Journal of Experimental & Theoretical Artificial Intelligence, Vol. 15, No. 4, October–December 2003, 489–511
Analogical projection in pattern perception MEHDI DASTANI1, BIPIN INDURKHYA2 and REMKO SCHA3 1
Institute of Information and Computing Sciences, Utrecht University, PO Box 80.089, 3508 TB, Utrecht, The Netherlands e-mail:
[email protected] 2 Tokyo University of Agriculture and Technology, Japan e-mail:
[email protected] 3 University of Amsterdam, The Netherlands e-mail:
[email protected] Abstract. This paper proposes a perceptually motivated method for solving proportional analogy problems involving sequential patterns. Our method is based on an algebraic model of pattern perception that determines the gestalt structure of sequential patterns. The gestalts of sequential patterns are represented as algebraic terms in the model. An analogical relation between sequential patterns appearing in proportional analogy is then formalized as a mapping between algebras that generate terms representing the gestalts of these patterns. Based on this formalism, an algorithm is proposed that solves proportional analogy problems: given three sequential patterns, the algorithm computes a fourth pattern such that the resulting two pairs of patterns are perceived as having an identical analogical relation. Keywords:
proportional analogy, analogy, gestalt perception, pattern perception, analogical projection, creative analogy, structural information theory
Received August 2002; revised June 2003; accepted July 2003 1. Introduction A fundamental activity underlying perception is that of integrating sensory stimuli into a system of concepts. Indeed, Gestalt psychologists since Wertheimer (1923) have sought to articulate principles that constrain and govern this integration process, which is also sometimes referred to as perceptual organization, though this term is rather ambiguous in that it refers both to the process of perceptual organization itself and its result (Kubovy and Gepshtein 2003). Needless to say, there are a number of different theories of perceptual organization, but they all generally acknowledge that although a perceptual stimulus can potentially be organized or structured in a number of different ways, the human perceptual system tends to prefer one or a few of these possible structures. This idea was originally articulated by Wertheimer (1923) as follows: When we are presented with a number of stimuli we do not as a rule experience ‘‘a number’’ of individual things, this one and that and that. Instead larger wholes separated from and related to one another are given in experience; their arrangement and division are concrete and definite. (Wertheimer 1923) Journal of Experimental & Theoretical Artificial Intelligence ISSN 0952–813X print/ISSN 1362–3079 online # 2003 Taylor & Francis Ltd http://www.tandf.co.uk/journals DOI: 10.1080/09528130310001626283
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Wertheimer coined the term gestalt to refer to this organized or structured ‘whole’. An important objective of the gestalt research was to determine a set of rules or principles that constrain the relationships between sensory stimuli and their perceptual organizations. These rules could then be considered as the rules of perception, which can be used to explain why certain types of stimuli are experienced (perceived) as having a certain organization. In particular, Wertheimer proposed five principles — known as proximity, similarity, continuity, closure and habit or past experience—to constrain the perceptual organization of sensory stimuli, and he claimed that these constitute the laws underlying the mechanism of human perception. (See also Kubovy and and Gepshtein 2000 for a recent perspective on the gestalt research.) For example, the line pattern shown in figure 1a has, among others, two possible structures shown in figures 1b and 1c. However, humans tend to prefer the structure corresponding to figure 1b rather than the one in figure 1c. This can be explained on the basis of the closure principle, which says that closed figures are preferred rather than open ones, and the similarity principle, which says that a structure containing similar parts is preferred rather than one containing dissimilar parts. Relatively recently, Leeuwenberg (1971) argued, following the notion of Pra¨gnanz introduced by Koffka (1935), that the gestalt principles can be explained by a more primitive principle of human perception. This principle is related to the structural regularity and the simplicity of perceptual structure of sensory stimuli, and, consequently, Leeuwenberg’s theory is referred to as the Structural Information Theory (SIT, henceforth). It introduces a notion of information complexity to measure the simplicity of perceptual structure of sensory stimuli and to argue that in organizing a sensory stimulus, the structures with lower information complexity are preferred by the human perceptual system. Another major aspect of perceptual organization, which also has been known since the beginning of the gestalt research, is that it is heavily influenced by the context. Presence of other stimuli or recently seen stimuli can easily affect how a certain stimulus is structured. This phenomenon can be most starkly demonstrated by proportional analogy problems represented as ‘A is to B as C is to D’, abbreviated as A : B :: C : D, such as the ones shown in figure 2.
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Figure 1. Visual pattern a has two potential structures b and c.
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Figure 2. Examples of proportional analogies based on letter strings. Proportional analogies like these illustrate the intricacies of the gestalt disambiguation and how it is influenced by the context. In these examples, the presence of patterns A and B, and the objective of trying to form an appropriate analogy forces one to assign structures to the pattern C that might not be preferred when it is the only stimulus present. However, theories of perceptual organization, including variants of gestalt approaches and SIT, have not been able to account for this context effect satisfactorily yet. One recent attempt at formally modelling the context effect in perceptual domain, which is based on embedding SIT in an algebraic framework, is outlined in Dastani and Indurkhya (2001). Though the gestalt research has primarily focused on perception, it has been argued that a similar phenomenon can be evidenced in cognition. For example, a context effect similar to that of figure 2 was demonstrated by Tversky (1977), who asked the participants to choose a country that is most similar to Austria from a given list of three countries. When the given list was Sweden, Hungary and Poland, Sweden was chosen over Hungary more often. But when the given list was Sweden, Hungary and Norway, Hungary tended to be preferred over Sweden. Hofstadter and his colleagues have made persuasive arguments to show that the traditional distinction between perception and cognition is very much misplaced: they coined a term, high-level perception, to refer to processes that play an active role in both perception and cognition (Chalmers et al. 1992). They consider contextual disambiguation inherent in such analogies to be not only one such high-level perceptual process, but also a key to creativity (Hofstadter 1995). Further arguments are echoed in Indurkhya (1991, 1992) where, focusing on creative analogies and metaphors, it is demonstrated how the descriptions or representations of objects need to change in order to create similarities. Finally, Hunter and Indurkhya (1998) have shown how principles of gestalt disambiguation are useful in modelling creative thinking in the rather abstract and conceptual domain of legal reasoning. With all this background, we propose here a model that integrates analogical projection with perceptual organization. The model posits that the structures assigned by human cognitive apparatus to sensory stimuli are determined by the interaction between analogical projection and the principles that govern human perceptual system. In this interaction view, the process of analogical projection between sensory stimuli depends on underlying principles of human perception, and vice versa, the perception of sensory stimuli depends on analogical projection. In our view, this capability of analogical projection together with other principles that govern human perception provide us an insight into the higher cognitive processes such as creativity and discovery. This is the prime motivating factor underlying our research programme. The integration of gestalt disambiguation with analogical projection is clearly evidenced in proportional analogies such as those of figure 2. In order to
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comprehend the analogies, the structures of A and B must be construed as analogous to each other; the structures of C and D must be construed as analogous to each other; and there must be an analogical projection from A to C and from B to D, respectively. Thus, proportional analogy problems are fairly complex: there are four patterns whose perceived structures mutually influence each other through analogical mappings. A proportional analogy problem is constructed by omitting one element in a proportional analogy relation. We write A : B :: C : X; the task is to find an X which is related to B in a similar way as C is related to A. Solving a proportional analogy problem requires: (1) a mutual contextualization of the terms A and B; and (2) construction of a mapping between A and C, which generalizes to a broader domain that includes B. For outlining our model, we focus on the proportional analogy problems because they have a clearly defined goal, namely generate an appropriate fourth term, given the other three terms of the analogy. There are two main principles underlying our model. The first is related to the context-sensitivity of the description of patterns involved in the analogy. It states that the description of each pattern must be such that there exists a mapping from A onto C that is applicable to B. The second principle is related to the gestalt of the patterns and states that the proper description of patterns must be in accordance with their perceptual structures. This principle suggests that the description of patterns must be as simple as possible. As a consequence of these two principles, the description of each individual pattern is not computed in isolation and at once, but interacts with the context and the gestalt of the other patterns. This consideration makes it possible for our approach to analyse a certain class of creative analogies deterministically. The aim of this paper is twofold. On one hand, the solution of proportional analogy problem is defined such that both context and gestalt are taken into account in a mathematically rigorous way. On the other hand, a deterministic method is introduced which solves a certain class of proportional analogy problems. In both these aspects our model is fundamentally different from the approach of Hofstadter (1995), whose models are indeterministic and are manifested as computer implementations that lack any independent (that is, independent of the computer programs) framework or set of principles. (We will elaborate this point further in section 7.) Our model takes the structural information theory proposed by Leeuwenberg (1971) as its starting point. We embed SIT in an algebraic framework and extend it to model the context effect (Dastani and Idurkhya 2001). We then formalize analogies as mappings between algebras and outline an algorithm to solve proportional analogy problems. We demonstrate the efficacy of our model by considering some examples from the Copycat domain of Horstadter (1984), such as the ones shown in figure 2. Other examples of analogy problems in the domain of line patterns are shown in figure 3. These analogies show the interaction of analogical projection and context disambiguation. We chose this domain for two main reasons. One is that it makes it easier to compare our approach to the existing research and to notice the similarities and differences. The other reason is that there exists an encoding method proposed by Leeuwenberg that allows line patterns such as the ones in figure 3 to be represented as character strings. This paper is organized as follows. In the next section we provide an overview of Leeuwenberg’s structural information theory (1971), which provides the starting
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Figure 4. A hierarchical arrangement of identical parts of line pattern A. It indicates that A consists of two identical diamonds (B) each of which consists of four identical line segments (C ). point of our research. In section 3, we introduce an algebraic model of pattern perception based on SIT. In section 4, the algebraic model of pattern perception is used to formalize proportional analogy. We demonstrate various features of our approach in Hofstadter’s Copycat domain, and show how proportional analogy problems like the one shown in figure 3 can be solved with it. In section 5, we discuss various factors that constrain the search for appropriate gestalts in proportional analogies. In section 6, we present an algorithm for solving proportional analogy problems and compare it with other computational approaches. Finally, in section 7, we summarize the major points of our paper and briefly discuss some future research issues. 2. An overview of structural information theory As we take Leeuwenberg’s structural information theory (1971) as the starting point of our research, we provide an overview of it here. SIT is a general theory of pattern perception that tries to explain why human perceivers prefer certain structures of sensory stimuli rather than an arbitrary one. The explanation is based on the assumption that the human perceptual system is sensitive to certain types of structural regularities of sensory stimuli. A structural regularity of a sensory stimulus is a certain hierarchical arrangement of identical parts of the stimulus. For example, the regularity in the string abab may be defined in terms of the identity of the first and the third characters a, the second and the fourth characters b, and the first and the second substring ab. Note that these identities are hierarchically ordered: the identity of the substrings ab implies the identities of the a’s and the b’s. The same type of structural regularities occur in visual line patterns as illustrated in figure 4. The regularity of this pattern can be defined in terms of identical diamonds, and the regularities of the diamonds can in turn be defined in terms of identical line segments.
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According to SIT, only certain types of structural regularities are perceptually relevant. These structural regularities are specified by the ISA operators: iteration, symmetry and alternation. The iteration operator captures the repetition of consecutive subpatterns within a pattern, e.g. the repetition of ab in string abab. The symmetry operator captures the reflection of subpatterns within a pattern, e.g. the reflection of a around b in string aba. Finally, the alternation operator captures, like the iteration operator, the repetition of subpatterns excepts that the subpatterns need not be consecutive subpatterns, e.g. the repetition of a in string abapaf . It is conjectured that these operators reflect innate principles of mental representation. A pattern can be described by means of the ISA operators in different ways. For example, string abab can be described by iteration operator as two times ab; by the symmetry operator as a reflected around b, followed by another b; or by the alternation operator as a alternated two times by b and b. The resulting descriptions indicate different gestalts of the pattern. In order to disambiguate the set of alternative gestalts and to decide on the preferred (or perceived) gestalt of the pattern, a complexity measure is introduced. The complexity measure is defined on pattern descriptions and it indicates the lack of perceptual regularity within a pattern. It is claimed that the description of a pattern with the minimum complexity value describes the pattern in the most simple and cognitively economical way. In other words, the description with the minimum complexity value captures the maximum amount of regularity within the pattern and indicates the preferred gestalt of the pattern. For example, the proposed complexity measure indicates that the description of string abab in terms of the iteration operator is simpler than its descriptions in terms of the symmetry or alternation operators and therefore it states that string abab is perceived as consisting of subpattern ab iterated twice. The idea that the simplest description of a pattern indicates the preferred perceptual structure of that pattern is called the simplicity principle. The set of ISA operators and the complexity measure, as originally proposed in Leeuwenberg (1971), have undergone several revisions until Van der Helm and Leeuwenberg articulated them in a criterion called accessibility that is based on a formal analysis of regularity and hierarchy (Van der Helm, 1988). A number of psychological experiments have been carried out to verify the simplicity principle and the accessibility criterion. (Buffart et al. 1981, Boselie 1988, Van Leeuwen et al. 1988, Boselie and Wouterlood 1989, Van der Vegt et al. 1989, Van Leeuwen and Buffart 1989, Stins and Van Leeuwen 1993).
3. An algebraic coding system for SIT Although structural information theory is intended as a general theory of pattern perception, it is built on a system for describing one-dimensional sequential patterns, e.g. string patterns (Leeuwenberg 1971, Van der Helm and Leeuwenberg 1991). In this section, we give an algebraic coding system that generates possible gestalts (hierarchical constituent structure) for one-dimensional string patterns. This algebraic version is based on a simplified version of the ISA operators that cover the relevant gestalts of string patterns (Dastani 1998). The algebraic tools we use in our formalization are the ones developed for specifying programming languages and abstract data types (Goguen et al. 1978, Rus and Halverson 1994). An algebra is essentially a domain of objects, and a number of operators (or functions) defined on these objects. An n-ary operator takes
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as input n objects, and results in another object in the domain. The operators of the algebra essentially endow the objects of its domain with structure in that they allow some objects to be combined into another object or, in another way of looking at it, allow an object to be decomposed into its component objects. In order to develop an algebraic version of SIT, we define the structural ISA operators of SIT as algebraic operators. We now demonstrate how to do this in Hofstadter and Mitchell’s (1988) Copycat domain. First, we define the domain of objects that are of interest to us. For the Copycat domain, this is simply the set of all finite non-null strings composed from letters fa, b, . . . , zg. Definition 1: The domain D of one-dimensional string patterns is defined as the set of all finite strings of length one or more composed from fa, b, . . . , zg. In a significant point of departure from SIT, we also allow domain-dependent operators in our model, which can interact with SIT operators so that more complex patterns can also be represented in terms of SIT operators. To understand this point, first notice that the perceptual structures (regularities) covered in SIT are based only on identities of pattern constituents. For example, aaaa is analyzed as a repetition of a, i.e. it is recognized that the four elements appearing in the pattern are identical. However, it is also possible to consider other relations that may be specific to the domain. For instance, in the domain of geometric figures, one may consider the operation rotate-left. Or in a domain with ordered elements, such as Copycat, we may consider generalized successor (succ) and predecessor (pred ) functions, which also apply to sequences of elements as follows: succðbÞ ¼ c, succðabcÞ ¼ bcd, and so on. Using these functions, we may notice a regularity in the sequence abc, namely that it is a successor sequence: the next element of the sequence is obtained by taking the successor of the previous element. This issue is explicitly addressed in the SIT literature (Van der Helm and Leeuwenberg 1991); however, it is argued there that such operators are cognitive and not perceptual, and therefore should not be incorporated in the perceptual coding system. But we feel that while operators requiring addition or otherwise complex operations may be left out, simple operators like ‘successor’ and ‘rotate-left’ do end up playing a critical role in perception, and ought to be included in the gestalts. In any case, we are interested in representing cognitive operators as well, and being able to model how they affect the structural description of perceptual stimuli. We denote domain-dependent operators by F D . For the time being we limit ourselves to one-place domain-dependent operators. Thus, each element in F D is a function from D to D. The ISA operators can now be defined over < D, F D >. We augment the iteration and alternation operators so that they take an extra argument that is an element of F D . Moreover, we represent the concatenation operation, tacitly assumed in SIT, explicitly in our algebra by introducing an operator called Con. Definition 2: Let X, Y, X1 , . . . , Xn be arbitrary string patterns from D; f be a one-place domain-dependent operator ð f 2 fid, succ, pred, . . .gÞ; and n 2 N . Then, the following operators are the ISA operators: Iter(X, f, n) Syme(X) Symo(X,Y) Altr(X, f, X1, . . . , Xn)
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Altl(X, f, X1, . . . , Xn) Con(X1, . . . , Xn) We are now ready to define the SIT algebra over the domain D as follows: Definition 3: A SIT-algebra over the domain D is a quadruple < D, N , F D , S >, where D is the domain of objects, N is the set of natural numbers; F D is a set of one-place domain-dependent operators (so that each f 2 F is a function from D to D); and S is the set of ISA operators. In the definitions above, the particular variable used in each argument position conveys its sort. For example, the first argument of Iter must come from D, the second argument from F D , and so on. The output of each operator belongs to D. Note that we need to introduce the identity operator id explicitly, whenever we want the iteration and alternation operators to use identity of elements in constructing gestalts. Because the definitions of the iteration and alternation operators take domain-dependent operators as argument, the SIT-algebra becomes a higher order algebra (Meinke 1992, Meinke and Steggles 1993). Now we can consider the gestalts of patterns as being represented by terms of the SIT-algebra. The structural description of an object shows how the object is built out of other elements. This corresponds to what is called a term or an -word of an algebra (Cohn 1981). Thus, the set of all terms of the SIT-algebra corresponds to the set of all possible gestalts for the universe D. Definition 4: The class of gestalts over the SIT-algebra, denoted by G , is recursively defined as follows: . For all X 2 D, X 2 G . If f 2 F D and X 2 D, then f ½X 2 G . If f 2 F D , n 2 N and X 2 G , then Iter ½X, f , n 2 G . If X 2 G , then Syme ½X 2 G . If X1 , X2 2 G , then Symo ½X1 , X2 2 G . If f 2 F D and X, X1 , . . . , Xn 2 G , then Altr ½X, f , X1 , . . . , Xn 2 G and Altl ½X, f , X1 , . . . , Xn 2 G . If X1 , . . . , Xn 2 G with n > 1, then Con ½X1 , . . . , Xn 2 G Notice that we use square brackets instead of round ones to emphasize that these gestalts are not evaluated. In fact, we can define an evaluation function that evaluates each gestalt into a string. In other words, we define a function E, which evaluates each gestalt from G to a member of D. Definition 5: Let < D, N , F D , S > be the SIT-algebra and Reflect : D ! D be a function that inverts the order of letters in a string, i.e. Reflectðx1 . . . xk Þ ¼ xk . . . x1 . We write f n ðXÞ to indicate that the one-place function f is applied n times to X. The evaluation function E from the class of gestalts G to D is defined as follows: . . . .
EðXÞ ¼ X if X 2 D Eð f ½XÞ ¼ f ðEðXÞÞ EðIter½X, f , nÞ ¼ f 0 ðEðXÞÞ f n1 ðEðXÞÞ EðSyme ½XÞ ¼ EðXÞReflectðEðXÞÞ
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EðSymo ½X1 , X2 Þ ¼ EðX1 ÞEðX2 ÞReflectðEðX1 ÞÞ EðAltr ½X, f , X1 , . . . , Xn Þ ¼ f 0 ðEðXÞÞEðX1 Þf 1 ðEðXÞÞEðX2 Þ f n1 ðEðXÞÞEðXn Þ EðAltl ½X, f , X1 , . . . , Xn Þ ¼ EðX1 Þf 0 ðEðXÞÞEðX2 Þf 1 ðEðXÞÞ EðXn Þ f n1 ðEðXÞÞ EðCon½X1 , . . . , Xn Þ ¼ EðX1 Þ EðXn Þ
Below are some examples of gestalts and their evaluations to string patterns. Iter ½ab, succ, 3 Con½Iter½a, succ, 3, Iter½i, succ, 3
! abbccd ! abcijk
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! abkba ! afbkct
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! kfu
Thus, in the SIT-algebra the string abc can be analysed as having iteration structure according to which it is considered a single chunk. However, the structural operators of the SIT-algebra do not specify any regularity in the pattern kfu; its perceptual chunking therefore consists of three subchunks, k, f and u. In order to decide on the preferred gestalts of patterns, a complexity measure, called information load, is introduced. In the original version of SIT (Van der Helm 1994 and Leeuwenberg 1971, Van der Helm 1994), the information load measures the complexity of a pattern description by computing the number of occurrences of primitive elements in the description. For example, the string pattern abba can be described as the symmetry of ab and therefore the information load of this analysis is 2. A detailed discussion on this complexity measure can be found in Van der Helm (1994) and Van der Helm and Leeuwenberg (1991). The extension of the string language with domain functions changes the structure of descriptions and therefore their structural complexities. We assume that domain functions, except the identity function, increase the complexity of descriptions since patterns will be preferably perceived in terms of identical parts. In general, we assume that different functions have different perceptual relevance. This assumption reflects the prediction that there is a preference ordering among functions in terms of which string patterns are described. For example, the string pattern abab may be described as algebraic terms Con ½ab, a, b or Iter½Iter½a, succ, 2, Id, 2; the first description contains four primitive elements a, b, a and b so that its information load is equal to 4, and the second description contains one primitive element a and one domain function succ so that its information load is equal to 2. Definition 6: Let X, X1 , . . . , Xn 2 D, L : D ! N be the length function that counts the number of letters in strings, and f 2 F D be a domain function. The information load I of algebraic terms can then be determined as follows. . IðXÞ ¼ LðXÞ . Ið f Þ ¼ 1 for f 2 F D nfidg . Ið f Þ ¼ 0 for f ¼ id . Ið f ½XÞ ¼ IðXÞ þ Ið f Þ . IðIter½X, f , nÞ ¼ IðXÞ þ Ið f Þ . IðSyme ½XÞ ¼ IðXÞ . IðSymo ½X1 , X2 Þ ¼ IðX1 Þ þ IðX2 Þ P . IðAltr ½X, f , X1 , . . . , Xn Þ ¼ IðXÞ þ P ni¼1 IðXi Þ þ Ið f Þ . IðAltl ½X, f , X1 , . . . , XnP Þ ¼ IðXÞ þ ni¼1 IðXi Þ þ Ið f Þ . IðCon½X1 , . . . , Xn Þ ¼ ni¼1 IðXi Þ
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The minimum principle states that the preferable gestalt of a string pattern is determined by its algebraic SIT term that has the lowest information load. Allowing domain dependent operators may make the set of possible gestalts infinitely large. The reason is that domain dependent operators can be inverse of each other, thereby allowing a pattern to have infinitely many descriptions. For example, given succ and pred as a pair of inverse operators over the string domain, string b has infinitely many descriptions such as b, succðpredðbÞÞ, predðpredðsuccðsuccðbÞÞÞÞ, etc. However, we may consider the information load of the worse structural descriptions of a pattern, in which only Con operator and no domain dependent operators occur, as the upper limit of its information load. Consequently, the set of all possible gestalts of a pattern with an upper limit on information load becomes a finite set.
4. Formalizing proportional analogy in SIT Given that gestalts are formalized as algebraic SIT terms, we can now turn to the problem of how to characterize analogies between different gestalts—in particular, how to characterize proportional analogy relations. One obvious approach is to use the notion of structural identity between gestalts. For example, Iter½a, succ, 3 has the same structure as Iter½ p, pred, 3. This can be formalized using term unification (Siekmann 1989) between tree-like structural descriptions (gestalts). There are two ways to do it. One is to define a notion of structural template, which is essentially a gestalt containing variables and then say that two gestalts g1 and g2 in G are analogous if there exists some structural template gst , and substitutions 1 and 2 such that 1 gst ¼ g1 and 2 gst ¼ g2 . An equivalent definition is to allow inverse substitutions—so that if is a substitution replacing certain variables with terms, 1 would replace the terms with the corresponding variables—and say that two gestalts g1 and g2 are analogous if there exists substitutions 1 and 2 such that 11 g1 ¼ 21 g2 . This approach, however, seems too rigid because trivial changes in the structural description tree destroy the analogy. For example, the operator Con is associative such that agk could be written as Con½a, Con½g, k or Con½Con½a, g, k. However, the former gestalt (i.e. Con½a, Con½g, k) would be considered analogous to Con½Iter½m, id, 3, Con½Syme ½pq, w but the latter would not. To make the characterization of analogy invariant with respect to such trite changes in structural descriptions, we can allow a unification theory to be specified and used in the unification algorithm. The unification theory is essentially a set of axioms, where each axiom specifies a difference that may exist (or is allowed) between the terms that have to be unified. These differences may be concerned with, for instance, arity, recursion depth, order of arguments, etc. For example, to reflect the associativity of Con, we could include the axiom Con½X, Con½Y, Z ¼ Con½Con½X, Y, Z in the unification theory. Now both the above mentioned gestalts of agk would unify with the above gestalt for mmmpqqpw. It should be pointed out, however, that adding axioms makes the unification problem in most cases computationally intractable (Baxter 1977). Another approach to this problem is to follow the framework presented in (Indurkhya 1991, 1992). Here the source and the target domains are formalized as algebras, and the analogical relations between them are defined as correspondences over their respective subalgebras, which are relations that preserve algebraic structure. A correspondence, however, allows many-to-many relations and does
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not have any directionality. For the proportional analogy problems we are interested in modelling here, there is an implicit direction from the known elements to the unknown. Therefore, we restrict the correspondences to be homomorphisms, which are structure-preserving mappings from one algebra to another as defined below: Definition 7: A homomorphism from an algebra < D1 , F1 > to another algebra < D2 , F2 >, is a pair < , > where: 1. 2. 3. 4.
D1 D2 , ðnÞ ðF1 ðnÞ F2 ðnÞÞ for all n, where n refers to the arity of a function, whenever ðx, yÞ and ðx, y0 Þ are both in or in then y ¼ y0 , and if ða1 , b1 Þ, . . . , ðan , bn Þ 2 and ð!, Þ 2 ðnÞ then ð!ða1 . . . an Þ, ðb1 . . . bn ÞÞ 2
In particular, a homomorphism is itself an algebra. The elements of this algebra are pairs of elements, one from each participating algebra, and the operators of this algebra are pairs of operators, also one from each participating algebra. The main reason for favouring an algebraic approach is that we can use ‘generativity’ of algebras to represent and compute homomorphisms based on small subsets. To explain this, we first introduce the notions of closure and representation system. Definition 8: Given an algebra < D, F > and a pair < E, G > such that E D, GðnÞ FðnÞ for all n, the closure of < E, G > is E 0 such that E E 0 ; whenever e1 , . . . , en 2 E 0 and g 2 G, then gðe1 , . . . , en Þ 2 E 0 ; and nothing else is in E 0 . Intuitively, in taking the closure of a set of objects, given a set of functions, we include all those objects that can be generated from the given set of objects by repeatedly applying any sequence of functions from the given set. Notice that in the above definition, the pair < E 0 , G > forms an algebra; in particular, it is a subalgebra of < D, F >. Next we define a representation system for an object or a set of objects. Definition 9: Given an algebra < D, F >, a representation system is a pair < E, G > such that E D, G F, E and G are both finite. Any set of objects S D is said to be represented by < E, G > if S is a subset of the closure of < E, G >. In this case, we can also say that S is represented by < E, G >. In the above definition, if the closure of < E, G > includes D, then we say that < E, G > generates the algebra < D, G >, or < D, G > is the algebra corresponding to the representation system < E, G >. Though the concept of a representation system is defined with respect to an algebra (it is the algebra that specifies how the operators in the representation system act on the objects), we will often omit mentioning the algebra explicitly when it is obvious from the context. For instance, in this paper, we are primarily concerned with SIT algebras, so we will not mention it every time. In the following, we represent the SIT algebra < D, N , F D , S > as representation system < D, F D [ S > by omitting the natural numbers N and taking the union of domain dependent and structural operators. We assume that the analogical relation is not characterized between operators of different types, i.e. between a domain-dependent and a structural operator. Notice that, for any given set of objects of an algebra, a representation system is not unique. Also, while the notion of minimality can be easily defined for representation systems of a set of objects, a minimal representation system is also not unique. For instance, the string ‘abc’ can be generated by < fag, fsucc, Cong >, or
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by < fa, b, cg, fCong >, both of which are minimal in the sense that neither one is included in the other. The representation systems have the property that a homomorphism between two algebras can be represented as a mapping between a representation system of one algebra to a representation system of another. Using these definitions, we model proportional analogies of the form ‘A is to B as C is to D’ as a homomorphism between two algebras (each of which is a subalgebra of the SIT algebra), one containing the terms A and B, and the other containing the terms C and D. For example, consider the proportional analogy abc : abd :: ijk : ijl. The algebra corresponding to the representation system < fab, cg, fCon, succg > generates some descriptions for both abc and abd. These descriptions might be Con½ab, c and Con½ab, succ½c, respectively. Similarly, the algebra corresponding to the representation system < fij, kg, fCon, succg > generates (among others) the descriptions Con½ij, k and Con½ij, succ½k, respectively, for ijk and ijl. The proportional analogy is then represented by the following mapping between the two representation systems, which represents a homomorphism between the corresponding algebras: < fðab, ijÞ, ðc, kÞg, fðCon, ConÞ, ðsucc, succÞg >. This characterization is quite resilient because for any given pair of gestalts deemed analogous by this definition, any change in one of them resulting from the semantic properties of the domain-dependent operators can be reflected in an analogous change in the other gestalt. Now in order to compute the homomorphisms to solve proportional analogy problems, we can use a key theorem in universal algebra that says that a mapping from the representation system of an algebra into another algebra can be uniquely extended into a homomorphism from the algebra generated by the representation system to the other algebra (Cohn 1981). However, this standard theorem needs some slight adaptation here because we allow non-minimal representation systems, and subsets of algebraic functions. For this, we introduce the notion of local homomorphisms, which are functions that preserve the algebraic structure within a certain restricted set of objects and functions, and with only single applications of functions. Definition 10: Let < D1 , F1 > and < D2 , F2 > be two algebras. A relation < , > between < D1 , F1 > and < D2 , F2 > is a local homomorphism if and only if, . if ðx, yÞ and ðx, y0 Þ are both in or in then y ¼ y0 ; and . if ða1 , b1 Þ, . . . , ðan , bn Þ 2 , ð f , gÞ 2 ðnÞ, and there is y such that . ð f ða1 , . . . , an Þ, yÞ 2 then y ¼ gðb1 , . . . , bn Þ Thus, a mapping from one representation system to another that preserves the algebraic structure within the representation system becomes a local homomorphism between the corresponding algebras. Using this, we model proportional analogies of the form ‘A is to B as C is to D’ as a local homomorphism between two representation systems, one generating the terms A and B and the other generating the terms C and D. 5. Introducing constraints on gestalts Given a SIT-algebra, there are many possible representation systems generating different algebras (each of which is a subalgebra of the SIT-algebra) and, given any two representation systems, there may be many possible local homomorphisms between them. In this section, we present various factors that constrain the
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generation of representation algebras and therefore the generation of gestalts for proportional analogy problems. As mentioned in section 3, SIT assigns a complexity value called information load to each gestalt of a pattern and applies the minimum principle to select the preferred gestalt of the pattern. However, the minimum principle ignores the mutual contextualization effect in proportional analogies. This is because the lowest complexity gestalts, taken in isolation, of two individual patterns may not always be the gestalts that are perceived when they are presented together. In order to capture this context effect we introduce the notion of collective information load, which counts the information loads of the shared substructures of different patterns only once. In order to illustrate the notion of collective information load, consider proportional analogy abccba : ccabbacc :: pqrrqp : rrpqqprr. Using the information load as defined in Def. 6 and applying the minimum principle, the first pattern, namely abccba, considered out of context has the preferred gestalt g1 ¼ Syme ½Iter½a, succ, 3, but within the context of the current analogy, it would be preferable to see it as an odd symmetry structure g2 ¼ Symo ½Iter½a, succ, 2, Syme ½c, even though it has a higher information load. The reason is that g2 shares common substructures with the preferred gestalt g3 ¼ Symo ½Iter½c, id, 2, Syme ½ab of ccabbacc. The rule that if the gestalts of two simultaneously present patterns share substructures then their collective information load is lower will be called the simplicity constraint. In order to incorporate this feature (i.e. to count the information load of any shared substructure between simultaneously presented patterns only once), we define a complexity ordering on representation systems, which is determined by the number of elements in it: the more the elements, the higher is the complexity. The underlying idea here is that when two patterns have gestalts that overlap, the representation system that generates these two gestalts will have a lower complexity. For instance, the minimal representation system that generates the gestalts g1 and g3 for the first two patterns of the proportional analogy relation mentioned above is < fa, ab, cg, fsuccg >, which has four elements. But if we consider the gestalt g2 for abccba, then g2 and g3 can be generated by the representation system < fab, cg, ; >, with only two elements. For proportional analogy, there is an additional constraint, which we refer to as the projectability constraint, namely that it must be possible to construct an analogical mapping between the corresponding terms of the analogy relation. The lowest complexity gestalts of two patterns may not be appropriate for constructing analogical relations between them. For example, consider the analogy relation abccba : abcccccba :: ppqrpp : ppqrstupp. The first term, the same as in the example above, has a preferred gestalt Syme ½Iter½a, succ, 3, but this does not form any appropriate analogical mapping with the preferred gestalt for the third term (ppqrpp), namely, Symo ½pp, qr. To discover the mapping underlying this analogy, we may consider a higher information load gestalt for the first term, namely Symo ½ab, cc. Thus, to sum up this discussion, the preferred gestalts for patterns occurring in a proportional analogy relation ought to have a low information load, ought to be generated by a simple representation system (simplicity constraint) and ought to induce an appropriate analogical mapping (projectability constraint). Notice that we are saying ‘ought to’ because these three constraints interact with and influence each
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other. We will now present one heuristic approach that tries to combine these constraints. 6. Computational modelling of proportional analogy In this section, we propose a heuristic algorithm based on our formalism for solving proportional analogy problems. We first define the notion of ‘projectability’ that imposes an additional constraint on the formalism due to computational requirements. Then, in section 6.2, we present an algorithm to solve proportional analogy problems. This algorithm can be seen as an extension of the Van der Helm and Leeuwenberg’s (1986) algorithm, which computes the preferred context independent gestalts of patterns. To be precise, our algorithm extends Van der Helm and Leeuwenberg’s algorithm to compute the preferred context dependent gestalts of patterns that are presented in the proportional analogy context. Our algorithm uses a ‘test for projectability’ module to generate a mapping between two representation systems. The algorithm for this module is explained in section 6.3. We will present two simple examples in section 6.4 to illustrate the working of our algorithm. Finally, a limitation of our algorithm will be discussed in section 6.5. 6.1. Formal preliminaries: projectability In section 4, we defined proportional analogy ‘A is to B as C is to D’ as a local homomorphism between two representation systems. In this section, we define the concept of projectability that was informally explained in the previous section. Intuitively, a triple of gestalts ðt1 , t2 , t3 Þ is said to be projectable if and only if there exists a representation system that generates gestalts t1 and t2, a representation system that generates gestalts t3, and a local homomorphism from the first system to the second which, when applied to t1, yields t3. Definition 11: A triple of gestalts ðt1 , t2 , t3 Þ is said to be projectable if and only if there exists a representation system < D1 , F1 > such that t1 and t2 are in G , a representation system < D2 , F2 > such that t3 is in G , and a local homomorphism < , > from < D1 , F1 > to < D2 , F2 > such that < , > ðt1 Þ ¼ t3 . 6.2. An algorithm to solve proportional analogies We now present an algorithm to solve proportional analogies in our algebraic framework. This algorithm can be seen as an extension of Van der Helm and Leeuwenberg’s algorithm to incorporate the mutual contextualization effect and the projectability constraint. Let us assume that the sensory stimuli A, B and C of the proportional analogy problem A : B :: C : X are given, and the goal is to find the pattern X. We will utilize the simplicity and projectability criteria to direct the search process among possible gestalts of the given patterns, determine the appropriate representation systems and, finally, decide the fourth pattern X. The top-level structure of our algorithm is as follows: (1) For each of the sensory stimuli A, B and C separately, generate the set of possible gestalts (algebraic terms).1 (2) Order triples of gestalts for A, B and C according to their collective information load. This yields a list of sets of triples where elements of each set have the same collective information load.
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(3) Test for the simplicity constraint—iterate through the list of sets of triples in the order of increasing information load until the end of the list is reached; and for each set do: (a) Pick an arbitrary triple from the selected set. (b) Test the triple for projectability condition—check if there are two minimal representation systems such that: (i) A representation system S that generates gestalts of A and B. (ii) A representation system T that generates gestalt of C. (iii) A local homomorphism M between S and T exists that maps A to C. (c) If the end of the list is reached and no correspondence is found, return Fail. (4) Apply the local homomorphism M to the gestalt of B (in the triple that satisfied the projectability condition in the previous step) to get a gestalt for X. Notice that we select the possible triples first according to the simplicity criterion and then according to the projectability criterion. Obviously, the inverse order of applying these criteria is also possible but would be very inefficient. Our algorithm incorporates the assumption that those descriptions that represent the most salient features of patterns have the best chance of being the intended descriptions within the context of the proportional analogy. Moreover if the context-free lowest information load descriptions of the patterns are not appropriate for the proportional analogy, then our algorithm would choose different descriptions. Therefore, our model can construct creative as well as non-creative analogies (Indurkhya 1992). 6.3. An algorithm for testing the projectability condition As mentioned above, a triple of gestalts ðt1 , t2 , t3 Þ is considered to be projectable if we can find a representation system that generates t1 and t2, a representation system that generates t3, and a local homomorphism from the first system to the second such that it maps t1 into t3. Moreover, in keeping with the constraints introduced in section 5, we would like to keep the complexity of the two representation systems and of the mapping to a minimum. In this section, we present a heuristic algorithm for generating the two representation systems and the mapping. The algorithm returns ‘Fail’ if the triple is not projectable. In the proposed algorithm, a function called ‘projection’, starts by initializing and both to be empty sets. It then recursively examines the hierarchical structure of the three gestalts t1 , t2 and t3, starting from the outermost operator, and gradually adds the necessary elements to and . In examining the outer structure of the gestalts t1 , t2 and t3, we distinguish between four cases, each of which is shown in figure 5. In the cases shown in figures 5a and 5b, all the three gestalts are primitive terms, meaning that they all are either strings (i.e. belong to D ¼ fa, abd, . . .g) or domain-dependent operators (i.e. belong to F ¼ fsucc, pred, . . .g or natural numbers (i.e. belong to N ¼ f1, 2, . . .g). In the case shown in figure 5a, t1 and t2 are identical. In this case, we increment or by adding the mapping ðt1 , t3 Þ to it. In the case shown in figure 5b, t1 and t3 are identical, and we add the two identity elements ðt1 , t1 Þ and ðt2 , t2 Þ to or . In the cases shown in figures 5c and 5d, all three gestalts have some SIT operator at the outermost level. In the case shown in figure 5c, t1 and t2 have the same outermost operator, say f, and t3 has a different outermost operator, say g, but of the
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Figure 6. The algorithm that computes the Projection function.
(1) does not make it so that a single element (an object or an operator) is mapped to two different elements, and (2) does not violate the algebraic structure of the two representation systems. Though the first of these condition is easily checked, the second one requires the functions in the combined mapping be applied to the objects in the combined mapping to see that the algebraic structure is preserved. However, here also the fact that the mappings are added incrementally can be used to develop
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Figure 7. The algorithm that computes the Join function.
a more efficient algorithm. Figure 7 shows the join function. Here M1 and M2 are two mappings that are known to be local homomorphisms. If M1 and M2 can be combined into a local homomorphism, then their union is returned as M, otherwise ‘Fail’ is returned. Note that in this algorithm is applied to and it is checked if the local homomorphism condition is violated. However, because < 1 , 1 > is already known to be a local homomorphism, we can reduce some of the tests. So, when 1 is applied to 1, we do not need to check for it in 1, but only in 2 (which are new elements that are being added.) Similarly, when 2 is applied to 2, we do not need to check for it in 2, but only in 1. But when we apply 1 to 2, we need to check in both 1 and 2 (hence ), and when we apply 2 to 1, we need to check in both 2 and 1 (hence ). Regarding the constraints mentioned in section 5, namely that the complexity of the two representation systems and the mapping should be kept at a minimum, we would like to note that our incremental approach starts with empty systems and adds elements only as necessary to generate the given gestalts. Similarly, the mapping is
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also started an empty set, and pairs of elements are added only when necessary and as far as possible identically. This turns our approach into a kind of greedy algorithm. This is essentially a heuristic approach, which does not always guarantee an optimal solution. 6.4. Two examples In this section, we present two proportional analogy problems and show their underlying homomorphisms as computed by our projection algorithm of the last section. Consider the proportional analogy problem abc : abcd :: zyx : X. If we consider Iter½a, succ, 3, Iter½a, succ, 4, and Iter½z, pred, 3 to be the gestalts for the first three terms, respectively, then the local homomorphism induced by them is obtained as shown in figure 8. In the first step, the outer structures of the gestalts are compared as shown in figure 8a, to which the case of figure 5d applies. As a result, the local homomorphism, initialized to be ð;, ;Þ, is set to < ;, fðIter, IterÞg >, and the projection function is recursively called on three triples formed by taking the arguments in the same position from each of the three terms, as shown in figures 8b, c and d. The situations in figures 8b and c correspond to the case shown in figure 5a, and the one in figure 8d corresponds to figure 5b. Thus, the final value of the local homomorphism becomes < fða, zÞ, ð3, 3Þ, ð4, 4Þg, fðIter, IterÞ, ðsucc, predÞg >. This homomorphism maps the second element of the proportional analogy problem (i.e. abcd) to zyxw which is considered as the solution (i.e. the fourth term) of the problem. As a second example, consider the proportional analogy problem abc : abd :: iijjkk : X. Assume that the selected gestalts are Con½ab, id½c, Con½ab, succ½c, and Con½iijj, id½kk, respectively. The local homomorphism induced by these gestalts for A, B and C is recursively obtained as shown in figure 9. As in the last example, the outer structure of the three terms, shown in figure 9a, corresponds to figure 5d, and so the local homomorphism is set to < ;, fðCon, ConÞg > and the projection function is called with the two sets of triples formed by taking the first and the second arguments, respectively, from each of the three terms. The first of these calls, shown in figure 9b, corresponds to the case shown in figure 5a, and contributes the pair ðab, iijjÞ to the local homomorphism. The second call, shown in figure 9c, corresponds again to figure 5d, and makes another recursive call to the projection function with the set of triples formed by the single argument of each of the three terms. At this point, the value of the local homomorphism is < fðab, iijjÞg, fðCon, ConÞ, ðid, idÞ, ðsucc, succÞg >. In the last step, shown in figure 9d, which corresponds to the case shown in figure 5a, the element ðc, kkÞ is added to the local homomorphism, resulting in a final value of < fðab, iijjÞ, ðc, kkÞg, fðCon, ConÞ, ðid, idÞ, ðsucc, succÞg >. This homomorphism maps the second element of the proportional analogy problem (i.e. abd) to iijjll, which is considered as a solution (i.e. the fourth term) of the problem. 6.5. A limitation We would like to point out here that there is an assumption built into our formalism: namely that the same object occurring at different places in an algebraic term is the same. However, there are proportional analogies that do not satisfy this
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assumption. For example, consider the algebraic term Con½Iter½Con½a, b, id, 2, a which is a description of the first term of the proportional analogy ababa : abbaa :: cdcdg : cddcg. In order to construct a reasonable analogical mapping for this example, one must distinguish between the two occurrences of the letter ‘a’ in the mentioned algebraic term. In fact, the first ‘a’ should be mapped to ‘c’ and the second ‘a’ should be mapped to ‘g’. This is a general phenomenon that exists for proportional analogies involving visual patterns as well. In order to cover such analogies, one needs a mechanism to distinguish different occurrences of identical elements. In our formalization, one solution is to assign indices to the elements that occur at different places in algebraic terms: such indices can be generated on the basis of the positions of the elements in the algebraic term. For example, we may rewrite the above algebraic term as Con1, 1 ½Iter2, 1 ½Con3, 1 ½a4, 1 , b4, 2 , id, 2, a2, 2 . Notice that the two occurrences of the letter ‘a’ can now be distinguished. The assignment of indices to algebraic terms and the construction of algebraic correspondences are further elaborated by Dastani (1998).
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7. Conclusions and future research issues In this article, we have outlined an algebraic approach to modelling a process by which new gestalts of a perceptual stimulus can emerge in the context of proportional analogy. Our approach extends Leeuwenberg’s structural information theory in two significant ways. One is to embed it in an algebraic framework so that regularities based on domain-dependent relations are allowed to play a role in structural descriptions. The other is to modify the minimality principle so that mutual contextualization effect can be modelled. That is, when two (or more) patterns are presented together, we prefer those perceptual gestalts for the patterns that minimize the overall complexity; the gestalts of the patterns that minimize their individual complexity in isolation do not always result in the minimum overall complexity. This effect is modelled by introducing the concept of representation systems, which generate a class of patterns, and defining a measure of complexity on them. Then mutual contextualization is modelled by examining SIT-algebras for all the patterns that are to be considered together, and choosing one that is minimal and also minimizes the complexity of all the patterns together. We also formalized the notion of analogical mapping in our algebraic framework and showed how all these constraints can be integrated in an algorithm for solving proportional analogy problems. We claim that our algorithm is superior to other approaches to modelling proportional analogies in that we are able to incorporate the mutual contextualization effect, and the principles underlying our algorithm are clearly and formally explicated. We would like to make some remarks now comparing our algorithm to the other existing approaches to solving proportional analogy problems. In the early days of artificial intelligence research, Evans (1968) implemented a system for solving proportional analogy problems in the geometric figures domain. However, in Evans’ system, the representations of the figures (terms of the analogy relation) were determined first and then the mappings were computed. Thus, Evans’ system could not model the mutual contextualization effect, though he was quite aware of it. Moreover, even the context-free gestalts for individual figures were computed in a somewhat ad hoc fashion in Evans’ system. In our system, on the other hand, our main goal has been to model the contextualization effect, and also because our system is based on the structural information theory, for which a considerable empirical support has been found, we feel that it is much less ad hoc. More recently, Mitchell (1993) implemented the Copycat system, which was expressly designed to model the creativity phenomenon in proportional analogies as we discussed in the introduction. Consequently, in Copycat, representations of the terms are constructed hand in hand with the mappings, and thus the mutual contextualization effect is fully taken into account. However, many of the features of Copycat are not clearly, or formally, specified, so it is not clear how its approach can be applied to other domains. For example, consider the concept of temperature, which plays a key role in the Copycat architecture. Intuitively, the idea is that the ‘deeper’ or more cognitively appealing an analogy, the lower its temperature (which is based on an analogy with thermodynamics). However, nowhere in the Copycat architecture, or in its discussion, one finds any principles or rules or any explicit description for computing the temperature of an analogy relation. In fact, as the concept of information load in SIT-algebra can be considered analogous to Copycat’s temperature, this reveals starkly the contrast between our two approaches,
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for the focus of our research has been on explicating the principles that constrain the gestalts of a pattern, both in isolation and in context. Another point to emphasize is that our algebraic model is aimed at modelling only the input–output functionality of the human perceptual process. That is, we do not claim that people carry algebraic descriptions in their heads, or that the algorithm presented in section 6 mirrors in anyway how humans solve proportional analogy. By contrast, Copycat implicitly claims to model the human perceptual processes. Needless to say, there are many other open issues that still need to be investigated, and we would like to mention a few of those here. First of all, structural information theory, and our extension of it presented here, both consider only one-dimensional patterns. While the coding system of Leeuwenberg allows some two-dimensional regularities to be captured in one-dimensional patterns, there are many other regularities that cannot be so captured. We are currently working on extending SIT along this direction (Dastani 1998). Finally, we would like to note that, as mentioned in the introduction, our ultimate goal is to be to able model how creative insights occur in cognitive domains as well. A cognitive domain where analogies play a key role, and where creative aspects of analogy can be glimpsed, is legal reasoning (Indurkhya 1997). At this point, however, it remains an open issue whether and how our model of gestalt perception and disambiguation that we outlined in this paper would apply to creativity in cognitive domains such as legal reasoning. This, nevertheless, remains our long-term research dream. Acknowledgements We are grateful to Cees van Leeuwen and Peter van Emde Boas for many valuable discussions and suggestions during the research presented here. Note Index (1) Note that these sets are finite if we consider the information load of the worst structural description of each pattern as the upper limit of information load of its poosible gestalts. References Baxter, L. D., 1977, The complexity of unification. Technical Report Internal Report CS-77-25, University of Waterloo, Faculty of Mathematics, Ontario, Canada. Boselie, F., 1988, Local versus global minima in visual pattern completion. Perception & Psychophysics, 43: 431–445. Boselie, F., and Wouterlood, D., 1989, The minimum principle and visual pattern completion. Psychological Research, 51: 93–101. Buffart, H., Leeuwenberg, E., and Restle, F., 1981, Coding theory of visual pattern completion. Journal of Experimental Psychology: Human Perception and Performance, 7: 241–274. Chalmers, D., French, R., and Hofstadter, D., 1992, High-level perception representation, and analogy. a critique of artificial intelligence methodology. Journal of Experimental and Theoretical Artificial Intelligence (JETAI), 4(3): 185–211. Cohn, P. M., 1981, Universal algebra, revised edn (Dordrecht : D. Reidel). Dastani, M., 1998, Languages of Perception. PhD thesis, University of Amsterdam, The Nertherlands. Dastani, M., and Indurkhya, B., 2001, Modeling context effect in perceptual domains. In Proceedings of the CONTEXT’01: Third International Conference on Modeling and Using Context. Lecture Notes in Artificial Intelligence (LNAI), Springer Verlag, pp. 129–142. Evans, T. G., 1968, A program for the solution of a class of geometric-analogy intelligence-test questions. In M. Minsky (ed.) Semantic Information Processing (Cambridge, MA MIT Press), pp. 271–353.
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