A behavioural theory consists of machine-independent postulates characterizing a particular class of algorithms or systems, an abstract machine model that provably satisfies these postulates, and a rigorous proof that any algorithm or system stipulated by the postulates is captured by the abstract machine model. The class of interest in this article is that of (synchronous) parallel algorithms. For this class a behavioural theory has already been developed by Blass and Gurevich, which unfortunately, though mathematically correct, fails to be convincing, as it is not intuitively clear that the postulates really capture the essence of (synchronous) parallel algorithms.In this article we present a much simpler (and presumably more convincing) set of four postulates for (synchronous) parallel algorithms, which are rather close to those used in Gurevich's celebrated sequential ASM thesis, i.e. the behavioural theory of sequential algorithms. The key difference is made by an extension of the bounded exploration postulate using multiset comprehension terms instead of ground terms formulated over the signature of the states. In addition, all implicit assumptions are made explicit, which amounts to considering states of a parallel algorithm to be represented by meta-finite first-order structures.The article first provides the necessary evidence that the axiomatization presented in this article characterizes indeed the whole class of (synchronous) parallel algorithms, then formally proves that parallel algorithms are captured by Abstract State Machines (ASMs). The proof requires some recourse to methods from finite model theory, by means of which it can be shown that if a critical tuple defines an update in some update set, then also every other tuple that is logically indistinguishable defines an update in that update set.
@cite_1: We examine sequential algorithms and formulate a sequential-time postulate, an abstract-state postulate, and a bounded-exploration postulate . Analysis of the postulates leads us to the notion of sequential abstract-state machine and to the theorem in the title. First we treat sequential algorithms that are deterministic and noninteractive. Then we consider sequential algorithms that may be nondeterministic and that may interact with their environments.
The sequential ASM thesis @cite_1 characterizes sequential algorithms in terms of three postulates: sequential time, abstract state and bounded exploration. Moreover, it establishes the characterization theorem for sequential algorithms stating that algorithms defined this way are exactly captured by sequential ASMs, i.e. a well-defined abstract machine model that relies on the parallel execution of updates on the abstract states, if certain conditions guarding the updates are satisfied. Thus, also sequential ASMs support bounded parallelism, where the bound is a priori fixed by the algorithm and does not depend on the state.