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Computational Implementation of a Model of Category-Theoretic Metaphor Comprehension
Fumitaka Iwaki, Miho Fuyama, Hayato Saigo, Tatsuji Takahashi
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This paper presents an improved computational implementation of the Theory of Indeterminate Natural Transformation (TINT) for metaphor comprehension. By replacing deterministic 'hardmax' selection with a probabilistic 'softmax' approach and refining the relation-based algorithm to better utilize category-theoretic triangle structures, the authors demonstrate superior performance in data fitting, systematicity, and novelty compared to previous models.
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Fumitaka Iwaki â authored â Computational Implementation of a Model of Category-Theoretic Metaphor Comprehension
confidence 100% ¡ Computational Implementation of a Model of Category-Theroretic Metaphor Comprehension Fumitaka Iwaki
TINT â utilizes â Category Theory
confidence 100% ¡ The theory of indeterminate natural transformation (TINT) which is based on category-theoretic concepts
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confidence 90% ¡ In this study, we propose a method to replace the deterministic selection... with a probabilistic (softmax) operation.
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Abstract
Abstract:In this study, we developed a computational implementation for a model of metaphor comprehension based on the theory of indeterminate natural transformation (TINT) proposed by Fuyama et al. We simplified the algorithms implementing the model to be closer to the original theory and verified it through data fitting and simulations. The outputs of the algorithms are evaluated with three measures: data-fitting with experimental data, the systematicity of the metaphor comprehension result, and the novelty of the comprehension (i.e. the correspondence of the associative structure of the source and target of the metaphor). The improved algorithm outperformed the existing ones in all the three measures.
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- Source: https://arxiv.org/abs/2604.10035v1
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Computational Implementation of a Model of Category-Theroretic Metaphor Comprehension Fumitaka Iwaki School of Science and Technology, Tokyo Denki University Miho Fuyama College of Letters, Ritsumeikan University Hayato Saigo ZEN University Tatsuji Takahashi School of Science and Technology, Tokyo Denki University Abstract In this study, we developed a computational implementation for a model of metaphor comprehension based on the theory of indeterminate natural transformation (TINT) proposed by Fuyama et al. We simplified the algorithms implementing the model to be closer to the original theory and verified it through data fitting and simulations. The outputs of the algorithms are evaluated with three measures: data-fitting with experimental data, the systematicity of the metaphor comprehension result, and the novelty of the comprehension (i.e. the correspondence of the associative structure of the source and target of the metaphor). The improved algorithm outperformed the existing ones in all the three measures. Keywords: analogy; category theory; natural transformation; functor; theory of indeterminate natural transformation (TINT) 11footnotetext: F. Iwaki Ishizaka, Hatoyama, Hiki, Saitama, 350-0394, JAPAN E-mail: 25udj01@ms.dendai.ac.jp 22footnotetext: M. Fuyama 56-1, Toujiin Kitamachi Kita-ku, Kyoto-Shi, Kyoto, 603-8577, JAPAN E-mail: mifuyama@fc.ritsumei.ac.jp 33footnotetext: H. Saigo 3-12-11, Shinjuku, Zushi, Kanagawa, 249-0007, JAPAN E-mail: hayato_saigo@zen.ac.jp 44footnotetext: T. Takahashi E-mail: tatsuji.takahashi@gmail.com 1 Introduction The creativity of the human brain is exemplified by the ubiquity and centrality of metaphor in our thoughts [2]. The recent, dramatic progress of artificial intelligence has not been, however, due to replicating the flexibility in human imagination. The intelligence of computers has been acquired by absorbing a vast number of problems solved in the past. On the other hand, the key of human intelligence would be to find patterns where there seems to be no structure. Modern artificial intelligences has not yet achieved finding correspondences or metaphors in and between the complex physical and psychological domains. Understanding metaphor creation and comprehension processes is thus important for revealing the keys for flexible cognition. Category theory, which can mathematically formulate structureal correspondences, has been applied in various fields outside mathematics in recent years (see e.g. [4]). Some researchers expect that the tools of category theory such as functors and natural transformations, can contribute to elucidating the mechanisms of deft transfer learning and analogical reasoning of humans [8]. The theory of indeterminate natural transformation (TINT) which is based on category-theoretic concepts was proposed as a hypothesis for exploring the process of metaphor comprehension and creating meanings [5]. Ikeda et al. computationally implemented TINT as algorithms and verified how well it describes human metaphor interpretation data [8]. TINT takes an analogy position for metaphor comprehension [7], considering a comprehension of metaphor as building a mapping from the semantic structure of the source of a metaphor to that of the target. A representative model and algorithm of this position is structure-mapping theory [6] and structure-mapping engine [3], respectively. While they assume the well-organized knowledge structures (tree structures with labeled predicate relations) for the analogical reasoning (see [1] for a criticism), TINT merely assumes a network structure of association, which is even âweakerâ than the vector space embeddings of words, hence is potentially more general. In TINT, the meaning of a word (in general, an âimageâ) is represented by a âcoslice categoryâ which is the local associative structure around the image. The comprehension process of a metaphor such as âButterflies are dancersâ is described by the interaction between the coslice categories of the two images, âbutterflyâ (target) and âdancerâ (source). In the implementation by Ikeda et al., the comprehension-correspondences between images are searched deterministically (using the argmax operator), which is not realistic considering the nature of association and multistability of human output. Also, their implementation had some problems in terms of the type of the output correspondences that made the interpretation of the results difficult and the comparison by quantitative evaluation impossible. In this paper, we introduce new algorithms implementing TINT that solves these problems. As for the evaluation of the algorithms, we test the data-fit according to the data and methods introduced in Ikeda et al. We evaluate the algorithms in terms of a kind of systematicity of the comprehension-correpondences and the novelty of them. 2 Theory of Indeterminate Natural Transformation 2.1 The Concept of TINT TINT is the theory that describes the process of understanding metaphors as the interaction between semantic structures of the source and the target [5]. In TINT, the meaning of the images is the whole relationship around the images. The theory describes the interaction between these structures of images. In this case, the creation of a new meaning is considered to be the creation of new associative relationships between images. This corresponds to the process of understanding a new metaphor in which the source and the target, which were not related before, are related by the metaphorical relationship. For example, when the metaphor âButterflies are dancersâ is given, the association from âbutterflyâ to âdancerâ appears. Then the meaning of âbutterflyâ is updated by this change. In this way, understanding a new metaphor can cause creation of a new meaning. TINT attempts to better formalize the interaction between the meanings as the local structures around the images. 2.2 Basic Concepts of Category Theory Here we introduce four basic notions of category theory used in formalizing TINT: category, functor, natural transformation, and coslice category111For an illustrated exposition of the category-theoretic concepts and their importance in cognitive science, see [5]. A category C consists of objects objâ()obj(C) and arrows arrâ()arr(C). Each arrow fâarrâ()f (C) has the domain domâ(f)=Xâobjâ()dom(f)=X (C) and the codomain codâ(f)=Yâobjâ()cod(f)=Y (C), and denoted as f:XâYf:Xâ Y. Arrows f,gf,g can be composed to gâfg f under some condition (codâ(f)=domâ(g)cod(f)=dom(g)), and satisfy the associative and unit laws (the units for composition are identity arrows such as 1X1_X which exist uniqutely for each object X). A functor F:âF:C is a structure-preserving mapping between category C and D, which maps each object/arrow in C to the corresponding object/arrow in D, satisfying three conditions: (1) It maps f:XâśYf:X Y in C to Fâ(f):Fâ(X)âśFâ(Y)F(f):F(X) F(Y) in D. (2) Fâ(fâg)=Fâ(f)âFâ(g)F(f g)=F(f) F(g) for any composable pair of f,gf,g in C. (3) For each X in C, Fâ(1X)=1Fâ(X)F(1_X)=1_F(X). Let F,GF,G be functors from category C to category D. A natural transformation Ď from F to G satisfies the following conditions: (1) Ď maps each object X in C to the corresponding morphism ĎX:Fâ(X)âśGâ(X) _X:F(X) G(X) in D. (2) For any f:XâśYf:X Y in ,ĎYâFâ(f)=Gâ(f)âĎX.C, _Y F(f)=G(f) _X. Coslice category X , where X is an object in C, is defined as follows. Objects in X are morphisms f,g,âŚf,g,... in C from X (domâ(f)=domâ(g)=âŻ=Xdom(f)=dom(g)=¡s=X). A morphism between two objects f1:XâśX1f_1:X X_1 and f2:XâśX2f_2:X X_2 is the triple of (f1,f2,g)(f_1,f_2,g), where g:X1âśX2g:X_1 X_2 and f2=gâf1f_2=g f_1. In other words, the objects and arrows of X are arrows from X and commutative triangle diagrams, respectively. 2.3 Modeling TINT via Category Theory TINT defines two categories, the latent category of images, C, and the elicited category of associations. The latent category is a thin category (up to one arrow from an object to another) with an additional structure which is that each arrow f has a weight Îźfâ[0,1] _fâ[0,1]. As stated in Experiment, it can be a weighted complete directed graph on images. The elicited category is also a normal thin category with no weight. First, we define the category of images C as the semantic network where metaphor comprehension happens. For an object X in C, the âmeaningâ of X is represented by the coslice category X . We model the process of metaphor comprehension as exploration of functors (structure-preserving mappings) between coslice categories (the meanings of the target and the source of a metaphor) using natural transformations, as follows. Here we take as example of comprehension a metaphor âButterflies are dancersâ as in Figure 2 Figure 1: Schematic diagram of natural transformation search in TINT. Based on the canonical functor derived from the occurrence of association f, new functors such as F1F_1 and F2F_2 are searched as the construction of natural transformations Ď1 _1 and Ď2 _2, respectively. a. The metaphor âButterflies are dancersâ is given, and the association from image âbutterflyâ to âdancerâ occurs (an arrow f:ââbutterflyâ˛âââdancerâ˛f:`butterfly â`dancer is added to C, if it did not exist). Category C may change by this arrow f, and we denote it by â˛C . b. Due to f, the unique functor f\â˛f from B\â˛B to A\â˛A is created. It maps an arrow bib_i in B\â˛B to biâfb_i f in A\â˛A (composition functor) and is called the base-of-metaphor functor (BMF). It is that, from associations âDancer â Nightâ and âDancer â Danceâ, BMF creates âButterfly â Dancer â Nightâ and âButterfly â Dancer â Danceâ BMF is not clear in the interpretation nor non-trivial as an interpretation of a metaphor. c. Search for a new functor modeling a more natural metaphor interpretation by constructing a natural transformation from BMF. The two actual exploration procedures are described in the next section. This is equivalent to searching for a relationship (association) which corresponds to âButterfly â Dancer â Danceâ, such as âButterfly â Flyâ, and the correspondence from the former to the latter is a component of the natural transformation Ďb1 _b_1. This leads to the discovery of a new functor F which is defined by Ď:BMFâF :BMF F. d. F represents a correspondence such as âto dance for dancer is to fly for butterflyâ and âstage for dancer is sky for butterflyâ. As a result, images directly associated with B and that of A correspond to each other, and the meaning of the metaphor is interpreted. Figure 2: The specific example of an exploration in TINT, beginning with âA butterfly is like a dancer.â 2.4 TINT Algorithms [8] proposed two algorithms that implement TINT. One is an object-based algorithm which considers only the objects of the coslice category and corresponds only the objects of each coslice category. It does not consider the relationships (the commutative arrows of the coslice category). The other is relation-based in that it performs correspondence considering the commutative triangle structure. The triangle structure is a triplet which is an arrow of the coslice categories. These algorithms require the associative probabilities between images. Ikeda et al. conducted cognitive experiments asking people about the associative strength between images and set the results as the weights on the arrows in the latent category C. In the following, the images associated from a certain image X (that become the codomains of the arrows of the coslice category X ) the initial images. 2.4.1 The Object-based Exploration Figure 3: Construction of a natural transformation (at the same time a functor) by the object-based method Figure 4: A correspondence between images via the objective method Here we explain the algorithm for exploring natural transformation/functor based directly on the associations between images. The algorithm is described in four steps (from 3-a to 3-d) as in Figure 3. 3-a Set the initial state of the coslice categories representing the meaning of the source and the target. List the images associated from the source B as B1,B2,B3,âŚB_1,B_2,B_3,... and set them as the initial images of B (denoted as BiB_i). Excite the arrows from B to BiB_i bi:BâBib_i:Bâ B_i and set them as the initial state. Similarly, set the initial state (images) for the target A. Here, the associative relationship âA is Bâ occurs, and the arrow f:AâBf:Aâ B from A to B is elicited. 3-b Construct the trivial functor BMF:B\â˛âA\â˛BMF:B â A , from the coslice category of the source to that of the target. Due to the elicitation of the arrow f, f and bib_i are composed, and the composite arrow biâf:AâBib_i f:Aâ B_i from A to BiB_i occurs. In general, BMF:biâźbiâfBMF:b_i b_i f. 3-c Explore a new functor by constructing a natural transformation Ď from BMF. Consider to where B1B_1 is mapped by the new functor, as an example. Acquire the association weights Îź11,Îź12,Îź13,⌠_11, _12, _13,... from B1B_1 to AiA_i (i=1,2,3,âŚi=1,2,3,...) from the latent category. Each arrow from B1B_1 to AiA_i is elicited at the probability of Îźi _i. Among the elicited arrows, select the arrow with the largest Îź1âi _1i. If Îź11 _11 is the largest, the new functor maps the arrow a1:AâA1a_1:Aâ A_1 to b1b_1, and is adopted as an element Ďb1 _b_1 of the natural transformation Ď from BMF to F. If no arrow from B1B_1 to AiA_i is elicited, then the arrow b1b_1 is mapped to nothing (and F is a partial functor). 3-d Carry out this operation for the remaining initial images of the source B2,B3,âŚB_2,B_3,.... Suppose b2b_2 correspond to a2a_2 and b3b_3 to a3a_3. In this case, F is a functor with Fâ(bi)=aiâ(i=1,2,3)F(b_i)=a_i~~~(i=1,2,3). If there is a natural transformation Ď:BMFâF :BMFâ F, it means that F is the metaphor is obtained. F maps the objects bi:BâBib_i:Bâ B_i in B\â˛B (note that they are arrows in C) to the objects ai:AâAia_i:Aâ A_i in A\â˛A . The content of the metaphor comprehension is âBiB_i for Bâ is âAiA_i for Aâ (for i=1,2,3i=1,2,3). 2.4.2 The Relation-based Exploration Figure 5: The algorithms of the triangle method Figure 6: The distance between the triangle structures The relation-based algorithm constructs a mapping from an arrow in the coslice category (B\â˛B ) to another (A\â˛A )222The previous relation-based algorithm proposed in [8] output multiple functors, not a single functor. It made it difficult to compare the algorithm with the object-based one or others. To address this issue, we developed a more natural relation-based algorithm here. . The arrows in coslice categories are not just arrows in C which is an association, but the commutative triangle structures (triplets). The mapping is searched using the distance between an arrow (triangle) in B\â˛B and arrows (triangles) in A\â˛A . It is that the local associative structures of the meaning of the target and source are considered, unlike the object-based method that ignores any structure. The algorithm again is described in four steps that correspond to Figure 5-a to 5-d. 5-a Set the initial state of the coslice categories representing the meaning of the source and the target, in the same way as the object-based method (3-a). Then, all the arrows between the initial images for each coslice category are elicited. 5-b Construct BMF as in the object-based method (3-b). 5-c Explore a functor through a natural transformation from BMF, based on the triangle structures. Consider the triangle structure formed by the arrows b1,b2b_1,b_2, and b4b_4 on the source side. The associative probabilities in the latent category determine whether the arrows from B1,B2B_1,B_2 to AiA_i are excited. All possible triangle structures that can be constructed from AiA_i with an arrow from B1B_1, and AiA_i with an arrow from B2B_2, are the candidates. Among them, adopt the triangle structure with the smallest difference in associative probabilities between the corresponding arrows. In the example of Figure 5, the difference in the triangle structure is d=âi=1,2,4(ÎźbiâÎźai)2d= _i=\1,2,4\( _b_i- _a_i)^2. If the difference d with the triangle structure formed by a1,a2,a4a_1,a_2,a_4 is the smallest, b1b_1 corresponds to a1a_1 and b2b_2 corresponds to a2a_2. 5-d Perform this operation for all the triangle structures on the target side. In the example of the Figure 5, there are 3Ă2=63Ă 2=6 structures (as much as the ordered pairs of initial imgaes of the source). If we follow the above procedure for all the triangle structures for the source, it is possible that a source side arrow bib_i has correspondences with multiple arrows on the target side. To determine the correspondence uniquely, compare the difference d of the triangle structure when the correspondence is chosen, and adopt the smallest one as an element of the natural transformation from BMF to F. Finally, we obtain the functor F which is the interpretation of the metaphor. 2.5 Introducing Stochastic Correspondence In this study, we propose a method to replace the deterministic selection of elements of natural transformations in the simulations of the object-based and the relation-based in Ikeda et al, with a probabilistic (softmax) operation. It is because the selection should reflect the (coslice category) structures that are supposed to represent the meaning of the target and source, while the deterministic selection ignores most of the information of the structures. In the object-based method, we selected the arrow with largest weight, Îź, out of the arrows, excited from the objects of the vehicleâs coslice category to that of the targetâs coslice category in Figure (3-c), as elements of natural transformations based on the associative probabilities. We replaced this part with a method that stochastically selects elements by a softmax function with each Îź as input, instead of selecting the one in the âgreedyâ way. In the relation-based method, we adopted the triangle structure with the smallest difference d as a candidate for natural transformations in Figure (5-c), using d in the same softmax selection as the object-based method. 3 Experiment We tested the new TINT algorithms in the same environment as the simulations by [8]. We chose the metaphor âButterflies are dancersâ from the metaphor stimulus-interpretation set by [11]. We define the association weights in the latent category by the data collected by Ikeda et al. in a cognitive experiment. In this experiment, participants answered the the associative strength of âA associates Bâ for all pairs of images on a 5-point scale (1: not associated, ⌠5: strongly associated). After that, the association strength s was converted to the weight Îź by Îź=0.05+0.225â(sâ1)Îź=0.05+0.225(s-1). We also used the metaphor interpretation data collected by Ikeda et al. to evaluate the metaphor comprehensions constructed by the TINT algorithms. The metaphor interpretation data is the data in which people answered the associative strength of how much they agree with the metaphor of the image pair. We conducted 10,000 simulations of the correspondence from the source initial images to the targetâs initial images begining with the metaphor. We compared the simulation results between the previous two algorithms by Ikeda et al. as âhardmaxâ and the proposed method as âsoftmaxâ. Figure 7: The association weights assigned to the latent category. Only the weights from the initial images of the source to that of the target are shown. 4 Result We compare the four metaphor comprehension algorithms (relation-based or object-based) Ă (deterministic/hardmax or probabilistic/softmax) in three measures: data fit, systematicity of the comprehension, and the novelty of the comprehension. All the results are shown in Figure 8, with the free parameter β (inverse temparature) for the softmax method as a variable on the x-axis. 4.1 Data Fit We calculated the rank correlation coefficient between the metaphor interpretation data and the simulation results for each algorithm. We ranked the initial images of the target by the number of correspondences by TINT for each initial image of the source. For the metaphor interpretation data, we made a ranking in a similar way, according to the responses by the participants. We calculated the rank correlation coefficient between them.The results are shown in Figure 8 as the gray lines. The relation-based softmax (gray solid line) shows better datafit than the object-based (gray broken line) or the two methods by Ikeda et al. (gray triangle and circle) for most of the range for β. 4.2 Systematicity (the Width of the Functors) We considered it is important that the width of the functors as the metaphor comprehension, as the width corresponds to the systematicity in structure-mapping theory [6]. We defined the width of the functors as the number of objects in the targetâs coslice category that were mapped to. In other words, the width is the size of the range of the functor (where, the bigger the size is, the closer the functor is to an injective mapping). The number of objects in the coslice category is the number of initial images. The maximum width of the functors is 8. The results of comparing the average width for each trial of the simulation are shown in Figure 8 in orange. The width of the functor constructed with the relation-based methods (solid orange curve or orange triangle) exceeded that with the object-based. 4.3 The Novelty of the Metaphor Comprehension We examined the novelty of the metaphor comprehension output of TINT algorithms by comparison with the word embeddings of the images. If an algorithm can acquire a metaphor comprehension correspondence between pairs of images with lower cosine similarity on the word embeddings, we considered that the comprehension has a relative novelty. We used word2vec for the word embeddings [10, 9]. We employed the word embedding model that was pre-trained on the full text of the Japanese Wikipedia [12] to obtain the distributed representations of the images used in the experiment. Since the two words âbeauty (utsukushisa)â and âtransience (hakanasa)â were not included in the dictionary of the model, we adopted âbeautiful (utsukushii)â and âtransient (hakanai)â instead. Next, we calculated the cosine similarity between the images. We show the results in Figure 8 in blue. This is a discriminative index that is opposite to the correlation with the metaphor interpretation data. The relation-based methods showed a much lower correlation coefficient with the word embeddings than the object-based, which means that the relation-based can produce more novel metaphor comprehension. Figure 8: The output of the algorithms evaluated in 1. data fit (gray), 2. systematicity (orange), and 3. novelty (blue). The β value along the x-axis is the inverse temperature parameter for the softmax methods. The circles and triangles in three colors are of the algorithms proposed in Ikeda et al. (that adopted the absolute errors, instead of the square errors for the distance between triangle structures, d). 5 Discussion In Figure 8, we can see that the proposed softmax method is closer to the metaphor interpretation data than the conventional method hardmax, both in the object-based and the relation-based exploration methods. In hardmax, most of the correspondences are made with the pairs, initial images of the source and that of the target, that have high association weights. For example, the initial images of âdancerâ and âbutterflyâ have a common image of âwomanâ as shown in Figure 7 (the second column and sixth row). In this experiment, the associative probability of the identity arrow is set to 1, thus âwomanâ is always corresponded to âwomanâ in the object-based hardmax method. The softmax function in the proposed method could probabilistically disperse the correspondences even in such cases. We considered that the our method is closer to the metaphor interpretation data due to this feature. In other words, we suggest that the metaphor comprehension process in humans is based on more probabilistic procedures. The relation-based method tends to correlate better with humans metaphor interpretation. In terms of the treatment of analogy and metaphor on structure-mapping theory by Gentner et al., the object-based exploration loosely corresponds to the mapping between attributes, and the relation-based to the mapping between relations, respectively [6]. We compared the broadness of the destination of the objects in the coslice category by the functors. To express flexible metaphors of humans, the correspondence from initial images of a source to more initial images of a target preferred to that of to particular initial images of a target. We see in Figure 8 that the relation-based output consistently broader functors compared to the object-based. Finally, we examined the novelty of the obtained metaphor comprehension using word embeddings. Figure 8 shows that the relation-based methods gave far lower correlation with the word embeddings overall. The results suggested that the relation base could acquire more novel metaphor comprehensions compared to object-based. 6 Conclusion In this paper, we simplified and generalized the metaphor comprehension algorithms based on TINT.We proposed two new evaluation indices for comparing the methods, systematicity and novelty of the metaphor comprehension outputs. The new algorithms proposed in this paper showed better datafit, higher systematicity, and higher novelty. References [1] D. J. Chalmers, R. M. French, and D. R. Hofstadter (1992) High-level perception, representation, and analogy: A critique of artificial intelligence methodology. Journal of Experimental and Theoretical Artificial Intelligence 4 (3), p. 185â211. External Links: ISSN 13623079, Document Cited by: §1. [2] N. 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