Halting and Equivalence of Program Schemes in Models of Arbitrary Theories

Abstract

In this note we consider the following decision problems. Let S be a fixed first-order signature. (i) Given a first-order theory or ground theory T over S of Turing degree A, 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 first-order theory or ground theory T over S of Turing degree A 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 Sigma^A_1-complete and problem (ii) is Pi^A_2-complete. Both problems remain hard for their respective complexity classes even if 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 over models of theories of any Turing degree.