An Argument for Derivational Analogy

A common reason for the use of analogy in (computational) problem solving is the lack of appropriate object-level knowledge, e.g. rules, necessary to solve the problem from first principles. Hence, the absence of sufficient (object-level) domain knowledge is assumed in most case-based reasoning (CBR) systems. Even those CBR systems that combine rule-based and casebased reasoning rely on a similar assumption: if rules exist, then reason from first principles, otherwise use case-based reasoning [17, 18]. That is, the use of analogy as a search control strategy by transferring control knowledge, is hardly an issue in CBR research, except in case-based planning (CBP). As far as we know, the situation is similar in cognitive research on analogy. Why this? One reason might be that more often than not the problems chosen for cognitive experiments have single-step solutions rather than solutions with many steps as in planning and hence, search control does not matter much. For instance, the much investigated/standard problems “atom/solar system”, “water flow/ heat flow”, and Duncker’s radiation problem do not require a searchintensive multi-step solution process. As opposed to solutions of these problems, Newtonian physics problem solving [20] and especially mathematical theorem proving need a complex multi-step problem solving process, where search control is a central issue. The same is true for many computational planning problems. The problems to be solved by CBP may have complex and multi-step solutions, e.g., in mathematical theorem proving [12]. Therefore, CBP aims at reducing the search effort for finding a solution [5, 22, 1, 11]. This paper is centered around our experiences with problem solving for complex solutions that have multiple steps, where decisions as to which sequences of steps to explore are crucial. Here, problem solving by analogy can have the following purposes: Computational analogy tries to improve the exploitation of limited resources, in particular of the number of user interactions, run time, and of knowledge. Hence, the purpose of analogy can be, cf. [10] to use analogy in solving problems that cannot be solved otherwise because of lack of object-level knowledge (e.g., rules or lemmata) or lack of control knowledge; to save user interaction (which is an replacement for control knowledge in interactive systems); to use analogy to replace search-intensive subroutines at low cost. Similarly, for human problem solving by analogy VanLehn and Jones [20] suggest that at least good human problem solvers use analogical problem solving when no general (object-level) knowledge for solving a current problem is available, e.g., if a knowledge gap has to be detected and filled; when specific information from an example can be used in order to work more efficiently – in other words to save search.