Using graphs to understand PDL

Abstract

This paper begins with the problem of sharpening our understanding of PDL. The position we take here is that PDL, which is ordinarily defined using regular operations on programs, is better understood in terms of finite state automata. Accordingly we rederive some basic PDL results (finite model, deterministic exponential satisfiability) in terms of automata. As corollaries to this we obtain answers to the following open questions. (i) What is the time complexity of satisfiability for propositional flowgraph logic? (ii) Can regular expressions be axiomatized equationally as succinctly as they can be represented with automata? We also show how converse and test relate to flowgraph operations.

The evidence to date strongly suggested that problem (i) should require double exponential time. We give a deterministic one-exponential bound, tight to within a polynomial. Two novel aspects of our algorithm are that it solves the problem by translation to modal logic with minimization, and that the concept of state is abstracted out of the algorithm. The tractability of satisfiability can be traced to two key properties of the definition of flowgraph operations. For (ii) we give for each flowgraph of size n a complete axiomatization of size a polynomail in n, by showing how to axiomatize matrix transitive closure equationally.

Our treatment of converse and test shows that they enjoy the same two key properties as flowgraph operations, permitting a uniform treatment of the bulk of the major flow-of-control constructs.