To save content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about saving content to .
To save content items to your Kindle, first ensure email@example.com
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras.
In this paper, we address the problem of proving confluence for string diagram rewriting, which was previously shown to be characterised combinatorially as double-pushout rewriting with interfaces (DPOI) on (labelled) hypergraphs. For standard DPO rewriting without interfaces, confluence for terminating rewriting systems is, in general, undecidable. Nevertheless, we show here that confluence for DPOI, and hence string diagram rewriting, is decidable. We apply this result to give effective procedures for deciding local confluence of symmetric monoidal theories with and without Frobenius structure by critical pair analysis. For the latter, we introduce the new notion of path joinability for critical pairs, which enables finitely many joins of a critical pair to be lifted to an arbitrary context in spite of the strong non-local constraints placed on rewriting in a generic symmetric monoidal theory.
The paper is devoted to an analysis of the concurrent features of asynchronous systems. A preliminary step is represented by the introduction of a non-interleaving extension of barbed equivalence. This notion is then exploited in order to prove that concurrency cannot be observed through asynchronous interactions, i.e., that the interleaving and concurrent versions of a suitable asynchronous weak equivalence actually coincide. The theory is validated on some case studies, related to nominal calculi (π-calculus) and visual specification formalisms (Petri nets). Additionally, we prove that a class of systems which is deemed (output-buffered) asynchronous, according to a characterization that was previously proposed in the literature, falls into our theory.
In this paper we focus on the synthesis of labelled transition systems (LTSs) for process calculi using Mobile Ambients (MAs) as a testbed. Our proposal is based on a graphical encoding: a process is mapped into a graph equipped with interfaces such that the denotation is fully abstract with respect to the standard structural congruence. Graphs with interfaces are amenable to the synthesis mechanism based on borrowed contexts (BCs), which is an instance of relative pushouts (RPOs). The BC mechanism allows the effective construction of an LTS that has graphs with interfaces as states and labels, and such that the associated bisimilarity is a congruence. We focus here on the analysis of an LTS over processes as graphs with interfaces: we use the LTS on graphs to recover an LTS directly defined over the structure of MA processes and define a set of SOS inference rules capturing the same operational semantics.
We propose a graphical implementation for (possibly recursive) processes of the π-calculus, encoding each process into a graph. Our implementation is sound and complete with respect to the structural congruence for the calculus: two processes are equivalent if and only if they are mapped into graphs with the same normal form. Most importantly, the encoding allows the use of standard graph rewriting mechanisms for modelling the reduction semantics of the calculus.
We present a categorical formulation
of the rewriting of possibly cyclic term graphs, based on a
variation of algebraic 2-theories. We show that
this presentation is equivalent to the well-accepted
operational definition proposed by Barendregt et al. – but
case of circular redexes , for which we propose (and
justify formally) a different treatment.
The categorical framework allows us to model in a concise way
also automatic garbage collection
for sharing/unsharing and folding/unfolding of structures,
and to relate
term graph rewriting to other rewriting formalisms.
Email your librarian or administrator to recommend adding this to your organisation's collection.