3,057 research outputs found
Model Checking CTL is Almost Always Inherently Sequential
The model checking problem for CTL is known to be P-complete (Clarke,
Emerson, and Sistla (1986), see Schnoebelen (2002)). We consider fragments of
CTL obtained by restricting the use of temporal modalities or the use of
negations---restrictions already studied for LTL by Sistla and Clarke (1985)
and Markey (2004). For all these fragments, except for the trivial case without
any temporal operator, we systematically prove model checking to be either
inherently sequential (P-complete) or very efficiently parallelizable
(LOGCFL-complete). For most fragments, however, model checking for CTL is
already P-complete. Hence our results indicate that, in cases where the
combined complexity is of relevance, approaching CTL model checking by
parallelism cannot be expected to result in any significant speedup. We also
completely determine the complexity of the model checking problem for all
fragments of the extensions ECTL, CTL+, and ECTL+
Complexity of the Guarded Two-Variable Fragment with Counting Quantifiers
We show that the finite satisfiability problem for the guarded two-variable
fragment with counting quantifiers is in EXPTIME. The method employed also
yields a simple proof of a result recently obtained by Y. Kazakov, that the
satisfiability problem for the guarded two-variable fragment with counting
quantifiers is in EXPTIME.Comment: 20 pages, 3 figure
De Morgan Dual Nominal Quantifiers Modelling Private Names in Non-Commutative Logic
This paper explores the proof theory necessary for recommending an expressive
but decidable first-order system, named MAV1, featuring a de Morgan dual pair
of nominal quantifiers. These nominal quantifiers called `new' and `wen' are
distinct from the self-dual Gabbay-Pitts and Miller-Tiu nominal quantifiers.
The novelty of these nominal quantifiers is they are polarised in the sense
that `new' distributes over positive operators while `wen' distributes over
negative operators. This greater control of bookkeeping enables private names
to be modelled in processes embedded as formulae in MAV1. The technical
challenge is to establish a cut elimination result, from which essential
properties including the transitivity of implication follow. Since the system
is defined using the calculus of structures, a generalisation of the sequent
calculus, novel techniques are employed. The proof relies on an intricately
designed multiset-based measure of the size of a proof, which is used to guide
a normalisation technique called splitting. The presence of equivariance, which
swaps successive quantifiers, induces complex inter-dependencies between
nominal quantifiers, additive conjunction and multiplicative operators in the
proof of splitting. Every rule is justified by an example demonstrating why the
rule is necessary for soundly embedding processes and ensuring that cut
elimination holds.Comment: Submitted for review 18/2/2016; accepted CONCUR 2016; extended
version submitted to journal 27/11/201
Logic Meets Algebra: the Case of Regular Languages
The study of finite automata and regular languages is a privileged meeting
point of algebra and logic. Since the work of Buchi, regular languages have
been classified according to their descriptive complexity, i.e. the type of
logical formalism required to define them. The algebraic point of view on
automata is an essential complement of this classification: by providing
alternative, algebraic characterizations for the classes, it often yields the
only opportunity for the design of algorithms that decide expressibility in
some logical fragment.
We survey the existing results relating the expressibility of regular
languages in logical fragments of MSO[S] with algebraic properties of their
minimal automata. In particular, we show that many of the best known results in
this area share the same underlying mechanics and rely on a very strong
relation between logical substitutions and block-products of pseudovarieties of
monoid. We also explain the impact of these connections on circuit complexity
theory.Comment: 37 page
Consistency of circuit lower bounds with bounded theories
Proving that there are problems in that require
boolean circuits of super-linear size is a major frontier in complexity theory.
While such lower bounds are known for larger complexity classes, existing
results only show that the corresponding problems are hard on infinitely many
input lengths. For instance, proving almost-everywhere circuit lower bounds is
open even for problems in . Giving the notorious difficulty of
proving lower bounds that hold for all large input lengths, we ask the
following question: Can we show that a large set of techniques cannot prove
that is easy infinitely often? Motivated by this and related
questions about the interaction between mathematical proofs and computations,
we investigate circuit complexity from the perspective of logic.
Among other results, we prove that for any parameter it is
consistent with theory that computational class , where is one of
the pairs: and , and , and
. In other words, these theories cannot establish
infinitely often circuit upper bounds for the corresponding problems. This is
of interest because the weaker theory already formalizes
sophisticated arguments, such as a proof of the PCP Theorem. These consistency
statements are unconditional and improve on earlier theorems of [KO17] and
[BM18] on the consistency of lower bounds with
Complexity of Nested Circumscription and Nested Abnormality Theories
The need for a circumscriptive formalism that allows for simple yet elegant
modular problem representation has led Lifschitz (AIJ, 1995) to introduce
nested abnormality theories (NATs) as a tool for modular knowledge
representation, tailored for applying circumscription to minimize exceptional
circumstances. Abstracting from this particular objective, we propose L_{CIRC},
which is an extension of generic propositional circumscription by allowing
propositional combinations and nesting of circumscriptive theories. As shown,
NATs are naturally embedded into this language, and are in fact of equal
expressive capability. We then analyze the complexity of L_{CIRC} and NATs, and
in particular the effect of nesting. The latter is found to be a source of
complexity, which climbs the Polynomial Hierarchy as the nesting depth
increases and reaches PSPACE-completeness in the general case. We also identify
meaningful syntactic fragments of NATs which have lower complexity. In
particular, we show that the generalization of Horn circumscription in the NAT
framework remains CONP-complete, and that Horn NATs without fixed letters can
be efficiently transformed into an equivalent Horn CNF, which implies
polynomial solvability of principal reasoning tasks. Finally, we also study
extensions of NATs and briefly address the complexity in the first-order case.
Our results give insight into the ``cost'' of using L_{CIRC} (resp. NATs) as a
host language for expressing other formalisms such as action theories,
narratives, or spatial theories.Comment: A preliminary abstract of this paper appeared in Proc. Seventeenth
International Joint Conference on Artificial Intelligence (IJCAI-01), pages
169--174. Morgan Kaufmann, 200
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