175 research outputs found
Homogeneity and Homogenizability: Hard Problems for the Logic SNP
We show that the question whether a given SNP sentence defines a
homogenizable class of finite structures is undecidable, even if the sentence
comes from the connected Datalog fragment and uses at most binary relation
symbols. As a byproduct of our proof, we also get the undecidability of some
other properties for Datalog programs, e.g., whether they can be rewritten in
MMSNP, whether they solve some finite-domain CSP, or whether they define the
age of a reduct of a homogeneous Ramsey structure in a finite relational
signature. We subsequently show that the closely related problem of testing the
amalgamation property for finitely bounded classes is EXPSPACE-hard or
PSPACE-hard, depending on whether the input is specified by a universal
sentence or a set of forbidden substructures.Comment: 34 pages, 3 figure
Datalog and Constraint Satisfaction with Infinite Templates
On finite structures, there is a well-known connection between the expressive
power of Datalog, finite variable logics, the existential pebble game, and
bounded hypertree duality. We study this connection for infinite structures.
This has applications for constraint satisfaction with infinite templates. If
the template Gamma is omega-categorical, we present various equivalent
characterizations of those Gamma such that the constraint satisfaction problem
(CSP) for Gamma can be solved by a Datalog program. We also show that
CSP(Gamma) can be solved in polynomial time for arbitrary omega-categorical
structures Gamma if the input is restricted to instances of bounded treewidth.
Finally, we characterize those omega-categorical templates whose CSP has
Datalog width 1, and those whose CSP has strict Datalog width k.Comment: 28 pages. This is an extended long version of a conference paper that
appeared at STACS'06. In the third version in the arxiv we have revised the
presentation again and added a section that relates our results to
formalizations of CSPs using relation algebra
Linear Datalog and Bounded Path Duality of Relational Structures
In this paper we systematically investigate the connections between logics
with a finite number of variables, structures of bounded pathwidth, and linear
Datalog Programs. We prove that, in the context of Constraint Satisfaction
Problems, all these concepts correspond to different mathematical embodiments
of a unique robust notion that we call bounded path duality. We also study the
computational complexity implications of the notion of bounded path duality. We
show that every constraint satisfaction problem \csp(\best) with bounded path
duality is solvable in NL and that this notion explains in a uniform way all
families of CSPs known to be in NL. Finally, we use the results developed in
the paper to identify new problems in NL
Low-level dichotomy for Quantified Constraint Satisfaction Problems
Building on a result of Larose and Tesson for constraint satisfaction
problems (CSP s), we uncover a dichotomy for the quantified constraint
satisfaction problem QCSP(B), where B is a finite structure that is a core.
Specifically, such problems are either in ALogtime or are L-hard. This involves
demonstrating that if CSP(B) is first-order expressible, and B is a core, then
QCSP(B) is in ALogtime.
We show that the class of B such that CSP(B) is first-order expressible
(indeed, trivially true) is a microcosm for all QCSPs. Specifically, for any B
there exists a C such that CSP(C) is trivially true, yet QCSP(B) and QCSP(C)
are equivalent under logspace reductions
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