40 research outputs found
The model checking problem for intuitionistic propositional logic with one variable is AC1-complete
We show that the model checking problem for intuitionistic propositional
logic with one variable is complete for logspace-uniform AC1. As basic tool we
use the connection between intuitionistic logic and Heyting algebra, and
investigate its complexity theoretical aspects. For superintuitionistic logics
with one variable, we obtain NC1-completeness for the model checking problem.Comment: A preliminary version of this work was presented at STACS 2011. 19
pages, 3 figure
Fusions of Modal Logics Revisited
The fusion Ll ? Lr of two normal modal logics formulated in languages with disjoint sets of modal operators is the smallest normal modal logic containing Ll [ Lr. This paper proves that decidability, interpolation, uniform interpolation, and Halld?encompleteness are preserved under forming fusions of normal polyadic polymodal logics. Those problems remained open in [Fine & Schurz [3]] and [Kracht & Wolter [10]]. The paper defines the fusion `l ? `r of two classical modal consequence relations and proves that decidability transfers also in this case. Finally, these results are used to prove a general decidability result for modal logics based on superintuitionistic logics
Degrees of the finite model property: the antidichotomy theorem
A classic result in modal logic, known as the Blok Dichotomy Theorem, states
that the degree of incompleteness of a normal extension of the basic modal
logic is or . It is a long-standing open problem
whether Blok Dichotomy holds for normal extensions of other prominent modal
logics (such as or ) or for extensions of the intuitionistic
propositional calculus . In this paper, we introduce the notion
of the degree of finite model property (fmp), which is a natural variation of
the degree of incompleteness. It is a consequence of Blok Dichotomy Theorem
that the degree of fmp of a normal extension of remains or
. In contrast, our main result establishes the following
Antidichotomy Theorem for the degree of fmp for extensions of :
each nonzero cardinal such that or is realized as the degree of fmp of some extension of
. We then use the Blok-Esakia theorem to establish the same
Antidichotomy Theorem for normal extensions of and
The model checking problem for propositional intuitionistic logic with one variable is AC^1-complete
We investigate the complexity of the model checking problem for propositional intuitionistic logic. We show that the model checking problem for intuitionistic logic with one variable is complete for logspace-uniform AC^1, and for intuitionistic logic with two variables it is P-complete. For superintuitionistic logics with one variable, we obtain NC^1-completeness for the model checking problem and for the tautology problem
Coherence in Modal Logic
A variety is said to be coherent if the finitely generated subalgebras of its
finitely presented members are also finitely presented. In a recent paper by
the authors it was shown that coherence forms a key ingredient of the uniform
deductive interpolation property for equational consequence in a variety, and a
general criterion was given for the failure of coherence (and hence uniform
deductive interpolation) in varieties of algebras with a term-definable
semilattice reduct. In this paper, a more general criterion is obtained and
used to prove the failure of coherence and uniform deductive interpolation for
a broad family of modal logics, including K, KT, K4, and S4