A new thesis concerning synchronised parallel computing – simplified parallel ASM thesis

Abstract

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 Preprint submitted to Theoretical Computer ScienceDecember 12, 2017 presented in this article characterizes indeed the whole class of deterministic, 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.

Concurrent abstract state machines

“…4 our definition and the sequential as well as the concurrent ASM thesis work also with basic ASMs (whose moves alternate with environment moves [20,Def. 2.4.22]) and with parallel ASMs (see [7,11]) for which a conjecture of the second author in [59]-namely that for alg(a) to capture unbounded parallelism, only the bounded exploration witness T has to be changed to a finite set of list comprehension terms-could recently be proven and is available in [22] leading to a parallel ASM thesis).…”

A Logic for Non-deterministic Parallel Abstract State Machines

“…Therefore the update sets produced by these machines are restricted to be finite as well. This is implicitly assumed in the parallel ASM thesis of Blass and Gurevich [3,4] and it is made explicit in the new parallel ASM thesis that we propose in [6].…”

“…This is a straightforward consequence of the fact that the variable x in the definition of the forall rule ranges over the (finite) primary part of S, and it is also the case in the ASM thesis for parallel algorithms of Blass and Gurevich [3,4] where it is implicitly assumed that the forall rule in the parallel ASMs range over finite hereditary multisets. See our work in [6] for a detailed explanation. Regarding the set ∆(r, S, ζ) of update sets, we note that it might be infinite since the unbounded choice rule can potentially produce infinitely many update sets.…”

“…This builds upon the complete logic introduced in the work of Nanchen and Stärk [13] for reasoning about single steps of hierarchical ASMs. Hierarchical ASMs capture the class of synchronous and deterministic parallel algorithms in the precise sense of the ASM thesis of Blass and Gure-vich [3,4] (see also [6]). Our main contribution to this regard is the handling of non-determinism (both bounded and unbounded) within the logical formalism.…”

Towards functional safety and security for adaptive and flexible business processes

“…The formal background is provided by abstract state machines (ASMs), which are known as a formalism that captures different classes of algorithms at any abstraction level . Abstract state machines have demonstrated their strength in various domains, eg, specifying the semantics of programming and modelling languages or providing new tools (logics) to aid formal verification . They have also been used to develop safety‐critical systems, eg, the commuter rail service, the haemodialysis machine, and aeroplane components .…”