Halting and Equivalence of Schemes over Recursive Theories

Abstract

Let S be a fixed first-order signature. In this note we consider the following decision problems. (i) Given a recursive ground theory T over S, a program scheme p over S, and input values specified by ground terms t1,...,tn, does p halt on input t1,...,tn in all models of T? (ii) Given a recursive ground theory T over S and two program schemes p and q over S, are p and q equivalent in all models of T? When T is empty, these two problems are the classical halting and equivalence problems for program schemes, respectively. We show that problem (i) is r.e.-complete and problem (ii) is Pi-0-2-complete. Both these problems remain hard for their respective complexity classes even if T is empty and S is restricted to contain only a single constant, a single unary function symbol, and a single monadic predicate. It follows from (ii) that there can exist no relatively complete deductive system for scheme equivalence.