Tadeusz Litak

Topological Perspective on the Hybrid Proof Rules

We consider the non-orthodox proof rules of hybrid logic from the viewpoint of topological semantics. Topological semantics is more general than Kripke semantics. We show that the hybrid proof rule BG is topologically not sound. Indeed, among all topological spaces the BG rule characterizes those that can be represented as a Kripke frame (i.e., the Alexandroff spaces). We also demonstrate that, when the BG rule is dropped and only the Name rule is kept, one can prove a general topological completeness result for hybrid logics axiomatized by pure formulas. Finally, we indicate some limitations of the topological expressive power of pure formulas. All results generalize to neighborhood frames.

All Finitely Axiomatizable Tense Logics of Linear Time Flows Are CoNP-complete

We prove that all finitely axiomatizable tense logics with temporal operators for ‘always in the future’ and ‘always in the past’ and determined by linear fows time are coNP-complete. It follows, for example, that all tense logics containing a density axiom of the form ■n+1Fp → nFp, for some n ≥ 0, are coNP-complete. Additionally, we prove coNP-completeness of all ∩-irreducible tense logics. As these classes of tense logics contain many Kripke incomplete bimodal logics, we obtain many natural examples of Kripke incomplete normal bimodal logics which are nevertheless coNP-complete.

Coalgebraic predicate logic

We propose a generalization of first-order logic originating in a neglected work by C.C. Chang: a natural and generic correspondence language for any types of structures which can be recast as Set-coalgebras. We discuss axiomatization and completeness results for two natural classes of such logics. Moreover, we show that an entirely general completeness result is not possible. We study the expressive power of our language, contrasting it with both coalgebraic modal logic and existing first-order proposals for special classes of Set-coalgebras (apart for relational structures, also neighbourhood frames and topological spaces). The semantic characterization of expressivity is based on the fact that our language inherits a coalgebraic variant of the Van Benthem-Rosen Theorem. Basic model-theoretic constructions and results, in particular ultraproducts, obtain for the two classes which allow for completeness--and in some cases beyond that.

Algebraization of hybrid logic with binders

This paper introduces an algebraic semantics for hybrid logic with binders

Modal Incompleteness Revisited

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On the Termination Problem for Declarative XML Message Processing

. It is known that this formalism is a modal counterpart of the bounded fragment of the first-order logic, studied by Feferman in the 1960s. The algebraization process leads to an interesting class of boolean algebras with operators, called substitution-satisfaction algebras. We provide a representation theorem for these algebras and thus provide an algebraic proof of completeness of hybrid logic.

Completions of GBL-algebras: negative results

AbstractIn this paper, we are going to analyze the phenomenon of modal incompleteness from an algebraic point of view. The usual method of showing that a given logic L is incomplete is to show that for some Σ

An algebraic approach to incompleteness in modal logic

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