Heavy-Tailed Behavior and Randomization in Proof Planning

Proof Planning Proof planning considers mathematical theorems as planning problems. A proof planning problem is defined by an initial state specified by the proof assumptions, the goal state given by the theorem to be proved, and a set of planning operators called methods. Finding a proof corresponds therefore to searching for a sequence of planning operators that derive the theorem from the assumptions. In the proof planning system MEGA (Benzmueller et al. 1997) the traditional proof planning approach is enriched by incorporating mathematical knowledge into the planning process (see (Melis & Siekmann 1999) for details). In particular, methods represent mathematically meaningful inference steps and can be specific for a mathematical domain. We explore the domain of the residue classes over the integers ((Meier, Pollet, & Sorge 2000)) using a proof planning approach. We apply MEGA to solve large testbeds of algebraic problems of a residue class set (e.g. ) over the integers together with a binary operation Æ (e.g. ) such as is closed with respect to Æ, is associative with respect to Æ etc. The results of these proofs are in turn used to classify a given structure Æ in terms of the algebraic structure it forms, i.e., whether it is a semi-group, monoid etc. Moreover, another classification process divides given residue class structures into equivalence classes of isomorphic structures. During this classification process we have to prove proof obligations stating that two structures are isomorphic or not. Our experiments show that the hardest problem instances correspond to problems stating that two structures are not isomorphic (non-isomorphism problems). For some instances the planner generates long proofs, with long run times, while for other (similar) instances the planner generates short proofs, with short run times. Since we are not able to find a heuristic rule that enables us to control the unpredictability of the planner’s performance, we apply randomization and restart techniques to boost the search process and increase the solvability horizon for such non-isomorphism problems.