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The Caucal hierarchy of infinite graphs in terms of logic and higherorder pushdown automata
 IN FSTTCS’03, VOLUME 2914 OF LNCS
, 2003
"... In this paper we give two equivalent characterizations of the Caucal hierarchy, a hierarchy of infinite graphs with a decidable monadic secondorder (MSO) theory. It is obtained by iterating the graph transformations of unfolding and inverse rational mapping. The first characterization sticks to thi ..."
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Cited by 59 (8 self)
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In this paper we give two equivalent characterizations of the Caucal hierarchy, a hierarchy of infinite graphs with a decidable monadic secondorder (MSO) theory. It is obtained by iterating the graph transformations of unfolding and inverse rational mapping. The first characterization sticks to this hierarchical approach, replacing the languagetheoretic operation of a rational mapping by an MSOtransduction and the unfolding by the treegraph operation. The second characterization is noniterative. We show that the family of graphs of the Caucal hierarchy coincides with the family of graphs obtained as the εclosure of configuration graphs of higherorder pushdown automata. While the different characterizations of the graph family show their robustness and thus also their importance, the characterization in terms of higherorder pushdown automata additionally yields that the graph hierarchy is indeed strict.
Finite Presentations of Infinite Structures: Automata and Interpretations
 Theory of Computing Systems
, 2002
"... We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by modeltheoretic interpretations. ..."
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Cited by 42 (3 self)
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We study definability problems and algorithmic issues for infinite structures that are finitely presented. After a brief overview over different classes of finitely presentable structures, we focus on structures presented by automata or by modeltheoretic interpretations.
Finite Model Theory and Descriptive Complexity
, 2002
"... This is a survey on the relationship between logical definability and computational complexity on finite structures. Particular emphasis is given to gamebased evaluation algorithms for various logical formalisms and to logics capturing complexity classes. In addition to the ..."
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Cited by 24 (7 self)
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This is a survey on the relationship between logical definability and computational complexity on finite structures. Particular emphasis is given to gamebased evaluation algorithms for various logical formalisms and to logics capturing complexity classes. In addition to the
Regular sets of higherorder pushdown stacks
 In MFCS
, 2005
"... Abstract. It is a wellknown result that the set of reachable stack contents in a pushdown automaton is a regular set of words. We consider the more general case of higherorder pushdown automata and investigate, with a particular stress on effectiveness and complexity, the natural notion of regular ..."
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Cited by 17 (5 self)
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Abstract. It is a wellknown result that the set of reachable stack contents in a pushdown automaton is a regular set of words. We consider the more general case of higherorder pushdown automata and investigate, with a particular stress on effectiveness and complexity, the natural notion of regularity for higherorder stacks: a set of level k stacks is regular if it is obtained by a regular sequence of level k operations. We prove that any regular set of level k stacks admits a normalized representation and we use it to show that the regular sets of a given level form an effective Boolean algebra. In fact, this notion of regularity coincides with the notion of monadic second order definability over the canonical structure associated to level k stacks. Finally, we consider the link between regular sets of stacks and families of infinite graphs defined by higherorder pushdown systems.
TRANSFORMING STRUCTURES BY SET INTERPRETATIONS
"... We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic secondorder (WMSO) formulas with free set variables. They transform a given structure into a structure with a domain consisting of finite sets of eleme ..."
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Cited by 8 (2 self)
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We consider a new kind of interpretation over relational structures: finite sets interpretations. Those interpretations are defined by weak monadic secondorder (WMSO) formulas with free set variables. They transform a given structure into a structure with a domain consisting of finite sets of elements of the orignal structure. The definition of these interpretations directly implies that they send structures with a decidable WMSO theory to structures with a decidable firstorder theory. In this paper, we investigate the expressive power of such interpretations applied to infinite deterministic trees. The results can be used in the study of automatic and treeautomatic structures.
Logical Aspects of CayleyGraphs: The Group Case
 TO APPEAR IN ANNALS OF PURE AND APPLIED LOGIC
"... We prove that a finitely generated group is contextfree whenever its Cayleygraph has a decidable monadic secondorder theory. Hence, by the seminal work of Muller and Schupp, our result gives a logical characterization of contextfree groups and also proves a conjecture of Schupp. To derive this re ..."
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Cited by 6 (3 self)
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We prove that a finitely generated group is contextfree whenever its Cayleygraph has a decidable monadic secondorder theory. Hence, by the seminal work of Muller and Schupp, our result gives a logical characterization of contextfree groups and also proves a conjecture of Schupp. To derive this result, we investigate general graphs and show that a graph of bounded degree with a high degree of symmetry is contextfree whenever its monadic secondorder theory is decidable. Further, it is shown that the word problem of a finitely generated group is decidable if and only if the firstorder theory of its Cayleygraph is decidable.
Infinite State ModelChecking of Propositional Dynamic Logics
 In Proc. CSL 2006, LNCS 4207
, 2006
"... Abstract. Modelchecking problems for PDL (propositional dynamic logic) and its extension PDL ∩ (which includes the intersection operator on programs) over various classes of infinite state systems (BPP, BPA, pushdown systems, prefixrecognizable systems) are studied. Precise upper and lower bounds ..."
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Cited by 3 (3 self)
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Abstract. Modelchecking problems for PDL (propositional dynamic logic) and its extension PDL ∩ (which includes the intersection operator on programs) over various classes of infinite state systems (BPP, BPA, pushdown systems, prefixrecognizable systems) are studied. Precise upper and lower bounds are shown for the data/expression/combined complexity of these modelchecking problems. 1
Micromacro stack systems: A new frontier of decidability for sequential systems
 In 18th LICS, 381390
, 2003
"... We define the class of micromacro stack graphs, a new class of graphs modeling infinitestate sequential systems with a decidable modelchecking problem. Micromacro stack graphs are the configuration graphs of stack automata whose states are partitioned into micro and macro states. Nodes of the gr ..."
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Cited by 2 (2 self)
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We define the class of micromacro stack graphs, a new class of graphs modeling infinitestate sequential systems with a decidable modelchecking problem. Micromacro stack graphs are the configuration graphs of stack automata whose states are partitioned into micro and macro states. Nodes of the graph are configurations of the stack automaton where the state is a macro state. Edges of the graph correspond to the sequence of micro steps that the automaton makes between macro states. We prove that this class strictly contains the class of prefixrecognizable graphs. We give a direct automatatheoretic algorithm for model checking ¢calculus formulas over micromacro stack graphs. 1
Decidable Theories of Cayleygraphs
 PROCEEDINGS OF THE 20TH ANNUAL SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE (STACS 2003), BERLIN (GERMANY), NUMBER 2607 IN LECTURE NOTES IN COMPUTER SCIENCE
, 2003
"... We prove that a connected graph of bounded degree with only finitely many orbits has a decidable MSOtheory if and only if it is contextfree. This implies that a group is contextfree if and only if its Cayleygraph has a decidable MSOtheory. On the other hand, the rstorder theory of the Cayl ..."
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Cited by 2 (2 self)
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We prove that a connected graph of bounded degree with only finitely many orbits has a decidable MSOtheory if and only if it is contextfree. This implies that a group is contextfree if and only if its Cayleygraph has a decidable MSOtheory. On the other hand, the rstorder theory of the Cayleygraph of a group is decidable if and only if the group has a decidable word problem. For Cayleygraphs of monoids we prove the following closure properties. The class of monoids whose Cayleygraphs have decidable MSOtheories is closed under free products. The class of monoids whose Cayleygraphs have decidable firstorder theories is closed under general graph products. For the latter result on firstorder theories we introduce a new unfolding construction, the factorized unfolding, that generalizes the treelike structures considered by Walukiewicz. We show and use that it preserves the decidability of the firstorder theory. Most of
On factorisation forests And some applications
, 2007
"... infinite structures. Abstract. The theorem of factorisation forests shows the existence of nested factorisations — a la Ramsey — for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the importance of this approach in th ..."
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Cited by 1 (0 self)
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infinite structures. Abstract. The theorem of factorisation forests shows the existence of nested factorisations — a la Ramsey — for finite words. This theorem has important applications in semigroup theory, and beyond. The purpose of this paper is to illustrate the importance of this approach in the context of automata over infinite words and trees. We extend the theorem of factorisation forest in two directions: we show that it is still valid for any word indexed by a linear ordering; and we show that it admits a deterministic variant for words indexed by wellorderings. A byproduct of this work is also an improvement on the known bounds for the original result. We apply the first variant for giving a simplified proof of the closure under complementation of rational sets of words indexed by countable scattered linear orderings. We apply the second variant in the analysis of monadic secondorder logic over trees, yielding new results on monadic interpretations over trees. Consequences of it are new caracterisations of prefixrecognizable structures and of the Caucal hierarchy. 1