Progressive Collecting Semantics

We present a semantics of programs whose level of detail lie in between that of collecting semantics, which describe the set of states which appear at a program point, and of trace semantics, which additionally describe the sequence in which states appear. Collecting semantics, being simpler, have the advantage of being amenable to Hoare-style reasoning which operates on program syntax and is compositional. Trace semantics, on the other hand, detail the chronology of program behaviour and hence is more appropriate for reasoning about liveness and progress properties. The purpose of our work therefore is to provide a syntax-based and compositional semantics which naturally deals with liveness and progress properties. In what follows, we present Progressive Collecting Semantics (PCS) which describes how sets of states are sequenced at runtime. We then extend the traditional assertion, which is simply a set (and which specifies that the collecting semantics is a subset of this set), to become an indexed family of sets. Finally, we present a propagation mechanism, representing the basic deductive mechanism of PCS. Here we introduce a notion of conditional correctness. We employ the assertions simultaneously as assumptions as well as verification conditions. Each assumption may, in the verification process, become proved. In the end, a proven program is correct on the proviso that its unproven assertions are correct.