6,250 research outputs found
A Dichotomy Theorem for the Inverse Satisfiability Problem
The inverse satisfiability problem over a set of Boolean relations Gamma (Inv-SAT(Gamma)) is the computational decision problem of, given a set of models R, deciding whether there exists a SAT(Gamma) instance with R as its set of models. This problem is co-NP-complete in general and a dichotomy theorem for finite ? containing the constant Boolean relations was obtained by Kavvadias and Sideri. In this paper we remove the latter condition and prove that Inv-SAT(Gamma) is always either tractable or co-NP-complete for all finite sets of relations Gamma, thus solving a problem open since 1998. Very little of the techniques used by Kavvadias and Sideri are applicable and we have to turn to more recently developed algebraic approaches based on partial polymorphisms. We also consider the case when ? is infinite, where the situation differs markedly from the case of SAT. More precisely, we show that there exists infinite Gamma such that Inv-SAT(Gamma) is tractable even though there exists finite Delta is subset of Gamma such that Inv-SAT(Delta) is co-NP-complete
PSPACE Reasoning for Graded Modal Logics
We present a PSPACE algorithm that decides satisfiability of the graded modal
logic Gr(K_R)---a natural extension of propositional modal logic K_R by
counting expressions---which plays an important role in the area of knowledge
representation. The algorithm employs a tableaux approach and is the first
known algorithm which meets the lower bound for the complexity of the problem.
Thus, we exactly fix the complexity of the problem and refute an
ExpTime-hardness conjecture. We extend the results to the logic Gr(K_(R \cap
I)), which augments Gr(K_R) with inverse relations and intersection of
accessibility relations. This establishes a kind of ``theoretical benchmark''
that all algorithmic approaches can be measured against
Practical Reasoning for Very Expressive Description Logics
Description Logics (DLs) are a family of knowledge representation formalisms
mainly characterised by constructors to build complex concepts and roles from
atomic ones. Expressive role constructors are important in many applications,
but can be computationally problematical. We present an algorithm that decides
satisfiability of the DL ALC extended with transitive and inverse roles and
functional restrictions with respect to general concept inclusion axioms and
role hierarchies; early experiments indicate that this algorithm is well-suited
for implementation. Additionally, we show that ALC extended with just
transitive and inverse roles is still in PSPACE. We investigate the limits of
decidability for this family of DLs, showing that relaxing the constraints
placed on the kinds of roles used in number restrictions leads to the
undecidability of all inference problems. Finally, we describe a number of
optimisation techniques that are crucial in obtaining implementations of the
decision procedures, which, despite the worst-case complexity of the problem,
exhibit good performance with real-life problems
Linear Temporal Logic and Propositional Schemata, Back and Forth (extended version)
This paper relates the well-known Linear Temporal Logic with the logic of
propositional schemata introduced by the authors. We prove that LTL is
equivalent to a class of schemata in the sense that polynomial-time reductions
exist from one logic to the other. Some consequences about complexity are
given. We report about first experiments and the consequences about possible
improvements in existing implementations are analyzed.Comment: Extended version of a paper submitted at TIME 2011: contains proofs,
additional examples & figures, additional comparison between classical
LTL/schemata algorithms up to the provided translations, and an example of
how to do model checking with schemata; 36 pages, 8 figure
Recompression: a simple and powerful technique for word equations
In this paper we present an application of a simple technique of local
recompression, previously developed by the author in the context of compressed
membership problems and compressed pattern matching, to word equations. The
technique is based on local modification of variables (replacing X by aX or Xa)
and iterative replacement of pairs of letters appearing in the equation by a
`fresh' letter, which can be seen as a bottom-up compression of the solution of
the given word equation, to be more specific, building an SLP (Straight-Line
Programme) for the solution of the word equation.
Using this technique we give a new, independent and self-contained proofs of
most of the known results for word equations. To be more specific, the
presented (nondeterministic) algorithm runs in O(n log n) space and in time
polynomial in log N, where N is the size of the length-minimal solution of the
word equation. The presented algorithm can be easily generalised to a generator
of all solutions of the given word equation (without increasing the space
usage). Furthermore, a further analysis of the algorithm yields a doubly
exponential upper bound on the size of the length-minimal solution. The
presented algorithm does not use exponential bound on the exponent of
periodicity. Conversely, the analysis of the algorithm yields an independent
proof of the exponential bound on exponent of periodicity.
We believe that the presented algorithm, its idea and analysis are far
simpler than all previously applied. Furthermore, thanks to it we can obtain a
unified and simple approach to most of known results for word equations.
As a small additional result we show that for O(1) variables (with arbitrary
many appearances in the equation) word equations can be solved in linear space,
i.e. they are context-sensitive.Comment: Submitted to a journal. Since previous version the proofs were
simplified, overall presentation improve
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