A Discrete Geometric Model of Concurrent Program Execution

Abstract

Abstract. A trace of the execution of a concurrent object-oriented program can be displayed in two-dimensions as a diagram of a non-metric finite geometry. The actions of a programs are represented by points, its objects and threads by vertical lines, its transactions by horizontal lines, its communications and resource sharing by sloping arrows, and its partial traces by rectangular figures. We prove informally that the geometry satisfies the laws of Concurrent Kleene Algebra (CKA); these describe and justify the interleaved implementation of multithreaded programs on computer systems with a lesser number of concurrent processors. More familiar forms of semantics (e.g., verification-oriented and operational) can be derived from CKA. Programs are represented as sets of all their possible traces of execution, and non-determinism is introduced as union of these sets. The geometry is extended to multiple levels of abstraction and granularity; a method call at a higher level can be modelled by a specification of the method body, which is implemented at a lower level. The final section describes how the axioms and definitions of the geometry have been encoded in the interactive proof tool Isabelle, and reports on progress towards automatic checking of the proofs in the paper.

Generating Posets Beyond N

“…A natural approach to generating more expressive pomset languages is to “cut across” pomsets in more general ways when (de)composing them. This can be achieved by (de)composing along interfaces, and this idea can be traced back again to Winkowski [ 38 ]; see also [ 4 , 6 , 28 ] for interface-based compositions of graphs and posets, or [ 15 , 29 , 30 ] for recent interface-based graph models for . As a side effect, interfaces may yield notions of tests, assertions or modalities.…”

Posets with Interfaces for Concurrent Kleene Algebra

“…Notably they all satisfy a strict interchange law which is not in the spirit of CKA. Other interface-based graph models for CKA have been explored by Hoare et al in [22,35,36].…”

Generating Posets Beyond N

“…A natural approach to generating more expressive pomset languages is to "cut across" pomsets in more general ways when (de)composing them. This can be achieved by (de)composing along interfaces, and this idea can be traced back again to Winkowski [35]; see also [3,4,25] for interface-based compositions of graphs and posets, or [13,26,27] for recent interface-based graph models for CKA. As a side effect, interfaces may yield notions of tests or modalities.…”