Topological Semantics and Decidability

It is well-known that the basic modal logic of all topological spaces is $S4$. However, the structure of basic modal and hybrid logics of classes of spaces satisfying various separation axioms was until present unclear. We prove that modal logics of $T_0$, $T_1$ and $T_2$ topological spaces coincide and are S4$. We also examine basic hybrid logics of these classes and prove their decidability; as part of this, we find out that the hybrid logics of$T_1$and T_2$ spaces coincide.Comment: presentation changes, results about concrete structure adde

Quantum Harmonic Oscillator as a Zariski Geometry

We carry out a model-theoretic analysis of the Heisenberg algebra. To this end, a geometric structure is associated to the Heisenberg algebra and is shown to be a Zariski geometry. Furthermore, this Zariski geometry is shown to be non-classical, in the sense that it is not interpretable in an algebraically closed field. On assuming self-adjointness of the position and momentum operators, one obtains a discrete substructure of which the original Zariski geometry is seen as the complexification.Comment: some typos correcte

Topological Semantics and Decidability

It is well-known that the basic modal logic of all topological spaces is $S4$. However, the structure of basic modal and hybrid logics of classes of spaces satisfying various separation axioms was until present unclear. We prove that modal logics of $T_0$, $T_1$ and $T_2$ topological spaces coincide and are S4$. We also examine basic hybrid logics of these classes and prove their decidability; as part of this, we find out that the hybrid logics of$T_1$and T_2$ spaces coincide

Experiments in Theorem Proving for Topological Hybrid Logic

International audienceThis paper discusses two experiments in theorem proving for hybrid logic under the topological interpre-tation. We begin by discussing the topological interpretation of hybrid logic and noting what it adds to the topological interpretation of orthodox modal logic. We then examine two implemented proof methods. The first makes use of HyLoBan, a terminating theorem prover that searches for a winning search strategy in certain topologically motivated games. The second is a translation-based approach that makes use of HyLoTab, a tableaux-based theorem prover for hybrid logic under the standard relational interpretation. We compare the two methods, and note a number of directions for further work

Modal Languages for Topology: Expressivity and Definability

In this paper we study the expressive power and definability for (extended) modal languages interpreted on topological spaces. We provide topological analogues of the van Benthem characterization theorem and the Goldblatt-Thomason definability theorem in terms of the well established first-order topological language $L_t$

Hybrid logics of separation axioms.

International audienceWe study hybrid logics in topological semantics. We prove that hybrid logics of separation axioms are complete with respect to certain classes of finite topological models. This characterisation allows us to obtain several further results. We prove that aforementioned logics are decidable and PSPACE-complete, the logics of T1 and T2 coincide, the logic of T1 is complete with respect to two concrete structures: the Cantor space and the rational numbers

Hybrid definability in topological spaces

We present some results concerning definability of classes of topological spaces in hybrid languages. We use language Lt described in [9] to establish notion of “elementarity ” for classes of topological spaces. We use it to prove the analogue of Goldblatt-Thomason theorem in topological spaces for hybrid languages H(E) and H(@). We also prove a theorem that allows to reformulate definability result of Gabelaia ([10]) for modal logic in terms of elementary topological space classes.

Non Locally Modular Reducts of ACF

Non UBCUnreviewedAuthor affiliation: Hebrew University of JerusalemPostdoctora