On Davis-Putnam Reductions for Minimally Unsatisfiable Clause-Sets

Abstract

For investigations into the structure of MU, i.e., minimally unsatisfiable clause-sets or conjunctive normal forms, singular DP-reduction is a fundamental tool, applying DP-reduction -> DP_v(F) in case v occurs in one polarity only once. Recall, in general DP_v(F) replaces all clauses containing v by their resolvents on v (another name is variable elimination). We consider sDP(F), the set of all results of applying singular DP-reduction to in MU as long as possible, obtaining non-singular F' in MU with the same deficiency, i.e., delta(F') = delta(F). (In general, delta(F) is the difference c(F) - n(F) of the number of clauses and the number of variables.) Our main results are: 1. For all F', F in sDP(F) we have n(F') = n(F). 2. If is saturated (F in SMU), then we have |sDP(F)| = 1. 3. If is eventually saturated, that is, sDP(F) <= SMU, then for F', F in sDP(F) we have F' isomorphic F (establishing confluence modulo isomorphism). The results are obtained by a detailed analysis of singular DP-reduction for in MU. As an application we obtain that singular DP-reduction for in MU(2) (i.e., delta(F) = 2) is confluent modulo isomorphism (using the fundamental characterisation of MU(2) by Kleine Buening). The background for these considerations is the general project of the classification of MU in terms of the deficiency.