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InductiveDataType Systems
, 2002
"... In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgenera ..."
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Cited by 751 (22 self)
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In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the leI two authors presented a combined lmbined made of a (strongl normal3zG9 alrmal rewrite system and a typed #calA#Ik enriched by patternmatching definitions folnitio a certain format,calat the "General Schema", whichgeneral39I theusual recursor definitions fornatural numbers and simil9 "basic inductive types". This combined lmbined was shown to bestrongl normalIk39f The purpose of this paper is toreformul33 and extend theGeneral Schema in order to make it easil extensibl3 to capture a more general cler of inductive types, cals, "strictly positive", and to ease the strong normalgAg9Ik proof of theresulGGg system. Thisresul provides a computation model for the combination of anal"DAfGI specification language based on abstract data types and of astrongl typed functional language with strictly positive inductive types.
A Partial KArboretum of Graphs With Bounded Treewidth
 J. Algorithms
, 1998
"... 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 ..."
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Cited by 248 (34 self)
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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 monadic secondorder logic of graphs I. Recognizable sets of Finite Graphs
 Information and Computation
, 1990
"... The notion of a recognizable sef offinite graphs is introduced. Every set of finite graphs, that is definable in monadic secondorder logic is recognizable, but not vice versa. The monadic secondorder theory of a contextfree set of graphs is decidable. 0 19W Academic Press. Inc. This paper begins ..."
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Cited by 206 (16 self)
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The notion of a recognizable sef offinite graphs is introduced. Every set of finite graphs, that is definable in monadic secondorder logic is recognizable, but not vice versa. The monadic secondorder theory of a contextfree set of graphs is decidable. 0 19W Academic Press. Inc. This paper begins an investigation of the monadic secondorder 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 hyperedgelabelled 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 firstorder logic or in secondorder logic. It turns out that monadic secondorder 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 secondorder 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 secondorder logic) is recognizable. * This work has been supported by the “Programme de Recherches Coordonntes: Mathematiques et Informatique.”
The Expression Of Graph Properties And Graph Transformations In Monadic SecondOrder Logic
, 1997
"... By considering graphs as logical structures, one... ..."
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Cited by 143 (40 self)
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By considering graphs as logical structures, one...
Rewriting Logic as a Semantic Framework for Concurrency: a Progress Report
, 1996
"... . 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 mod ..."
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Cited by 82 (22 self)
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. 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...
An algebraic theory of graph reduction
, 1993
"... We show how membership in classes of graphs definable m monwhc secondorder]oglc and of bounded treewidth can be decided by finite sets of terminating reduction rules. The method is constructive in the sense that wc describe an algorlthm that wdl produce, from J formula in monxhc secondorder Ioglc ..."
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Cited by 68 (8 self)
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We show how membership in classes of graphs definable m monwhc secondorder]oglc and of bounded treewidth can be decided by finite sets of terminating reduction rules. The method is constructive in the sense that wc describe an algorlthm that wdl produce, from J formula in monxhc secondorder Ioglc and an mleger k such that the class dcfmed by the formul ~ IS of treewidth s k, a set of rewrite rules that rcducxs any member of the elms to one of’ firrltely many graphs, in a number of steps bounded by the size c~f the graph. This reduction syetem ymlds an algorithm that runs m time linear m the size of the graph. We illustrate our results with reduction systems that recognize some families of outerplanar and planar graphs.
An Algebraic Presentation of Term Graphs, via GSMonoidal Categories
 Applied Categorical Structures
, 1999
"... . We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the wellknown characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particula ..."
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Cited by 38 (25 self)
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. We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the wellknown 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 onetoone with the arrows of the free gsmonoidal 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, gsmonoidal categories, symmetric monoidal categories, term graphs. Mathematical Subject Clas...
The monadic secondorder logic of graphs V: on closing the gap between definability and recognizability
, 1991
"... ..."
Research Directions in Rewriting Logic
, 1998
"... 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 ..."
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Cited by 31 (12 self)
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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 userdefinable 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.
An Inductive View of Graph Transformation
 In Workshop on Algebraic Development Techniques
, 1998
"... . The dynamic behavior of rulebased 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 i ..."
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Cited by 30 (12 self)
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. The dynamic behavior of rulebased 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 dgsmonoidal category. We then apply 2categorical techniques, already known for term and term graph rewriting [29, 7], recasting in this framework the...