Hybrid languages are expansions of propositional modal languages which can refer to (or even quantify over) worlds. The use of strong hybrid languages dates back to at least [Pri67], but recent work (for example [BS98, BT98a, BT99]) has focussed on a more constrained system called H(#; @). We show in detail that H(#; @) is modally natural. We begin by studying its expressivity, and provide model theoretic characterizations (via a restricted notion of Ehrenfeucht-Frasse game, and an enriched notion of bisimulation) and a syntactic characterization (in terms of bounded formulas). The key result to emerge is that H(#; @) corresponds to the fragment of rst-order logic which is invariant for generated submodels. We then show that H(#; @) enjoys (strong) interpolation, provide counterexamples for its nite variable fragments, and show that weak interpolation holds for the sublanguage H(@). Finally, we provide complexity results for H(@) and other fragments and variants, and sh...

We study definability and complexity issues for automatic and w-automatic structures. These are, in general, infinite structures but they can be finitely presented by a collection of automata. Moreover, they admit effective (in fact automatic) evaluation of all first-order queries. Therefore, automatic structures provide an interesting framework for extending many algorithmic and logical methods from finite structures to infinite ones. We explain the notion of (w-)automatic structures, give examples, and discuss the relationship to automatic groups. We determine the complexity of model checking and query evaluation on automatic structures for fragments of first-order logic. Further, we study closure properties and definability issues on automatic structures and present a technique for proving that a structure is not automatic. We give model-theoretic characterisations for automatic structures via interpretations. Finally we discuss the composition theory of automatic structures and pro...

We identify the computational complexity of the satisfiability problem for FO², the fragment of first-order logic consisting of all relational first-order sentences with at most two distinct variables. Although this fragment was shown to be decidable a long time ago, the computational complexity of its decision problem has not been pinpointed so far. In 1975 Mortimer proved that FO² has the finite-model property, which means that if an FO²-sentence is satisfiable, then it has a finite model. Moreover, Mortimer showed that every satisfiable FO²-sentence has a model whose size is at most doubly exponential in the size of the sentence. In this paper, we improve Mortimer's bound by one exponential and show that every satisfiable FO²-sentence has a model whose size is at most exponential in the size of the sentence. As a consequence, we establish that the satisfiability problem for FO² is NEXPTIME-complete.

This paper is a survey and systematic presentation of decidability and complexity issues for modal and non-modal two-variable logics. A classical result due to Mortimer says that the two-variable fragment of firstorder logic, denoted FO 2 , has the finite model property and is therefore decidable for satisfiability. One of the reasons for the significance of this result is that many propositional modal logics can be embedded into FO 2 . Logics that are of interest for knowledge representation, for the specification and verification of concurrent systems and for other areas of computer science are often defined (or can be viewed) as extensions of modal logics by features like counting constructs, path quantifiers, transitive closure operators, least and greatest fixed points etc. Examples of such logics are computation tree logic CTL, the modal ¯-calculus L¯ , or popular description logics used in artificial intelligence. Although the additional features are usually not first-order...

this paper the internal dynamics of mental states, in particular states based on beliefs, desires and intentions, is formalised using a temporal language. A software environment is presented that can be used to specify, simulate and analyse temporal dependencies between mental states in relation to traces of them. If also relevant data on internal physical states over time are available, these can be analysed with respect to their relation to mental states as well

Abstract. We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a non-elementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group. • If H is a non-parabolic subgroup of a relatively hyperbolic group, and if any isometric H-action on an R-tree is trivial, then H is Hopfian. • If G is a non-elementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a non-elementary relatively hyperbolic quotient that is Hopfian. • Any finitely presented group is isomorphic to a finite index subgroup of Out(H) for some group H with Kazhdan property (T). (This sharpens a result of Ollivier-Wise). 1.

We articulate a dialectical argumentation framework for qualitative representation of epistemic uncertainty in scientific domains. The framework is grounded in specific philosophies of science and theories of rational mutual discourse. We study the formal properties of our framework and provide it with a game theoretic semantics. With this semantics, we examine the relationship between the snaphots of the debate in the framework and the long run position of the debate, and prove a result directly analogous to the standard (Neyman-Pearson) approach to statistical hypothesis testing. We believe this formalism for representating uncertainty has value in domains with only limited knowledge, where experimental evidence is ambiguous or conflicting, or where agreement between different stakeholders on the quantification of uncertainty is difficult to achieve. All three of these conditions are found in assessments of carcinogenic risk for new chemicals.

The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the # k relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k >= 7, the k relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that...

Contemporary or modern (mathematical) logic was born at the end of the 19th century. Its origin is connected with mathematics rather than philosophy, and my article will likewise be informed by a mathematical culture although I will try make connections with philosophy and the philosophy of

Abstract. We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups of homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological Rokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class (answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has a comeager conjugacy class (answering a question of Akin-Hurley-Kennedy). Finally, we study Polish groups that admit comeager conjugacy classes in any dimension (in which case the groups are said to admit ample generics). We show that Polish groups with ample generics have the small index property (generalizing results of Hodges-Hodkinson-Lascar-Shelah) and arbitrary homomorphisms from such groups into separable groups are automatically continuous. Moreover, in the case of oligomorphic permutation groups, they have uncountable cofinality and the Bergman property. These results in particular apply to automorphism groups of many ω-stable, ℵ0-categorical structures and of the random graph. In this connection, we also show that the infinite symmetric group S ∞ has a unique non-trivial separable group topology. For several interesting groups we also establish Serre’s properties (FH) and (FA). 1.