The authors present a theory of how relational inference and generalization can be accomplished within a cognitive architecture that is psychologically and neurally realistic. Their proposal is a form of symbolic connectionism: a connectionist system based on distributed representations of concept meanings, using temporal synchrony to bind fillers and roles into relational structures. The authors present a specific instantiation of their theory in the form of a computer simulation model, Learning and Inference with Schemas and Analogies (LISA). By using a kind of self-supervised learning, LISA can make specific inferences and form new relational generalizations and can hence acquire new schemas by induction from examples. The authors demonstrate the sufficiency of the model by using it to simulate a body of empirical phenomena concerning analogical inference and relational generalization. A fundamental aspect of human intelligence is the ability to form and manipulate relational representations. Examples of relational thinking include the ability to appreciate analogies between seemingly different objects or events (Gentner, 1983; Holyoak & Thagard, 1995) , the ability to apply abstract rules in novel situations (e.g., Smith, Langston, & Nisbett, 1992) , the ability to understand and learn language (e.g., Kim, Pinker, Prince, & Prasada, 1991) , and even the ability to appreciate perceptual similarities (e.g., Goldstone, Medin, & Gentner, 1991; Hummel, 2000; Hummel & Stankiewicz, 1996; Palmer, 1978) . Relational inferences and generalizations are so commonplace that it is easy to assume that the psychological mechanisms underlying them are relatively simple. But this would be a mistake. The capacity to form and manipulate explicit relational representations appears to be a late evolutionary development (Robin & Holyoak, 1995) , closely tied to the substantial increase in the size and complexity of the frontal cortex in the brains of higher primates, and most dramatically in humans (Stuss & Benson, 1986) . Waltz et al. (1999) found that the ability to represent and integrate multiple relations is profoundly impaired in patients with severe degeneration of prefrontal cortex, and neuroimaging studies have implicated dorsolateral prefrontal cortex in various tasks that require relational reasoning (Christoff et al., 2001; Duncan et al., 2000; Kroger et al., 2002; Prabhakaran, Smith, Desmond, Glover, & Gabrieli, 1997) , as well as in a variety of working-memory (WM) tasks (Cohen et al., 1997; . Our goal in this article is to present a theory of how relational inference and generalization could be accomplished within a cognitive architecture that is psychologically and neurally realistic. We begin by reviewing the characteristics of relational thinking and the logical requirements for realizing these properties in a computational architecture. We next present a theory of human relational reasoning motivated by these requirements and present a specific instantiation of the theory in the form of a computer simulation model, Learning and Inference with Schemas and Analogies (LISA). LISA was initially developed as a theory of analog retrieval and mapping (Hummel & Holyoak, 1997) , reflecting our assumption that the process of finding systematic correspondences between structured representations, which plays a central role in analogical reasoning, is fundamental to all relational reasoning (Hofstadter, 2001 ). Here we demonstrate that these same principles can be extended to account for both the ability to make specific inferences and the ability to form new relational generalizations, and hence to acquire new schemas by induction from examples. We demonstrate the sufficiency of the model by using it to simulate a body of empirical phenomena concerning analogical inference and relational generalization. (See Hummel & Holyoak, 1997, for simulations of data on retrieval and analogical mapping.) The power of relational reasoning resides in its capacity to generate inferences and generalizations that are constrained by the roles that elements play, rather than solely by the properties of the elements themselves. In the limit, relational reasoning yields universal inductive generalization from a finite and often very small set of observed cases to a potentially infinite set of novel instantiations. An important example of human relational reasoning is analogical inference. Suppose someone knows that Bill loves Mary, Mary loves Sam, and Bill is jealous of Sam (a source analog). The reasoner then observes that Sally loves Tom, and Tom loves Cathy (a target analog). A plausible analogical inference is that Sally will be jealous of Cathy. Mary corresponds to Tom on the basis of their shared roles in the two loves (x, y) relations; similarly, Bill corresponds to Sally and Sam to Cathy. John E. Hummel and Keith J. Holyoak, Department of Psychology, University of California, Los Angeles (UCLA). Preparation of this article was supported by National Science Foundation Grant SBR-9729023 and by grants from the UCLA Academic Senate and HRL Laboratories. We thank Miriam Bassok for valuable discussions and comments. Correspondence concerning this article should be addressed to John E. Hummel, Department of Psychology, UCLA, 405 Hilgard Avenue, Los Angeles, California 90095-1563. E-mail: jhummel@lifesci.ucla.edu Psychological Review Copyright 2003 by the American Psychological Association, Inc. 2003, Vol. 110, No. 2, 220 -264 0033-295X/03/$12.00 DOI: 10.1037/0033-295X.110.2.220 220 The analogical inference depends critically on these correspondences (e.g., it is not a plausible analogical inference that Tom will be jealous of Cathy). This same analysis, in which objects are mapped on the basis of common relational roles, can be extended to understand reasoning based on the application of schemas and rules. Suppose the reasoner has learned a schema or rule of the general form, "If person1 loves person2, and person2 loves person3, then person1 will be jealous of person3," and now encounters Sally, Tom and Cathy. By mapping Sally to person1, Tom to person2, and Cathy to person3, the inference that Sally will be jealous of Cathy can be made by extending the mapping to place Sally in the (unrequited) lover role (person1), Tom in the both-lover-and-beloved roles (person2), and Cathy in the beloved role (person3). When the schema has an explicit if-then form, as in this example, the inference can readily be interpreted as the equivalent of matching the left-hand side (the if clause) of a production rule and then "firing" the rule to create an instantiated form of the right-hand side (the then clause), maintaining consistent local variable bindings across the if and then clauses in the process (J. R. Anderson, 1993; Newell, 1973) . There is also a close connection between the capacity to make relational inferences and the capacity to learn variablized schemas or rules of the above sort. If we consider how a reasoner might first acquire the jealousy schema, a plausible bottom-up basis would be induction from multiple specific examples. For example, if the reasoner observed Bill being jealous of Sam, and Sally being jealous of Cathy, and was able to establish the relational mappings and observe the commonalities among the mapped elements (e.g., Bill and Sally are both people in love, and Sam and Cathy are both loved by the object of someone else's affection), then inductive generalization could give rise to the jealousy schema. The computational requirements for discovering these correspondences-and for making the appropriate inferences-include the following: 1. The underlying representation must bind fillers (e.g., Bill) to roles (e.g., lover). Knowing only that Bill, Mary, and Sam are in various love relations will not suffice to infer that it is Bill who will be jealous of Tom. 2. Each filler must maintain its identity across multiple roles (e.g., the same Mary is both a lover and a beloved), and each role must maintain its identity across multiple fillers (e.g., loves (x, y) is the same basic relation regardless of whether the lovers are Bill and Mary or Sally and Tom). In turn, such maintenance of identities implies that the representation of a role must be independent of its fillers, and vice versa (Holyoak & Hummel, 2000; Hummel, Devnich, & Stevens, 2003) . If the representation of loves (x, y) depended on who loved whom (e.g., with one set of units for Sally-as-lover and a completely separate set for Bill-as-lover), then there would be no basis for appreciating that Sally loving Tom has anything in common with Bill loving Mary. And if the representation of Mary depended on whether she is a lover or is beloved, then there would be no basis for inferring that Bill will be jealous of Sam: It would be as though Bill loved one person and someone else loved Sam. 3. Elements of the source and target can be mapped on the basis of their relational (i.e., role-based) correspondences. For example, we know that Sally maps to Bill because she also is a lover, despite the fact that she has a more direct similarity to Mary by virtue of being female. 4. The inference depends on an extension of the initial mapping. Because Sally maps to Bill on the basis of her role as an unrequited lover, it is inferred that she, like Bill, will also be jealous. 5. The symbols that compose these representations must have semantic content. In order to induce a general jealousy schema from the given examples, it is necessary to know what the people and relations involved have in common and how they differ. The semantic content of human mental representations manifests itself in countless other ways as well. Together, the first 4 requirements imply that the human cognitive architecture is a symbol system, in which relations and their arguments are represented independently but can be bound together to compose propositions. We take the 5th requirement to imply that the symbols in this system are distributed representations that explicitly specify the semantic content of their referents. The capacity to satisfy these requirements is a fundamental property of the human cognitive architecture, and it places strong constraints on models of that architecture (Hummel & Holyoak, 1997) . Our proposal is a form of symbolic connectionism: a cognitive architecture that codes relational structures by dynamically binding distributed representations of roles to distributed representations of their fillers. We argue that the resulting architecture captures the meanings of roles and their fillers-along with the natural flexibility and automatic generalization enjoyed by distributed connectionist representations-in a way that no architecture based on localist symbolic representations can (see also . At the same time, the architecture proposed here captures the power of human relational reasoning in a way that traditional connectionist architectures (i.e., nonsymbolic architectures, such as those that serve as the platform for backpropagation learning; see Marcus, 1998 Marcus, , 2001 cannot. Core Properties of a Biological Symbol System We assume that human reasoning is based on distributed symbolic representations that are generated rapidly (within a few hundred milliseconds per proposition) by other systems, including those that perform high-level perception (e.g., Hummel & Biederman, 1992 ) and language comprehension (e.g., Frazier & Fodor, 1978) . 1 The resulting representations serve both as the basis for relational reasoning in WM and the format for storage in long-term memory (LTM). 1 This estimate of a few hundreds of milliseconds per proposition is motivated by work on the time course of token formation in visual perception (Luck & Vogel, 1997) and cognition (Kanwisher, 1991; Potter, 1976) . Individual propositions. In LISA, propositions play a key role in organizing the flow of information through WM. A proposition is a four-tier hierarchical structure (see Figure 1A ). Each tier of the hierarchy corresponds to a different type of entity (semantic features, objects, and roles, individual role bindings, and sets of role bindings comprising a proposition), and our claim is that the cognitive architecture represents all four tiers explicitly. At the bottom of the hierarchy are distributed representations based on semantic units, which represent the semantic features of individual relational roles and their fillers. For example, the proposition loves (Bill, Mary) contains the semantic features of the lover and beloved roles of the loves relation (e.g., emotion, strong, positive), along with the features of Bill (e.g., human, adult, male) and Mary (e.g., human, adult, female). The three other layers of the hierarchy are defined over localist token units. At the second tier, the localist units represent objects (circles in Figure 1 ) and predicates (i.e., properties, as in red (x) or relational roles, as in the lover and beloved roles of loves (x, y); triangles in Figure 1 ). Each predicate unit represents one role of one relation (or property) and has bidirectional excitatory connections to the semantic units representing that role. Object units are just like predicate units except that they are connected to semantic units representing objects rather than roles. 2 At the third tier, subproposition (SP) units bind individual roles to their fillers (either objects or other propositions). For example, loves (Bill, Mary) would be represented by 2 For the purposes of learning, it is necessary to know which semantic units refer to roles and which refer to the objects filling the roles. For simplicity, LISA codes this distinction by designating one set of units to represent the semantics of predicate roles and a separate set to represent the semantic properties of objects. Figure 1 . A: A proposition is a four-tier hierarchy. At the bottom are the semantic features of individual object and relational roles (small circles). Objects and roles share semantic units to the extent that they are semantically similar (e.g., both Bill [B] and Mary [M] are connected to the semantic units human and adult). At the next level of the hierarchy are localist units for individual objects (large circles; e.g., Bill and Mary) and relational roles (triangles; e.g., lover and beloved). At the third tier of the hierarchy, subproposition (SP) units (rectangles) bind objects to relational roles (e.g., Bill to the lover role), and at the top of the hierarchy, proposition (P) units (ovals) bind separate role bindings into complete multiplace propositions (e.g., binding "Billϩlover" and "Maryϩbeloved" together to form loves (Bill, Mary)). B: To represent a hierarchical proposition, such as knows (Mary, loves (Bill, Mary)), the P unit for the lower level proposition (in the current example, loves (Bill, Mary)) serves in the place of an object unit under the appropriate SP of the higher level proposition (in the current example, the SP for knownϩloves (Bill, Mary)). HUMMEL AND HOLYOAK two SPs, one binding Bill to lover, and the other binding Mary to beloved (see Figure 1A ). SPs share bidirectional excitatory connections with their role and filler units. At the top of the hierarchy, proposition (P) units bind role-filler bindings into complete propositions. Each P unit shares bidirectional excitatory connections with the SPs representing its constituent role-filler bindings. This basic four-tier hierarchy can be extended to represent more complex embedded propositions, such as know (Mary, loves (Bill, Mary)) (see Figure 1B) . When a proposition serves as an argument of another proposition, the lower level P unit serves in the place of an object unit under the appropriate SP. The localist representation of objects, roles, role-filler bindings (SPs), and propositions in the upper three layers of the hierarchy is important because it makes it possible to treat each such element as a distinct entity (cf. Hofstadter, 2001; Hummel, 2001; Hummel & Holyoak, 1993; Page, 2001) . For example, if Bill in one analog maps to Sally in another, then this mapping will be represented as an excitatory connection linking the Bill unit directly to the Sally unit (as elaborated shortly). The localist nature of these units greatly facilitates the model's ability to keep this mapping distinct from other mappings. If the representation of Sally were distributed over a collection of object units (analogous to the way it is distributed over the semantic units), then any mapping learned for Sally (e.g., to Bill) would generalize automatically to any other objects sharing object units with Sally. Although automatic generalization is generally a strength of distributed representations, in this case it would be a severe liability: It simply does not follow that just because Sally corresponds to Bill, all other objects therefore correspond to Bill to the extent that they are similar to Sally. The need to keep entities distinct for the purposes of mapping is a special case of the familiar type-token differentiation problem (e.g., Kahneman, Treisman, & Gibbs, 1992; Kanwisher, 1991; Xu & Carey, 1996) . Localist coding is frequently used to solve this and related problems in connectionist (also known as neural network) models of perception and cognition (e.g., Feldman & Ballard, 1982; Hummel & Biederman, 1992; Hummel & Holyoak, 1997; McClelland & Rumelhart, 1981; Poggio & Edelman, 1990; Shastri & Ajjanagadde, 1993) . As in these other models, we assume that the localist units are realized neurally as small populations of neurons (as opposed to single neurons), in which the members of each population code very selectively for a single entity. All tokens (localist units) referring to the same type are connected to a common set of semantic units. For example, if Bill is mentioned in more than one analog, then all the various object units representing Bill (one in each analog) would be connected to the same semantic units. The semantic units thus represent types. It should be emphasized that LISA's representation of any complex relational structure, such as a proposition or an analog, is inevitably distributed over many units and multiple hierarchical levels. For example, the representation of love (Bill, Mary) in Figure 1A is not simply a single P unit, but the entire complex of token and semantic units. While the distributed semantic representation enables desirable forms of automatic generalization, the localist layers make it possible to treat objects, roles, and other structural units as discrete entities, each with potentially unique properties and mappings, and each with a fundamental identity that is preserved across propositions within an analog. Complete analogs. A complete analog (e.g., situation, story, or event) is encoded as a collection of token units, along with the connections to the semantic units. For example, the simple story "Bill loves Mary, Mary loves Sam, and Bill is jealous of Sam" would be represented by the collection of units shown in Figure 2 . Within an analog, a single token represents a given object, predicate, SP, or proposition, no matter how many times that structure is mentioned in the analog. Separate analogs are represented by nonoverlapping sets of tokens (e.g., if Bill is mentioned in a second analog, then a separate object unit would represent Bill in that analog). Although separate analogs do not share token units, all analogs are connected to the same set of semantic units. The shared semantic units allow separate analogs to communicate with one another, and thus bootstrap memory retrieval, analogical mapping, and all the other functions LISA performs. Mapping connections. The final element of LISA's representations is a set of mapping connections that serve as a basis for representing analogical correspondences (mappings) between tokens in separate analogs. Every P unit in one analog has the potential to learn a mapping connection with every P unit in every other analog; likewise, SPs can learn connections across analogs, as can objects and predicates. As detailed below, mapping connections are created whenever two units of the same type in separate analogs are in WM simultaneously; the weights of existing connections are incremented whenever the units are coactive, and decremented whenever one unit is active and the other inactive. By the end of a run, corresponding units will have mapping connections with large positive weights, and noncorresponding units will either lack mapping connections altogether or have connections with weights near zero. The mapping connections keep track of what corresponds to what across analogs, and they allow the model to use known mappings to constrain the discovery of additional mappings. A fundamental property of a symbol system is that it specifies the binding of roles to their fillers. For example, loves (Bill, Mary) explicitly specifies that Bill is the lover and Mary the beloved, rather than vice versa. For this purpose, it is not sufficient simply to activate the units representing the various roles and fillers making up the proposition. The resulting pattern of activation on the semantic units would create a bizarre concatenation of Bill, Mary, and loving, rather than a systematic relation between two separate entities. It is therefore necessary to represent the rolefiller bindings explicitly. The manner in which the bindings are represented is critical. Replacing the independent role and filler units (in the first two tiers of the hierarchy) with units for specific role-filler conjunctions (e.g., like SPs, with one unit for Bill-as- 3 We assume all tokens for a given object, such as Bill, are connected to a central core of semantic features (e.g., human, adult, and male, plus a unique semantic feature for the particular Bill in question-something such as Bill#47). However, different tokens of the same type may be connected to slightly different sets of semantics, as dictated by the context of the particular analog. For instance, in the context of Bill loving Mary, Bill might be connected to a semantic feature specifying that he is a bachelor; in the context of a story about Bill going to the beach, Bill might be connected to a semantic feature specifying that he is a surfer. lover, another unit for Bill-as-traveler, etc.) would specify how roles are bound to fillers, but it would necessarily violate rolefiller independence (Hummel et al., 2003; Hummel & Holyoak, 1997) . And simply adopting a combination of conjunctive and independent units would result in a system with a combination of their limitations (Hummel & Biederman, 1992) . Binding is something a symbol system does to symbols for roles and fillers; it is not a property of the symbols themselves. What is needed is an explicit "tag" that specifies how things are bound together but that permits the bound entities to remain independent of one another. The most common and neurally plausible approach to binding that satisfies this criterion is based on synchrony of firing. The basic idea is that when a filler is bound to a role, the corresponding units fire in synchrony with one another and out of synchrony with the units representing other role-filler bindings (Hummel & Biederman, 1992; Hummel & Holyoak, 1993 Shastri & Ajjanagadde, 1993; Strong & Whitehead, 1989; von der Malsburg, 1981) . As illustrated in Figure 3 , loves (Bill, Mary) would be represented by units for Bill firing in synchrony with units for lover, while units for Mary fire in synchrony with units for beloved. The proposition loves (Mary, Bill) would be represented by the very same units, but the synchrony relations would be reversed. There is a growing body of neurophysiological evidence for binding by synchrony in visual perception (e.g., in striate cortex; Eckhorn et al., 1988; Gray, 1994; Kšnig & Engel, 1995; Singer, 2000 ; see also Alais, Blake, & Lee, 1998; Lee & Blake, 2001; Usher & Donnelly, 1998) and in higher level processing dependent on frontal cortex (Desmedt & Tomberg, 1994; Vaadia et al., 1995) . Numerous neural network models use synchrony for binding. It has been applied in models of perceptual grouping (e.g., Eckhorn, Reitboeck, Arndt, & Dicke, 1990; Horn, Sagi, & Usher, 1992; von der Malsburg & Buhmann, 1992) , object recognition (Hummel, 2001; Hummel & Biederman, 1992; Hummel & Saiki, 1993; Hummel & Stankiewicz, 1996 , rule-based reasoning (Shastri & Ajjanagadde, 1993) , episodic storage in the hippocampal memory system (Shastri, 1997), and analogical reasoning (Hummel & Holyoak, 1992 . In contrast to bindings represented by conjunctive coding, bindings represented by synchrony of firing are dynamic in the sense that they can be created and destroyed on the fly, and that they permit the same units to participate in different bindings at different times. The resulting role-filler independence permits the meaning, interpretation, or effect of a role to remain invariant across different (even completely novel) fillers, and vice versa (see ). As noted previously, it is precisely this capacity to represent relational roles independently of their fillers that gives relational representations their expressive power and that permits relational generalization in novel contexts. Hummel et al. (2003) demonstrate mathematically that among the neurally plausible approaches to binding in the computational literature, synchrony of firing is unique in its capacity to represent bindings both explicitly and independently of the bound entities. The need to represent roles independently of their fillers does not imply that the interpretation of a predicate must never vary as a function of its arguments. For example, the predicate loves suggests somewhat different interpretations in loves (Bill, Mary) versus loves (Bill, rib roast). Rather, the point is that the architecture of the symbol system must permit roles and fillers to remain invariant across different bindings. Effects of the filler on the interpretation of a role, as in the case of loving a person versus a food, must stem from the system's prior knowledge, rather than being an inevitable side effect of the way it binds roles to fillers. Thus if Bill were a cannibal, then he might indeed love Mary and rib roast in the same way; but a symbol system that was incapable of representing loves (x, y) in the same way regardless of the argument bound to y would be incapable of understanding the similarity of Bill's fondness for rib roast and his fondness for the unfortunate Mary. Using synchrony for dynamic binding imposes a hierarchical temporal structure on LISA's knowledge representations. This structure is analogous to Cowan's (1988 Cowan's ( , 1995 embeddedprocesses model, with four levels of memory (see Figure 4) . Active memory (cf. long-term working memory; Ericsson & Kintsch, 1995) is the active subset of LTM, including the various modalityspecific slave systems proposed by Baddeley and Hitch (1974; Baddeley, 1986) . During analogical mapping, the two analogs and the emerging mapping connections between them are assumed to reside in active memory. Following Cowan (1995), we assume that active memory has no firm capacity limit, but rather is time limited, with activation decaying in roughly 20 s unless it is reactivated by allocation of attention. When a person is reasoning about large structures (e.g., mapping complex analogs), most of the relevant information will necessarily be first encoded into LTM, and then placed in active memory as needed. Within active memory, a very small number of role bindings can enter the phase set-that is, the set of mutually desynchronized role bindings. We use the term working memory to refer specifically to the phase set. Each "phase" in the set corresponds to one role binding; a single phase corresponds to the smallest unit of WM. The phase set corresponds to the current focus of attentionwhat the model is thinking about "right now"-and is the most significant bottleneck in the system. The size of the phase set-that is, the capacity of WM-is determined by the number of role-filler bindings (phases) it is possible to have simultaneously active but  HUMMEL AND HOLYOAK mutually out of synchrony. This number is necessarily limited, and its value is proportional to the length of time between successive peaks in a phase (the period of the oscillation) divided by the duration of each phase (at the level of small populations of neurons) and/or temporal precision (at the level of individual spikes; see also Lisman & Idiart, 1995) . There is evidence that binding is accomplished by synchrony in the 40-Hz (gamma) range, meaning a neuron or population of neurons generates one spike (or burst) approximately every 25 ms (see Singer & Gray, 1995 , for a review). According to Singer and Gray (1995) , the temporal precision of spike timing is in the range of 4 -6 ms. With a 5-ms duration-precision and a 25-ms period, these figures suggest a WM capacity of approximately 25/5 ϭ 5 role bindings (Singer & Gray, 1995; see also Luck & Vogel, 1997). LISA's architecture for dynamic binding thus yields a principled estimate of the maximum amount of information that can be processed together during analogical mapping: 4 -6 role bindings, or roughly 2-3 propositions. (As detailed in Appendix A, the algorithm LISA uses to allow multiple role bindings to time-share efficiently is inherently capacity limited.) Other researchers (e.g., Hummel & Biederman, 1992; Hummel & Holyoak, 1993 Hummel & Stankiewicz, 1996; Lisman & Idiart, 1995; Shastri & Ajjanagadde, 1993) have suggested similar explanations for the capacity limits of visual attention and WM. Behavioral work with human subjects suggests a remarkably similar figure for the capacity of visual WM (four to five items; Bundesen, 1998; Luck & Vogel, 1997; Sperling, 1960) and WM for relations (three to five items; Broadbent, 1975; Cowan, 1995 Cowan, , 2001 . Because of the strong capacity limit on the phase set, LISA's processing of complex analogs is necessarily highly sequential: LISA can hold at most only three propositions in WM simultaneously, so it must map large analogies in small pieces. The resulting algorithm-which serves as the basis of analog retrieval, mapping, inference, and schema induction-is analogous to a form of guided pattern recognition. At any given moment, one analog is the focus of attention and serves as the driver. One (or at most three) at a time, propositions in the driver become active, generating synchronized patterns of activation on the semantic units (one pattern per SP). In turn, these patterns activate propositions in one or more recipient analogs in LTM (during analog retrieval) or in active memory (during mapping, inference, and schema induction). Units in the recipient compete to respond to the semantic patterns in much the same way that units for words or objects  INFERENCE AND GENERALIZATION compete to respond to visual features in models of word recognition (e.g., McClelland & Rumelhart, 1981) and object recognition (e.g., Hummel & Biederman, 1992) . For example, consider a simplified version of our "jealousy" analogy in which males map to males and females to females. (In the analogy discussed previously, males map to females and vice versa. As described later, LISA can map such analogies. But for our current purposes, the example will be much clearer if we allow males to map to males and females to females.) As in the previous example, let the source analog state that Bill loves Mary, Mary loves Sam, and Bill is jealous of Sam. Let the target state that Tom loves Cathy and Cathy loves Jack ( Figure 5 ). Let the source be the driver and the target the recipient. When the proposition loves (Bill, Mary) fires in the driver, the patterns of activation on the semantic units will tend to activate the units representing loves (Tom, Cathy) in the recipient. Specifically, the semantics activated by Bill (which include human, adult, and male) will tend to activate Tom and Jack equally. The semantics of lover (activated by lover in the driver) will activate lover in the recipient. Lover will activate the SPs for both Tomϩlover and Cathyϩlover equally. Tom will excite Tomϩlover, and Jack will excite Jackϩbeloved. Note that the SP Tomϩlover is receiving two sources of input, whereas Cathyϩlover and Jackϩbeloved are each receiving one source of input. Tomϩlover will therefore inhibit Cathyϩlover and Jackϩbeloved to inactivity. Similar operations will cause Cathyϩbeloved to become active in the target when Maryϩbeloved fires in the source. Together, Tomϩlover and Cathyϩbeloved will activate the P unit for loves (Tom, Cathy). All these operations cause Tom to become active whenever Bill is active (i.e., Tom fires in synchrony with Bill), Cathy to become active whenever Mary is active, and the roles of loves (x, y) in the recipient to become active whenever the corresponding roles are active in the driver. These synchrony relations between the driver and the recipient drive the discovery of mappings: LISA assumes that units that fire together map to one another (accomplished by a simple form of Hebbian learning, Figure 4 . Illustration of the hierarchical structure of memory in Learning and Inference with Schemas and Analogies (LISA). We assume that long-term memory (LTM) lasts a lifetime (mostly) and has, for all practical purposes, unlimited capacity. Active memory (Cowan, 1988 (Cowan, , 1995 cf. long-term working memory; Ericsson & Kintsch, 1995) is the active subset of LTM. The two analogs and the emerging mapping connections between them are assumed to reside in active memory. We assume that active memory has no firm capacity limit but is time limited, with activation decaying in roughly 20 s, unless it is reactivated by attention (Cowan, 1995) . Working memory (WM) consists of the set of role bindings (phases) that are simultaneously active and mutually out of synchrony with one another. Its capacity is limited by the number of phases it is possible to keep mutually desynchronized (roughly 5 or 6). We assume that role bindings reside in WM for a period of time ranging from roughly half a second to a few seconds. The smallest unit of memory in LISA is one role binding (phase), which we assume oscillates in WM (although not necessarily in a regular periodic manner), with each peak lasting a few milliseconds. approx ϭ approximately. HUMMEL AND HOLYOAK detailed shortly). Thus, Bill maps to Tom, Mary to Cathy, lover to lover, and beloved to beloved. By the same operations, LISA learns that Sam maps to Jack, the only exception being that the mappings discovered previously aid in the discovery of this latter mapping. The sequential nature of mapping, as imposed by LISA's limited-capacity WM, is analogous to the sequential scanning of a visual scene, as imposed by the limits of visual WM (see, e.g., Irwin, 1992) . Just as the visual field typically contains more information than can be simultaneously foveated, so active memory will typically contain more information than can be simultaneously placed in WM. And just as eye movements systematically guide the flow of information through foveal vision, LISA's attentional mechanisms guide the flow of propositions through the phase set. As a general default, we assume that propositions in the driver fire in roughly the order they would naturally be stated in a text-that is, in accord with the principles of text coherence (see Hummel & Holyoak, 1997)-and that the most important propositions (i.e., pragmatically central propositions, such as those on the causal chain of the episode, and those pertaining to goals; see Spellman & Holyoak, 1996) tend to fire earlier and more often than less important propositions. Also by default, propositions are fired one at a time (i.e., with only one proposition in a phase set), reflecting the assumption that holding multiple propositions in WM requires greater cognitive effort than thinking about only one fact at a time. If multiple propositions need to be considered in the phase set simultaneously (e.g., in cases when the mapping would be structurally ambiguous on the basis of only a single proposition), we assume that propositions are more likely to be placed together into WM if they are connected in ways that create textual coherence (Kintsch & van Dijk, 1978) . In particular, propositions are more likely to enter WM together if they are arguments of the same higher order relation (especially a causal relation), or if they share arguments or predicates, than if they are unrelated (Hummel & Holyoak, 1997; Kubose, Holyoak, & Hummel, 2002) . In the original LISA (Hummel & Holyoak, 1997) , the firing order and grouping of propositions had to be specified by hand, and the constraints stated above served as guidelines for doing so. As elaborated under the section The Operation of LISA, the current model contains several mechanisms that allow it to set its own firing order and groupings in a manner that conforms to the above constraints. LISA's ability to discover a mapping between two analogs is strongly constrained by the order in which it fires propositions in the driver and by which propositions it puts in WM simultaneously. This is especially true for structurally complex or semantically ambiguous analogies (Kubose et al., 2002) . Mappings discovered early in the mapping process constrain the discovery of later mappings. If the first mappings to be discovered correspond to structurally "correct" mappings, then they can aid significantly in the discovery of subsequent mappings. But if the first mappings correspond to structurally incorrect mappings, then they can lead LISA down a garden path that causes it to discover additional incorrect mappings. Mapping performance is similarly affected by which propositions LISA puts together into WM. For example, recall the original analogy between two pairs of lovers: In one analog, Bill loves Mary, and Mary loves Sam; in the other, Sally loves Tom, and Tom loves Cathy. Although it is readily apparent that Bill corresponds to Sally, Mary to Tom, and Sam to Cathy, these mappings are in fact structurally ambiguous unless one considers both statements in a given analog simultaneously: By itself, loves (Bill, Mary) bears as much structural similarity to loves (Tom, Cathy) (the incorrect mapping) as it does to loves (Sally, Tom) (the correct mapping). (Indeed, the genders of the characters involved tend to favor mapping Bill to Tom and Mary to Cathy.) Only by considering loves (Bill, Mary) simultaneously  Our final set of core theoretical claims concern the relationship between analogical mapping, analogical and rule-based inference, and schema induction. We assume that both schema induction and analogical (and rule-based) inference depend on structure mapping and that exactly the same algorithm is used for mapping in these tasks as is used for mapping analogies. A related claim is that the processes of rule-and schema-based inference are governed by exactly the same algorithm as the process of analogical inference. That is, analogical, rule-, and schema-based inferences are all special cases of the same algorithm. These claims imply that brain damage and other factors (e.g., WM load imposed by a dual task) that affect reasoning in one of these domains should have similar effects on reasoning in all the others. Our approach is based on a handful of core theoretical claims (see Table 1 for a summary of these claims and their instantiation in the model): 1. The mind is a symbol system that represents relationships among entities as explicit propositions. 2. A proposition is a four-tier hierarchical structure that represents the semantic properties of objects and relational roles in a distributed fashion and simultaneously represents roles, objects, role-filler bindings, and propositions in a sequence of progressively more localist units. 3. These hierarchical representations are generated rapidly (i.e., in a few hundreds of milliseconds per proposition) in WM, and the resulting structure serves both to represent the proposition for the purposes of relational reasoning and to store it in LTM. 4. In the lowest layers of the hierarchy, roles are represented independently of their fillers and must therefore be bound together dynamically when the proposition enters WM. 5. The need for dynamic role-filler binding gives rise to the capacity limits of WM. 6. Relational representations are generated and manipulated in a hierarchy of temporal scales, with individual role-filler bindings (phases) at the bottom of the hierarchy, combining to form complete propositions (phase sets) at the next level of the hierarchy, and multiple propositions forming the representation of an analog in active memory and LTM. 7. Because of the capacity limits of WM, structure mapping is necessarily a sequential process. 8. Also because of capacity limits, mapping is directional, proceeding from a driver analog to the recipient(s). 9. As a result of the previous two claims (7 and 8), mapping is sensitive to factors that affect the order in which propositions are mapped and the groupings of propositions that enter WM together. 10. Principles of text coherence, pragmatic centrality, and expertise determine which propositions a reasoner is likely to think about together, and in what order. 11. Analog retrieval, mapping, inference, and schema induction, as well as rule-based reasoning, are all special cases of the same sequential, directional algorithm for mapping. The Operation of LISA Figure 6 illustrates our core claims about the relationship between analogical access, mapping, inference, and schema induction. First, analog access (using a target analog to retrieve an appropriate source from LTM) is accomplished as a form of guided pattern matching. Propositions in a target analog generate synchronized patterns of activation on the semantic units, which in turn activate propositions in potential source analogs residing in LTM. Second, the resulting coactivity of driver and recipient elements, augmented with a capacity to learn which structures in the recipient were coactive with which in the driver, serves as the basis for analogical mapping (discovering the correspondences between elements of the source and target). Third, augmented with a simple algorithm for self-supervised learning, the mapping algorithm serves as the basis for analogical inference (using the source to draw new inferences about the target). Finally, augmented with a simple algorithm for intersection discovery, selfsupervised learning serves as the basis for schema induction (inducing a more general schema or rule based on the common elements of the source and target). There is no necessary linkage between the driver-recipient distinction and the more familiar source-target distinction. However, a canonical flow of control involves initially using the target as the driver to access a familiar source analog in LTM. Once a source is in active memory, mapping can be performed in either direction (see Hummel & Holyoak, 1997) . After a stable mapping is established, the source is used to drive both the generation of inferences in the target and schema induction. When directed to set its own firing order, LISA randomly chooses propositions in the driver to fire (although it is still possible, and often convenient, to set the firing order by hand). The probability that a proposition will be chosen to fire is proportional to the proposition's importance and its support from other propositions, and it is inversely proportional to the recency with which it fired. Support relations (defined quantitatively below) allow propositions to influence the probability with which one another fire. To the extent that proposition i supports proposition j, then when i fires, it increases the probability that j will fire in the near future. The role of text coherence in firing order is captured by support relations that reflect the order in which propositions would 228 HUMMEL AND HOLYOAK be stated in a text (i.e., the first proposition supports the second, the second supports the third, etc.). The role of higher order relations (in both firing order and grouping of multiple proposition in WM) is captured by support relations between higher order propositions and their arguments and between propositions that are arguments of a common higher order relation. The role of shared predicates and arguments is captured by support relations between propositions that share predicates and/or arguments. Whereas support relations affect the conditional probability that proposition i will be selected to fire given that j has fired, pragmatic centrality-that is, a proposition's importance in the context of the reasoner's goals (Holyoak & Thagard, 1989; Spellman & Holyoak, 1996) -is captured in the base rate of i's firing: Pragmatically important propositions have a higher base rate of firing than do less important propositions, so they tend to fire earlier and more often than do less important propositions (Hummel & Ho- LISA represents knowledge as propositions that explicitly represent relations as predicates and explicitly represents the binding of relational roles to their arguments. 2. A proposition is a four-tier hierarchical structure that represents the semantic properties of objects and relational roles in a distributed fashion and simultaneously represents roles, objects, role-filler bindings, and propositions in a sequence of progressively more localist units. LISA represents each tier of the hierarchy explicitly, with units that represent the semantic features of roles and objects in a distributed fashion at the bottom of the hierarchy. Localist structure units represent tokens for relational roles and objects, role-filler bindings, and complete propositions. 3. These hierarchical representations are generated rapidly (i.e., in a few hundreds of milliseconds per proposition) in WM, and the resulting structure serves both to represent the proposition for the purposes of relational reasoning and to store it in LTM. The same hierarchical collection of units represents a proposition in WM for the purpose of mapping and for storage in LTM (with the exception that synchrony of firing dynamically binds roles to their fillers in WM but not in LTM). 4. In the lowest layers of the hierarchy, roles are represented independently of their fillers and must therefore be bound together dynamically when the proposition enters WM. At the level of semantic features and object and predicate units, LISA represents roles independently of their fillers, in the sense that the same units represent a role regardless of its filler and the same units represent an object regardless of the role(s) to which the object is bound. LISA uses synchrony of firing to dynamically bind distributed representations of relational roles to distributed representations of their fillers in WM. 5. The need for dynamic role-filler binding gives rise to the capacity limits of WM. The capacity of LISA's WM is given by the number of SPs it can fit into the phase set simultaneously; this number is necessarily limited. 6. Relational representations are generated and manipulated in a hierarchy of temporal scales, with individual role-filler bindings (phases) at the bottom of the hierarchy, combining to form complete propositions (phase sets) at the next level of the hierarchy, and multiple propositions forming the representation of an analog in active memory and LTM. In LISA, roles fire in synchrony with their fillers and out of synchrony with other role-filler bindings. Role-filler bindings belonging to the same phase set oscillate in closer temporal proximity than role-filler bindings belonging to separate phase sets. 7. Because of the capacity limits of WM, structure mapping is necessarily a sequential process. A single mapping may consist of many separate phase sets, which must be processed one at a time. 8. Mapping is directional, in the sense that it proceeds from the driver to the recipient(s). Activation starts in the driver and flows into the recipient, initially only through the semantic units, and later through both the semantic units and the emerging mapping connections. 9. Mapping is sensitive to factors that affect the order in which propositions are mapped and the groupings of propositions that enter WM together. LISA's mapping performance is highly sensitive to the order and groupings of propositions in the phase set. 10. Principles of text coherence, pragmatic centrality, and expertise determine which propositions a reasoner is likely to think about together, and in what order. Importance (pragmatic centrality) and support relations among propositions (reflecting constraints on firing order) determine the order in which LISA fires propositions in WM and which propositions (if any) it puts together in the phase set. 11. Analog retrieval, mapping, inference, and schema induction, as well as rule-based reasoning, are all special cases of the same sequential, directional algorithm for mapping. LISA uses the same mapping algorithm and self-supervised learning algorithm to perform all these functions. Note. LISA ϭ Learning and Inference with Schemas and Analogies; WM ϭ working memory; LTM ϭ long-term memory; SP ϭ subproposition. INFERENCE AND GENERALIZATION lyoak, 1997). When LISA is directed to set its own firing order, the probability that P unit i will be chosen to fire is given by the Luce choice axiom: where p i is the probability that i will be chosen to fire, j are the propositions in the driver, c i is a function of i's support from other propositions (s i ), its pragmatic importance (m i ), and its "readiness" to fire (r i ), which is proportional to the length of time that has elapsed since it last fired: A proposition's support from other propositions at time t is as follows: where ␦ s is a decay rate, j are other propositions in the driver, ij is the strength of support from j to i ( ij ʦ Ϫϱ. . .ϱ), and f j (t -1) is 1 if unit j fired at time t -1 (i.e., j was among the P units last selected to fire) and 0 otherwise. If i is selected to fire at time t, then s i (t) is initialized to 0 (because the unit has fired, satisfying its previous support); otherwise it decays by half its value on each iteration until i fires (i.e., ␦ s ϭ 0.5). A P unit's readiness at time t is as follows: where ␥ r (ϭ 0.1) is the growth rate of a proposition's readiness to fire. Together, Equations 1-4 make a proposition more likely to fire to the extent that it is either pragmatically central or supported by other propositions that have recently fired, and less likely to fire to the extent that it has recently fired. In the course of retrieving a relevant source analog from LTM, propositions in the driver activate propositions in other analogs (as described below), but the model does not keep track of which propositions activate which. (Given the number of potential matches in LTM, doing so would require it to form and update an unreasonable number of mapping connections in WM.) Hummel and Holyoak (1997) show that this single difference between access and mapping allows LISA to simulate observed differences in human sensitivity to various types of similarity in these two processes (e.g., Ross, 1989 ). In the current article, we are concerned only with analogical mapping and the subsequent processes of inference and schema induction. In the discussions and simulations that follow, we assume that access has already taken place, so that both the driver and recipient analogs are available in active memory. Each SP unit, i, is associated with an inhibitory unit, I i , which causes its activity to oscillate. When the SP becomes active, it excites the inhibitor, which after a delay, inhibits the SP to inactivity. While the SP is inactive, the inhibitor's activity decays, allowing the SP to become active again. In response to a fixed excitatory input, an SP's activation will therefore oscillate between 0 and (approximately) 1. (See Appendix A for details.) Each SP excites the role and filler to which it is connected and strongly inhibits other SPs. The result is that roles and fillers that are bound together oscillate in synchrony with one another, and separate role-filler bindings oscillate out of synchrony. SPs also excite P units "above" themselves, so a P unit will tend to be active whenever any of its SPs is active. Activity in the driver starts in the SPs of the propositions in the phase set. Each SP in the phase set receives an external input of 1.0, which we assume corresponds to attention focused on the SP. (The external input is directed to the SPs, rather than the P units, to permit the SPs to "time-share," rather than one proposition's SPs dominating all the others, when multiple propositions enter WM together.) Because of the within-SP excitation and between-SP inhibition in the driver, the proposition(s) in the phase set generate a collection of distributed, mutually desynchronized patterns of activation on the semantic units, one for each role-filler binding. In the recipient analog(s), object and predicate units compete to respond Figure 6 . Illustration of our core claims about the relationship between analogical access, mapping, inference, and schema induction. Starting with distributed symbolic-connectionist representations (see Figures 1-3 ), analog retrieval (or access) is accomplished as a form of guided pattern matching. Augmented with a basis for remembering which units in the driver were active in synchrony with which units in the source (i.e., by a form of Hebbian learning), the retrieval algorithm becomes a mapping algorithm. Augmented with a simple basis for self-supervised learning, the mapping algorithm becomes an algorithm for analogical inference. And augmented with a basis for intersection discovery, the inference algorithm becomes an algorithm for schema induction. HUMMEL AND HOLYOAK to these patterns. In typical winner-take-all fashion, units whose connections to the semantic units best fit the pattern of activation on those units tend to win the competition. The object and predicate units excite the SPs to which they are connected, which excite the P units to which they are connected. Units in the recipient inhibit other units of the same type (e.g., objects inhibit other objects). After all the SPs in the phase set have had the opportunity to fire once, P units in the recipient are permitted to feed excitation back down to their SPs, and SPs feed excitation down to their predicate and argument units. 4 As a result, the recipient acts like a winner-take-all network, in which the units that collectively best fit the synchronized patterns on the semantic units become maximally active, driving all competing units to inactivity (Hummel & Holyoak, 1997) . All token units update their activations according to where ␥ is a growth rate, n i is the net (i.e., excitatory plus inhibitory) input to unit i, and ␦ is a decay rate. Equation 5 embodies a standard "leaky integrator" activation function. Any recipient unit whose activation crosses a threshold at any time during the phase set is treated as "retrieved" into WM for the purposes of analogical mapping. When unit i in the recipient is first retrieved into WM, mapping hypotheses, h ij , are generated for it and every selected unit, j, of the same type in the driver (i.e., P units have mapping hypotheses for P units, SPs for SPs, etc.). Units in one analog have mapping connections only to units of the same type in other analogs-for instance, object units do not entertain mapping hypotheses to SP units. In the following, this type-restriction is assumed: When we refer to the mapping connections from unit j in one analog to all units i in another, we mean all i of the same type as j. Mapping hypotheses are units that accumulate and compare evidence that a given unit, j, in the driver maps to a given unit, i, in the recipient. We assume that the mapping hypotheses, like the mapping connection weights they will ultimately inform, are not literally realized as synaptic weights but rather correspond to neurons in prefrontal cortex with rapidly modifiable synapses (e.g., Asaad, Rainer, & Miller, 1998; Fuster, 1997) . For example, the activations of these neurons might gate the communication between the neurons whose mapping they represent. The mapping hypothesis, h ij , that unit j in the driver maps to unit i in the recipient grows according to the simple Hebbian rule, shown below: where a i and a j are the activations of units i and j. That is, the strength of the hypothesis that j maps to i grows as a function of the extent to which i and j are active simultaneously. At the end of the phase set, each mapping weight, w ij , is updated competitively on the basis of the value of the corresponding hypothesis, h ij , and on the basis of the values of other hypotheses, h ik , referring to the same recipient unit, i (where k are units other than unit j in the driver), and hypotheses, h lj , referring to the same driver unit, j (where l are units other than unit i in the recipient). Specifically, the value of a weight goes up in proportion to the value of the corresponding mapping hypothesis and goes down in proportion to the value of other hypothesis in the same "row" and "column" of the mapping connection matrix. This kind of competitive weight updating serves to enforce the one-to-one mapping constraint (i.e., the constraint that each unit in the source analog maps to no more than one unit in the recipient; Falkenhainer, Forbus, & Gentner, 1989; Holyoak & Thagard, 1989) . In contrast to the algorithm described by Hummel and Holyoak (1997), mapping connection weights in the current version of LISA are constrained to take values between 0 and 1. That is, there are no inhibitory mapping connections in the current version of LISA. In lieu of inhibitory mapping connections, each unit in the driver transmits a global inhibitory input to all recipient units. The value of the inhibitory signal is proportional to the driver unit's activation and to the value of the largest mapping weight leading out of that unit; the effect of the inhibitory signal on any given recipient unit is also proportional to the maximum mapping weight leading into that unit (see Equation A10 in Appendix A). This global inhibitory signal (see also Wang, Buhmann, & von der Malsburg, 1990) has exactly the same effect as inhibitory mapping connections (as in the 1997 version of LISA) but is more "economical" in the sense that it does not require every driver unit to have a mapping connection to every recipient unit. After the mapping hypotheses are used to update the weights on the mapping connections they (the hypotheses) are initialized to zero. Mapping weights are thus assumed to reside in active memory (long-term WM), but mapping hypotheses-which can only be entertained and updated while the driver units to which they refer are active-must reside in WM. The capacity of WM therefore sharply limits the number of mapping hypotheses it is possible to entertain simultaneously. In turn, this limits the formal complexity of the structure mappings LISA is capable of discovering (Hummel & Holyoak, 1997; Kubose et al., 2002) . Recall the original jealousy example from the introduction, which requires the reasoner to map males to females and vice versa, and recall that finding the correct mapping requires LISA to put two propositions into the same phase set. (See Figure 7 for an illustration of placing separate propositions into separate phase sets [see Figure 7A ] vs. placing multiple propositions into the same phase set [see Figure 7B ].) Using a similarity-judgment task that required participants to align structurally complex objects (such as pictures of "butterflies" with many parts), Medin, Goldstone, and 4 If units in the recipient are permitted to feed activation downward before all the SPs in the phase set have had the opportunity to fire, then the recipient can fall into a local energy minimum on the basis of the first SPs to fire. For example, imagine that P1 in the recipient best fits the active driver proposition overall, but P2 happens to be a slightly better match based on the first SP to fire. If the recipient is allowed to feed activation downward on the basis of the first SP to fire, it may commit incorrectly to P2 before the second driver SP has had the opportunity to fire. Suspending the top-down flow of activation in the recipient until all driver SPs have fired helps to prevent this kind of premature commitment (Hummel & Holyoak, 1997) . 5 If a connection already exists between a driver unit and a recipient unit, then that connection is simply used again; a new connection is not created. Gentner (1994) showed that people place greater emphasis on object features than on interfeature relations early in mapping (and similarity judgment), but greater emphasis on relations than on surface features later in mapping (and similarity judgment). More generally, as emphasis on relations increased, emphasis on features decreased, and vice versa. Accordingly, whenever LISA places multiple propositions into WM, it ignores the semantic features of their arguments. This convention is based on the assumption, noted previously, that people tend by default to think about only a single proposition at a time. In general, the only reason to place multiple propositions into WM simultaneously is to permit higher order (i.e., interpropositional) structural constraints to guide the mapping: That is, placing multiple propositions into WM together represents a case of extreme emphasis on relations over surface features in mapping. 6 (One limitation of the present implementation is that the user must tell LISA when to put multiple propositions together in the phase set, although the model can select for itself which propositions to consider jointly.) When LISA places loves (Bill, Mary) and loves (Mary, Sam) into the same phase set and ignores the semantic differences between objects, it correctly maps Bill to Sally, Mary to Tom and Sam to Cathy. It does so as follows (see Table 2 ). When loverϩBill fires in the driver, it activates loverϩSally and loverϩTom equally, strengthening the mapping hypotheses for lover D 3lover R , Bill3Sally, and Bill3Tom ( (Table 2 , row 5). When the mapping weights are updated at the end of the phase set, Mary3 Tom therefore takes 6 Ignoring semantic differences between objects entirely in this situation is doubtless oversimplified. It is probably more realistic to assume that emphasis on object semantics and structural constraints increase and decrease in a graded fashion. However, implementing graded emphasis would introduce additional complexity (and free parameters) to the algorithm without providing much by way of additional functionality. Once LISA knows that Mary maps to Tom, the discovery of the remaining mappings is straightforward and does not require LISA to place more than one proposition into WM at a time. When LISA fires loves (Bill, Mary), the mapping connection from Mary to Tom causes loves (Bill, Mary) to activate loves (Sally, Tom) rather than loves (Tom, Cathy), establishing the mapping of Bill to Sally (along with the SPs and P units of the corresponding propositions). Analogous operations map Sam to Cathy when loves (Mary, Sam) fires. This example illustrates how a mapping connection established at one time (here, the connection between Mary and Tom, established when the two driver propositions were in the phase set together) can influence the mappings LISA discovers subsequently. The correspondences that develop as a result of analogical mapping serve as a useful basis for initiating analogical inference. LISA suspends analogical inference until a high proportion (in simulations reported here, at least 70%) of the importanceweighted structures (propositions, predicates, and objects) in the target map to structures in the source. This constraint, which is admittedly simplified, 7 is motivated by the intuitive observation that the reasoner should be unwilling to make inferences from a source to a target until there is evidence that the two are meaningfully analogous. If the basic criterion for mapping success is met (or if the user directs LISA to license analogical inference regardless of whether it is met), then any unmapped unit in the source (driver) becomes a candidate for inferring a new unit in the target (recipient). LISA uses the mapping-based global inhibition as a cue for detecting when a unit in the driver is unmapped. Recall that if unit i in the recipient maps to some unit j in the driver, then i will be globally inhibited by all other driver units k j. Therefore, if some driver unit, k, maps to nothing in the recipient, then when it fires, it will inhibit all recipient units that are mapped to other driver units and it will not excite any recipient units. The resulting inhibitory input, unaccompanied by excitatory input, is a reliable cue that nothing in the recipient corresponds to driver unit k. It therefore triggers the recipient to infer a unit to correspond to k. (Naturally, we do not assume the brain grows a new neuron to correspond to k. Instead, we assume the element corresponding to k is generated by the same process that generates structure units at the time of encoding, as discussed previously.) Units inferred in this manner connect themselves to other units in the recipient by simple Hebbian learning: Newly inferred P units learn connections to active SPs, newly inferred SPs learn connections to active predicate and object units, and newly inferred predicate and object units learn connections to active semantic units (the details of the learning algorithm are given in Appendix A, Equations A15 and A16 and the accompanying text). Together, these operations-inference of new units in response to unmatched global inhibition, and Hebbian learning of connections to inferred units-allow LISA to infer new objects, new relations among objects, and new propositions about existing and new relations and objects. To illustrate, let us return to the familiar jealousy example (see Figure 2 ). Once LISA has mapped Bill to Sally, Mary to Tom, and Sam to Cathy (along with the corresponding predicates, SPs and P units), every unit in the target (i.e., the analog about Sally) will be mapped to some unit in the source. The target therefore satisfies LISA's 70% criterion for analogical inference. When the proposition jealous (Bill, Sam) fires, Bill and Sam will activate Sally and Cathy, respectively, via their mapping connections. However, jealous1 (the agent role of jealous) and jealous2 (the patient role) map to nothing in the recipient, nor do the SPs jealous1ϩBill or jealous2ϩSam or the P unit for the proposition as a whole (P3). As a result, when jealous1ϩBill fires, the predicate unit jealous1 will inhibit all predicate units in the recipient; jealous1ϩBill will inhibit all the recipient's SPs, and P3 will inhibit all its P units. In response, the recipient will infer a predicate to correspond to jealous1, an SP to correspond to jealous1ϩBill, and a P unit to correspond to P3. By convention, we refer to inferred units by using an asterisk, followed by the name of the driver unit that signaled the recipient to infer it, or in the case of inferred SPs, an 7 As currently implemented, the constraint is deterministic, but it could easily be made stochastic. We view the stochastic algorithm as more realistic, but for simplicity we left it deterministic for the simulations reported here. A more important simplification is that the algorithm considers only the mapping status of the target analog as a whole. A more realistic algorithm would arguably generate inferences based on the mapping status of subsets of the target analog-for example, as defined by higher order relations (Gentner, 1983) . Note. SP ϭ subproposition; ⌬ ϭ the change in the value of the mapping hypothesis due to the active driver SP; h ϭ cumulative value of the mapping hypothesis. asterisk followed by the names of the its role and filler units. Thus, the inferred units in the recipient are *jealous1, *jealous1ϩSally, and *P3. Because jealous1, jealous1ϩBill, and Bill are all firing in synchrony in the driver-and because Bill excites Sally directly via their mapping connection-*jealous1, *jealous1ϩSally, and Sally will all be firing in synchrony with one another in the recipient. This coactivity causes LISA to learn excitatory connections between *jealous1 and *jealous1ϩSally and between Sally and *jealous1ϩSally. *Jealous1 also learns connections to active semantic units, that is, those representing jealous1. Similarly, *P3 will be active when *jealous1ϩSally is active, so LISA will learn an excitatory connection between them. Analogous operations cause the recipient to infer *jealous2 and *jealous2ϩCathy, and to connect them appropriately. That is, in response to jealous (Bill, Sam) in the driver, the recipient has inferred jealous (Sally, Cathy). This kind of learning is neither fully supervised, like backpropagation learning (Rumelhart, Hinton, & Williams, 1986) , in which exquisitely detailed error correction by an external teacher driver drives learning, nor fully unsupervised (e.g., Kohonen, 1982; Marshall, 1995; von der Malsburg, 1973) , in which learning is driven exclusively by the statistics of the input patterns. Rather, it is best described as self-supervised, because the decision about whether to license an analogical inference, and if so what inference to make, is based on the system's knowledge about the mapping between the driver and recipient. Thus, because LISA knows that Bill is jealous of Sam and because it knows that Bill maps to Sally and Sam to Cathy, it infers relationally, that Sally will be jealous of Cathy. This kind of self-supervised learning permits LISA to make inferences that lie well beyond the capabilities of traditional supervised and unsupervised learning algorithms. The same principles, augmented with the capacity to discover which elements are shared by the source and target, allow LISA to induce general schemas from specific examples (Hummel & Holyoak, 1996) . In the case of the current example, the schema to be induced is roughly "If person x loves person y, and y loves z, then x will be jealous of z." LISA induces this schema as follows. The schema starts out as an analog containing no token units. Tokens are created and interconnected in the schema according to exactly the same rules in the target: If nothing in the schema maps to the currently active driver units, then units are created to map to them. Because the schema is initially devoid of tokens, they are created every time a token fires for the first time in the driver. The behavior of the semantic units plays a central role in LISA's schema induction algorithm. In contrast to token units, semantic units represent objects and roles in a distributed fashion, so it is neither necessary nor desirable for them to inhibit one another. Instead, to keep their activations bounded between 0 and 1, semantic units normalize their inputs divisively: where n i is the net input to semantic unit i, and n in the net input to any semantic unit. That is, a semantic unit takes as its activation its own input divided by the maximum of the absolute value of all inputs or 1, whichever is greater. Albrecht and Geisler (1991), Bonds (1989) , and Heeger (1992) present physiological evidence for broadband divisive normalization in the visual system of the cat, and Foley (1994) and Thomas and Olzak (1997) present evidence for divisive normalization in human vision. Because of Equation 7, a semantic unit's activation is a linear function of its input (for any fixed value in the denominator). During mapping, units in the recipient feed activation back to the semantic units, so semantic units shared by objects and predicates in both the driver and recipient receive about twice as much input-and therefore become about twice as active-as semantic units unique to one or the other. As a result, object and predicate units in the schema learn stronger connections to shared semantic units than to unshared ones. That is, the schema performs intersection discovery on the semantics of the objects and relations in the examples from which it is induced. In the case of the jealousy example, the induced schema is "If person x loves person y, and y loves z, then x will be jealous of z," where x, y, and z are all connected strongly to semantic units for "person" but are connected more weakly (and approximately equally) to units for "male" and "female": They are "generic" people. (See Appendix B for detailed simulation results.) This algorithm also applies a kind of intersection discovery at the level of complete propositions. If LISA fails to fire a given proposition in the driver (e.g., because its pragmatic importance is low), then that proposition will never have the opportunity to map to the recipient. As a result, it will neither drive inference in the recipient nor drive the generation of a corresponding proposition in the emerging schema. In this way, LISA avoids inferring many incoherent or trivial details in the recipient and it avoids incorporating such details into the emerging schema. LISA's schema induction algorithm can be applied iteratively, mapping new examples (or other schemas) onto existing schemas, thereby inducing progressively more abstract schemas. Because the algorithm learns weaker connections to unique semantics than to shared semantics but does not discard unique semantics altogether, objects and predicates in more abstract schemas will tend to learn connections to semantic units with weights proportional to the frequency with which those semantics are present in the examples. For example, if LISA encountered 10 love triangles, of which 8 had a female in the central role-that is, y in loves (x, y) and loves (y, z)-with males in the x and z roles, then the object unit for y in the final schema induced would connect almost as strongly to female as to person, and x and z would likewise be biased toward being male. In this section, we illustrate LISA's ability to simulate a range of empirical phenomena concerning human relational inference and generalization. Table 3 provides a summary of 15 phenomena that we take to be sufficiently well-established that any proposed theory will be required to account for them. We divide the phenomena into those related to the generation of specific inferences about a target, those related to the induction of more abstract relational schemas, and those that involve interactions between schemas and inference. In LISA, as in humans, these phenomena are highly interconnected. Inferences can be made on the basis of prior knowledge (representations of either specific cases or more 234 HUMMEL AND HOLYOAK general schemas); schemas can be induced by bottom-up learning from comparison of examples, and existing schemas can modulate the encoding of further examples, thereby exerting top-down influence on subsequent schema induction. We make no claim that these phenomena form an exhaustive set. Our focus is on analogical inference as used in comprehension and problem solving, rather than more general rule-based reasoning, or phenomena that may be specific to language (although the core principles motivating LISA may well have implications for these related topics). Following previous analyses of the subprocesses of analogical thinking (Holyoak, Novick, & Melz, 1994) , we treat aspects of inference that can be viewed as pattern completion but exclude subsequent evaluation and adaptation of inferences (as the latter processes make open-ended use of knowledge external to the given analogs). We focus on schema-related phenomena that we believe require use of relational WM (reflective reasoning), rather than phenomena that likely are more automatic (reflexive reasoning; see Holyoak & Spellman, 1993; Shastri & Ajjanagadde, 1993) . Our central concern is with schemas that clearly involve relations between elements, rather than (possibly) simpler schemas or prototypes based solely on features of a single object (as have been discussed in connection with experimental studies of categorization; e.g., Posner & Keele, 1968) . Possible extensions of LISA to a broader range of phenomena, including reflexive use of schemas during comprehension, are sketched in the General Discussion section. Basic phenomena. Phenomena 1 and 2 are the most basic characteristics of analogical inference (see Holyoak & Thagard, 1995 , for a review of evidence). Inferences can be made using a single source analog (Phenomenon 1) by using and extending the mapping between source and target analogs (Phenomenon 2). The basic form of analogical inference has been called "copy with substitution and generation," or CWSG see also Falkenhainer et al., 1989), and involves constructing target analogs of unmapped source propositions by substituting the corresponding target element, if known, for each source element (Phenomenon 2a), and if no corresponding target element exists, postulating one as needed (Phenomenon 2b). This procedure gives rise to two important corollaries concerning inference errors. First, if critical elements are difficult to map (e.g., because of strong representational asymmetries, such as those that hinder mapping a discrete set of elements to a continuous variable; Bassok & Holyoak, 1989; , then no inferences can be constructed (Phenomenon 2c). Second, if elements are mismapped, predictable inference errors will result (Phenomenon 2d; Reed, 1987; see Holyoak et al., 1994). All major computational models of analogical inference use some variant of CWSG (e.g., Falkenhainer et al., 1989; Halford et al., 1994; Hofstadter & Mitchell, 1994; Holyoak et al., 1994; Keane & Brayshaw, 1988; Kokinov, 1994) , and so does LISA. Increasing confidence in the mapping from source to target (e.g., by increasing similarity of mapped elements) increases confidence in inferences from source to target. 4. An unmapped higher order relation in the source, linking a mapped with an unmapped proposition, increases the propensity to make an inference based on the unmapped proposition. 5. Pragmatically important inferences (e.g., those based on goal-relevant causal relations) receive preference. 6. When one-to-many mappings are possible, multiple (possibly contradictory) inferences are sometimes made (by different individuals or by a single individual). 7. Nonetheless, even when people report multiple mappings of elements, each individual inference is based on mutually consistent mappings. 8. Schemas preserve the relationally constrained intersection of specific analogs. 9. Schemas can be formed from as few as two examples. 10. Schemas can be formed as a side effect of analogical transfer from one source analog to a target analog. 11. Different schemas can be formed from the same examples, depending on which relations receive attention. 12. Schemas can be further generalized with increasing numbers of diverse examples. Interaction of schemas with inference 13. Schemas evoked during comprehension can influence mapping and inference. 14. Schemas can enable universal generalization (inferences to elements that lack featural overlap with training examples). 15. Schemas can yield inferences about dependencies between role fillers. CWSG is critically dependent on variable binding and mapping; hence models that lack these key elements (e.g., those based on back-propagation learning) fail to capture even the most basic aspects of analogical inference. LISA's ability to account for Phenomena 1 and 2 is apparent in its performance on the "jealousy" problem illustrated previously. As another example, consider a problem adapted from St. John (1992). The source analog states that Bill has a Jeep, Bill wanted to go to the beach, so he got into his Jeep and drove to the beach; the target states that John has a Civic, and John wanted to go to the airport. The goal is to infer that John will drive to the airport in his Civic. This problem is trivial for people to solve, but it poses a serious difficulty for models that cannot dynamically bind roles to their fillers. This difficulty is illustrated by the performance of a particularly sophisticated example of such a model, the Story Gestalt model of story comprehension developed by St. John (1992; St. John & McClelland, 1990) . In one computational experiment (St. John, 1992, Simulation 1), the Story Gestalt model was first trained with 1,000,000 short texts consisting of propositions based on 136 different constituent concepts. Each story instantiated a script such as "ϽpersonϾ decided to go to Ͻdesti-nationϾ; ϽpersonϾ drove ϽvehicleϾ to ϽdestinationϾ" (e.g., "George decided to go to a restaurant; George drove a Jeep to the restaurant"; "Harry decided to go to the beach; Harry drove a Mercedes to the beach"). After the model had learned a network of associative connections based on the 1,000,000 examples, St. John (1992) tested its ability to generalize by presenting it with a text containing a new proposition, such as "John decided to go to the airport." Although the proposition as a whole was new, it referred to people, objects, and places that had appeared in other propositions used for training. St. John reported that when given a new proposition about deciding to go to the airport, the model would typically activate the restaurant or the beach (i.e., the destinations in prior examples of the same script) as the destination, rather than making the contextually appropriate inference that the person would drive to the airport. This type of error, which would appear quite unnatural in human comprehension, results from the model's inability to learn generalized linking relationships (e.g., that if a person wants to go location x, then x will be the person's destination-a problem that requires the system to represent the variable x as well as its value, independently of its binding to the role of desired location or destination). As St. John noted, "developing a representation to handle role binding proved to be difficult for the model" (p. 294). LISA solves this problem using self-supervised learning (see Appendix B for the details of this and other simulations). As summarized in Table 4 , the source analog states that Bill has a Jeep (P1); Bill wanted (P3) to go to the beach (P2); therefore (P5), Bill drove his Jeep to the beach (P4). The target states that John has a Civic (P1) and John wanted (P3) to go to the airport (P2). When P2, go-to (Bill, beach), fires in the driver, it maps to P2, go-to (John, airport), in the recipient, mapping Bill to John, beach to airport, and the roles of go-to to their corresponding roles in the recipient. Likewise, has (Bill, Jeep) in the driver maps to has (John, Civic), and want (Bill, P2) maps to want (John, P2). However, when drive-to (Bill, beach) fires, only Bill and beach map to units in the recipient; the P unit (P4), its SPs, and the predicate units drive-to1 and drive-to2 do not correspond to any existing units in the recipient. Because of the mappings established in the context of the other propositions (P1-P3), every unit in the recipient has a positive mapping connection to some unit in the driver, but none map to P4, its SPs, or to the predicate units drive-to1 and drive-to2. As a result, these units globally inhibit all P, SP, and predicate units in the recipient. In response, the recipient infers new P, SP, and predicate units, and connects them together (into a proposition) via Hebbian learning. The resulting units represent the proposition drive-to (John, airport). That is, LISA has inferred that John will drive to the airport. Other inferences (e.g., the fact that John's decision to go to the airport caused him to drive to the airport) are generated by exactly the same kinds of operations. By the end, the recipient is structurally isomorphic with the driver and represents the idea "John wanted to go to the airport, so he drove his Civic to the airport" (see Appendix B for details). We have run this simulation many times, and the result is always the same: LISA invariably infers that John's desire to go to the airport will cause him to drive his Civic to the airport. As illustrated here, LISA makes these inferences by analogy to a specific episode (i.e., Bill driving to the beach). By contrast, most adult human reasoners would likely make the same inference on the basis of more common knowledge, namely, a schema specifying that when person x wishes to go to destination y, then x will drive to y (this is perhaps especially true for residents of Los Angeles). When the "Bill goes to the beach" example is replaced with this schema, LISA again invariably infers that John's desire to go to the airport will cause him to drive to the airport. Constraints on selection of source propositions for inference. Although all analogy models use some form of CWSG, it has been widely recognized that human analogical inference involves additional constraints (e.g., Clement & Gentner, 1991; Holyoak et al., 1994; Markman, 1997) . If CWSG were unconstrained, then any unmapped source proposition would generate an inference about the target. Such a loose criterion for inference generation would lead to rampant errors whenever the source was not isomorphic to the target (or a subset of the target), and such isomorphism will virtually never hold for problems of realistic complexity. Phenomena 3-5 highlight human constraints on the selection of source propositions for generation of inferences about the target. All these phenomena (emphasized in Holyoak & Thagard's, 1989 , multiconstraint theory of analogy) were demonstrated in a study by Lassaline (1996; see also Clement & Gentner, 1991; Spellman & Holyoak, 1996) . Lassaline had college students read analogs (mainly describing properties of hypothetical animals) and then rate various possible target inferences for "the probability that the conclusion is true, given the information in the premise" (p. 758). Participants rated potential inferences as more probable Table 4 The Airport/Beach Analogy  HUMMEL AND HOLYOAK when the source and target analogs shared more attributes, and hence mapped more strongly (Phenomenon 3). In addition, their ratings were sensitive to structural and pragmatic constraints. The presence of a higher order linking relation in the source made an inference more credible (Phenomenon 4). For example, if the source and target animals were both described as having an acute sense of smell, and the source animal was said to have a weak immune system that "develops before" its acute sense of smell, then the inference that the target animal also has a weak immune system would be bolstered relative to stating only that the source animal had an acute sense of smell "and" a weak immune system. The benefit conveyed by the higher order relation was increased (Phenomenon 5) if the relation was explicitly causal (e.g., in the source animal, a weak immune system "causes" its acute sense of smell), rather than less clearly causal ("develops before"). Two properties of the LISA algorithm permit it to account for these constraints on the selection of source propositions for generating inferences about the target. One is the role of importance and support relations that govern the firing order of propositions. More important propositions in the driver are more likely to be chosen to fire and are therefore more likely to drive analogical inference. Similarly, propositions with a great deal of support from other propositions-for example, by virtue of serving as arguments of a higher order causal relation-are more likely to fire (and therefore to drive inference) than those with little support. The second relevant property of the LISA algorithm is that inference is only attempted if at least 70% of the importance-weighted structures in the target map to structures in the source (see Equation  A15 in Appendix A). We simulated the findings of Lassaline (1996) by generating three versions of her materials, corresponding to her strong, medium, and weak constraint conditions. In all three simulations, the source analog stated that there was an animal with an acute sense of smell and that this animal had a weak immune system; the target stated that there was an animal with an acute sense of smell. Both the source and target also contained three propositions stating additional properties of this animal, plus three propositions stating the properties of an additional distractor animal. The propositions about the distractor animal served to reduce the probability of firing the various propositions about the critical animal. In the strong constraint simulation, the source stated that the crucial animal's acute sense of smell caused it to have a weak immune system. The propositions stating the acute sense of smell (P1), the weak immune system (P2), and the causal relation between them (P3) were all given an importance value of 5; the other six propositions had importance values of 1 (the default value; henceforth, when importance values are not stated, they are 1). In addition, propositions P1-P3 mutually supported one another with strength 5. No other propositions supported one another (the default value for support is 0). To simulate Lassaline's (1996) manipulation of the similarity relations among the animals, the critical animals in the source and target shared three properties in addition to their acute sense of smell (properties A, B, and C, stated in propositions P4 -P6 in the source and in propositions P2-P4 in the target). In the target, all the propositions describing the critical animal (P1-P4) had an importance value of 5. The remaining propositions (P5-P7) had an importance value of 1 and stated the properties of the distractor animal. The medium and weak constraint simulations were identical to the strong constraint simulation with the following changes. In the medium constraint simulation, the causal relation between acute smell and the weak immune system (P3) was replaced by the less constraining relation that the acute sense of smell preceded the weak immune system. The importance of the three statements (P1-P3) was accordingly reduced from 5 to 2, as was the support between them. In addition, the similarity between the critical animals in the source and target was reduced from three additional features (A, B, and C) to two (A and B) by giving feature C to the distractor animal. In the target, the propositions stating properties of the critical animal had an importance value of 2. In the weak constraint simulation, the causal relation between the acute sense of smell and the weak immune system (P3) was removed from the source altogether, and the importance of P1 (stating the acute sense of smell) and P2 (stating the weak immune system) were accordingly reduced to the default value of 1. In the target, features A, B, and C (propositions P2-P4) were assigned to the distractor animal rather than the critical animal, and their importance was dropped to 1. In all other respects (except a small difference in firing order, noted below), both the weak and medium constraint simulations were identical to the strong constraint simulation. We ran each simulation 50 times. On each run, the target was initially the driver, the source was the recipient, and LISA fired five propositions, randomly chosen according to Equation 1. Then the source was designated the driver and the target was designated the recipient. In the strong and medium constraint conditions, LISA first fired P1 (the acute sense of smell) in the source, and then fired five randomly chosen propositions. In the weak constraint condition, the source fired only the five randomly chosen propositions. This difference reflects our assumption that the acute sense of smell is more salient in the strong and medium conditions than in the weak condition. This kind of "volleying," in which the target is initially the driver, followed by the source, is the canonical flow of control in LISA (Hummel & Holyoak, 1997). As described above, LISA "decided" for itself whether to license analogical inference on the basis of the state of the mapping. LISA's inferences in the 50 runs of each simulation are summarized in Table 5 . In the strong constraint simulation, LISA inferred both that the critical target animal would have a weak immune system and that this condition would be caused by its Note. Values in the "Both" column represent the number of simulations (out of 50) on which Learning and Inference with Schemas and Analogies (LISA) inferred both the weak immune system and the higher order cause (strong constraint simulation) or preceed (medium constraint simulation) relation. Values in the "Weak," "Higher," and "Neither" columns represent the number of simulations on which LISA inferred the weak immune system only, the higher order relation only, and neither the weak immune system nor the higher order relation, respectively. INFERENCE AND GENERALIZATION acute sense of smell on 35 of the 50 runs; it inferred the weak immune system without inferring it would be caused by the acute sense of smell on 2 runs; on 9 runs it inferred that the acute sense of smell would cause something but failed to infer that it would cause the weak immune system. On 4 runs it failed to infer anything. LISA made fewer inferences, and notably fewer higher order inferences, in the medium constraint simulation. Here, on 15 runs it made both inferences; on 11 it inferred the weak immune system only; on 13 it inferred that the acute sense of smell would precede something but failed to infer what it would precede, and on 11 runs it made no inferences at all. LISA made the fewest inferences in the low-constraint simulation. On 14 runs it inferred the weak immune system (recall that the source in this simulation contains no higher order relation, so none can be inferred in the target), and on the remaining 36 runs it made no inferences. These simulations demonstrate that LISA, like Lassaline's (1996) subjects, is sensitive to pragmatic (causal relations), structural (higher order relations), and semantic (shared features) constraints on the selection of source propositions for generation of analogical inferences about the target. Inferences with one-to-many mappings. Phenomena 6 -8 involve further constraints on analogical inference that become apparent when the analogs suggest potential one-to-many mappings between elements. LISA, like all previous computational models of analogical mapping and inference, favors unique (one-to-one) mappings of elements. The one-to-one constraint is especially important in analogical inference, because inferences derived by using a mix of incompatible element mappings are likely to be nonsensical. For example, suppose the source analog "Iraq invaded Kuwait but was driven out" is mapped onto the target "Russia invaded Afghanistan; Argentina invaded the Falkland Islands." If the one-to-one constraint on mapping does not prevail over the positive evidence that Iraq can map to both Russia and Argentina, and Kuwait to both Afghanistan and the Falklands, then the CWSG algorithm might form the target inference, "Russia was driven out of the Falkland Islands." This "inconsistent" inference is not only false, but bizarre, because the target analog gives no reason to suspect that Russia had anything to do with the Falklands. Although all analogy models have incorporated the one-to-one constraint into their mapping algorithms, they differ in whether the constraint is inviolable. Models such as the structural mapping engine (SME; Falkenhainer et al., 1989) and the incremental analogy machine (IAM; Keane & Brayshaw, 1988) compute strict one-to-one mappings; indeed, without this hard constraint, their mapping algorithms would catastrophically fail. In contrast, models based on soft constraint satisfaction, such as the analogical constraint mapping engine (ACME; Holyoak & Thagard, 1989) and LISA (Hummel & Holyoak, 1997) , treat the one-to-one constraint as a strong preference rather than an inviolable rule. Empirical evidence indicates that when people are asked to map homomorphic analogs and are encouraged to give as many mappings as they consider appropriate, they will sometimes report multiple mappings for an element (Markman, 1997; Spellman & Holyoak, 1992 , 1996 . However, when several mappings form a cooperative set, which is incompatible with a "rival" set of mappings (much like the perceptual Necker cube), an individual reasoner will generally give only one consistent set of mappings; if one set is more relevant to the reasoner's goal, that set will be strongly preferred (Spellman & Holyoak, 1996) . Models with a "soft" one-to-one constraint can readily account for the tendency for people to settle on a single set of consistent mappings (see Spellman & Holyoak, 1996 , for simulations using ACME, and Hummel & Holyoak, 1997, for comparable simulations using LISA). Even if the analogy provides equal evidence in support of alternative sets of mappings, randomness in the mapping algorithm (coupled with inhibitory competition between inconsistent mappings) will tend to yield a consistent set of winning mappings. When people are asked to make inferences based on such homomorphic analogs, different people will find different sets of mappings, and hence may generate different (and incompatible) inferences (Phenomenon 6); but any single individual will generate consistent inferences (Phenomenon 7; Spellman & Holyoak, 1996) . LISA provides a straightforward account of these phenomena. LISA can discover multiple mappings for a single object, predicate or proposition, in the sense that it can discover different mappings on different runs with an ambiguous analogy (such as those of Spellman & Holyoak, 1996) . However, on any given run, it will always generate inferences that are consistent with the mappings it discovers: Because inference is driven by mapping, LISA is incapable of making an inference that is inconsistent with the mappings it has discovered. To explore LISA's behavior when multiple mappings are possible, we have extended the simulations of the results of Holyoak (1996, as cited in Hummel & Holyoak, 1997) to include inference as well as mapping. In Spellman and Holyoak's Experiment 3, participants read plot summaries for two elaborate soap operas. Each plot involved various characters interconnected by professional relations (one person was the boss of another), romantic relations (one person loved another), and cheating relations (one person cheated another out of an inheritance; see Table  6 ). Each of these relations appeared once in Plot 1 and twice in Plot 2. Plot 1 also contained two additional relations not stated in Plot 2: courts, which follows from the loves relation, and fires, which follows from the bosses relation. Solely on the basis of the bosses and loves relations, Peter and Mary in Plot 1 are four-ways ambiguous in their mapping to characters in Plot 2: Peter could map to any of Nancy, John, David, or Lisa, as could Mary. If the bosses relation is emphasized over the loves relation (i.e., if a processing goal provides a strong reason to map bosses to bosses and hence the characters in the bosses propositions to one another), then Peter should map to either Nancy or David and Mary would correspondingly map to either John or Lisa. There would be no way to choose between these two alternative mappings (ignoring gender, which in the experiment was controlled by counterbalancing). But if the cheats relations are considered, then they provide a basis for selecting unique mappings for Peter and Mary: Peter maps to Nancy, so Mary must map to John. Similarly, if the loves propositions are emphasized (absent cheats), then Peter should map to John or Lisa and Mary would map to Nancy or David; if the cheats propositions are considered in addition to loves, then Peter should map to Lisa and hence Mary to David. Spellman and Holyoak (1996, as cited in Hummel & Holyoak, 1997) had participants make plot extensions of Plot 2 on the basis of Plot 1. That is, participants were encouraged to indicate which specific characters in Plot 2 were likely to court or fire one another, 238 HUMMEL AND HOLYOAK on the basis of their mappings to the characters in Plot 1. Spellman and Holyoak manipulated participants' processing goals by encouraging them to emphasize either the bosses or the loves relations in extending Plot 2 on the basis of Plot 1; the cheating relations were always irrelevant. Participants' preferred mappings were revealed on this plot-extension (i.e., analogical inference) task by their choice of Plot 2 characters to play roles analogous to those played by Peter and Mary in Plot 1. Afterwards, participants were asked directly to provide mappings of Plot 2 characters onto those in Plot 1. Spellman and Holyoak's (1996) results are presented in the left column of Figure 8 , which depicts the percentage of mappings for the pair Peter-Mary that were consistent with respect to the processing goal (collapsing across focus on professional vs. romantic Table 6 LISA's Schematic Propositional Representations of the Relations Between Characters in Spellman and Holyoak's (1996)  INFERENCE AND GENERALIZATION relations), inconsistent with the processing goal, or some other response (e.g., a mixed mapping). The goal-consistent and goalinconsistent mappings are further divided into those consistent versus inconsistent with the irrelevant cheating relation. The plotextension and mapping tasks were both sensitive to the processing goal, in that participant's preferred mappings in both tasks tended to be consistent with the goal-relevant relation. This effect was more pronounced in the plot-extension than the mapping task. In contrast, consistency with the cheating relation was more pronounced in the mapping than in the plot-extension task. To simulate these findings, we gave LISA schematic representations of the two plots based on the propositions in Table 6 (see Appendix B for details). We generated four levels of pragmatic focus: High emphasis on the professional relation (in which LISA was strongly encouraged to focus on the bosses relation over the loves relation; P-high); low emphasis on the professional relation (in which LISA was weakly encouraged to focus on bosses over loves; P-low); high emphasis on the romantic relation (in which LISA was strongly encouraged to focus on loves over bosses; R-high); and low emphasis on the romantic relation (in which LISA was weakly encouraged to focus on loves over bosses; R-low). The high-emphasis simulations (P-high and R-high) were intended to simulate Spellman and Holyoak's (1996) plot extension task, and the low-emphasis simulations (P-low and R-low) were intended to simulate their mapping task. In all four simulation conditions, propositions in the source (Plot 1) and target (Plot 2) were assigned support relations reflecting their causal connectedness and shared predicates and arguments. In the source, P1 and P4 supported one another with strength 100 (reflecting their strong causal relation), as did P2 and P5. These support relations were held constant across all simulations, reflecting our assumption that the relations among statements remain relatively invariant under the reasoner's processing goals. The only factor we varied across conditions was the importance values of individual propositions (see Appendix B). The Spellman and materials are substantially more complex than the Lassaline (1996) materials, in the sense that the Spellman and Holyoak materials form multiple, mutually inconsistent sets of mappings. Accordingly, we set the importance and support relations to more extreme values in these simulations in order to strongly bias LISA in favor of discovering coherent sets of mappings. The corresponding theoretical claim is that human reasoners are sensitive to the pragmatic and causal interrelations between elements of analogs and can use those interrelations to divide complex analogs (such as the Spellman and Holyoak, 1996, materials) into meaningful subsets. (Actually, the differences in the parameter values are less extreme than they may appear because LISA's behavior is a function of the propositions' relative values of support and importance, rather than the absolute values of these parameters; recall Equations 1-4.) In the P-high simulations, bosses (Peter, Mary) in Plot 1 was given an importance value of 20, as were bosses (Nancy, John) and bosses (David, Lisa) in Plot 2; all other propositions in both plots were given the default value of 1. In the P-low simulation, all the bosses propositions (i.e., bosses (Peter, Mary) in the source and the corresponding propositions in the target) had an importance of 2 (rather than 20, as in P-high). In the R-high and R-low simulations, the importance values were precisely analogous to those in P-high and P-low, respectively, except that the emphasis was on the romantic relations rather than the professional ones. We ran each simulation 48 times. LISA's mapping performance in the high-and low-focus conditions (summed over romantic and professional relations) is shown in the second column of Figure 8 . We did not attempt to produce a detailed quantitative fit between the simulation results and human data, but a close qualitative correspondence is apparent. In the high-pragmatic-focus simulations, LISA's Peter-Mary mappings were almost always goalconsistent but were independent of the irrelevant cheating relation. In the low-focus simulations, LISA was again more likely to produce mappings that were goal-consistent, rather than goalinconsistent, but this preference was weaker than in the high-focus simulations. However, in the low-pragmatic-focus simulations LISA more often selected mappings that were consistent with the cheating relation. Thus LISA, like people, produced mappings that were dominated by the goal-relevant relation alone when pragmatic focus was high but that were also influenced by goalirrelevant relations when pragmatic focus was low. LISA's most notable departure from the human data is that it was less likely to report "other" mappings-that is, mappings that are inconsistent with the structure of the analogy. We take this departure from the human data to reflect primarily random factors in human performance (e.g., lapses of attention, disinterest in the task) to which LISA is immune. It is important to note that LISA's inferences (i.e., plot extensions) in the high-focus conditions were always consistent with its mappings. In the P-high simulations, if LISA mapped Peter to David and Mary to Lisa, then it always inferred that David would fire Lisa; if it mapped Peter to Nancy and Mary to John, then it inferred that Nancy would fire John. In the R-high simulations, if it mapped Peter to John and Mary to Nancy, then it inferred that John would court Nancy, and if it mapped Peter to Lisa and Mary to David, then it inferred that Lisa would court David. These results highlight three important properties of LISA's mapping and inference algorithm. First, as illustrated by the results of the high-pragmatic-focus simulations, LISA will stochastically produce different mappings, consistent with the goal-relevant relation but less dependent on the goal-irrelevant relations, when the latter are only given a limited opportunity to influence mapping. Second, as illustrated by the low-pragmatic-focus simulations, differences in selection priority are sufficient (when no single relation is highly dominant) to bias LISA in one direction or another in resolving ambiguous mappings. And third, even though LISA is capable of finding multiple different mappings in ambiguous situations, its analogical inferences are always consistent with only a single mapping. We now consider how LISA accounts for several phenomena related to the induction of relational schemas. There is a great deal of empirical evidence that comparison of multiple analogs can result in the induction of relationally constrained generalizations, which in turn facilitate subsequent transfer to additional analogs (Phenomenon 8). The induction of such schemas has been demonstrated in both adults (Catrambone & Holyoak, 1989; Gick & Holyoak, 1983; Ross & Kennedy, 1990) and young children (Brown, Kane, & Echols, 1986; Holyoak, Junn, & Billman, 1984) . We have previously shown by simulation that LISA correctly predicts that an abstract schema will be retrieved more readily than a remote analog (Hummel & Holyoak, 1997) , by using the convergence analogs discussed by Gick and Holyoak (1983) . Here we extend those simulations to show how LISA induces schemas from specific analogs. People are able to induce schemas by comparing just two analogs to one another (Phenomenon 9; Gick & Holyoak, 1983) . Indeed, people will form schemas simply as a side effect of applying one solved source problem to one unsolved target problem (Phenomenon 10; Novick & Holyoak, 1991; Ross & Kennedy, 1990) . We illustrated LISA's basic capacity for schema induction previously, with the example of the jealousy schema. We have also simulated the induction of a schema using the airport/beach analogy taken from St. John and McClelland (1990) , discussed earlier. For this example, the schema states (roughly): "Person x owns vehicle y. Person x wants to go to destination z. This causes person x to drive vehicle y to destination z." Because both people in the examples from which this schema was induced were male, person x is also male. But with a few examples of females driving places, the schema would quickly become gender neutral. The previous simulations are based on simple "toy" analogies. As a more challenging test of LISA's ability to induce schemas from examples, we gave it the task of using a source "convergence" problem-the "fortress" story of Holyoak (1980, 1983)-to solve Duncker's (1945) "radiation" problem, and then we examined the schema formed in the aftermath of analogical transfer. The simulations started with representations of the radiation and fortress problems that have been used in other simulations of the fortress/radiation analogy (e.g., Holyoak & Thagard, 1989; Kubose et al., 2002) . The details of the representations are given in Appendix B. The fortress analog states that there is a general who wishes to capture a fortress that is surrounded by friendly villages. The general's armies are powerful enough to capture the fortress, but if he sends all his troops down any single road leading to the fortress, they will trigger mines and destroy the villages. Smaller groups of his troops would not trigger the mines, but no small group by itself would be powerful enough to capture the fortress. The general's dilemma is to capture the fortress without triggering the mines. The solution, provided in the statement of the problem, is for the general to break his troops into many small groups and send them down several roads to converge on the fortress in parallel, thereby capturing the fortress without triggering the mines. The radiation problem states that a doctor has a patient with an inoperable tumor. The doctor has a device that can direct a beam of radiation at the tumor, but any single beam powerful enough to destroy the tumor would also destroy the healthy tissue surrounding it; any beam weak enough to spare the healthy tissue would also spare the tumor. The doctor's dilemma is to destroy the tumor without also destroying the healthy tissue. The solution, which is not provided in the statement of the problem, is to break the beam into several smaller beams which, converging on the tumor from multiple directions, will spare the healthy tissue but destroy the tumor. Given this problem, LISA induced a more general schema based on the intersection of the fortress and radiation problems. The detailed representation of the resulting schema is given in Appendix B. In brief, the schema states that there is an object (the fortress/tumor) surrounded by another object (the villages/healthy tissue), and a protagonist (general/doctor) who wants to destroy the inner object without harming the surrounding object. The protagonist can form small forces (units/beams), and if he uses small forces, he will destroy the inner object. Given the simplified representation of the problem provided to LISA, this is the correct schema to have induced from the examples. Although people can form schemas in the aftermath of analogical transfer, the schemas formed will differ depending on which aspects of the analogs are the focus of attention (Phenomenon 11). In the case of problem schemas, more effective schemas are formed when the goal-relevant relations are the focus rather than incidental details (Brown et al., 1986; Brown, Kane, & Long, 1989; Gick & Holyoak, 1983) . Again borrowing from the materials used by Kubose et al. (2002) , we gave LISA the fortress/ radiation analogy, but this time we biased the simulation against LISA's finding an adequate mapping between them. Specifically, we used the unsolved radiation problem (rather than the solved fortress problem) as the driver, and we set the propositions' importance and support relations so that LISA's random firing algorithm would be unlikely to fire useful propositions in a useful order (see Appendix B and Kubose et al., 2002). On this simulation, LISA did not infer the correct solution to the general problem, and it did not induce a useful schema (see Appendix B). It is interesting to note that it inferred that there is some solution to the convergence problem, but it was unable to figure out what that solution is: P unit P15 in the schema states if-then (P16, P4), where P4 was a statement of the goal, *destroyed (*tumor), and P16 presumably should specify an act that will satisfy the goal. However, P16 has no semantic content: It is a P unit without roles or fillers. Two examples can suffice to establish a useful schema, but people are able to incrementally develop increasingly abstract schemas as additional examples are provided (Phenomenon 12; Brown et al., 1986 Brown et al., , 1989 Catrambone & Holyoak, 1989) . We describe a simulation of the incremental acquisition of a schema in the next section. As schemas are acquired from examples, they in turn guide future mappings and inferences. Indeed, one of the basic properties ascribed to schemas is their capacity to fill in missing information (Rumelhart, 1980) . Here we consider a number of phenomena involving the use of a relationally constrained abstraction to guide comprehension of new examples that fit the schema. Schema-guided mapping and inference. Bassok, Wu, and Olseth (1995) have shown that people's mappings and inferences are guided by preexisting schemas (Phenomenon 13). These investigators used algebra word problems (permutation problems) similar to ones studied previously by Ross (1987 Ross ( , 1989 . Participants were shown how to compute permutations using an example (the source problem) in which items from a set of n members were assigned to items from a set of m members (e.g., how many different ways can you assign computers to secretaries if there are n computers and m secretaries?). They were then tested for their ability to transfer this solution to new target problems. The critical basis for transfer hinged on the assignment relation in each analog-that is, what kinds of items (people or inanimate objects) served in the roles of n and m. In some problems (type OP) objects (n) were assigned to people (m; e.g., "computers were assigned to secretaries"); in 241 INFERENCE AND GENERALIZATION others (PO) people (n) were assigned to objects (m; "secretaries were assigned to computers"); and in others (PP) people (n) were assigned to other people (m) of the same type ("doctors were assigned to doctors"). Solving a target problem required the participant to map the elements of the problem appropriately to the variables n and m in the equation for calculating the number of permutations. Although all the problems were formally isomorphic, argued that people will typically interpret an "assign" relation between an object and a person as one in which the person "gets" the object (i.e., assign (object, person) 3 get (person, object)). It is important to note that the get schema favors interpreting the person as the recipient of the object no matter which entity occupies which role in the stated "assign" relation. As a result, assign (person, object), although less natural than the reversed statement, will also tend to be interpreted as implying get (person, object). In contrast, because people do not typically own or "get" other people, an "assign" relation involving two people is likely to be interpreted in terms of a symmetrical relation such as "pair" (i.e., assign (person1, person2) 3 pair (person1, person2)). These distinct interpretations of the stated "assign" statement yielded systematic consequences for analogical mapping. found that, given an OP source analog, people tended to link the "assigned" object to the "received" role (rather than the "recipient" role) of the get schema, which in turn was then mapped to the mathematical variable n, the number of "assigned" objects. As a result, when the target analog also had an OP structure, transfer was accurate (89%); but when the target was in the reversed PO structure, the set inferred to be n was systematically wrong (0% correct)-that is, the object set continued to be linked to the "received" role of "get," and hence erroneously mapped to n. When the source analog was in the less natural PO form, transfer was more accurate to a PO target (53%) than to an OP target (33%). The overall reduction in accuracy when the source analog had the PO rather than the OP form was consistent with people's general tendency to assume that objects are assigned to people, rather than vice versa. A very different pattern of results obtained when the source analog was in the PP form, so as to evoke the symmetrical pair schema. These symmetrical problems did not clearly indicate which set mapped to n; hence accuracy was at chance (50%) when the target problem was also of type PP. When the target problem involved objects and people, n was usually determined by following the general preference that the assigned set be objects, not people. This preference led to higher accuracy for OP targets (67%) than for PO targets (31%). We simulated two of Bassok et al.'s (1995) conditionsnamely, target OP and target PO, both with source OP-as these conditions yielded the clearest and most striking difference in performance (89% vs. 0% accuracy, respectively). Following the theoretical interpretation of Bassok et al., the simulations each proceeded in two phases. In the first phase, the target problem (OP or PO) was mapped onto the get schema, and the schema was allowed to augment the representation of the target (by building additional propositions into the target). In the second phase, the augmented target was mapped onto the source problem in order to map objects and people into the roles of n and m. The goal of these simulations was to observe how LISA maps the objects and people in the stated problem onto n and m. As LISA cannot actually solve permutations problems, we assumed that mapping objects and people to n and m in the mathematically correct fashion corresponds to finding the correct solution to the problem, and mapping them incorrectly corresponds to finding the incorrect solution. In all the simulations, the get schema contained two propositions. P1 stated that an assigner assigned an object to a recipientthat is, assign-to (assigner, object, recipient)-where the second (object) and third (recipient) roles of the assignment relation are semantically very similar (sharing four of five semantic features; see Appendix B), but the arguments object and recipient are semantically very different. The semantic features of object strongly bias it to map to inanimate objects rather than people, and the features of recipient bias it to map to people rather than inanimate objects. As a result, an assignment proposition, assign-to (A, B, C) , in a target analog will tend to map B to object if it is inanimate and to recipient if it is a person; likewise, C will tend to map to object if it is inanimate and to recipient if it is a person. Because the second and third roles of assign-to are semantically similar, LISA is only weakly biased to preserve the role bindings stated in the target proposition (i.e., mapping B in the second role to object in the second role, and C in the third to recipient in the third). It is thus the semantics of the arguments of the assign-to relation that implement Bassok et al.'s (1995) assumption that people are biased to assign objects to people rather than vice versa. The second proposition in the get schema, P2, states that the recipient will get the object-that is, get (recipient, object). Whichever argument of the target's assign-to relation maps to recipient in the schema's assign-to will also map to recipient in the context of gets. 8 Likewise, whichever argument maps to object in the context of assign-to will also map to object in get. For example, the target proposition assign-to (boss, computers, secretaries) will map computers to object, and secretaries to recipient, leading to the (correct) inference get (secretaries, computers). However, the target proposition assign-to (boss, secretaries, computers)-in which computers "get" secretaries-will also map computers to object, and secretaries to recipient, leading to the incorrect inference get (secretaries, computers). In both the OP and PO simulations, the source problem (an OP problem) contained four propositions. A teacher assigned prizes to students (P1 is assign-to (teacher, students, prizes)), and students got prizes (P2 is get (students, prizes)). Thus, prizes instantiate the variable n (P3 is instantiates (prizes, n)), and students instantiate the variable m (P4 is instantiates (students, m); see Appendix B for details). In the OP simulation, the target started with the single proposition assign-to (boss, secretaries, computers). In the first phase of the simulation, this proposition was mapped onto the get schema (with the target serving as the driver), mapping secretaries to recipient and computers to object. The schema was then mapped back onto the target, generating (by self-supervised learning) the proposition get (secretaries, computers) in the target. In the second phase of the simulation, the augmented target was mapped onto the 8 Recall that, within an analog, the same object unit represents an object in all propositions in which that object is mentioned. Thus, in the schema, the object unit that represents recipient in assign-to (assigner, recipient, object) also represents recipient in gets (recipient, object). Therefore, by mapping to recipient in the context of assign-to, an object in the target automatically maps to recipient in gets as well. HUMMEL AND HOLYOAK source. As a result of their shared roles in the get relation, secretaries mapped to students and computers to prizes. The source was then mapped back onto the target, generating the propositions instantiates (computers, n) and instantiates (secretaries, m). That is, in this case, LISA correctly assigned items to variables in the target problem by analogy to their instantiation in the source problem. The PO simulation was directly analogous to the OP simulation except that the target problem stated that the boss assigned secretaries to computers-that is, assign-to (boss, computers, secretaries)-reversing the assignment of people and objects to n and m. As a result, in the first phase of the simulation, the get schema augmented the target with the incorrect inference get (secretaries, computers). When this augmented target was mapped to the source, LISA again mapped secretaries to students and computers to prizes (based on their roles in get). When the source was mapped back onto the target, it generated the incorrect inferences instantiates (computers, n) and instantiates (secretaries, m). Both the OP and PO simulations were repeated over 20 times, and in each case LISA behaved exactly as described above: Like Bassok et al. 's (1995) subjects, LISA's ability to correctly map people and objects to n and m in the equation for permutations was dependent on the consistency between the problem as stated (i.e., whether objects were assigned to people or vice versa) and the model's expectations as embodied in its get schema. These simple simulations demonstrate that LISA provides an algorithmic account of how people's mappings and inferences are guided by preexisting schemas (Phenomenon 13). Universal generalization. As we have stressed, the defining property of a relational representation (i.e., symbol system) is that it specifies how roles and fillers are bound together without sacrificing their independence. This property of symbol systems is important because it is directly responsible for the capacity for universal generalization (Phenomenon 14): If and only if a system can (a) represent the roles that objects play explicitly and independently of those objects, (b) represent the role-filler bindings, and (c) respond on the basis of the roles objects play, rather than the semantic features of the objects themselves, then that system can generalize responses learned in the context of one set of objects universally to any other objects. We illustrate this property by using LISA to simulate the identity function (see also Holyoak
Although LISA REF can model some analogical inferences, the number of relations (see Table 3 ) in these analogies is beyond the number of relationships it can currently handle (2 or 3).
A symbolic-connectionist theory of relational inference and generalization
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