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42
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
Describing groups
 Bull. Symb. Logic
"... Abstract. Two ways of describing a group are considered. 1. A group is finiteautomaton presentable if its elements can be represented by strings over a finite alphabet, in such a way that the set of representing strings and the group operation can be recognized by finite automata. 2. An infinite f.g ..."
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Cited by 10 (3 self)
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Abstract. Two ways of describing a group are considered. 1. A group is finiteautomaton presentable if its elements can be represented by strings over a finite alphabet, in such a way that the set of representing strings and the group operation can be recognized by finite automata. 2. An infinite f.g. group is quasifinitely axiomatizable if there is a description consisting of a single firstorder sentence, together with the information that the group is finitely generated. In the first part of the paper we survey examples of FApresentable groups, but also discuss theorems restricting this class. In the second part, we give examples of quasifinitely axiomatizable groups, consider the algebraic content of the notion, and compare it to the notion of a group which is a prime model. We also show that if a structure is biinterpretable in parameters with the ring of integers, then it is prime and
Bounds in ωregularity
"... We consider an extension of ωregular expressions where two new variants of the Kleene star L ∗ are added: L B and L S. These exponents act as the standard star, but restrict the number of iterations to be bounded (for L B) or to tend toward infinity (for L S). These expressions can define languages ..."
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Cited by 8 (4 self)
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We consider an extension of ωregular expressions where two new variants of the Kleene star L ∗ are added: L B and L S. These exponents act as the standard star, but restrict the number of iterations to be bounded (for L B) or to tend toward infinity (for L S). These expressions can define languages that are not ωregular. We develop a theory for these languages. We study the decidability and closure questions. We also define an equivalent automaton model, extending Büchi automata. This culminates with a — partial — complementation result. 1
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.
Cardinality and counting quantifiers on omegaautomatic structures
 In Proceedings of the 25th International Symposium on Theoretical Aspects of Computer Science, STACS 2008
, 2008
"... Abstract. We investigate structures that can be represented by omegaautomata, so called omegaautomatic structures, and prove that relations defined over such structures in firstorder logic expanded by the firstorder quantifiers ‘there exist at most ℵ0 many’, ’there exist finitely many ’ and ’the ..."
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Cited by 8 (0 self)
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Abstract. We investigate structures that can be represented by omegaautomata, so called omegaautomatic structures, and prove that relations defined over such structures in firstorder logic expanded by the firstorder quantifiers ‘there exist at most ℵ0 many’, ’there exist finitely many ’ and ’there exist k modulo m many ’ are omegaregular. The proof identifies certain algebraic properties of omegasemigroups. As a consequence an omegaregular equivalence relation of countable index has an omegaregular set of representatives. This implies Blumensath’s conjecture that a countable structure with an ωautomatic presentation can be represented using automata on finite words. This also complements a very recent result of Hjörth, Khoussainov, Montalban and Nies showing that there is an omegaautomatic structure which has no injective presentation. 1.
Runtime Monitoring of Metric Firstorder Temporal Properties
"... ABSTRACT. We introduce a novel approach to the runtime monitoring of complex system properties. Inparticular,wepresentanonlinealgorithmforasafetyfragmentofmetricfirstordertemporal logic that is considerably more expressive than the logics supported by prior monitoring methods. Ourapproach,basedonau ..."
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Cited by 7 (0 self)
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ABSTRACT. We introduce a novel approach to the runtime monitoring of complex system properties. Inparticular,wepresentanonlinealgorithmforasafetyfragmentofmetricfirstordertemporal logic that is considerably more expressive than the logics supported by prior monitoring methods. Ourapproach,basedonautomaticstructures,allowstheunrestricteduseofnegation,universaland existential quantification over infinite domains, and the arbitrary nesting of both past and bounded future operators. Moreover, we show how to optimize our approach for the common case where structuresconsistofonlyfiniterelations,overpossiblyinfinitedomains. Underanadditionalrestriction, we prove that the space consumed by our monitor is polynomially bounded by the cardinality of the data appearing intheprocessed prefixof thetemporal structure being monitored.
Invariants of Automatic Presentations and SemiSynchronous Transductions With Appendix
 IN STACS 2006, LNCS
, 2006
"... Automatic structures are countable structures finitely presentable by a collection of automata. We study questions related to properties invariant with respect to the choice of an automatic presentation. We give a ..."
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Cited by 6 (2 self)
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Automatic structures are countable structures finitely presentable by a collection of automata. We study questions related to properties invariant with respect to the choice of an automatic presentation. We give a
On Decidability Properties of OneDimensional Cellular Automata
, 2009
"... In a recent paper Sutner proved that the firstorder theory of the phasespace SA = (Q Z, −→) of a onedimensional cellular automaton A whose configurations are elements of Q Z, for a finite set of states Q, and where − → is the “next configuration relation”, is decidable [Sut08b]. He asked whether ..."
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Cited by 5 (0 self)
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In a recent paper Sutner proved that the firstorder theory of the phasespace SA = (Q Z, −→) of a onedimensional cellular automaton A whose configurations are elements of Q Z, for a finite set of states Q, and where − → is the “next configuration relation”, is decidable [Sut08b]. He asked whether this result could be extended to a more expressive logic. We prove in this paper that this is actuallly the case. We first show that, for each onedimensional cellular automaton A, the phasespace SA is an ωautomatic structure. Then, applying recent results of Kuske and Lohrey on ωautomatic structures, it follows that the firstorder theory, extended with some counting and cardinality quantifiers, of the structure SA, is decidable. We give some examples of new decidable properties for onedimensional cellular automata. In the case of surjective cellular automata, some more efficient algorithms can be deduced from results of [KL08a] on structures of bounded degree. On the other hand we show that the case of cellular automata give new results on automatic graphs.
The Isomorphism Problem On Classes of Automatic Structures
"... Several undecidability results on isomorphism problems for automatic structures are shown: (i) The isomorphism problem for automatic equivalence relations is Π0 1complete. (ii) The isomorphism problem for automatic trees of height n ≥ 2 is Π0 2n−3complete. (iii) The isomorphism problem for automat ..."
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Cited by 4 (2 self)
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Several undecidability results on isomorphism problems for automatic structures are shown: (i) The isomorphism problem for automatic equivalence relations is Π0 1complete. (ii) The isomorphism problem for automatic trees of height n ≥ 2 is Π0 2n−3complete. (iii) The isomorphism problem for automatic linear orders is not arithmetical. 1