Day 5: Applications and Extensions The precise topics for the course will depend in part on the interests and background of the students attending the lectures. This document contains an introduction to the course including some pointers to relevant literature. The focus is on the basics of neighborhood semantics with only pointers to literature relating to the more advanced topics. The website for the course is staff.science.uva.nl/∼epacuit/nbhd esslli.html On this website you will find the lecture slides (and possibly course notes) updated daily. Two other sources will also be useful:

Neighborhood Semantics for Modal Logic

We present a set of SAT-based decision procedures for various classical modal logics. By SAT based, we mean built on top of a SAT solver. We show how the SAT-based approach allows for a modular implementation for these logics. For some of the logics we deal with, we are not aware of any other implementation. For the others, we define a testing methodology that generalizes the 3CNFi>K methodology by Giunchiglia and Sebastiani. The experimental evaluation shows that our decision procedures perform better than or as well as other state-of-the-art decision procedures.

SAT-Based Decision Procedures for Classical Modal Logics

We give two generic proofs for cut elimination in propositional modal logics, interpreted over coalgebras. We first investigate semantic coherence conditions between the axiomatisation of a particular logic and its coalgebraic semantics that guarantee that the cut-rule is admissible in the ensuing sequent calculus. We then independently isolate a purely syntactic property of the set of modal rules that guarantees cut elimination. Apart from the fact that cut elimination holds, our main result is that the syntactic and semantic assumptions are equivalent in case the logic is amenable to coalgebraic semantics. As applications we present a new proof of the (already known) interpolation property for coalition logic and newly establish the interpolation property for the conditional logics LCK and LCKID.

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Cut elimination in coalgebraic logics

Proof Theory for Non-normal Modal Logics: The Neighbourhood Formalism and Basic Results.

This paper develops sequent calculi for several classical modal logics. Utilizing a polymodal translation of the standard modal language, we are able to establish a base system for the minimal classical modal logic E from which we generate extensions (to include M, C, and N) in a modular manner. Our systems admit contraction and cut admissibility, and allow a systematic proof-search procedure of formal derivations.

Modular Sequent Calculi for Classical Modal Logics

In this paper we show how to extend KEM, a tableau-like proof system for normal modal logic, in order to deal with classes of non-normal modal logics, such as monotonic and regular, in a uniform and modular way.

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Alessandro Luppi

Papers

Labelled Tableaux for Non-normal Modal Logics

On data-driven stabilization of systems with nonlinearities satisfying quadratic constraints