Axiomatic Rewriting Theory I: A Diagrammatic Standardization Theorem

By extending nondeterministic transition systems with concurrency and copy mechanisms, Axiomatic Rewriting Theory provides a uniform framework for a variety of rewriting systems, ranging from higher-order systems to Petri nets and process calculi. Despite its generality, the theory is surprisingly simple, based on a mild extension of transition systems with independence: an axiomatic rewriting system is defined as a 1-dimensional transition graph G equipped with 2-dimensional transitions describing the redex permutations of the system, and their orientation. In this article, we formulate a series of elementary axioms on axiomatic rewriting systems, and establish a diagrammatic standardization theorem. Foreword by the author Many concepts of Rewriting Theory started in the -calculus — which is by far the most studied rewriting system in history. A remarkable illustration is the confluence theorem. The theorem was formulated by A. Church and J.B. Rosser in the early years of the -calculus [7]. The theorem was then generalized and applied extensively to other rewriting systems. It became eventually an object of study in itself, in a line of research pioneered by H.-B. Curry and R. Feys in their book on Combinatory Logic (1958). This culminated in a series of beautiful papers by G. Huet, J. W. Klop, and J.-J. Lévy published at the end of the 1970s and beginning of the 1980s. Today, more than half a century after its appearance in the -calculus, the confluence property is universally accepted as the theoretical principle underlying deterministic computations. The article is concerned with another key property of the -calculus: the standardization theorem, which was discovered by A. Church and J.B. Rosser quite at the same time as the confluence property. We advocate in this article that, in the same way as confluence underlies deterministic computations, standardization guides causal computations. It is worth clarifying here what kind of causality we have in mind, since the concept has been used in so many different ways. First of all, by computation, we mean a rewriting path M1 u1 !M2 u2 !M3 ! !Mn 1 un !Mn in which every term Mk describes a particular state of the system, and in which every redex uk describes a particular transition on states, for 1 k n. Then, by causal