13 research outputs found
Axiomatizability of propositionally quantified modal logics on relational frames
Propositional modal logic over relational frames is naturally extended with propositional quantifiers by letting them range over arbitrary sets of worlds of the relevant frame. This is also known as second-order propositional modal logic. The propositionally quantified modal logic of a class of relational frames is often not axiomatizable, although there are known exceptions, most notably the case of frames validating the strong modal logic S5 . Here, we develop new general methods with which many of the open questions in this area can be answered. We illustrate the usefulness of these methods by applying them to a range of examples, which provide a detailed picture of which normal modal logics define classes of relational frames whose propositionally quantified modal logic is axiomatizable. We also apply these methods to establish new results in the multimodal case
Finite Axiomatizability of Transitive Logics of Finite Depth and of Finite Weak Width
This paper presents a study of the finite axiomatizability of transitive
logics of finite depth and finite weak width. We prove the finite
axiomatizability of each transitive logic of finite depth and of weak width
that is characterized by rooted transitive frames in which all antichains
contain at most irreflexive points. As a negative result, we show that
there are non-finitely-axiomatizable transitive logics of depth and of weak
width for each and
Decidability and complexity via mosaics of the temporal logic of the lexicographic products of unbounded dense linear orders
This article considers the temporal logic of the lexicographic products of unbounded dense linear orders and provides via mosaics a complete decision procedure in nondeterministic polynomial time for the satisfiability problem it gives rise to
On the proof complexity of logics of bounded branching
We investigate the proof complexity of extended Frege (EF) systems for basic
transitive modal logics (K4, S4, GL, ...) augmented with the bounded branching
axioms . First, we study feasibility of the disjunction property
and more general extension rules in EF systems for these logics: we show that
the corresponding decision problems reduce to total coNP search problems (or
equivalently, disjoint NP pairs, in the binary case); more precisely, the
decision problem for extension rules is equivalent to a certain special case of
interpolation for the classical EF system. Next, we use this characterization
to prove superpolynomial (or even exponential, with stronger hypotheses)
separations between EF and substitution Frege (SF) systems for all transitive
logics contained in or under some
assumptions weaker than . We also prove analogous
results for superintuitionistic logics: we characterize the decision complexity
of multi-conclusion Visser's rules in EF systems for Gabbay--de Jongh logics
, and we show conditional separations between EF and SF for all
intermediate logics contained in .Comment: 58 page
All finitely axiomatizable normal extensions of K4.3 are decidable
We use the apparatus of the canonical formulas introduced by Zakharyaschev [10] to prove that all finitely axiomatizable normal modal logics containing K4.3 are decidable, though possibly not characterized by classes of finite frames. Our method is purely frame-theoretic. Roughly, given a normal logic L above K4.3, we enumerate effectively a class of (possibly infinite) frames with respect to which L is complete, show how to check effectively whether a frame in the class validates a given formula, and then apply a Harropstyle argument to establish the decidability of L, provided of course that it has finitely many axioms
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established