The notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies. This survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class. Also, some mutual relations between such classes are discussed.

The notion of a recognizable sef offinite graphs is introduced. Every set of finite graphs, that is definable in monadic second-order logic is recognizable, but not vice versa. The monadic second-order theory of a context-free set of graphs is decidable. 0 19W Academic Press. Inc. This paper begins an investigation of the monadic second-order logic of graphs and of sets of graphs, using techniques from universal algebra, and the theory of formal languages. (By a graph, we mean a finite directed hyperedge-labelled hypergraph, equipped with a sequence of distinguished vertices.) A survey of this research can be found in Courcelle [ 111. An algebraic structure on the set of graphs (in the above sense) has been proposed by Bauderon and Courcelle [2,7]. The notion of a recognizable set of finite graphs follows, as an instance of the general notion of recognizability introduced by Mezei and Wright in [25]. A graph can also be considered as a logical structure of a certain type. Hence, properties of graphs can be written in first-order logic or in secondorder logic. It turns out that monadic second-order logic, where quantifications over sets of vertices and sets of edges are used, is a reasonably powerful logical language (in which many usual graph properties can be written), for which one can obtain decidability results. These decidability results do not hold for second-order logic, where quantifications over binary relations can also be used. Our main theorem states that every definable set of finite graphs (i.e., every set that is the set of finite graphs satisfying a graph property expressible in monadic second-order logic) is recognizable. * This work has been supported by the “Programme de Recherches Coordonntes: Mathematiques et Informatique.”

By considering graphs as logical structures, one...

. This paper surveys the work of many researchers on rewriting logic since it was first introduced in 1990. The main emphasis is on the use of rewriting logic as a semantic framework for concurrency. The goal in this regard is to express as faithfully as possible a very wide range of concurrency models, each on its own terms, avoiding any encodings or translations. Bringing very different models under a common semantic framework makes easier to understand what different models have in common and how they differ, to find deep connections between them, and to reason across their different formalisms. It becomes also much easier to achieve in a rigorous way the integration and interoperation of different models and languages whose combination offers attractive advantages. The logic and model theory of rewriting logic are also summarized, a number of current research directions are surveyed, and some concluding remarks about future directions are made. Table of Contents 1 In...

Rewriting logic expresses an essential equivalence between logic and computation. System states are in bijective correspondence with formulas, and concurrent computations are in bijective correspondence with proofs. Given this equivalence between computation and logic, a rewriting logic axiom of the form t \Gamma! t 0 has two readings. Computationally, it means that a fragment of a system 's state that is an instance of the pattern t can change to the corresponding instance of t 0 concurrently with any other state changes; logically, it just means that we can derive the formula t 0 from the formula t. Rewriting logic is entirely neutral about the structure and properties of the formulas/states t. They are entirely user-definable as an algebraic data type satisfying certain equational axioms. Because of this ecumenical neutrality, rewriting logic has, from a logical viewpoint, good properties as a logical framework, in which many other logics can be naturally represented. And, computationally, it has also good properties as a semantic framework, in which many different system styles and models of concurrent computation and many different languages can be naturally expressed without any distorting encodings. The goal of this paper is to provide a relatively gentle introduction to rewriting logic, and to paint in broad strokes the main research directions that, since its introduction in 1990, have been pursued by a growing number of researchers in Europe, the US, and Japan. Key theoretical developments, as well as the main current applications of rewriting logic as a logical and semantic framework, and the work on formal reasoning to prove properties of specifications are surveyed.

. We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the well-known characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particular, we show that term graphs over a signature \Sigma are one-to-one with the arrows of the free gs-monoidal category generated by \Sigma. Such a category satisfies all the axioms for Cartesian categories but for the naturality of two transformations (the discharger ! and the duplicator r), providing in this way an abstract and clear relationship between terms and term graphs. In particular, the absence of the naturality of r and ! has a precise interpretation in terms of explicit sharing and of loss of implicit garbage collection, respectively. Keywords: algebraic theories, directed acyclic graphs, gs-monoidal categories, symmetric monoidal categories, term graphs. Mathematical Subject Clas...

. The dynamic behavior of rule-based systems (like term rewriting systems [24], process algebras [27], and so on) can be traditionally determined in two orthogonal ways. Either operationally, in the sense that a way of embedding a rule into a state is devised, stating explicitly how the result is built: This is the role played by (the application of) a substitution in term rewriting. Or inductively, showing how to build the class of all possible reductions from a set of basic ones: For term rewriting, this is the usual definition of the rewrite relation as the minimal closure of the rewrite rules. As far as graph transformation is concerned, the operational view is by far more popular: In this paper we lay the basis for the orthogonal view. We first provide an inductive description for graphs as arrows of a freely generated dgs-monoidal category. We then apply 2-categorical techniques, already known for term and term graph rewriting [29, 7], recasting in this framework the...

This thesis proposes a general framework for equational logic programming, called categorybased equational logic by placing the general principles underlying the design of the programming language Eqlog and formulated by Goguen and Meseguer into an abstract form. This framework generalises equational deduction to an arbitrary category satisfying certain natural conditions; completeness is proved under a hypothesis of quantifier projectivity, using a semantic treatment that regards quantifiers as models rather than variables, and regards valuations as model morphisms rather than functions. This is used as a basis for a model theoretic category-based approach to a paramodulation-based operational semantics for equational logic programming languages. Category-based equational logic in conjunction with the theory of institutions is used to give mathematical foundations for modularisation in equational logic programming. We study the soundness and completeness problem for module imports i...

. Regular tree grammars can be used to generate graphs, by generating expressions that denote graphs. Top-down and bottom-up tree transducers are a tool for proving properties of such graph generating tree grammars. 1 Introduction The most well-known way of generating graphs is to replace a subgraph by another subgraph, analogous to string rewriting systems. The resulting theory of graph grammars has been developed since the seventies (see, e.g., [41, 20, 19, 32, 7]). Of particular interest are the context-free graph grammars, in which the replaced subgraph is a single edge or node. Such grammars model graph properties that can be defined in a recursive way. A conceptually completely different way of generating graphs is to use symbolic computation, as follows. First a set of operations on graphs is selected (such as disjoint union of graphs, addition of edges, identification of nodes, graph substitution, etc.). Then one uses a tree grammar to generate expressions over these oper...

This paper proposes a declarative paradigm in which parallelism is implicit and machine-independent, and the programs so developed are intrinsically parallel. This paradigm is obtained by generalizing the notion of rewriting to make it more widely applicable and capable of expressing not only functional computations but also a wide variety of parallel computations that are highly nonfunctional in nature. The generalization in question is provided by rewriting logic, a logic of change in which the states of a system are understood as algebraically axiomatized data structures, and the basic local changes that can concurrently occur in a system are axiomatized as rewrite rules that correspond to local patterns that, when present in the state of a system, can change into other patterns. Simple Maude, a carefully designed sublanguage of rewriting logic supporting three types of rewriting -- term, graph, and object-oriented --, is then proposed as a machine-independent parallel programming ...