. We expand rst order models with a tolerance relation on the domain. Intuitively, two elements stand in this relation if they are \cognitively close" for the agent who holds the model. This simple notion turns out to be very powerful. It leads to a semantic characterization of the guarded fragment of Andreka, van Benthem and Nemeti, and highlights the strong analogies between modal logic and this fragment. Viewing the resulting logic |tolerance logic| dynamically it is a resource{conscious information processing alternative to classical rst order logic. The dierences are indicated by several examples. Keywords: Guarded fragments, Relativised rst order logic 1. Introduction Out of the joint work of Johan van Benthem and the Hungarian group round Hajnal Andreka, Istvan Nemeti and Ildiko Sain and their PhD students, two approaches for taming a logic evolved. With taming a logic we mean changing the logic in such a way that it becomes decidable. For rst order logic, they too...

A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary.

The connection between some modularity properties and interpolation is revisited and restated in a general "logicindependent " framework. The presence of uniform interpolants is shown to assist in certain proof obligations, which suffice to establish the composition of refinements. The absence of the desirable interpolation properties from many logics that have been used in refinement, motivates a thorough investigation of methods to expand a specification formalism orthogonally, so that the critical uniform interpolants become available. A potential breakthrough is outlined in this paper. 1. A refinement paradigm Let us consider program development by means of stepwise refinements. One postulates some abstract data typelike specification 1 (ADT), suitable for the problem at hand, which has to be implemented on the available system. The end product consists of (the text of) an abstract program manipulating the postulated ADT, together with a suite of (texts of) modules implementin...

Relativization is one of the central topics in the study of algebras of relations (i.e. relation and cylindric algebras). Relativized representable relation algebras behave much nicer than the original class RRA: for instance, one obtains finite axiomatizability, decidability and amalgamation by relativization. The properties of the class obtained by relativizing RRA depend on the kind of element with which is relativized. We give a systematic account of all interesting choices of relativizing RRA, and show that relativizing with transitive elements forms the borderline where all above mentioned three properties switch from negative to positive. In algebraic logic, relativized cylindric and relation algebras have been studied relatively deeply (cf. e.g., Henkin et al. (Henkin et al., 1981), Maddux (Maddux, 1982) and Resek-- Thompson (Resek and Thompson, 1991)). The emphasis, however was different from the perspective we will take here. As the name "relativization" indicates, the non--...

Contents Acknowledgments vii 1 Introduction. 1 2 General preliminaries. 7 2.1 Logics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Languages and structures. . . . . . . . . . . . . . . . . . . 7 2.1.2 Syntax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.3 Semantics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.4 Translations. . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.5 Derivability in Basic Modal Logic. . . . . . . . . . . . . . . 10 2.2 Bisimulation and the like. . . . . . . . . . . . . . . . . . . . . . . 12 2.3 A brief introduction to the -Calculus. . . . . . . . . . . . . . . . 17 2.4 The family of graded modal logics and their semantics. . . . . . . 22 2.5 Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5.1 Uniform interpolation. . . . . . . . . . . . . . . . . . . . . 29 2.5.2 Elementary interpolation. . . . . . . . . . . . . . . . . . . 30 2.6 N

this dissertation. More precisely, we are referring to what is called the orthodox version of these logics in these works

In this technical report, we collect the results of our investigations about the two-variable fragment of modal logics of relations interpreted on local squares, LC2 . It contains a full system description of the PROLOG implementation and the complete proofs for the theoretical results published in [9] and [10]. A labelled tableau calculus is presented and its soundness and completeness are proven. Further, a termination proof enables us to use the calculus as a theorem prover. The prover has been implemented in Prolog, and we give the full system description. The paper also contains examples for how the system works including translations from other modal logics into LC2 . 1 Contents 1 Local cubic modal logic of two dimensions 3 2 Simulating other logics in LC 2 4 2.1 Normal modal logic K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Normal modal logic KT, KB and KTB . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Multi-modal logic K multi - De...

This is a review of those aspects of the theory of varieties of Boolean algebras with operators (BAO's) that emphasise connections with modal logic and structural properties that are related to natural properties of logical systems.

The aim of this paper is to apply the mosaic method for proving complexity, Hilbert-style and tableau completeness results for Prior's temporal logic over linear ows of time. We also show how to implement the mosaic idea for automated theorem-proving. Finally we indicate the modifications required to achieve similar results for special linear flows of time and for the more expressive logic of until and since.

this paper only concerned ourselves with the class of all models for our language. Uniform interpolation could also be studied for more restricted classes of models. Things are different there: for instance, uniform interpolation in the class of transitive, reflexive models (S4) does not hold for modal logic ([22], [9]) Does it hold for the ¯-calculus? Or even for PDL (as in that case non-wellfoundedness can be expressed, namely by ?)?