Ulam Game, the Logic of Maxsat, and Many-Valued Partitions

Abstract

The maximum satisfiability problem MAXSAT asks whether a set of Boolean clauses C ₁,..., C _m contains a satisfiable subset of cardinality ≥k. Trivially, there exists a Turing machine Μ recognizing MAXSAT in nondeterministic polynomial time: in fact, the problem is NP-complete, [Garey and Johnson, 1979]. Furthermore, there is a Turing machine Τ working in deterministic polynomial time, such that, for any input instance (C ₁,..., C _m , k) of MAXSAT, Τ outputs a Boolean formula D = D(C ₁,..., C _m , k) which is satisfiable iff (C ₁,..., C _m , k) ∈ MAXSAT. In its actual form, as given by Cook’s theorem, D is only vaguely reminiscent of the input clauses C _j : most of the variables in D take care of the description of an accepting computation of (C ₁,..., C _m , k) by Μ. Incorporation of the numerical parameter k into the Boolean formula D takes its toll.