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© Paul Thagard and Keith J. Holyoak, 1997
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| Introduction |
| Similarity, Structure, Purpose |
| Analogical Thinking in Everyday Life |
| Computational Models of Analogical Coherence |
| Future Directions |
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Abstract
We examine the use of analogy in human thinking from the perspective of a multiconstraint theory, which postulates three basic types of constraints: similarity, structure and purpose. The operation of these constraints is apparent in both laboratory experiments on analogy and in naturalistic settings, including politics, psychotherapy, and scientific research. We sketch how the multiconstraint theory can be implemented in detailed computational simulations of the analogical human mind.
Many parents know that young children take comfort in getting a kiss on an injury to "make it better". Little Aaron, aged 24 months, would routinely come to his mother saying things like, "I bump my head. Kiss it." But one morning, for the first time ever, the tables turned. While his mother was dressing him, she realized she had a bruise on her hand. Without really thinking she said, "Ow, my hand hurts." Aaron immediately responded, "I kiss it." His mother then put her hand in front of Aaron's face and received a kiss from him. (1)
Aaron's reaction provides a small example of thinking by analogy: trying to reason and learn about a new situation (the target analog) by relating it to a more familiar situation (the source analog) that can be viewed as stucturally parallel. Aaron's source is the knowledge that when he had been hurt in the past, his mother's kiss had made it better; this source is now evoked by the target situation of his mother's bruised hand (the access or retrieval step in analogy use). The child goes on to find the correspondences between the source and target (mapping step). Note that he does not simply use the superficial mapping of his mother to herself (if he had, he would have simply told mom to kiss her own hand!). Rather, Aaron maps his mother to himself (for she is now the injured one), and himself to her. Based on these mappings, he finds a solution to the target problem: his kiss will ease her pain (inference step). Although we don't know for sure, it is quite possible that Aaron's use of analogy also led him to learn something more general, a kind of abstraction of the commonalities shared by the source and target (learning step). Roughly, he may have induced a schema or rule along the lines, "If a person is injured, a kiss from a loved one will ease the pain." Our description of Aaron as analogizing from treatment of his own injury to treatment of his mother's assumes that he had not previously formed this general rule.
Aaron at age two had an analogical mind. The remarkable thing about this example of a child's reasoning is that it is not especially exceptional. Young children, before they enter school, without any specialized tutoring from their parents or elders, develop a capacity for analogical thinking (e.g., Gentner, 1977; Goswami & Brown, 1989; Inagaki & Hatano, 1987; Holyoak, Junn, & Billman, 1984). The analogical mind is simply the mind of a normal human being. Indeed, to a limited but impressive degree, it is the mind of at least a few other primates, most notably chimpanzees that have received extensive training in symbol manipulation (Gillan, Premack & Woodruff, 1981). Analogical thinking can be traced from these early phylogenetic and ontogenetic beginnings to an extraordinarily diverse range of uses by human adults, including generation of metaphors for the self, decision making in politics, business and law, and scientific discovery.
Our aim in this paper is to provide an overview of analogical thinking from a perspective we have termed the multiconstraint theory (Holyoak & Thagard, 1989, 1995). As its name implies, the multiconstraint theory assumes that people's use of analogy is guided by a number of general constraints that jointly encourage coherence in analogical thinking. We will first describe these constraints in qualitative terms, illustrating them with examples from psychological studies. We will then survey additional examples of naturalistic uses of analogy that can be understood in terms of the constraints. Finally, we will discuss approaches to implementing the multiconstraint theory in computational models that simulate the human analogical mind. (For a more thorough discussion of these issues see Holyoak & Thagard, 1995.)
Three broad classes of constraints form the basis of the multiconstraint theory. Each of these constraints can be illustrated with young Aaron's analogy. First, the analogy is guided to some extent by direct similarity of the elements involved. We noted that Aaron did not simply map his mother to herself (which would have maximized one local similarity between mapped objects). However, the analogy clearly depended on similarity of key relations between objects: the source and target both involve an injury sustained by a loved one. In general, similarity of concepts at any level of abstraction contributes to analogical thinking, particularly in the initial access step (e.g., Keane, 1986; Ross, 1989; Seifert, McKoon, Abelson, & Ratcliff, 1986).
Second, the analogy is guided by a pressure to identify consistent structural parallels between the roles in the source and target domain (Gentner, 1983). The key structural constraint underlying analogical mapping and inference is a pressure to establish an isomorphism -- a set of consistent, one-to-one correspondences -- between the elements of the source and target. Thus once Aaron had decided to place the source and target "injuries" into correspondence (based on similarity of relations), structural consistency required that the person injured in the source (child) be mapped onto the person injured in the target (mother), because each is playing the same relational role. In this case the constraint of maintaining consistent correspondences apparently dominated the rival similarity constraint, which by itself would encourage mapping the mother to herself. In the subsequent inference stage, consistency further required that it be the child in the target (now mapped to mother in the source) who provides the soothing kiss.
Third, the constraint of purpose implies that analogical thinking is guided by the reasoner's goals -- what the analogy is intended to achieve. Why did Aaron even consider the analogy with the kissing ritual? It appears that his mother's expression of pain gave rise to the goal of alleviating it; this goal in turn caused the child's attention to focus on those aspects of the target situation that were relevant to achieving a solution. Once his attention was biased so as to favor goal-relevant aspects of the situation, Aaron was led to access source analogs involving injuries, rather than (for example) earlier instances of being dressed by his mother.
These three kinds of constraints -- similarity, structure, and purpose -- do not operate like rigid rules dictating the interpretation of analogies. Instead they function more like the various pressures that guide an architect engaged in creative design, with some forces converging, others in opposition, and their constant interplay pressing toward some satisfying compromise that is internally coherent. When we describe computational models of analogy, we will suggest how such local contradictions between constraints can be resolved by a process of constraint satisfaction. First, however, we will briefly review some examples of experimental tests that reveal the operation of the constraints in the analogical thinking of college students.
The analogy between the Persian Gulf War and World War II, which in 1991 figured prominently in debates about whether the United States should make a military response to Iraq's invasion of Kuwait, provided a striking historical example of the role of analogy in shaping public opinion. In addition, this analogy illustrates the interactions between the multiple constraints that guide construction of a mapping. During the first two days of the counterattack in January 1991, Spellman and Holyoak (1992) asked a group of undergraduates at the University of California, Los Angeles, a few questions to find out how they interpreted the analogy between the Persian Gulf situation and World War II. The two situations were by no means completely isomorphic; in fact, the analogy was messy and ambiguous. Similarity at the object level favored mapping the United States of 1991 to the United States of World War II, simply because it was the same country, which would in turn support mapping Bush to Roosevelt. On the other hand, the United States did not go to war until it was bombed by Japan, well after Hitler had marched through much of Europe. One might therefore argue that the United States of 1991 mapped to Great Britain of World War II, and that Bush mapped to Winston Churchill (because Bush, like Churchill, led his nation and Western allies in early opposition to aggression). However, other relational similarities supported mappings to the United States and Roosevelt; for example, the United States was the major supplier of arms and equipment for the Allies, a role parallel to that played by the United States in the Persian Gulf situation. These conflicting pressures made the mappings ambiguous.
The pressure to maintain structural consistency -- a central component of our multiconstraint theory -- implies that people who mapped the United States to Britain should also tend to map Bush to Churchill; whereas those who mapped the United States to the United States should instead map Bush to Roosevelt. Notice this is not a logical requirement -- a person might think the United States should map to itself, but Bush should map to Churchill because each was the dominant leader of the war effort. Indeed, nothing prevented people from giving both mappings as answers. However, the multiconstraint theory predicts that people should prefer one-to-one mappings -- Bush to either Churchill or Roosevelt, but not to both -- and mappings that maximize structural consistency, by keeping leaders and their countries together.
At the same time, the multiconstraint theory allows the possibility of mappings that violate the one-to-one constraint when enough evidence favors multiple mappings. For example, several European nations (Austria, Czechoslovakia and Poland) were targets of German aggression prior to the outbreak of World War II, and Kuwait might be mapped to more than one of them. Similarly, Saudi Arabia played a role somewhat similar to that played by Great Britain in World War II (staging area for the counterattack, target of missile attacks) and also somewhat similar to that played by France (under threat from Germany at the time Britain responded to the invasion of Poland).
The undergraduates were asked to suppose that Saddam Hussein was analogous to Hitler. Regardless of whether they thought the analogy was appropriate, they were then asked to write down the most natural match in the World War II situation for Iraq, the United States, Kuwait, Saudi Arabia and George Bush. For those students who gave evidence that they knew the basic facts about World War II, the majority produced mappings that fell into one of two patterns, as depicted in Figure 1. Those students who mapped the United States to itself also mapped Bush to Roosevelt; these same students also tended to map Saudi Arabia to Great Britain. Other students, in contrast, mapped the United States to Great Britain and Bush to Churchill, which in turn (by the one-to-one constraint) forced Saudi Arabia to map to some country other than Britain (usually France). The mapping for Kuwait (which did not depend on the choice of mappings for Bush, the United States, or Saudi Arabia) was usually to one or two of the early victims of Germany in World War II, usually Austria or Poland (or to a grouping such as "countries Hitler took over").
Figure 1. A bistable mapping: If Bush is FDR (Franklin Delano Roosevelt) then the US-'91 (United States during the Persian Gulf War) is the US-WW2 (United States during World War II) and Saudi Arabia is Great Britain (dotted lines); if Bush is Churchill, then the US-'91 is Great Britain and Saudi Arabia is France (dashed lines). In either case, Kuwait maps to Poland and/or Austria (solid lines). From Spellman and Holyoak (1992). Copyright (1992) by the American Psychological Association. Reprinted by permission.
The analogy between the Persian Gulf situation and World War II thus generated a "bistable" mapping: people tended to provide mappings based on either of two coherent but mutually incompatible sets of correspondences. Spellman and Holyoak (1992) went on to perform a second study, using a different group of undergraduates, to show that people's preferred mappings could be pushed around by manipulating their knowledge of the source analog, World War II. Because many undergraduates were lacking in knowledge about the major participants and events in World War II, it proved possible to "guide" them to one or the other mapping pattern by having them first read a slightly biased summary of events in World War II. The various summaries were all historically "correct", in the sense of providing only information taken directly from history books, but each contained somewhat different information and emphasized different points. Some versions emphasized the personal role of Churchill and the national role of Britain; other versions placed greater emphasis on what Roosevelt and the United States did to further the war effort. After reading one of these summaries of World War II, the undergraduates were asked the same mapping questions as had been used in the previous study. The same bistable mapping patterns emerged as before, but this time the summaries influenced which of the two coherent patterns of responses students tended to give. People who read a "Churchill" version tended to map Bush to Churchill and the United States to Great Britain, whereas those who read a "Roosevelt" version tended to map Bush to Roosevelt and the United States to the United States. Even summaries that had been written to support a "crossed" mapping (for example, making Churchill the most important leader but the United States the most important country) tended instead to produce one of the two patterns in which the mapping kept the leader and his country together. It appears that even when an analogy is messy and ambiguous, the constraints on analogical coherence produce predictable interpretations of how the source and target fit together.
The research we have described so far demonstrates the impact of both structure and similarity on mapping. What about the purpose of the analogy? It is generally accepted that people seek and use analogies to achieve their goals. However, it has been less clear whether the purpose can actually change the mappings that people generate, rather than just the initial selection of a source or the later adaptation of a solution. One way to investigate this issue is to have people draw analogies between situations for which the mapping is ambiguous, and then see if their goals will alter people's preferred mappings. Spellman and Holyoak (in press, Experiment 3) performed an experiment of this sort in which college students were asked to map the characters in two soap-opera plots. The students were told to pretend that they were writers of a successful new soap opera, and that they were in court trying to prove that writers from another soap opera had stolen their ideas. Each soap opera involved the entanglements of multiple characters. In the first soap opera, set at a university, an ex-alcoholic professor named Peter was in love with his research assistant, Mary, and had cheated his brother out of his inheritance. These characters were connected by three types of relations: professional (Peter was Mary's boss), romantic (Peter was in love with Mary), and inheritance (Peter cheated his brother). The second soap opera was set in a city, and involved two fairly distinct sets of characters. The "lawyer set" included Nancy, an ex-addict entertainment lawyer, and John, a young lawyer working at her law firm who had often filled in for her. The "doctor set" included David, a prominent physician who had become an alcoholic, and Lisa, an intern who was now treating most of David's patients. Nancy and David were half-siblings and John and Lisa were cousins. Both pairs had aging relatives ready to leave them money in a will; in one version of the story Nancy and Lisa (the women) cheat David and John (the men), respectively, out of their shares of the inheritance, and in the other version the men cheat the women out of their shares. From this description the object mappings are ambiguous; for example, if the women are the cheaters then Peter seems to map equally to Nancy and Lisa.
To manipulate the purpose of using the analogy, the students were told that the judge in the plagiarism trial wanted them to predict what would happen in the next episode of the city soap opera. If they could figure out who would do what to whom in the episode, this would be solid proof that the writers of the city soap opera had plagiarized ideas from the university soap opera. Half the students were told that the crucial episode was "just like" what had happened in an episode of the university soap opera in which Peter had tried to steal credit for Mary's ideas. For these students, the professional relations between the characters were therefore most important for the plot development. The other half of the students were asked to predict an episode "just like" one in which Peter had tried to seduce Mary, in which case the romantic relations were critical. The inheritance relations did not play any direct role in either of the two episodes. After they had written extensions of the plot, all the students were directly asked to select the best match for each character in the university soap opera from among the characters of the city soap opera. Thus the experiment measured people's preferred mappings both indirectly by which characters were used to extend the plot, and directly by the mapping task.
Table 1. Optimal mappings for the source characters based on pragmatic manipulation and gender of cheater in the Inheritance relation
| Professional plot extension | Romantic plot extension | |||||
| Gender of cheater | Gender of cheater | |||||
| Male | Female | Role | Source characters | Role | Male | Female |
David Lisa |
Nancy John |
Boss Underling |
Peter Mary |
Persuer Persued |
John Nancy |
Lisa David |
(from Spellman & Holyoak, in press, Experiment 3). Adapted with permission of Academic Press.
So, which characters in the city soap opera correspond to Peter the professor and Mary his assistant? Without taking the goal into account, the mapping is actually four-ways ambiguous, as schematized in Table 1. The basic ambiguity is that Peter is somebody's boss, as are Nancy and David, and he pursues someone, as do John and Lisa. But consider how the mapping would be expected to shift if people place greater weight on the relations that are most pragmatically central in extending the plot to predict the crucial new episode. If the episode hinges on the professional relations, then Peter will seem more like Nancy or David (the bosses) than like John and Lisa (the underlings). Suppose that we are dealing with the version of the city soap opera in which the women cheat the men out of their inheritance. If people place at least some weight on the incidental inheritance relations, the mapping of Peter to Nancy will be preferred over the mapping to David (because Nancy, like Peter, cheated someone out of an inheritance). To be consistent with this mapping for Peter, Mary would then be mapped to John. (Similarity of the characters' gender was controlled in the experiment by counterbalancing, and therefore will be ignored in our discussion.)
Now consider the situation from the point of view of someone who had to predict the plot focusing on the romantic relations. Peter would now map best to either John or Lisa, who shared the role of pursuer. Of these two possibilities, the mapping to Lisa will be preferred if people are sensitive to the inheritance relations as well as the romantic relations. Consistency would then make Mary tend to map to David.
By seeing how the students actually mapped Peter and Mary as a pair, we can determine whether their mappings were sensitive to the students' purpose in using the analogy. Those students who gave greater emphasis to the type of relation that was pragmatically important for extending the plot (either the professional or the romantic relations) would give one of the two mappings consistent with the goal-relevant relations. In addition, those students who also gave at least some weight to the inheritance relations -- even though these was not relevant to the plot extension -- would select a mapping in which Peter mapped to a cheater.
Figure 2. Top panel: Percentages of participants in the plot-extension task of Experiment 3 of Spellman and Holyoak (in press) who mapped characters in accord with the goal-relevant relation (either Professional or Romantic) and in accord with the incidental Inheritance relation. Other = participants who did not write an analogous plot-extension or whose analogous plot-extension included two sets of characters. Bottom panel: Percentages of participants in the mapping task of Experiment 3 who mapped characters in accord with the goal-relevant relation (either Professional or Romantic) and in accord with the incidental Inheritance relation. Other = participants who did not write an analogous plot-extension, included two sets of characters in the plot-extension, or did not map to a congruous Peter/Mary pair. Reprinted with permission of Academic Press.
The top panel of Figure 2 displays the results for the plot-extension task. The great majority of the students developed a sensible plot extension in which Peter and Mary mapped consistently to one of the two character pairs that matched on the important type of relation. Of these two possibilities, there was a weak preference for the pair that also matched on the incidental inheritance relations. The goal clearly had a strong inference on people's choice of characters.
The bottom panel of Figure 2 displays the results for the explicit mapping task. This task, unlike the plot-extension task, did not actually require people to focus on the type of relation needed to write the new episode. Nonetheless, people may continue to give greater weight to whichever type of relation had been relevant to their goal. And in fact, people preferred to map Peter and Mary on the basis of the goal-relevant relation (left two bars) rather than the opposing relation, although this preference was weaker than it had been in the plot-extension task. In addition, people tended to prefer a mapping that was consistent with the inheritance relations.
Notice that in both the plot-extension and explicit mappings tasks, the majority of the students mapped Peter and Mary to some consistent pair of characters (that is, two people who interacted with each other), rather than splitting the mapping in some way (the "other" responses, indicated by the bar at the right of each panel). Spellman and Holyoak's (in press) experiment thus shows that people are sensitive to all three of the basic constraints we have been talking about -- structure (making consistent mappings for the pair of characters), similarity (mapping professional relations to professional, romantic to romantic), and purpose (resolving ambiguous mappings on the basis of whichever type of relation is most relevant to the person's goal in using the analogy).
People's sensitivity to analogical constraints of similarity, structure, and purpose is vividly exhibited outside the laboratory. Conventional wisdom has it that people at the beginning of their careers -- a young professional in training, such as a medical student, an articling lawyer, or a graduate student working on a Ph.D. thesis -- can benefit substantially from having role models. Consider Jane, an intelligent, hard-working student who aspires to be a clinical psychologist. She may encounter an older, successful clinical psychologist on whom she can model, at least partly, her own choices concerning career and personal life. Using a role model in this way is a kind of analogical thinking, in which Jane's own career is the target problem and the role model's career becomes a potential source of insight. (2)
We are not aware of any empirical research that addresses the question of how people choose their role models and apply them to inform their lives. From the perspective of the multiconstraint theory of analogy, choosing a role model is a kind of analog selection, and applying the role model is a kind of analogical mapping. For example, Jane may select her role model by remembering an established clinical psychologist who is similar to her in many respects such as race, gender, personality, and so on. We conjecture that, like analogical retrieval, role model selection is primarily dominated by such salient similarities, although subtler aspects of relational structure (e.g., how the older psychologist has been connected with other people and institutions) and purpose (e.g., Jane's career and personal goals) may also affect her selection of a role model. Once she has identified a role model, however, Jane's analogical thinking will be much more affected by structure and purpose than by more superficial similarities. To make decisions in her own life (e.g., whether to emphasize therapy or research, whether to marry another psychologist, or whether to have a baby in graduate school), Jane may be able to take into account the positive and negative results of similar decisions in her role model's life. Jane's thinking might implicitly proceed along the following lines: "I don't know whether I should get put a lot of energy into my Ph.D. research instead of getting more clinical experience. My role model Alice is similar to me in that she is a woman who was interested in both therapy and research. She completed a fine Ph.D. thesis that yielded several publications which helped her get a good placement that started her off on a very successful career as a clinical psychologist. So maybe I should also see research as furthering my career aspirations." At this point in Jane's thinking, the fact that Alice is also a woman will be important to the extent that the structure of her life maps onto Joan's and suggests to Joan how she might accomplish her goals. For example, Alice's being a woman may turn out to affect structure and purpose if Joan's situation involves gender-related impediments to accomplishing career goals.
If Jane becomes a practicing clinical psychologist, she may find herself noticing frequent use of analogies by both her clients and herself. According to Meichenbaum (1994, pp. 112-113), victims of post-traumatic stress disorder frequently use analogies and metaphors to describe their own situations. Here are some of the metaphors used by people recovering from severe psychological trauma:
All of these examples involve analogical mappings from a familiar situation to the situation of the patient. (For discussion of the relation between metaphor and analogy, see Holyoak & Thagard, 1995, ch. 9.)
Meichenbaum (1994, p. 115) describes how changes in metaphors used by clients can mark improvements in their conditions. Recovering trauma victims replace metaphors such as those in the list in the previous paragraph by metaphors such as the following:
The use of these hopeful metaphors for clients' problems suggests that people can map themselves to persons or situations in ways that suggest solutions to personal problems. Recognition of patients' changing metaphors can therefore be a useful part of Jane's clinical practice. There is as yet no experimental evidence that metaphor change plays a causal role in the patients' improvements, but clinical observations of patients suggest that metaphor change is an integral part of healing, not just a reflection of emotional states before and after treatment.
Jane is likely to find herself not only noticing her patients' analogies, but also using them herself. One of the major predictors of therapeutic success is empathy, the extent to which Jane is able to understand her client's emotions by putting herself in the client's shoes and getting a sense of how she would feel if she were in the client's situation (Dawes, 1994). Barnes and Thagard (in press) argue that empathy is essentially a process of analogical mapping, in which the empathizer is able to produce a structured comparison that produces transfer, not just of verbal information, but also of an emotional state. For example, if Jane's client is a rape victim, she can imagine how she would feel if she had experienced the same trauma and then map this emotion back to the client in order to understand more deeply the ongoing distress. Empathic therapy may also involve analog retrieval or construction, when therapists have to work hard to find situations in their own lives that are semantically and structurally similar to that of the clients.
The use of empathy in therapy involves an analogy between the client and the therapist, but the therapist may also find useful analogies between the client and someone else, perhaps a previous client with a similar problem. Such a mapping may help the therapist better understand the current client, and may also be used to provide a role model that the client can use as a source analog to suggest steps toward recovery. Barker (1985) provides additional examples of the use of analogies and metaphors in psychotherapy.
In addition to her life as a therapist, Jane's career as a researcher may benefit from analogical thinking. When she designs her experiments, she may look carefully at experiments already done in the area in which she has interested. An experiment conducted previously may provide a source analog to suggest in part how she should structure her own experiments. (See Dunbar, in press, for analyses of the use of analogy in working laboratories within the field of molecular biology.) Which experiments Jane selects to guide her own design, and how she maps them to produce her own experiment, should depend primarily on structure and purpose, although other similarities may get carried over as well.
Theoretical ideas in psychology and other fields often arise by analogy from related fields. In current cognitive science, the analogy between thinking and computation is the major source of theories of mind (Thagard, in press). Conceiving the mind as a rule-based computer, or a neural-network-like computer, or a chaotic computer, makes possible precise specification of mental mechanisms that may explain people's psychological capacities. Current theories of analogical thinking have been heavily influenced by such computational analogies.
Our development of the multiconstraint theory has depended heavily on computational models designed to simulate aspects of human analogical thinking. ARCS (Analog Retrieval by Constraint Satisfaction; Thagard et al., 1990) and ACME (Analogical Mapping by Constraint Satisfaction; Holyoak & Thagard, 1989) address the steps of access and mapping, respectively. ACME has also been extended to make inferences based on the mappings it computes (Holyoak, Novick & Melz, 1994). As their names imply, these systems in essence attempt to find the optimal "fit" to the constraints postulated by the multiconstraint theory. The models attempt to make use of the strengths of both the symbolic and the connectionist approaches to modeling cognition (Barnden, 1994), combining symbolic representations of explicit knowledge with connectionist processing.
The structures in a parallel constraint satisfaction model consists of elements and various kinds of constraints among them. We can classify constraints as being either internal or external: Internal constraints involve only relations among the elements, while external constraints come from outside the system of elements. In addition, constraints can be either positive or negative, depending on whether they imply that two elements are compatible or incompatible.
We will illustrate the general approach by focusing on the ACME model of analogical mapping, which specifies how the constraints of similarity, structure, and purpose can be jointly optimized to yield a coherent set of correspondences between a source and target (see Holyoak & Thagard, 1989, for a full description). Consider a simplified version of the Persian Gulf analogy that includes only the information that Saddam was president of Iraq which invaded Kuwait, and Hitler was führer of Germany which occupied Austria:
Target |
Source |
|---|---|
| president-of (Saddam, Iraq) | führer-of (Hitler, Germany) |
| invade (Iraq, Kuwait) | occupy (Germany, Austria). |
Considered in isolation the mappings for these fragments are obvious; but in the context of more realistic representations of people's knowledge about the two wars the computational difficulty of the task would be apparent. The ACME model shows how multiple constraints make mapping possible.
If we focus on structure, we can constrain the mapping problem considerably by mapping predicates only to predicates and objects to objects, so that the correspondence invade <--> Hitler will never even be considered. The elements in our constraint-satisfaction theory of analogical mapping include only hypotheses that relate analog components of similar types: predicate-predicate hypotheses such as invade <--> occupy and invade <--> führer-of, and object-object hypotheses such as Saddam <--> Hitler and Saddam <--> Germany. We can also ignore hypotheses that involve objects that never fill corresponding slots, such as Saddam <--> Austria.
Among the hypotheses worth considering, two further kinds of structural constraints can be applied: the positive constraint of structural consistency and the negative constraint of one-to-one mapping. For example, structural consistency requires that the hypothesis invade <--> occupy should encourage and be encouraged by the mappings Iraq <--> Germany and Kuwait <--> Austria. Similarly, one-to-one mapping requires that the hypothesis Iraq <--> Germany should discourage and be discouraged by Iraq <--> Hitler. In ACME, structural consistency and one-to-one mappings are both "soft" constraints, encouraging mappings but not insisting on them, whereas ruling out mappings between objects and predicates is a hard, inviolable constraint.
Similarity and purpose are both treated as external constraints on mapping. We want to favor mappings that involve semantically similar components such as "invade" and "occupy", not ones involving elements as different as "invade" and "führer-of". Again this is a soft constraint, as we want the system to be able to discover correspondences between elements that were not previously seen as related to each other. Similarly, the purpose will favor mapping hypotheses that fit with the goals of the analogist: If the point of the analogy is to show that Saddam is evil like Hitler, then the mapping hypothesis Saddam <--> Hitler will be encouraged by the soft constraint that mappings should serve the purpose of the analogy.
Now we can move from the constraint theory to the computational model, in which elements are represented by units, positive and negative constraints are represented respectively by excitatory and inhibitory links, external constraints are represented by links to special units, and parallel constraint satisfaction is achieved by algorithms for updating activations of the units based on their links to other units. In the simple example above, we need eleven units to represent all the mapping hypotheses. (For simplicity, we ignore hypotheses about mappings between propositons.) These units will be interconnected by excitatory and inhibitory links to represent the positive constraint of structural consistency and the negative constraint of one-to-one mapping. To implement the external constraints, we need two special units, one for semantic similarity and the other for purpose. A special unit will be linked with each unit that represents a mapping hypothesis that satisfies the constraints of either semantic similarity or relevance to purpose (or both). For example, ACME produces a link from the special "similarity" unit to the unit representing the president-of <--> führer-of correspondence but not the president-of <--> occupy correspondence. Of course, satisfying a constraint can be a matter of degree. For example, the concept of being a president is somewhat similar to that of being a führer, but perhaps less so than to that of, say, being a prime minister. The magnitude of the positive or negative weight on each link reflects the degree to which the corresponding constraint is satisfied or violated.
Figure 3 depicts the network created by ACME when it is given as input the source and target represented above. Once this network is created, a simple "relaxation" algorithm updates the activation of each unit in parallel to determine which mapping hypotheses should be accepted (see Rumelhart, Hinton & McClelland, 1986). All units start with activations near 0, except for the special units for semantic similarity and purpose, which start with and retain full activation of 1. These units start to activate the units with which they are linked; then activation spreads throughout the system, fostered by excitatory links and suppressed by inhibitory links.
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Figure3 is not available
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Figure 3. The structure of the network created by ACME for the simplified Saddam example. Solid lines indicate excitatory links and dashed lines indicate inhibitory links. Units representing mappings between whole propositions are not shown. From Holyoak and Thagard (1995). Reprinted with permission of MIT Press.
ACME is capable of exploiting higher-order relations (i.e., relations that take propositions as arguments; see Gentner, 1983) to provide much deeper mappings. An enhanced representation of our Persian Gulf target and World War II source might include the information that Saddam's being president of Iraq was a cause of Iraq's invading Kuwait, just as Hitler's being führer of Germany was a cause of Germany's occupying Austria. ACME would then map the two "cause" relations together and create additional mapping hypotheses, putting entire propositions into correspondence with each other.
We have tested ACME on dozens of examples, and have showed how ACME can closely mimic human mapping behavior in a variety of psychological experiments. For example, ACME can find human-like mappings between the Persian Gulf and World War II analogs when given propositions that capture an elaborate summary of each (Spellman & Holyoak, 1992). Like humans, ACME is sensitive to the "Necker-cube" quality of this ambiguous analogy, settling into one of two sets of coherent but mutually exclusive correspondences: President Bush and the United Sates tend to be mapped to Roosevelt and the United States, or to Churchill and Great Britain, but not to a "mixed" combination of a leader and a country, such as Churchill and the United States. Also like people, ACME occasionally will tolerate a one-to-many mapping, such as that between Kuwait and Austria/Poland. Similarly, ACME is able to simulate the manner in which the processing goal guides the resolution of ambiguous mappings (Spellman & Holyoak, in press). In general, the model seems to capture the human ability to find coherent relationships between complex and imperfectly understood situations, based on the interplay between the constraints of structure, similarity, and purpose.
Despite ACME's successful simulation of many analogies, we do not believe that it or other current computational models provide the final word on analogical thinking. Human use of analogies and metaphors still far surpasses existing computational models in semantic richness and flexibility of application. One promising direction for future progress in computational understanding of analogy involves the use of distributed representations of meaning. ACME is a localist connectionist model in which each neuron-like unit represents a mapping hypothesis linking pairs of predicates and objects. A distributed representation, in contrast, represents predicates, objects, and propositions by complexes of units, just as concepts seem to be distributed over a large set of neurons in the brain. We are currently exploring different ways of introducing distributed representations into our analogy models, and finding that they do indeed enable greater flexibility than ACME affords (Hummel & Holyoak, in press; Eliasmith & Thagard, forthcoming). Models based on distributed representations can capture more subtle interactions among the constraints on analogical thinking. In addition, distributed representations make it much easier to understand the connection between analogical thinking and learning of abstract schemas. These newer models are nonetheless instantiations of the multiconstraint theory of analogy, as they perform analogical mapping using the three basic constraints of similarity, structure, and purpose. We anticipate that continued development of more sophisticated computational models will lead to deeper understanding of the analogical mind.
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Preparation of this article was supported by NSF Grant SBR-9511504 to John Hummel and Keith Holyoak and by a grant from the Natural Sciences and Engineering Research Council of Canada to Paul Thagard. The sections reviewing mapping experiments and the ACME model are adapted from Holyoak and Thagard (1995), with permission of MIT Press. Requests for reprints may be directed to Keith Holyoak, Department of Psychology, UCLA, Los Angeles, CA 90095-1563.
1. We thank Aaron Novick for producing this example, and his mother Laura for providing it to us along with an interpretation of the episode as an instance of analogical thinking (personal communication, June 14, 1993).
2. We are grateful to Ziva Kunda for ideas about role models as analogies.