Annals of Mathematics and Artificial Intelligence 18 (1996) 51-67 51 The complexity of minimum partial truth assignments and implication in negation-free formulae James R Delgrande and Arvind Gupta School of" Computing Science, Simon Fraser Universit3.; Burnaby B.C., Canada VSA IS6 E-mail: {jim,arvind} @cs.sfu.ca In Artificial Intelligence there has been a great deal of interest in the tradeoff between expressiveness and tractability for various areas of symbolic reasoning. Here we present several complexity theory results for two areas, wherein we restrict the application of negation. First, we consider the problem of determining a minimum satisfying assignment for a (restricted) propositional sentence. We show that the problem of determining a minimum satisfying assignment for a sentence in negation-free CNF, even with no more than two disjuncts per clause, is NP-complete. We also show that unless P = NP, no polynomial time approximation scheme can exist for this problem. However, the problem is in polynomial time if either each clause contains exactly one negative and one positive literal or we use exclusive-OR in the clauses instead of the more standard inclusive-OR. Second, the problem of determining logical implication between sentences composed solely of conjunctions and disjunctions is shown to be as difficult as that between arbitrary sentences. We also study this problem when the sentences are restricted to being in CNF or DNE Determining whether a CNF sentence logically implies a DNF sentence is co-NP-complete, but in all other cases this problem is polynomial time. We argue that these results are relevant, first to areas where a least solution (in some fashion) is desired, and second, to limited deductive systems. 1. Introduction In Artificial Intelligence (AI) a great many problems expressed using a propo- sitional language are known to be NP-hard. The classic example is the problem of determining whether there exists a satisfying assignment to the set of literals in a propositional sentence; this, of course, is also the original example of a NP-complete problem [6]. Consequently, the most efficient known algorithms for such problems exhibit exponential worst-case behaviour. As a result, there has been a great deal of interest in AI in determining tractable or computationally reasonable subsets of these problems. In this paper we examine this tradeoff by considering two classes of problems. The first deals with finding a minimum satisf#ing assignment to a set of variables; the second concerns the relation of logical implication between two formulae. Our strategy © J.C. Baltzer AG, Science Publishers