5 research outputs found
Countermodel Construction via Optimal Hypersequent Calculi for Non-normal Modal Logics
International audienceWe develop semantically-oriented calculi for the cube of non-normal modal logics and some deontic extensions. The calculi manipulate hypersequents and have a simple semantic interpretation. Their main feature is that they allow for direct countermodel extraction. Moreover they provide an optimal decision procedure for the respective logics. They also enjoy standard proof-theoretical properties, such as a syntactical proof of cut-admissibility
Sequent calculi and interpolation for non-normal modal and deonticlogics
G3-style sequent calculi for the logics in the cube of non-normal modal
logics and for their deontic extensions are studied. For each calculus we prove
that weakening and contraction are height-preserving admissible, and we give a
syntactic proof of the admissibility of cut. This implies that the subformula
property holds and that derivability can be decided by a terminating proof
search whose complexity is in PSPACE. These calculi are shown to be equivalent
to the axiomatic ones and, therefore, they are sound and complete with respect
to neighbourhood semantics. Finally, it is given a Maehara-style proof of
Craig's interpolation theorem for most of the logics considered
Intuitionistic Non-Normal Modal Logics: A general framework
We define a family of intuitionistic non-normal modal logics; they can bee
seen as intuitionistic counterparts of classical ones. We first consider
monomodal logics, which contain only one between Necessity and Possibility. We
then consider the more important case of bimodal logics, which contain both
modal operators. In this case we define several interactions between Necessity
and Possibility of increasing strength, although weaker than duality. For all
logics we provide both a Hilbert axiomatisation and a cut-free sequent
calculus, on its basis we also prove their decidability. We then give a
semantic characterisation of our logics in terms of neighbourhood models. Our
semantic framework captures modularly not only our systems but also already
known intuitionistic non-normal modal logics such as Constructive K (CK) and
the propositional fragment of Wijesekera's Constructive Concurrent Dynamic
Logic.Comment: Preprin
Intuitionistic non-normal modal logics: A general framework
International audienceWe define a family of intuitionistic non-normal modal logics; they can be seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only Necessity or Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. We thereby obtain a lattice of 24 distinct bimodal logics. For all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then define a semantic characterisation of our logics in terms of neighbourhood models containing two distinct neighbourhood functions corresponding to the two modalities. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K (CK) and the propositional fragment of Wijesekera's Constructive Concurrent Dynamic Logic
Hypersequent calculi for non-normal modal and deontic logics: Countermodels and optimal complexity
We present some hypersequent calculi for all systems of the classical cube
and their extensions with axioms , , , and, for every , rule
. The calculi are internal as they only employ the language of the
logic, plus additional structural connectives. We show that the calculi are
complete with respect to the corresponding axiomatisation by a syntactic proof
of cut elimination. Then we define a terminating root-first proof search
strategy based on the hypersequent calculi and show that it is optimal for
coNP-complete logics. Moreover, we obtain that from every saturated leaf of a
failed proof it is possible to define a countermodel of the root hypersequent
in the bi-neighbourhood semantics, and for regular logics also in the
relational semantics. We finish the paper by giving a translation between
hypersequent rule applications and derivations in a labelled system for the
classical cube