47,643 research outputs found
Logics for complexity classes
A new syntactic characterization of problems complete via Turing reductions
is presented. General canonical forms are developed in order to define such
problems. One of these forms allows us to define complete problems on ordered
structures, and another form to define them on unordered non-Aristotelian
structures. Using the canonical forms, logics are developed for complete
problems in various complexity classes. Evidence is shown that there cannot be
any complete problem on Aristotelian structures for several complexity classes.
Our approach is extended beyond complete problems. Using a similar form, a
logic is developed to capture the complexity class which very
likely contains no complete problem.Comment: This article has been accepted for publication in Logic Journal of
IGPL Published by Oxford University Press; 23 pages, 2 figure
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
Algorithmic complexity of monadic multimodal predicate logics with equality over finite Kripke frames
The paper investigates algorithmic complexity of monadic multimodal predicate
logics with equality over finite Kripke frames or classes of finite Kripke
frames. Precise complexity bounds for monadic logics of classes of Kripke
frames with finitely many possible worlds are obtained.Comment: Semantical and Computational Aspects of Non-Classical Logics (Moscow
+ Online, June 13-17, 2023), Steklov International Mathematical Center,
Moscow, 202
The Descriptive Complexity of the Deterministic Exponential Time Hierarchy
AbstractIn Descriptive Complexity, we investigate the use of logics to characterize computational complexity classes. Since 1974, when Fagin proved that the class NP is captured by existential second-order logic, considered the first result in this area, other relations between logics and complexity classes have been established. Well-known results usually involve first-order logic and its extensions, and complexity classes in polynomial time or space. Some examples are that the first-order logic extended by the least fixed-point operator captures the class P and the second-order logic extended by the transitive closure operator captures the class PSPACE. In this paper, we will analyze the combined use of higher-order logics of order i, HOi, for i⩾2, extended by the least fixed-point operator, and we will prove that each level of this hierarchy captures each level of the deterministic exponential time hierarchy. As a corollary, we will prove that the hierarchy of HOi(LFP), for i⩾2, does not collapse, that is, HOi(LFP)⊂HOi+1(LFP)
Frame definability in finitely-valued modal logics
In this paper we study frame definability in finitely valued modal logics and establish two main results via suitable translations: (1) in finitely valued modal logics one cannot define more classes of frames than are already definable in classical modal logic (cf. [27, Thm. 8]), and (2) a large family of finitely valued modal logics define exactly the same classes of frames as classical modal logic (including modal logics based on finite Heyting and MV-algebras, or even BL-algebras). In this way one may observe, for example, that the celebrated Goldblatt–Thomason theorem applies immediately to these logics. In particular, we obtain the central result from [26] with a much simpler proof and answer one of the open questions left in that paper. Moreover, the proposed translations allow us to determine the computational complexity of a big class of finitely valued modal logics
Complexity of Prioritized Default Logics
In default reasoning, usually not all possible ways of resolving conflicts
between default rules are acceptable. Criteria expressing acceptable ways of
resolving the conflicts may be hardwired in the inference mechanism, for
example specificity in inheritance reasoning can be handled this way, or they
may be given abstractly as an ordering on the default rules. In this article we
investigate formalizations of the latter approach in Reiter's default logic.
Our goal is to analyze and compare the computational properties of three such
formalizations in terms of their computational complexity: the prioritized
default logics of Baader and Hollunder, and Brewka, and a prioritized default
logic that is based on lexicographic comparison. The analysis locates the
propositional variants of these logics on the second and third levels of the
polynomial hierarchy, and identifies the boundary between tractable and
intractable inference for restricted classes of prioritized default theories
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