A programming language for the inductive sets, and applications
Information and Control
Dynamic Logic
Computability and completeness in logics of programs (Preliminary Report)
STOC '77 Proceedings of the ninth annual ACM symposium on Theory of computing
| Hi-index | 0.00 |
In this note we consider the following decision problems. Let Σ be a fixed first-order signature. (i) Given a first-order theory or ground theory T over Σ of Turing degree α, a program scheme p over Σ, 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 Σ of Turing degree α and two program schemes p and q over Σ, 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 Σα1 -complete and problem (ii) is Πα2 -complete. Both problems remain hard for their respective complexity classes even if Σ 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.