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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.”
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 147 (39 self)
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By considering graphs as logical structures, one...
A Comparison of Tree Transductions defined by Monadic Second Order Logic and by Attribute Grammars
, 1998
"... . Two wellknown formalisms for the specication and computation of tree transductions are compared: the mso graph transducer and the attributed tree transducer with lookahead, respectively. The mso graph transducer, restricted to trees, uses monadic second order logic to dene the output tree in ..."
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Cited by 25 (8 self)
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. Two wellknown formalisms for the specication and computation of tree transductions are compared: the mso graph transducer and the attributed tree transducer with lookahead, respectively. The mso graph transducer, restricted to trees, uses monadic second order logic to dene the output tree in terms of the input tree. The attributed tree transducer is an attribute grammar in which all attributes are trees; it is preceded by a lookahead phase in which all attributes have nitely many values. The main result is that these formalisms are equivalent, i.e., that the attributed tree transducer with lookahead is an appropriate implementation model for the tree transductions that are speciable in mso logic. This result holds for mso graph transducers that produce trees with shared subtrees. If no sharing is allowed, the attributed tree transducer satises the single use restriction. 1 Introduction Formulas of monadic second order (mso) logic can be used to express properti...
Algebraic Approaches to Nondeterminism  an Overview
 ACM Computing Surveys
, 1997
"... this paper was published as Walicki, M.A. and Meldal, S., 1995, Nondeterministic Operators in Algebraic Frameworks, Tehnical Report No. CSLTR95664, Stanford University ..."
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Cited by 23 (3 self)
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this paper was published as Walicki, M.A. and Meldal, S., 1995, Nondeterministic Operators in Algebraic Frameworks, Tehnical Report No. CSLTR95664, Stanford University
Logics For ContextFree Languages
, 1995
"... We define matchings, and show that they capture the essence of contextfreeness. More precisely, we show that the class of contextfree languages coincides with the class of those sets of strings which can be defined by sentences of the form 9 b', where ' is first order, b is a binary predicate sym ..."
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Cited by 23 (5 self)
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We define matchings, and show that they capture the essence of contextfreeness. More precisely, we show that the class of contextfree languages coincides with the class of those sets of strings which can be defined by sentences of the form 9 b', where ' is first order, b is a binary predicate symbol, and the range of the second order quantifier is restricted to the class of matchings. Several variations and extensions are discussed.
Decidability of the Finiteness of Ranges of Tree Transductions
 Inform. and Comput
, 1996
"... . The finiteness of ranges of tree transductions is shown to be decidable for TBY + , the composition closure of macro tree transductions. Furthermore, TBY + definable sets and TBY + computable relations are considered, which are obtained by viewing a tree as an expression that denotes an elem ..."
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Cited by 17 (7 self)
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. The finiteness of ranges of tree transductions is shown to be decidable for TBY + , the composition closure of macro tree transductions. Furthermore, TBY + definable sets and TBY + computable relations are considered, which are obtained by viewing a tree as an expression that denotes an element of a given algebra. A sufficient condition on the considered algebra is formulated under which the finiteness problem is decidable for TBY + definable sets and for the ranges of TBY + computable relations. The obtained result applies in particular to the class of string languages that can be defined by TBY + transductions via the yield mapping. This is a large class which is proved to form a substitutionclosed full AFL. 1 Introduction The finiteness problem is one of the classical decidability problems in formal language theory. For a given language of interest, one usually does not wish to know whether that language is finite (because it usually is not), but rather whether the l...
Recognizability equals Monadic SecondOrder definability, for sets of graphs of bounded treewidth.
 In Proc. STACS'98, volume 1373 of LNCS
, 1998
"... We prove that for each k, there exists a MSOtransduction that associates with every graph of treewidth at most k one of its treedecompositions of width at most k. Courcelle proves in (The Monadic secondorder logic of graphs, I: Recognizable sets of finite graphs) that every set of graphs is recogn ..."
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Cited by 16 (2 self)
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We prove that for each k, there exists a MSOtransduction that associates with every graph of treewidth at most k one of its treedecompositions of width at most k. Courcelle proves in (The Monadic secondorder logic of graphs, I: Recognizable sets of finite graphs) that every set of graphs is recognizable if it is definable in Counting Monadic SecondOrder logic. It follows that every set of graphs of bounded treewidth is CMSOdefinable if and only if it is recognizable. A fundamental theorem by Buchi [2] states that a language of words is recognizable iff it is definable by some formula in a monadic secondorder logic (MSOL). This result is extended to finite ranked ordered trees by Doner [7], and to sets of finite unranked unordered trees by Courcelle [3]. This last result uses an extension of MSOL, called counting monadic secondorder logic (CMSOL), that allows counting of cardinality of sets modulo fixed integers. These results relate an algebraic aspect, namely recognizab...
TreeBased Picture Generation
 Theoretical Computer Science
, 2000
"... . The concept of treebased picture generation is introduced. It is shown that there are equivalent treebased definitions of four picturegenerating devices known from the literature, namely collage grammars, mutually recursive function systems, contextfree chaincode grammars, and 0Lsystems with ..."
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Cited by 14 (7 self)
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. The concept of treebased picture generation is introduced. It is shown that there are equivalent treebased definitions of four picturegenerating devices known from the literature, namely collage grammars, mutually recursive function systems, contextfree chaincode grammars, and 0Lsystems with turtle interpretation. Furthermore, generalisations of each of these systems are discussed. 1 Introduction During the last two decades picture generation has become a large field whose manyfold aspects are studied in mathematics as well as in practical and theoretical computer science. It attracts the interest of numerous researchers from diverse directions, which is no surprise because one can find in this area a great number of intellectually appealing mathematical and computational problems, interesting applications like the modelling of plant development and, one should not forget to mention this, an astonishing variety of beautiful pictures. In this paper picture generation is studied...
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