Propositional Dynamic Logic of Flowcharts

Following a suggestion of Pratt, we consider propositional dynamic logic in which programs are nondeterministic finite automata over atomic programs and tests (i.e., flowcharts), rather than regular expressions. While the resulting version of PDL, call it APDL, is clearly equivalent in expressive power to PDL, it is also (in the worst case) exponentially more succinct. In particular, deciding its validity problem by reducing it to that of PDL leads to a double exponential time procedure, although PDL itself is decidable in exponential time. We present an elementary combined proof of the completeness of a simple axiom system for APDL and decidability of the validity problem in exponential time. The results are thus stronger than those for PDL, since PDL can be encoded in APDL with no additional cost, and the proofs simpler, since induction on the structure of programs is virtually eliminated. Our axiom system for APDL relates to the PDL system just as Floyd's proof method for partial correctness relates to Hoare's.