713 research outputs found
Introducing LoCo, a Logic for Configuration Problems
In this paper we present the core of LoCo, a logic-based high-level
representation language for expressing configuration problems. LoCo shall allow
to model these problems in an intuitive and declarative way, the dynamic
aspects of configuration notwithstanding. Our logic enforces that
configurations contain only finitely many components and reasoning can be
reduced to the task of model construction.Comment: In Proceedings LoCoCo 2011, arXiv:1108.609
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Descriptive complexity of graph spectra.
Two graphs are cospectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are cospectral with it are isomorphic to it. We consider these properties in relation to logical definability. We show that any pair of graphs that are elementarily equivalent with respect to the three-variable counting first-order logic are cospectral, and this is not the case with , nor with any number of variables if we exclude counting quantifiers. We also show that the class of graphs that are determined by their spectra is definable in partial fixed-point logic with counting. We relate these properties to other algebraic and combinatorial problems.OZ was supported by CONACyT-Mexico Grant 384665, SS was supported by EPSRC and The Royal Society
Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy
It is shown that for any fixed , the -fragment of
Presburger arithmetic, i.e., its restriction to quantifier alternations
beginning with an existential quantifier, is complete for
, the -th level of the weak EXP
hierarchy, an analogue to the polynomial-time hierarchy residing between
and . This result completes the
computational complexity landscape for Presburger arithmetic, a line of
research which dates back to the seminal work by Fischer & Rabin in 1974.
Moreover, we apply some of the techniques developed in the proof of the lower
bound in order to establish bounds on sets of naturals definable in the
-fragment of Presburger arithmetic: given a -formula
, it is shown that the set of non-negative solutions is an ultimately
periodic set whose period is at most doubly-exponential and that this bound is
tight.Comment: 10 pages, 2 figure
Dichotomies in Ontology-Mediated Querying with the Guarded Fragment
We study the complexity of ontology-mediated querying when ontologies are
formulated in the guarded fragment of first-order logic (GF). Our general aim
is to classify the data complexity on the level of ontologies where query
evaluation w.r.t. an ontology O is considered to be in PTime if all (unions of
conjunctive) queries can be evaluated in PTime w.r.t. O and coNP-hard if at
least one query is coNP-hard w.r.t. O. We identify several large and relevant
fragments of GF that enjoy a dichotomy between PTime and coNP, some of them
additionally admitting a form of counting. In fact, almost all ontologies in
the BioPortal repository fall into these fragments or can easily be rewritten
to do so. We then establish a variation of Ladner's Theorem on the existence of
NP-intermediate problems and use this result to show that for other fragments,
there is provably no such dichotomy. Again for other fragments (such as full
GF), establishing a dichotomy implies the Feder-Vardi conjecture on the
complexity of constraint satisfaction problems. We also link these results to
Datalog-rewritability and study the decidability of whether a given ontology
enjoys PTime query evaluation, presenting both positive and negative results
Trees over Infinite Structures and Path Logics with Synchronization
We provide decidability and undecidability results on the model-checking
problem for infinite tree structures. These tree structures are built from
sequences of elements of infinite relational structures. More precisely, we
deal with the tree iteration of a relational structure M in the sense of
Shelah-Stupp. In contrast to classical results where model-checking is shown
decidable for MSO-logic, we show decidability of the tree model-checking
problem for logics that allow only path quantifiers and chain quantifiers
(where chains are subsets of paths), as they appear in branching time logics;
however, at the same time the tree is enriched by the equal-level relation
(which holds between vertices u, v if they are on the same tree level). We
separate cleanly the tree logic from the logic used for expressing properties
of the underlying structure M. We illustrate the scope of the decidability
results by showing that two slight extensions of the framework lead to
undecidability. In particular, this applies to the (stronger) tree iteration in
the sense of Muchnik-Walukiewicz.Comment: In Proceedings INFINITY 2011, arXiv:1111.267
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