A Modal Logic for Termgraph Rewriting

We propose a modal logic tailored to describe graph transformations and discuss some of its properties. We focus on a particular class of graphs called termgraphs. They are first-order terms augmented with sharing and cycles. Termgraphs allow one to describe classical data-structures (possibly with pointers) such as doubly-linked lists, circular lists etc. We show how the proposed logic can faithfully describe (i) termgraphs as well as (ii) the application of a termgraph rewrite rule (i.e. matching and replacement) and (iii) the computation of normal forms with respect to a given rewrite system. We also show how the proposed logic, which is more expressive than propositional dynamic logic, can be used to specify shapes of classical data-structures (e.g. binary trees, circular lists etc.)

PORGY: Strategy-Driven Interactive Transformation of Graphs

This paper investigates the use of graph rewriting systems as a modelling tool, and advocates the embedding of such systems in an interactive environment. One important application domain is the modelling of biochemical systems, where states are represented by port graphs and the dynamics is driven by rules and strategies. A graph rewriting tool's capability to interactively explore the features of the rewriting system provides useful insights into possible behaviours of the model and its properties. We describe PORGY, a visual and interactive tool we have developed to model complex systems using port graphs and port graph rewrite rules guided by strategies, and to navigate in the derivation history. We demonstrate via examples some functionalities provided by PORGY.Comment: In Proceedings TERMGRAPH 2011, arXiv:1102.226

Constraint solving in non-permutative nominal abstract syntax

Nominal abstract syntax is a popular first-order technique for encoding, and reasoning about, abstract syntax involving binders. Many of its applications involve constraint solving. The most commonly used constraint solving algorithm over nominal abstract syntax is the Urban-Pitts-Gabbay nominal unification algorithm, which is well-behaved, has a well-developed theory and is applicable in many cases. However, certain problems require a constraint solver which respects the equivariance property of nominal logic, such as Cheney's equivariant unification algorithm. This is more powerful but is more complicated and computationally hard. In this paper we present a novel algorithm for solving constraints over a simple variant of nominal abstract syntax which we call non-permutative. This constraint problem has similar complexity to equivariant unification but without many of the additional complications of the equivariant unification term language. We prove our algorithm correct, paying particular attention to issues of termination, and present an explicit translation of name-name equivariant unification problems into non-permutative constraints

Graph rewriting with POLARIZED CLONING

We tackle the problem of graph transformation with a particular focus on node cloning. We propose a graph rewriting framework where nodes can be cloned zero, one or more times. A node can be cloned together with all its incident edges, with only the outgoing edges, with only the incoming edges or without any of the incident edges. We thus subsume previous works such as the sesqui-pushout, the heterogeneous pushout and the adaptive star grammars approaches. A rule is defined as a span L l ← K r → R where the right-hand side R is a multigraph, the left-hand side L and the interface K are polarized multigraphs. A polarized multigraph is a multigraph endowed with some cloning annotations on nodes and edges. We introduce the notion of polarized multigraphs and define a rewriting step as pushback followed by a pushout in the same way as in the sesqui-pushout approach

Logical foundations for reasoning about transformations of knowledge bases

This paper is about transformations of knowledge bases with the aid of an imperative programming language which is non-standard in the sense that it features conditions (in loops and selection statements) that are description logic (DL) formulas, and a non-deterministic assignment statement (a choice operator given by a DL formula). We sketch an operational semantics of the proposed programming language and then develop a matching Hoare calculus whose pre- and post-conditions are again DL formulas. A major difficulty resides in showing that the formulas generated when calculating weakest preconditions remain within the chosen DL fragment. In particular, this concerns substitutions whose result is not directly representable. We therefore explicitly add substitution as a constructor of the logic and show how it can be eliminated by an interleaving with the rules of a traditional tableau calculu

Adjunction for Garbage Collection with Application to Graph Rewriting

Abstract. We investigate garbage collection of unreachable parts of rooted graphs from a categorical point of view. First, we define this task as the right adjoint of an inclusion functor. We also show that garbage collection may be stated via a left adjoint, hence preserving colimits, followed by two right adjoints. These three adjoints cope well with the different phases of a traditional garbage collector. Consequently, our results should naturally help to better formulate graph transformation steps in order to get rid of garbage (unwanted nodes). We illustrate this point on a particular class of graph rewriting systems based on a double pushout approach and featuring edge redirection. Our approach gives a neat rewriting step akin to the one on terms, where garbage never appears in the reduced term.