Model Theory. That is, we have a non-empty family I of partial isomorphisms between two models M and N, which is closed under taking restrictions to smaller domains, and where the standard Back-and-Forth properties are now restricted to apply only to partial isomorphisms of size at most k. Proof. (A complete argument is in [16].) An outline is reproduced here, for convenience. First, k-variable formulas are preserved under partial isomorphism, by a simple induction. More precisely, one proves, for any assignment A and any partial isomorphism I 2 I which is defined on the A-values for all variables x 1 ; : : : ; x k , that M;A j= OE iff N; I ffi A j= OE: The crucial step in the induction is the quantifier case. Quantified variables are irrelevant to the assignment, so that the relevant partial isomorphism can be restricted to size at most k \Gamma 1, whence a matching choice for the witness can be made on the opposite side. This proves "only if". Next, "if" has a proof analogous to...

We show how labelled deductive systems can be combined with a logical framework to provide a natural deduction implementation of a large and well-known class of propositional modal logics (including K, D, T , B, S4, S4:2, KD45, S5). Our approach is modular and based on a separation between a base logic and a labelling algebra, which interact through a fixed interface. While the base logic stays fixed, different modal logics are generated by plugging in appropriate algebras. This leads to a hierarchical structuring of modal logics with inheritance of theorems. Moreover, it allows modular correctness proofs, both with respect to soundness and completeness for semantics, and faithfulness and adequacy of the implementation. We also investigate the tradeoffs in possible labelled presentations: We show that a narrow interface between the base logic and the labelling algebra supports modularity and provides an attractive proof-theory (in comparision to, e.g., semantic embedding) but limits th...

Transition systems can be viewed either as process diagrams or as Kripke structures. The first perspective is that of process theory, the second that of modal logic. This paper shows how various formalisms of modal logic can be brought to bear on processes. Notions of bisimulation can not only be motivated by operations on transition systems, but they can also be suggested by investigations of modal formalisms. To show that the equational view of processes from process algebra is closely related to modal logic, we consider various ways of looking at the relation between the calculus of basic process algebra and propositional dynamic logic. More concretely, the paper contains preservation results for various bisimulation notions, a result on the expressive power of propositional dynamic logic, and a definition of bisimulation which is the proper notion of invariance for concurrent propositional dynamic logic. Keywords: modal logic, transition systems, bisimulation, process algebra 1 In...

. We define cut-free display calculi for nominal tense logics extending the minimal nominal tense logic (MNTL) by addition of primitive axioms. To do so, we use a translation of MNTL into the minimal tense logic of inequality (MTL 6= ) which is known to be properly displayable by application of Kracht's results. The rules of the display calculus ffiMNTL for MNTL mimic those of the display calculus ffiMTL 6= for MTL 6= . Since ffiMNTL does not satisfy Belnap's condition (C8), we extend Wansing's strong normalisation theorem to get a similar theorem for any extension of ffiMNTL by addition of structural rules satisfying Belnap's conditions (C2)-(C7). Finally, we show a weak Sahlqvist-style theorem for extensions of MNTL, and by Kracht's techniques, deduce that these Sahlqvist extensions of ffiMNTL also admit cut-free display calculi. 1 Introduction Background: The addition of names (also called nominals) to modal logics has been investigated recently with different motivations; see...

We present a (sound and complete) tableau calculus for Quantified Hybrid Logic (QHL). QHL is an extension of orthodox quantified modal logic: as well as the usual # and # modalities it contains names for (and variables over) states, operators @s for asserting that a formula holds at a named state, and a binder that binds a variable to the current state. The first-order component contains equality and rigid and non rigid designators. As far as we are aware, ours is the first tableau system for QHL. Completeness

We present an approach to a coherent program synthesis system which integrates a variety of interactively controlled and automated techniques from theorem proving and algorithm design at different levels of abstraction. Besides providing an overall view we summarize the individual research results achieved in the course of this development.

It is known that the tiling technique can be used to give simple proofs of undecidability of various two-dimensional modal and temporal logics. However, up until now, the simplest two-dimensional temporal logic, the compass logic of Venema, has eluded such treatment. We present a new coding of an enumeration of the tiling plane which enables us to show that the compass logic is undecidable.

We provide a general method which can be used in an algorithmic manner to reduce certain classes of 2nd-order circumscription axioms to logically equivalent 1st-order formulas. The algorithm takes as input an arbitrary 2nd-order formula and either returns as output an equivalent 1st-order formula, or terminates with failure. In addition to demonstrating the algorithm by applyingittovarious circumscriptive theories, we analyze its strength and provide formal subsumption results based on comparison with existing approaches. 1

The main goal of this paper is to explain the link between the algebraic and the Kripke-style models for certain classes of propositional logics. We start by presenting a Priestley-type duality for distributive lattices endowed with a general class of well-behaved operators. We then show that finitely-generated varieties of distributive lattices with operators are closed under canonical embedding algebras. The results are used in the second part of the paper to construct topological and non-topological Kripke-style models for logics that are sound and complete with respect to varieties of distributive lattices with operators in the above-mentioned classes. Introduction In the study of non-classical propositional logics (and especially of modal logics) there are two main ways of defining interpretations or models. One possibility is to use algebras -- usually lattices with operators -- as models. Propositional variables are interpreted over elements of these algebraic models, an...

. We point out a simple but hitherto ignored link between the theory of updates and counterfactuals and classical modal logic: update is a classical existential modality, counterfactual is a classical universal modality, and the accessibility relations corresponding to these modalities are inverses. The Ramsey Rule (often thought esoteric) is simply an axiomatisation of this inverse relationship. We use this fact to translate between postulates for updates and postulates for counterfactuals. Thus, Katsuno/Mendelzon's postulates U1--U8 are translated into counterfactual postulates C1--C8 (table VII), and many of the familiar counterfactual postulates are translated into postulates for updates (table VIII). Our conclusions are summarised in table V. We also present a syntactic condition which is sufficient to guarantee that a translation from update to counterfactual (or vice versa) is possible. 1. Introduction Background. An intuitive connection between theory change and counterfactuals...