Complexity, convexity and combinations of theories

Abstract

We restrict our attention to decidable quantifier-free theories, such as the quantifier-free theory of integers under addition, the quantifier-free theory of arrays under storing and selecting, or the quantifier-free theory of list structure under cons, car and cdr. We consider combinations of such theories: theories whose sets of symbols are the union of the sets of the symbols of the individual theories and whose set of axioms is the union of the sets of axioms of the individual theories. We give a general technique for determining the complexity of decidable combinations of theories, and show, for example, that the satisfiability problem for the quantifier-free theory of integers, arrays, list structure and uninterpreted function symbols under +, ≤, store, select, cons, car and cdr is NP-complete. We next consider the complexity of the satisfiability problem for formulas already in disjunctive normal form: why some combinations of theories admit deterministic polynomial time decision procedures while for others the problem is NP-hard. Our analysis hinges on the question of whether the theories being combined are convex; that is, whether any conjunction of literals in the theory can entail a proper disjunction of equalities between variables. This leads to a discussion of the role that case analysis plays in deciding combinations of theories.