Analogy--Watershed or Waterloo? 
Structural alignment and the development of 
connectionist models of analogy 
Dedre Gentner Arthur B. Markman 
Department of Psychology Department of Psychology 
Northwestern University Northwestern University 
2029 Sheridan Rd. 2029 Sheridan Rd. 
Evanston, IL 60208 Evanston, IL 60208 
ABSTRACT 
Neural network models have been criticized for their inability to make 
use of compositional representations. In this paper, we describe a series 
of psychological phenomena that demonstrate the role of structured 
representations in cognition. These findings suggest that people 
compare relational representations via a process of structural alignment. 
This process will have to be captured by any model of cognition, 
symbolic or subsymbolic. 
1.0 INTRODUCTION 
Pattern recognition is central to cognition. At the perceptual level, we notice key features 
of the world (like symmetry), recognize objects in front of us and identify the letters on a 
printed page. At a higher level, we recognize problems we have solved before and 
determine similarities--including analogical similarities--between new situations and old 
ones. Neural network models have been successful at capturing sensory pattern 
recognition (e.g., Sabourin & Mitiche, 1992). In contrast, the determination of higher 
level similarities has been well modeled by symbolic processes (Falkenhainer, Forbus, & 
Gentner, 1989). An important question is whether neural networks can be extended to 
high-level similarity and pattern recognition. 
In this paper, we will summarize the constraints on cognitive representations suggested 
by the psychological study of similarity and analogy. We focus on three themes: (1) 
structural alignment; (2) structural projection; and (3) flexibility. 
855 
856 Gentner and Markman 
2.0 STRUCTURAL ALIGNMENT IN SIMILARITY 
Extensive psychological research has examined the way people compare pairs of items to 
determine their similarity. Mounting evidence suggests that the similarity of two 
complex items depends on the degree of match between their component objects (common 
and distinctive attributes) and on the degree of match between the relations among the 
component objects. Specifically, there is evidence that (1) similarity involves structured 
pattern matching, (2) similarity involves structured pattern completion, (3) comparing the 
same item with different things can highlight different aspects of the item and (4) even 
comparisons of a single pair of items may yield multiple interpretations. We will 
examine these four claims in the following sections. 
2.1 SIMILARITY INVOLVES STRUCTURED PATTERN MATCHING 
The central idea underlying structured pattern matching is that similarity involves an 
alignment of relational structure. For example, in Figure la, configuration A is clearly 
more similar to the top configuration than configuration B, because A has similar objects 
taking part in the same relation (above), while B has similar objects taking part in a 
different relation (next-to). This determination can be made regardless of whether the 
objects taking part in the relations are similar. For example, in Figure lb configuration 
A is also more similar to the top configuration than is configuration B, because A shares 
a relation with the top configuration, while B does not. As a check on this intuition, 10 
subjects were asked to tell us which configuration (A or B) went best with ti'/e top 
configuration for the triads in Figures la and lb. All 10 subjects chose configuration A 
for both triads. This example demonstrates that relations (such as the common above 
relation) are important in similarity processing. 
A 
A B A 
(^) 
Figure 1. 
Examples of structural alignment in perception. 
B 
The inportance of relations was also demonstrated by Palmer (1978) who asked subjects 
to rate the similarity of pairs of configurations like those in Figure 2. The pair in Figure 
2a shares the global property that both are open figures, while the pair in Figure 2b does 
not. As would be expected if subjects attend to relations when determining similarity, 
Structural alignment &: the development of connectionist models of analogy 857 
higher similarity ratings were given to pairs like the one in Figure 2a than to pairs like 
the one in Figure 2b. This finding can only be explained by appealing to structural 
similarity, because both pairs of configurations share the same number of local line 
segments. Consistent with this result, Palmer also found that subjects were faster to say 
that the items in Figure 2b are different than that the items in Figure 2a are different. A 
similar result was obtained by Lockhead and King (1977). 
Figure 2. 
Structured pattern matching in a study by Palmer (1978). 
Further research suggests that common bindings between relations and the items they 
relate are also central to similarity. For example, Clement and Gentner (1991) presented 
subjects with pairs of analogous stories. One story described organisms called Tams that 
ate rocks, while the other described robots that collected data on a planet. In each story, 
one matching fact also had a matching causal antecedent. For example, the Tams' 
exhausting the minerals on the rock CAUSED them to move to another rock, while 
the robots' exhausting the data on a planet CAUSED them to move to another 
planet. A second matching fact did not have a matching causal antecedent. For 
example, the Tams' underbelly could not function on a new rock and the 
robots' probe could not function on a new planet, but the causes of these facts 
did not match. Subjects were asked which of the two pairs of key facts (shown in bold) 
was more important to the stories. Subjects selected the pair that had the matching causal 
antecedent, suggesting that their mappings preserved the relational connections in the 
stories. 
2.2 STRUCTURED PATTERN COMPLETION 
Pattern completion has long been a central feature of neural network models (Anderson, 
Silverstein, Ritz, & Jones, 1977; Hopfield, 1982). For example, in the BSB model of 
Anderson et al., vectors in which some units are below their maximum value are filled in 
by completing a pattern based on the vector similarities of the current activation pattern 
to previously learned patterns. 
The key issue here is the kind of information that guides pattern completion in humans. 
Data from psychological studies suggests that subjects' pattern matching ability is 
controlled by structural similarities rather than by geometric similarities. For example, 
Medin and Goldstone presented subjects with pairs of objects like those in Figure 3 
(Medin, Goldstone & Gentner, in press). The left-hand figure in both pairs is somewhat 
ambiguous, but the right-hand figure is not. Subjects who were asked to list the 
commonalities of the pair in Figure 3a said that both figures had three prongs, while 
subjects who were asked to list the commonalities of the pair in Figure 3b said that both 
figures had four prongs. This finding was obtained for 15 of 16 triads tested, and suggests 
858 Gentner and Markman 
that subjects were mapping the structure from the unambiguous figure onto the 
ambiguous one. Of course, in order for the mapping to take place, the underlying 
structure of the figures has to be readily alignable, and there must be ambiguity in the 
target figure. In the pair in Figure 3c, the left hand item cannot be viewed as having four 
prongs, and so this mapping is not made. 
Figure 3: 
(c) 
Example of structured pattern completion. 
Structured pattern completion also occurs in conceptual structures. Clement and Gentner 
(1991) extended the study described above by deleting the key matching facts from one of 
the stories (e.g., the bold facts from the robot story). Subjects read both stories, then 
predicted one new fact about the robot story. Subjects were free to predict anything at all, 
but 50% of the subjects predicted the fact with the matching causal antecedent, while only 
28% of the subjects predicted the fact with no matching causal antecedent. By 
comparison, a control group that made predictions about the target story without seeing 
the base predicted both facts at the same rate (about 5%). This finding underlines the 
importance of connectivity in pauern completion. People's predictions were determined 
not just by the local information, but by whether it was connected to matching 
information. Thus, pattern completion is structure-sensitive. 
2.3 DIFFERENT COMPARISONS-DIFFERENT INTERPRETATIONS 
Comparison is flexible. When an item takes part in many comparisons, it may be 
interpreted differently in each comparison. For example, in Figure 3a, the left figure is 
interpreted as having 3 prongs, while in Figure 3b, it is interpreted as having 4 prongs. 
Similarly, the comparison 'My surgeon is a butcher' conveys a clumsy surgeon, but 
'Genghis Khan was a butcher' conveys a ruthless killer (Glucksberg and Keysar, 1992). 
This type of flexibility is also evident in an example presented by Spellman and Holyoak 
(1992). They pointed out that some politicians likened the Gulf War to World War II, 
implying that the United States was acting as the world's policeman to stop a tyrant. 
Other politicians compared Operation Desert Storm to Vietnam, implying that the United 
States entered into a potentially endless conflict between two other nations. Clearly, 
different comparisons highlighted different features of the Gulf War. 
Structural alignment &: the development of connectionist models of analogy 859 
2.4 SAME COMPARISON-DIFFERENT INTERPRETATIONS 
Even a single comparison can yield more than one distinct interpretation. This situation 
may arise when the items are richly represented, with many different clusters of 
knowledge. It can also arise when the comparison permits more than one alignment, as 
when the similarities of the objects in an item suggest different correspondences than do 
the relational similarities (i.e. components are cross-mapped (Gentner & Toupin, 1986)). 
Markman and Gentnet (in press) presented subjects with pairs of scenes like those 
depicting the perceptual higher order relation monotonic increase in size shown in Figure 
4. In Figure 4, the circle with the arrow over it in the left-hand figure is the largest circle 
in the array. It is cross-mapped, since it is the same size as the middle circle in the right- 
hand figure, but plays the same relational role as the left (largest) circle. Subjects were 
given a mapping task in which they were asked to point to the object in the right-hand 
figure that went with with the cross-mapped circle in the left-hand figure. In this task, 
subjects chose the circle that looked most similar 91% of the time. However, a second 
group of subjects, who rated the similarity of the pair before doing this mapping, selected 
the object playing the same relational role 61% of the time. In both tasks, when subjects 
were asked whether there were any other good choices, they generally described the other 
possible mapping. These results show that the same comparison can be aligned in 
different ways, and that similarity comparisons promote structural alignment. 
Figure 4: 
Stimuli with a cross-mapping from Markman and Gentner (in press). 
Goldstone (personal communication) has demonstrated that, not only are comparisons 
flexible, but subjects can attend to attribute and relation matches selectively. He 
presented subjects with triads like the one in Figure 5. Subjects were asked to choose 
either the bottom figure with the most attribute similarity to the top one, or the bottom 
figure with the most relational similarity to the top one. In this study, and other pilot 
studies, subjects were highly accurate at both task, suggesting flexibility to attend to 
different kinds of similarity. 
75O 
 
A B 
Figure 5: Sample stimuli from study by Goldstone. 
860 Gentner and Markman 
Similar flexibility can also be found in stimuli with conceptual relations. Gentner (1988) 
presented children with double metaphors that can have two meanings, one based on 
attribute similarities and a second based on relational similarities. For example, the 
metaphor ?lant stems are like drinking straws' can mean that both are round and skinny, 
or that both transport fluids from low places to high places. Gentner found that young 
children (age 5-6) made the attribute-based interpretation, while older children (age 9-10) 
and adults could make either interpretation (but preferred the relation-based interpretation). 
There are limits to this flexibility. People prefer to make structurally consistent 
mappings (Gentner, 1983). For example, Spellman and Holyoak (1992) told subjects to 
map Operation Desert Storm onto World War II. When they asked subjects to find a 
correspondence for George Bush given that Saddam Hussein corresponded to Hitler, 
subjects generally chose either FDR or Churchill. Then, subjects were asked to make a 
mapping for the United States in 1991. Interestingly, subjects who mapped Bush to 
FDR almost always mapped the US in 1991 to the US during World War II. In contrast, 
subjects who mapped Bush to Churchill almost always mapped the US in 1991 to Britain 
during World War II. Thus, subjects maintained structurally consistent mappings. 
This type of flexibility adds significant complexity to the comparison process, because a 
system cannot simply be trained to search for relational correspondences or be taught to 
prefer only attribute matches. Rather, the comparison process must determine both 
attribute and relation matches and must be able to keep different mappings distinct from 
each other. 
2.5 SUMMARY OF EMPIRICAL EVIDENCE 
These findings suggest that comparisons of both perceptual and conceptual materials 
involve structural alignment. Further, structural alignment promotes structure sensitive 
pattern completion. Finally, comparisons allow for multiple interpretations of a single 
item in different comparisons or multiple interpretations of a single comparison. Any 
model of human cognition that involves comparison must exhibit these properties. 
3.0 IMPLICATIONS FOR COGNITIVE MODELS 
Many of the questions concerning the adequacy of connectionist models and neural 
networks for high-level cognitive tasks have centered on linguistic processing and the 
crucial role of compositional relational structures in sentence comprehension (Fodor & 
McLaughlin, 1990; Fodor & Pylyshyn, 1988). Recent work has addressed this problem 
by examining ways to represent hierarchical structure in connectionist models, 
implementing stacks and binary trees to model variable binding and recursive sentence 
processing (e.g., Elman, 1990; Pollack, 1990; Smolensky, 1990; see also Quinlan, 1991 
for a review). It is too soon to tell how successful these methods will be, or whether 
they can be extended to the general case of structural alignment. 
The results summarized here underline the need for representations that permit structural 
alignment. How should this be done? As van Gelder (1990) discusses, symbolic systems 
traditionally use concatenative representation, in which symbol names are concatenated to 
build a compositional representation. For example, a circle above a triangle could be 
Structural alignment & the development of connectionist models of analogy 861 
represented by the assertion above(circle,triangle). Such symbolic representations have 
been used to model the analogy and similarity phenomena described here with some 
success (Falkenhainer, et al., 1989). Van Gelder (1990) suggests a weaker criterion of 
functional compositionality. In functionally compositional representations, tokens for 
the symbols are not directly present in the representation, but they can be extracted from 
the representation via some process. Van Gelder suggests that the natural representation 
used by neural networks is functionally compositional. Analogously, the question of 
whether connecfionist models can model the phenomena described here should be couched 
in terms of functional alignability: whether the representations can be decomposed and 
aligned, rather than whether the structure is transparently present. 
Along this track, an intriguing question is whether the surface form of functionally 
compositional representations will be similar to the degree that the structures they 
represent are similar. If so, the alignment process could take place simply by comparing 
activation vectors. As yet, there are no networks that exhibit this behavior. Further, 
given the evidence that geometric representations are insufficient to model human 
similarity comparisons (see Tversky (1977) for a review), we are pessimistic about the 
prospects that this type of model will be developed. 
In conclusion, substantial psychological evidence suggests that determining the similarity 
of two items requires a flexible alignment of structured representations. We suspect that 
connectionist models of cognitive processes that involve comparisons will have to 
exhibit concatenative compositionality in order to capture the flexibility inherent in 
comparisons. However, we leave open the possibility that systems exhibiting functional 
alignability will be successful. 
Acknowledgments 
This research was sponsored by ONR grant BNS-87-20301. We thank Jon Handler, Ed 
Wisniewski, Phil Wolff and the whole Similarity and Analogy group for comments on 
this work. We also thank Laura Kotovsky, Catherine Kreiser and Russ Poldrack for 
running the pilot studies described above. 
References 
Anderson, J. A., Silverstein, J. W., Ritz, S. A., & Jones, R. S. (1977). Distinctive 
features, categorical perception and probability leaming: Some applications of a neural 
model. Psychological Review, 84, 413-451. 
Clement, C. A., & Gentner, D. (1991). Systematicity as a selection constraint in 
analogical mapping. Cognitive Science, 15, 89-132. 
Elman, J.L. (1990). Finding structure in time. Cognitive Science, 14(2), 179-212. 
Falkenhainer, B., Forbus, K. D., & Gentner, D. (1989). The structure-mapping engine: 
Algorithm and examples. Artificial Intelligence, 41(1), 1-63. 
Fodor, J., & McLaughlin, B. (1990). Connectionism and the problem of systematicity: 
Why Smolensky's solution doesn't work. Cognition, 35., 183-204. 
862 Gentner and Markman 
Fodor, J. A., & Pylyshyn, Z. W. (1988). Connectionism and cognitive architecture: A 
critical analysis. Cognition, 28, 3-71. 
Gentner, D. (1983). Structure mapping: A theoretical framework for analogy. Cognitive 
Science, 7, 155-170. 
Gentner, D. (1988). Metaphor as structure mapping: The relational shift. Child 
Development, 59, 47-59. 
Gentner, D., & Toupin, C. (1986). Systematicity and surface similarity in the 
development of analogy. Cognitive Science, 10, 277-300. 
Glucksberg, S. & Keysar, B. (1990). Understanding metaphorical comparisons: Beyond 
similarity. _Psychological Review, 97(1), 3-18. 
Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective 
computational abilities. Proceedings of the National Academy of S .ciences, 79, 2554- 
2558. 
Lockhead, G. R., & King, M. C. (1977). Classifying integral stimuli. Journal of 
Experimental Psychology: Human Perception and Performance, 3(3), 436-443. 
Markman, A. B., & Gentner, D. (in press). Structural alignment during similarity 
comparisons. Cognitive Psychology. 
Medin, D. L., Goldstone, R. L., & Gentner, D. (in press). Respects for similarity. 
Psychological Review. 
Palmer, S. E. (1978). Structural aspects of visual similarity. Memory and Cognition, 
6(2), 91-97. 
Pollack, J. B. (1990). Recursive distributed representations. Artificial Intelligence, 46(1- 
2), 77-106. 
Quinlan, P.T. (1991). Connectionism and Psychology: A psychological perspective on 
new connectionist research. Chicago: The University of Chicago Press. 
Sabourin, M. & Mitiche, A. (1992). Optical character recognition by a neural network. 
Neural Networks, 5(5), 843-852. 
Smolensky, P. (1990). Tensor product variable binding and the representation of 
symbolic structures in connectionist systems. Artificial Intelligence, 46, 159-216. 
Spellman, B. A., & Holyoak, K. J. (1992). If Saddam is Hitler then who is George 
Bush? Analogical mapping between systems of social roles. Journal of Personality and 
Social Psychology, 62(6), 913-933. 
Tversky, A. (1977). Features of similarity. Psychological Review, 84(4), 327-352. 
van Gelder, T. (1990). Compositionality: A connectionist variation on a classical theme. 
Cognitive Science, 14(3), 355-384. 
