An Algebraic Approach to Modeling Analogical Projection in Pattern Perception

In this paper we propose an algebraic method for solving proportional analogy problems involving sequential patterns. In this approach, sequential patterns are considered as elements of an algebra. The gestalt that is assigned to a pattern is represented by the term of the algebra that denotes it. Most patterns are ambiguous: there are diierent terms that correspond to a pattern, representing diierent gestalts of it. When a pattern is perceived as part of a proportional analogy, it may assume a diierent gestalt than when it is viewed in isolation. We consider analogical relations as mappings between algebras and give a formal deenition of the preferred gestalts for an analogy in terms of the simplicity of algebraic terms that denote the patterns involved in the proportional analogy. We conjecture that these terms correspond to the gestalts that are perceived by a human who solves the proportional analogy problem. Finally, we describe an algorithm to solve the proportional analogy problems, which computes the simplest gestalts for the three given terms | taking into account the mutual context eeect | on the basis of which the fourth term of the proportional analogy is determined. Creativity consists in the generation of ideas that are experienced as new | that is, not implied by systematic or routine cognitive processes; ideas that require a leap of imagination. Some approaches to modeling creativity have been presented in Hofstadter (1995) and Indurkhya (1992), where it is observed that new ideas and perspectives are often created when one realm of experience is interpreted using structures from a diierent realm: this mechanism is often referred to as analogical or metaphorical projection. We conjecture that the cognitive mechanism of metaphorical projection is essentially the same as analogical projection between perceptual structures.