Analogy in Context

Context appears in two forms in analogy. First, there is the context in which the analogy is performed. Second, each statement being compared is done so within the context of each analogue's overall description. By explicitly taking context into account, we can provide a more robust account of analogy. Central to this account is that each element of an analogue's description has an identifiable role, corresponding to the dependencies it satisfies or its relevant properties in the given context . The relations comprising each analogue are placed in correspondence by virtue of their filling corresponding roles . This work makes three principle contributions . First, it extracts a set of core principles underlying reasoning from similarity which enables correspondences between syntactically distinct expressions and the controlled introduction of many-to-many mappings . Second, it uses these principles to provide a unifying look at a variety of recent analogical and case-based reasoning systems. Third, it introduces the map and analyze cycle, a general computational framework for computing similarity correspondences and producing useful inferences . The work is demonstrated with implemented examples, including classification, causal explanation, and a number of examples from the literature . When plotting the perfect murder, how is an icicle analogous to a knife? When packing an ice-chest, how is an icicle analogous to an ice-cube? Icicles, ice cubes, knives, and spoons have numerous properties . Their relative similarities vary according to the context in which the similarity is assessed . Analogical reasoning is a process in which similarities between two situations are identified and used to suggest that knowledge of one situation may also apply to the other . Yet, by what criterion should similarity be judged? How does similarity lend credibility to this kind of knowledge transfer? Analogical similarity is typically defined in terms of matching patterns in the descriptions of two analogues, seeking both isomorphic structures and features described by the same terms [16, 21, 25, 33, 44] . Thus, if our icicle and knife were explicitly de scribed in their respective narratives as hard and pointed objects, their similarity would be identified via these common terms . This view of analogy is amenable to simple computational frameworks and has demonstrated some success on a variety of tasks . It rests on the fundamental assumption that corresponding aspects of two analogues are represented in the same manner and may be identified using simple, syntactic matching criteria . As task complexity and representational sophistication grow, this assumption breaks down. As we will demonstrate, similarity criteria based on one-to-one mappings between identical terms are too restrictive and brittle. Furthermore, existing solutions based on the conceptual closeness of terms [5, 27, 43, 44] (e.g ., generalization hierarchies) are too weak; they can produce unmotivated and incorrect similarity correspondences. Determining similarity requires understanding which aspects of the terms being compared are relevant in a given context. This insight was crucial to the success of PHINEAS [12, 13, 14], a program that constructs causal explanations of observed phenomena based on their similarity to understood phenomena. As we will show, similar insights may be found in several other recently developed systems as well [3, 30, 32, 34] . This paper uses the insights gained from recent efforts to reexamine the principles underlying analogical similarity. It describes contextual structure-mapping (CSM), a knowledge-level account of similarity that attempts to explain the basic principles by which reasoning from similarity is meaningful and identify the implicit assumptions embodied in existing systems. CSM provides a general characterization of similarity that allows the controlled introduction of many-to-many mappings and matches between syntactically distinct expressions . This account is based on the notion that each element of an analogue's description has an identifiable role, corresponding to the dependencies it satisfies or its relevant properties in the given context. The relations comprising each analogue correspond by virtue of their filling corresponding roles . The notion of role includes a component's participation in a given behavior, design decisions achieving a design's rationale, an actor's actions and traits achieving a story's plot, and an antecedent justifying its consequents in a logical proof. For example, in the context of a murder mystery, an icicle is analogous to a knife in the role of being an effective murder weapon because they have the requisite properties, not because they are conceptually close along some dimension. In the context of packing an ice chest, different properties become relevant and an icicle may become analogous to an ice cube. The second part of this paper shows how contextual structure-mapping serves to unify and explain some of the intuitions behind a variety of recently developed approaches, including derivational replay [6, 37], tweaking [30, 29], knowledge-based pattern match ing [3], and my work on analogical explanation [12, 13] . It shows that each technique is an attempt to exploit some aspect of role and context in analogy. The third part of this paper uses this characterization of similarity to examine the computational requirements of an analogical reasoning system. It describes the map and analyze cycle, a general computational framework for computing correspondences and producing useful inferences . Initial correspondences between the two analogue's given descriptions are found using simple pattern matching operations . These correspondences are extended by analyzing incomplete portions of the analogy, seeking ad1 .1 Objectives (a) (b) Figure 1 : Two thermostats . ditional information about the roles of unmatched items. An implementation using the structure-mapping engine (SME) [15, 16] is described. While noting similarities between two situations is a central aspect of analogy, we are most interested in its use for problem solving, in which knowledge of one analogue (the base) is used to answer questions about the other (the target) . Analogy begins when a base analogue that has potentially relevant similarities to the target is retrieved (or directly supplied, as by a teacher) . Ultimately, it may lead to memory reorganization, comparative evaluation of competing hypotheses, and continued use of the base analogue to further explore its consequences for the target domain . This paper is concerned with what lies in between how similarities are elaborated and used to solve an open problem once a candidate base analogue has been identified .' It presents a general model, applicable to across domain comparisons (with within-domain comparisons as a special case), that may be used to either accelerate problem solving or produce plausible inferences to overcome an incomplete domain theory . For example, consider the thermostat shown in Figure 1(a) . This device regulates a heater to maintain a specified temperature . Temperature deviation is sensed by the coil, which expands and contracts as it heats up and cools down, respectively. The furnace is on when the mercury in the glass tube electrically connects the two terminals, which keeps the valve open . As the environment temperature increases, the coil expands and the glass tube's angle increases . Eventually, the temperature reaches a point where the mercury moves to the right, disconnecting the terminals and closing the valve . The 'See [13, 14] for a description of PHIHEAS, a complete analogical reasoning system that uses CSM in conjunction with retrieval and hypothesis evaluation mechanisms . lever position controls the device's goal temperature by shifting the relationship between temperature and position . By analogy, how does device (b) operate as a thermostat? Intuitively, our understanding of device (a) should make this explanation easier . Like the coil, the bimetallic rod senses temperature deviation via expansion and contraction . When in the on position, gas flows past its end. As the temperature increases, the rod lengthens and acts as a valve to shut off the gas flow to the heater. The dial acts like device (a)'s lever to control the equilibrium temperature . The spring stabilizes the dial by increasing the friction on its threads. 1 .2 Analogy in context What are the principles and methods that enable the explanation of device (a) to assist, and perhaps make possible, the process of explaining device (b)? Clearly, one aspect is identifying possible correspondences between their descriptions . In comparing two analogues, most methods only match expressions that use the same predicate (e.g ., the two furnaces). However, differences between cases or domains often lead to different predicates describing analogous concepts . If changes are to be allowed in response to differences in the two analogues' descriptions, what constitutes an acceptable change? The common approach is to measure conceptual closeness, using a generalization hierarchy [44, 5, 25, 43] or a-priori similarity score [27] . When comparing the two thermostats, this approach might suggest a correspondence between device (a)'s lever and device (b)'s dial (both adjustment knobs) . However, it might also suggest a correspondence between device (a)'s coil and device (b)'s spring (both springs) . Yet, their similarity is irrelevant and misses the coil's role with respect to the device's components and teleology. The coil's relevant aspect in this context is its thermal expansion characteristics the same aspect that is relevant to understanding device (b)'s bimetallic rod . When an expression is used in some context (e.g ., in a chain of inference), it denotes certain characteristics about the world important for that context. Without knowing which effects are relevant, there is no way to know which generalizations or similarities are appropriate. Thus, objects and relations are identified as being similar by examining their roles in their respective analogue descriptions. To do this, we must clearly understand what a role is, how role information affects similarity assessment, and how incomplete role information affects the validity of an analogically derived conclusion . In addition to predicate similarity, constraints on analogical mapping tend to require that the mapping be one-to-one . However, in device (a), the coil and the valve provide temperature sensing and gas control, respectively, while both functions are provided by device (b)'s rod . Here, an isomorphic mapping fails to fully capture the correspondence ; a many-to-one mapping from {coil, valve} to {rod} is needed . Due to function sharing, many-to-many mappings are common in physical systems . Yet, allowing these mappings can dramatically increase computational cost and lead to incoherent mappings . Thus, the conditions under which the one-to-one restriction should be relaxed must be precisely defined . These conditions come from explicitly considering the roles of compared items . If an expression has more than one role, then each role may lead to a different analogical correspondent . Finally, several mappings may be possible and some criteria must be used to select the "best" one. The thermostats are relatively unambiguous, but at least two interpretations are conceivable: one in which the coil is mapped to the spring and one in which the coil is mapped to the rod. Evaluation criteria proportional to match size, such as Gentner's [21] systematicity, could prefer the former interpretation if the thermostats' descriptions included detailed descriptions of the coil and spring . However, the purpose of the analogy in this context is to explain the devices' overall behavior as thermostats, and only the second interpretation achieves this . Current goals should have a strong influence over which matches are preferred . The process of computing similarities is traditionally depicted As a form of pattern matching between base and target descriptions . However, in realistic memories, not all that is inferrably known about the base and target is explicit in their initial descriptions . Furthermore, they may not be described using the same terminology (e.g ., coil and rod rather than sensor) . Thus, the process of elaborating correspondences andadapting elements of the base to fit the target situation often requires inferring additional information in response to mapping impasses. Because pattern matching alone can be expensive, and there are many (potentially infinite) inferences that can be made to extend either analogue, this process must be tightly focused and goal directed. Therefore, we decompose it to form a map and analyze cycle (Figure 2) : use simple matching criteria to determine the best, initial mapping between the analogues, analyze the results and seek additional relevant information about unmatched areas, reexamine the mapping to determine the information's impact on the mapping (i .e ., extensions or complete shifts), analyze the new results, etc . In this manner, only when impasses arise, such as an expression having no apparent correspondent in the other analogue, is the domain theory consulted and more detail about that expression and its role sought . AdditionBase analogue Target case AB2 Base analogue Target case Map ` ` espa"iow 8( Analyze Cycle Identify direct matches Analyse unmapped relation's roles to adapt to target case Figure 2 : Map and analyze cycle. Analogy involves comparing two representation structures . Obvious correspondences may motivate additional correspondences according to the corresponding elements' roles. The task is complicated by abstracted or implicit reasoning steps in the analogue descriptions, which often must be further analyzed to decompose an expression's roles into finer levels of detail . The map and analyze cycle focuses more in-depth reasoning only where and when needed to converge to a point where the match can no longer be extended .