132 research outputs found
Characterizing the NP-PSPACE Gap in the Satisfiability Problem for Modal Logic
There has been a great of work on characterizing the complexity of the
satisfiability and validity problem for modal logics. In particular, Ladner
showed that the validity problem for all logics between K, T, and S4 is {\sl
PSPACE}-complete, while for S5 it is {\sl NP}-complete. We show that, in a
precise sense, it is \emph{negative introspection}, the axiom \neg K p \rimp K
\neg K p, that causes the gap. In a precise sense, if we require this axiom,
then the satisfiability problem is {\sl NP}-complete; without it, it is {\sl
PSPACE}-complete.Comment: 6 page
PSPACE Bounds for Rank-1 Modal Logics
For lack of general algorithmic methods that apply to wide classes of logics,
establishing a complexity bound for a given modal logic is often a laborious
task. The present work is a step towards a general theory of the complexity of
modal logics. Our main result is that all rank-1 logics enjoy a shallow model
property and thus are, under mild assumptions on the format of their
axiomatisation, in PSPACE. This leads to a unified derivation of tight
PSPACE-bounds for a number of logics including K, KD, coalition logic, graded
modal logic, majority logic, and probabilistic modal logic. Our generic
algorithm moreover finds tableau proofs that witness pleasant proof-theoretic
properties including a weak subformula property. This generality is made
possible by a coalgebraic semantics, which conveniently abstracts from the
details of a given model class and thus allows covering a broad range of logics
in a uniform way
Does Treewidth Help in Modal Satisfiability?
Many tractable algorithms for solving the Constraint Satisfaction Problem
(CSP) have been developed using the notion of the treewidth of some graph
derived from the input CSP instance. In particular, the incidence graph of the
CSP instance is one such graph. We introduce the notion of an incidence graph
for modal logic formulae in a certain normal form. We investigate the
parameterized complexity of modal satisfiability with the modal depth of the
formula and the treewidth of the incidence graph as parameters. For various
combinations of Euclidean, reflexive, symmetric and transitive models, we show
either that modal satisfiability is FPT, or that it is W[1]-hard. In
particular, modal satisfiability in general models is FPT, while it is
W[1]-hard in transitive models. As might be expected, modal satisfiability in
transitive and Euclidean models is FPT.Comment: Full version of the paper appearing in MFCS 2010. Change from v1:
improved section 5 to avoid exponential blow-up in formula siz
Complexity results for modal logic with recursion via translations and tableaux
This paper studies the complexity of classical modal logics and of their
extension with fixed-point operators, using translations to transfer results
across logics. In particular, we show several complexity results for
multi-agent logics via translations to and from the -calculus and modal
logic, which allow us to transfer known upper and lower bounds. We also use
these translations to introduce a terminating tableau system for the logics we
study, based on Kozen's tableau for the -calculus, and the one of Fitting
and Massacci for modal logic. Finally, we show how to encode the tableaux we
introduced into -calculus formulas. This encoding provides upper bounds
for the satisfiability checking of the few logics we previously did not have
algorithms for.Comment: 43 pages. arXiv admin note: substantial text overlap with
arXiv:2209.1037
- …