forall (?x) ( (instance-of phase-chemical-composition ?x) (and (binary-relation ?x) (subclass-of exact-domain ?x chemical-substance) (subclass-of exact-range ?x fraction) (not (empty ?x)) (= 1 (pair-range-sum ?x)) (forall (?p ?q ?v ?w) (=> (and (holds ?x ?p ?v) (holds ?x ?q ?w) (/= ?v ?w)) (/= ?p ?q)))))))
(define-relation chemical-phase (?pcc ?as) "Any combination of a phase-chemical-composition and an aggregation state qualifies as a bona fide phase in the chemical sense of the word. The string phase is not used for this class because that string is a reserved word in KIF." :iff-def (and (instance-of ?pcc phase-chemical-composition) (instance-of ?as aggregation-state)))
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;;; SYSTEMS ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; (define-class homogeneous-system (?hos) "A homogeneous system consists by definition of a single phase. Phase and homogeneous system have to be distinguished because a phase is defined as a part of a system while a homogeneous system is a system. The present definition tries to make the distinction by having a particular homogeneous system be an instance of the class homogeneous-system while a phase is a member of the set of tuples defined by the relation chemical-phase. I am not sure, however, whether I have done this correctly." :axiom-def (forall (?x) ( (instance-of homogeneous-system ?x) (and (unary-relation ?x) (instance-of chemical-phase ?x)))))
(define-class heterogeneous-system-composition (?hsc) "A heterogeneous system is defined by its composition and its microstructure. The present class defines the composition of a heterogeneous system as a non-empty set of ordered pairs, where the first member is a phase and the second member a fraction specifying the proportion of the phase in the system. The requirement that the sum of the fractions for all ordered pairs in the set be one has been formalised using the pair-range-sum function. As was the case several times before, a heterogeneoussystem-composition is ill-defined if a phase occurs in more than one ordered pair." :axiom-def (forall (?x) ( (instance-of heterogeneous-system-composition ?x) (and (binary-relation ?x) (subclass-of exact-domain ?x chemical-phase) (subclass-of exact-range ?x fraction) (not (empty ?x)) (= 1 (pair-range-sum ?x)) (forall (?p ?q ?v ?w) (=> (and (holds ?x ?p ?v) (holds ?x ?q ?w) (/= ?v ?w)) (/= ?p ?q))))))) (define-relation heterogeneous-system (?hsc ?mi) "Any combination of a heterogeneous-system-composition and a specification of microstructure qualifies as a bona fide heterogeneous system." :iff-def (and (instance-of ?hsc heterogeneous-system-composition) (instance-of ?mi microstructure-specification)))
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;;; MATERIALS ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; ;;; GLASSES ;;; (define-class glass (?gl) "A material is either a glass or a predominantly solid heterogeneous system. A glass is a homogeneous system of which the only constituent phase has the aggregation state vitreous." :iff-def (and (instance-of ?gl homogeneous-system) (forall (?x ?y) (=> (holds ?gl ?x ?y) (= ?y vitreous)))))
;;; PREDOMINANTLY SOLID HETEROGENEOUS SYSTEMS ;;; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;
The definition of the concept of predominantly solid heterogeneous system proceeds in stages, for clarity. First, the set of phasetuples is exhaustively partitioned into the set of solid-phase-tuples and that of non-solid-phase-tuples. Then, two classes are defined: that of solid constituents of a heterogeneous system and that of non-solid constituents of a heterogeneous system. Finally, the composition of a heterogeneous system as the union of those two sets. For a predominantly solid heterogeneous system, the total of the fractions for the solid constituents must add up to a number greater that or equal to some number called threshold (which is itself specified in background knowledge). The fractions for all constituents must add up to one. We re-start the exercise at the point where chemical-phase was defined.
(define-relation solid-phase (?pcc ?sas) "For the conceptualisation of a material as a predominantly solid heterogeneous system, we need to be able to build a heterogeneous system out of phases that are recognisably solid or non-solid. A phase is solid if its aggregation state is a solid state" :iff-def (and (instance-of ?pcc phase-chemical-composition) (instance-of ?sas solid-aggregation-state)))
(define-relation non-solid-phase (?pcc ?nas) "For the conceptualisation of a material as a predominantly solid heterogeneous system, we need to be able to build a heterogeneous system out of phases that are recognisably solid or non-solid. A phase is non-solid if its aggregation state is one of the non-solid states." :iff-def (and (instance-of ?pcc phase-chemical-composition) (instance-of ?nas non-solid-aggregation-state)))
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(define-class solid-heterogeneous-system-part (?shp) "For the conceptualisation of a material as a predominantly solid heterogeneous system, we need to be able to build a heterogeneous system out of phases that are recognisably solid or non-solid. The solid-heterogeneous-system-part assembles all solid phases making up a heterogeneous system, including the fractions of each phase in the total heterogeneous system. The fractions for the solid parts have to add up to a number between zero and 1." :axiom-def (forall (?x) ( (instance-of solid-heterogeneous-system-part ?x) (and (binary-relation ?x) (subclass-of exact-domain ?x solid-phase) (subclass-of exact-range ?x fraction) (not (empty ?x)) (instance-of fraction (pair-range-sum ?x)) (forall (?p ?q ?v ?w) (=> (and (holds ?x ?p ?v) (holds ?x ?q ?w) (/= ?v ?w)) (/= ?p ?q)))))))
(define-class non-solid-heterogeneous-system-part (?nshp) "For the conceptualisation of a material as a predominantly solid heterogeneous system, we need to be able to build a heterogeneous system out of phases that are recognisably solid or non-solid. The non-solid-heterogeneous-system-part assembles all non-solid phases making up a heterogeneous system, including the fractions of each phase in the total heterogeneous system. The fractions for the non-solid parts have to add up to a number between zero and 1." :axiom-def (forall (?x) ( (instance-of non-solid-heterogeneous-system-part ?x) (and (binary-relation ?x) (subclass-of exact-domain ?x non-solid-phase) (subclass-of exact-range ?x fraction) (not (empty ?x)) (instance-of fraction (pair-range-sum ?x)) (forall (?p ?q ?v ?w) (=> (and (holds ?x ?p ?v) (holds ?x ?q ?w) (/= ?v ?w)) (/= ?p ?q)))))))
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(define-instance threshold (fraction) "Threshold is a value to be set in a particular application. It serves below to conceptualise the notion of predominantly solid heterogeneous system. It is the minimal value of the sums of fractions that specify the proportions of solid phases that make up a heterogeneous system. In other words, if that sum is smaller than threshold, the system does not count as a predominantly heterogeneous system. In our opinion, its actual value has to be specified in an actual application rather than in an ontology. In Plinius, the background knowledge base is the place where the value of threshold is given.")
(define-class pred-solid-heterog-system (?dhsc) "For the conceptualisation of a material as a predominantly solid heterogeneous system, we need to be able to build a heterogeneous system out of phases that are recognisably solid or non-solid. A pred-solid-heterog-system specifies the composition of a heterogeneous system as consisting exhaustively of one instance of the class of solid heterogeneous system parts and one instance of the class of non-solid heterogeneous system parts. The fractions of the two specifications have to add up to one. The sum of the fractions that specify the proportions of the solid phases has to be larger than the value specified by threshold." :axiom-def (forall (?d) ( (instance-of pred-solid-heterog-system ?d) (forall (?x ?y) (and (exhaustive-subclass-partition pred-solid-heterog-system (setof ?x ?y)) (instance-of solid-heterogeneous-system-part ?x) (instance-of non-solid-heterogeneous-system-part ?y) (>= (pair-range-sum ?x) threshold) (= 1 (+ (pair-range-sum ?x) (pair-range-sum ?y))))))))
;;; END ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
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