# Deterministic process logic is elementary

## Abstract

Process logic (PL) is a language for reasoning about the behavior of a program during a computation, while propositional dynamic logic (PDL) can only reason about the input-output states of a program. Nevertheless, it is shown that to each PL model M, there corresponds in a natural way a PDL model Mt such that each path in M is represented by a state in Mt. Moreover, to every PL formula p, there corresponds a PDL formula pt, whose length is linear in that of p, such that p is true of a path in M iff pt is true of the state which represents that path in Mt. Then it is shown that p is satisfiable iff pt is satisfiable in a finite PDL model with special properties which is called a pseudomodel. The size of the pseudomodel is in general nonelementary but is bounded by both the depth of nesting of the suf operator and the alternation of the suf and diamond operators. However, for DPL, a deterministic version of PL, the pseudomodel has exponential size, giving a deterministic exponential time procedure for deciding DPL validity. These results suggest that it is the interaction between nondeterministic programs and the suf operator that makes the general decision problem for PL so difficult. © 1983 Academic Press, Inc.