Reversing P/T Nets

Abstract

Petri Nets are a well-known model of concurrency and provide an ideal setting for the study of fundamental aspects in concurrent systems. Despite their simplicity, they still lack a satisfactory causally reversible semantics. We develop such semantics for Place/Transitions Petri Nets (P/T nets) based on two observations. Firstly, a net that explicitly expresses causality and conflict among events, e.g., an occurrence net, can be straightforwardly reversed by adding reversal for each of its transitions. Secondly, the standard unfolding construction associates a P/T net with an occurrence net that preserves all of its computation. Consequently, the reversible semantics of a P/T net can be obtained as the reversible semantics of its unfolding. We show that such reversible behaviour can be expressed as a finite net whose tokens are coloured by causal histories. Colours in our encoding resemble the causal memories that are typical in reversible process calculi.

An Axiomatic Approach to Reversible Computation

“…The main aim of this paper is to take an abstract model, namely labelled transition systems with independence equipped with reverse transitions (Section 2), and to show that the properties above (as well as others) can be derived from a small set of simple axioms (Sections 3, 4, 5). This is in sharp contrast with the large part of works in the literature, which consider specific frameworks such as CCS [6], CCS with broadcast [26], CCB [14], π-calculus [5], higher-order π [18], Klaim [11], Petri nets [25], μOz [22] and Erlang [20], and all give similar but formally unrelated proofs of the same main results. Such proofs will become instances of our general results.…”

“…Since this the setting is very similar to the one of HOπ (both calculi have a reduction semantics and a coinitial notion of independence defined on enriched labels), we get the same results as for HOπ, including CC, and CS and CL. [25,24] [25,Section 3] which is defined between arbitrary firings (transitions). Hence, we get a general LTSI.…”

“…PL and CC require several pages of proofs [24]. The causal safety and causal liveness properties are not considered in [25,24].…”

“…Matters are less clear when it comes to reversible computation in the concurrent setting. Indeed, various reversible concurrent models have been studied, most notably in the areas of process calculi [6,29,18], event structures [34], Petri nets [1,25] and programming languages such as Erlang [20].…”

Foundations of Reversible Computation

“…Another special case of bounded nets are occurrence nets, that is 1-safe and acyclic nets without backward conflicts, where one can always use strict reverses. This property of occurrence nets and their infinite extensions was used as an intermediate step in [132], described later on.…”