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26
Infinitary Logic and Inductive Definability over Finite Structures
 Information and Computation
, 1995
"... The extensions of firstorder logic with a least fixed point operator (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [Abi ..."
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Cited by 56 (6 self)
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The extensions of firstorder logic with a least fixed point operator (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [Abiteboul and Vianu, 1991b] investigated the relationship of these two logics in the absence of an ordering, using a machine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each formula has a bounded number of variables, L ! 1! (see, for instance, [Kolaitis and Vardi, 1990]). We investigate this logic of finite structures and provide a normal form for it. We also present a treatment of the results in [Abiteboul and Vianu, 1991b] from this point of view. In particular, we show that we can write a formula of FO + LFP that defines ...
Feasible Computation through Model Theory
, 1993
"... The computational complexity of a problem is usually defined in terms of the resources required on some machine model of computation. An alternative view looks at the complexity of describing the problem (seen as a collection of relational structures) in a logic, measuring logical resources such as ..."
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Cited by 36 (7 self)
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The computational complexity of a problem is usually defined in terms of the resources required on some machine model of computation. An alternative view looks at the complexity of describing the problem (seen as a collection of relational structures) in a logic, measuring logical resources such as the number of variables, quantifiers, operators, etc. A close correspondence has been observed between these two, with many natural logics corresponding exactly to independently defined complexity classes. For the complexity classes that are generally identified with feasible computation, such characterizations require the presence of a linear order on the domain of every structure, in which case the class PTIME is characterized by an extension of firstorder logic by means of an inductive operator. No logical characterization of feasible computation is known for unordered structures. We approach this question from two directions. On the one hand, we seek to accurately characterize the expre...
Modal Model Theory
 ANNALS OF PURE AND APPLIED LOGIC
, 1995
"... This paper contributes to the model theory of modal logic using bisimulations as the fundamental tool. A uniform presentation is given of modal analogues of wellknown definability and preservation results from firstorder logic. These results include algebraic characterizations of modal equivalen ..."
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Cited by 9 (5 self)
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This paper contributes to the model theory of modal logic using bisimulations as the fundamental tool. A uniform presentation is given of modal analogues of wellknown definability and preservation results from firstorder logic. These results include algebraic characterizations of modal equivalence, and of the modally definable classes of models; the preservation results concern preservation of modal formulas under submodels, unions of chains, and homomorphisms.
Axiomatising Various Classes of Relation and Cylindric Algebras
 Logic Journal of the IGPL
, 1997
"... We outline a simple approach to axiomatising the class of representable relation algebras, using games. We discuss generalisations of the method to cylindric algebras, homogeneous and complete representations, and atom structures of relation algebras. 1 Introduction Relation algebras are to bina ..."
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Cited by 8 (5 self)
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We outline a simple approach to axiomatising the class of representable relation algebras, using games. We discuss generalisations of the method to cylindric algebras, homogeneous and complete representations, and atom structures of relation algebras. 1 Introduction Relation algebras are to binary relations what boolean algebras are to unary ones. They are used in artificial intelligence, where, for example, the AllenKoomen temporal planning system checks the consistency of given relations between time intervals. In mathematics, they form a part of algebraic logic. The history of this goes back to the nineteenth century, the early workers including Boole, de Morgan, Peirce, and Schroder; it was studied intensively by Tarski's group (including, at various times, Chin, Givant, Henkin, J'onsson, Lyndon, Maddux, Monk, N'emeti) from around the 1950s, and currently we know of active groups in Amsterdam, Budapest, Rio de Janeiro, South Africa, and the U.S., among other places. Abstract...
Finite Variable Logics
, 1993
"... In this survey article we discuss some aspects of finite variable logics. We translate some wellknown fixedpoint logics into the infinitary logic L ! 1! , discussing complexity issues. We give a game characterisation of L ! 1! , and use it to derive results on Scott sentences. In this conne ..."
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Cited by 7 (0 self)
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In this survey article we discuss some aspects of finite variable logics. We translate some wellknown fixedpoint logics into the infinitary logic L ! 1! , discussing complexity issues. We give a game characterisation of L ! 1! , and use it to derive results on Scott sentences. In this connection we consider definable linear orderings of types realised in finite structures. We then show that the Craig interpolation and Beth definability properties fail for L ! 1! . Finally we examine some connections of finite variable logic to temporal logic. Credits and references are given throughout. 1 Some extensions of firstorder logic Quisani: Hello. Who are you? I am Yuri's imaginary student, and I usually talk to him at this time. Author: I'm afraid he may be a bit late. I am a computer scientist from London, England. I have some imaginary students myself, so maybe I can help. I was reading your earlier conversation on 01 laws [Gu3]. Quisani: I remember it. We examined the...
A modal proof theory for final polynomial coalgebras. Theoret
 Comput. Sci
"... An infinitary proof theory is developed for modal logics whose models are coalgebras of polynomial functors on the category of sets. The canonical model method from modal logic is adapted to construct a final coalgebra for any polynomial functor. The states of this final coalgebra are certain “maxim ..."
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Cited by 4 (1 self)
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An infinitary proof theory is developed for modal logics whose models are coalgebras of polynomial functors on the category of sets. The canonical model method from modal logic is adapted to construct a final coalgebra for any polynomial functor. The states of this final coalgebra are certain “maximal ” sets of formulas that have natural syntactic closure properties. The syntax of these logics extends that of previously developed modal languages for polynomial coalgebras by adding formulas that express the “termination ” of certain functions induced by transition paths. A completeness theorem is proven for the logic of functors which have the Lindenbaum property that every consistent set of formulas has a maximal extension. This property is shown to hold if if the deducibility relation is generated by countably many inference rules. A counterexample to completeness is also given. This is a polynomial functor that is not Lindenbaum: it has an uncountable set of formulas that is deductively consistent but has no maximal extension and is unsatisfiable, even though all of its countable subsets are satisfiable. 1
Enumerations in computable structure theory
 Ann. Pure Appl. Logic
"... Goncharov, Harizanov, Knight, Miller, and Solomon gratefully acknowledge NSF support under binational ..."
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Cited by 4 (1 self)
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Goncharov, Harizanov, Knight, Miller, and Solomon gratefully acknowledge NSF support under binational
Model Theoretic Complexity of Automatic Structures
 PROC. TAMC ’08, LNCS 4978
, 2008
"... We study the complexity of automatic structures via wellestablished concepts from both logic and model theory, including ordinal heights (of wellfounded relations), Scott ranks of structures, and CantorBendixson ranks (of trees). We prove the following results: 1) The ordinal height of any autom ..."
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Cited by 4 (2 self)
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We study the complexity of automatic structures via wellestablished concepts from both logic and model theory, including ordinal heights (of wellfounded relations), Scott ranks of structures, and CantorBendixson ranks (of trees). We prove the following results: 1) The ordinal height of any automatic wellfounded partial order is bounded by ωω; 2) The ordinal heights of automatic wellfounded relations are unbounded below ωCK 1, the first noncomputable ordinal; 3) For any computable ordinal α, there is an automatic structure of Scott rank at least α. Moreover, there are automatic structures of Scott rank ωCK 1, ωCK 1 +1; 4) For any computable ordinal α, there is an automatic successor tree of CantorBendixson rank α.
Classification from a computable viewpoint
 The Bulletin of Symbolic Logic
"... Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism ..."
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Cited by 4 (0 self)
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Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism