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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 208 (14 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.”
A Hidden Agenda
 Theoretical Computer Science
, 2000
"... This paper publicly reveals, motivates, and surveys the results of an ambitious hidden agenda for applying algebra to software engineering. The paper reviews selected literature, introduces a new perspective on nondeterminism, and features powerful hidden coinduction techniques for proving behaviora ..."
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Cited by 110 (23 self)
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This paper publicly reveals, motivates, and surveys the results of an ambitious hidden agenda for applying algebra to software engineering. The paper reviews selected literature, introduces a new perspective on nondeterminism, and features powerful hidden coinduction techniques for proving behavioral properties of concurrent systems, especially renements; some proofs are given using OBJ3. We also discuss where modularization, bisimulation, transition systems and combinations of the object, logic, constraint and functional paradigms t into our hidden agenda. 1 Introduction Algebra can be useful in many dierent ways in software engineering, including specication, validation, language design, and underlying theory. Specication and validation can help in the practical production of reliable programs, advances in language design can help improve the state of the art, and theory can help with building new tools to increase automation, as well as with showing correctness of the whole e...
TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
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A Functorial Semantics for MultiAlgebras and Partial Algebras, With Applications to Syntax
, 2000
"... Multialgebras allow for the modeling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multialgebras and partial algebras, analogous to the classica ..."
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Cited by 14 (7 self)
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Multialgebras allow for the modeling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multialgebras and partial algebras, analogous to the classical presentation of algebras over a signature as cartesian functors from the algebraic theory over to Set. We introduce two dierent notions of theory over a signature, both having a structure weaker than cartesian, and we consider functors from them to Rel or Pfn, the categories of sets and relations or partial functions, respectively. Next we discuss how the functorial presentation provides guidelines for the choice of syntactical notions for a class of algebras, and as an application we argue that the natural generalization of usual terms are \conditioned terms" for partial algebras, and \term graphs" for multialgebras. Contents 1 Introduction 2 2 A short recap on multialgebras 4 3...
On logically defined recognizable tree languages
 In: Proc. FST TCS 2003
, 2003
"... Abstract. We provide an algebraic characterization of the expressive power of various naturally defined logics on finite trees. These logics are described in terms of Lindström quantifiers, and particular cases include firstorder logic and modular logic. The algebraic characterization we give is ex ..."
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Cited by 14 (6 self)
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Abstract. We provide an algebraic characterization of the expressive power of various naturally defined logics on finite trees. These logics are described in terms of Lindström quantifiers, and particular cases include firstorder logic and modular logic. The algebraic characterization we give is expressed in terms of a new algebraic structure, finitary preclones, and uses a generalization of the block product operation. 1
Rationality in Algebras With a Series Operation
 Information and Computation
, 2000
"... . This paper considers languages in a free algebra which has a binary associative operation called the series product. We define automata operating in these algebras and rational expressions, and we show that their expressive powers coincide (a Kleene theorem). We also show that this expressive p ..."
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Cited by 14 (4 self)
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. This paper considers languages in a free algebra which has a binary associative operation called the series product. We define automata operating in these algebras and rational expressions, and we show that their expressive powers coincide (a Kleene theorem). We also show that this expressive power equals that of algebraic recognizability (a MyhillNerode theorem). This generalizes the work of Thatcher and Wright. The first equivalence continues to hold when conditions such as associativity and commutativity are imposed on the term operations, but recognizability is weaker when one of the term operations (other than the series product) is associative. We also consider languages which have a bound on the number of nested occurrences of certain designated term operations and get both the equivalences mentioned above. This generalizes our earlier work and answers a question left open therein. 1 Automata form one of the most commonly used computing device, from their histor...
Universal regular path queries
 HigherOrder and Symbolic Computation
, 2003
"... Given are a directed edgelabelled graph G with a distinguished node n0, and a regular expression P which may contain variables. We wish to compute all substitutions φ (of symbols for variables), together with all nodes n such that all paths n0 → n are in φ(P). We derive an algorithm for this proble ..."
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Cited by 12 (1 self)
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Given are a directed edgelabelled graph G with a distinguished node n0, and a regular expression P which may contain variables. We wish to compute all substitutions φ (of symbols for variables), together with all nodes n such that all paths n0 → n are in φ(P). We derive an algorithm for this problem using relational algebra, and show how it may be implemented in Prolog. The motivation for the problem derives from a declarative framework for specifying compiler optimisations. 1 Bob Paige and IFIP WG 2.1 Bob Paige was a longstanding member of IFIP Working Group 2.1 on Algorithmic Languages and Calculi. In recent years, the main aim of this group has been to investigate the derivation of algorithms from specifications by program transformation. Already in the mideighties, Bob was way ahead of the pack: instead of applying transformational techniques to wellworn examples, he was applying his theories of program transformation to new problems, and discovering new algorithms [16, 48, 52]. The secret of his success lay partly in his insistence on the study of general algorithm design strategies (in particular
Categories of Relational Structures
, 1998
"... . The paper characterises compositional homomorphims of relational structures. A detailed study of three categories of such structures  viewed as multialgebras  reveals the one with the most desirable properties. In addition, we study analogous categories with homomorphisms mapping elements to s ..."
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Cited by 11 (3 self)
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. The paper characterises compositional homomorphims of relational structures. A detailed study of three categories of such structures  viewed as multialgebras  reveals the one with the most desirable properties. In addition, we study analogous categories with homomorphisms mapping elements to sets (thus being relations). Finally, we indicate some consequences of our results for partial algebras which are special case of multialgebras. 1 Introduction In the study of universal algebra, the central place occupies the pair of "dual" notions of congruence and homomorphism: every congruence on an algebra induces a homomorphism into a quotient and every homomorphism induces a congruence on the source algebra. Categorical approach attempts to express all (internal) properties of algebras in (external) terms of homomorphisms. When passing to relational structures, however, the close correspondence of these internal and external aspects seems to get lost. The most common, and natural, gene...
Solving Optimisation Problems with Catamorphisms
, 1992
"... . This paper contributes to an ongoing effort to construct a calculus for deriving programs for optimisation problems. The calculus is built around the notion of initial data types and catamorphisms which are homomorphisms on initial data types. It is shown how certain optimisation problems, which a ..."
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Cited by 11 (3 self)
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. This paper contributes to an ongoing effort to construct a calculus for deriving programs for optimisation problems. The calculus is built around the notion of initial data types and catamorphisms which are homomorphisms on initial data types. It is shown how certain optimisation problems, which are specified in terms of a relational catamorphism, can be solved by means of a functional catamorphism. The result is illustrated with a derivation of Kruskal's algorithm for finding a minimum spanning tree in a connected graph. 1 Introduction Efficient algorithms for solving optimisation problems can sometimes be expressed as homomorphisms on initial data types. Such homomorphisms, which correspond to the familiar fold operators in functional programming, are called catamorphisms. In this paper, we give conditions under which an optimisation problem can be solved by a catamorphism. Our results are a natural generalisation of earlier work by Jeuring [5, 6], who considered the same problem ...
On the MyhillNerode Theorem for Trees
 Bulletin of the EATCS
, 1992
"... Science Department, Cornell University, Ithaca, New York 14853, USA mention of the equivalence of the tree analogs of (i) and (ii) seems to be by Brainerd [2, 3] and Eilenberg and Wright [4], although the latter claim that their paper "contains nothing that is essentially new, except perhaps for a ..."
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Cited by 9 (1 self)
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Science Department, Cornell University, Ithaca, New York 14853, USA mention of the equivalence of the tree analogs of (i) and (ii) seems to be by Brainerd [2, 3] and Eilenberg and Wright [4], although the latter claim that their paper "contains nothing that is essentially new, except perhaps for a point of view" [4]. A relation on trees analogous to jR was defined and clause (iii) added explicitly by Arbib and Give'on [1, Definition 2.13], although it is also essentially implicit in work of Brainerd [2, 3]. All the cited papers from the 1960s involve heavy use of universal algebra and/or category theory. In these papers, a tree automaton is a finite \Sigmaalgebra, and the map b ffi (see below) is a \Sigmaalgebra homomorphism. Although exceedingly elegant, this approach renders the result less accessible to the average computer science undergraduate. Fulop and V'agvolgyi take a somewhat different approach, appealing to t