Syntactic Regions for Concurrent Programs

In order to gain a better understanding of the state space of programs, with the aim of making their verification more tractable, models based on directed topological spaces have been introduced, allowing to take in account equivalence between execution traces, as well as translate features of the execution (such as the presence of deadlocks) into geometrical situations. In this context, many algorithms were introduced, based on a description of the geometrical models as regions consisting of unions of rectangles. We explain here that these constructions can actually be performed directly on the syntax of programs, thus resulting in representations which are more natural and easier to implement. In order to do so, we start from the observation that positions in a program can be described as partial explorations of the program. The operational semantics induces a partial order on positions, and regions can be defined as formal unions of intervals in the resulting poset. We then study the structure of such regions and show that, under reasonable conditions, they form a boolean algebra and admit a representation in normal form (which corresponds to covering a space by maximal intervals), thus supporting the constructions needed for the purpose of studying programs. All the operations involved here are given explicit algorithmic descriptions.