The extensions of first-order 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 ...

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 first-order 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...

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 Allen--Koomen 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...

In this survey article we discuss some aspects of finite variable logics. We translate some well-known fixed-point 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 first-order 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 0--1 laws [Gu3]. Quisani: I remember it. We examined the...

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 well-known definability and preservation results from first-order 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.

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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 counter-example 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

this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language L, then, clearly, any definable subset S ae M (i.e., a subset S = fa j M j= '(a)g defined by some formula ') is invariant under all automorphisms of M . The same is of course true for subsets of M

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Abstract. We prove that if ZF is consistent then ZFC + GCH is consistent with the following statement: There is for every k < ω a model of cardinality ℵ1 which is L∞ω1-equivalent to exactly k non-isomorphic models of cardinality ℵ1. In order to get this result we introduce ladder systems and colourings different from the “standard ” counterparts, and prove the following purely combinatorial result: For each prime number p and positive integer m it is consistent with ZFC + GHC that there is a “good ” ladder system having exactly pm pairwise nonequivalent colourings. 1.