Dennis de Champeaux

An Improved Bidirectional Heuristic Search Algorithm

A modification of Pohls bidirectional heuristic search algorithm is described together with a simplified implementation. Theorems are proved about conditions yielding shortest paths. The results are given of a worst-case analysis of different algorithms, suggesting a rank order of their quality. Results that Pohl had obtained with a unidirectional heuristic search algorithm on the 15-puzzle are compared with the results obtained with the new — simplified — algorithm.

Bidirectional Heuristic Search Again

Extensive modifications of a previously published bidirectional heuristic search algorithm (see D. de Champeaux and L. Sint, IBID., vol.24, no.2, p.177-91, 1977) are presented in order to preserve the property that shortest solutions are found under appropriate circumstances. Theoretical results encompass the generalisation of the unidirectional optimality theorem. 5 references.

Subproblem finder and instance checker, two cooperating modules for theorem provers

Properties are proved about INSTANCE, a theorem prover module that recognizes that a formula is a special case and/or an alphabetic variant of another formula, and about INSURER, another theorem prover module that decomposes a problem, represented by a formula, into independent subproblems, using a conjunction. The main result of INSTANCE is soundness; the main result of INSURER is a maximum decomposition into subproblems (with some provisos). Experimental results show that a connection graph theorem prover extended with these modules is more effective than the resolution-based connection graph theorem prover alone.

Challenge problem 1 without search

The S IGART Newsletter #72, July 1980, has a description of L. Henschen et al. on their treatment of this problem. Their resolution prover found several solutions due to different P.C.-C.N.F. translation techniques. A straightforward translation would produce 1024 clauses, probably mostly of length 8. They describe tricks yielding respectively 86 and 26 clauses for the refutation machinery. As to be expected the 86-set of input clauses required more effort than the smaller one. The number of clauses to be generated before a contradiction was obtained, was respectively 1052 and 478.

A note on resolution, connection graphs, and subsumption

Completeness is endangered by the combination in connection graph of subsumption deletion rules and the run time pure literal rule. With some care we can restore completeness.