2,751 research outputs found
HoCHC: A Refutationally Complete and Semantically Invariant System of Higher-order Logic Modulo Theories
We present a simple resolution proof system for higher-order constrained Horn
clauses (HoCHC) - a system of higher-order logic modulo theories - and prove
its soundness and refutational completeness w.r.t. the standard semantics. As
corollaries, we obtain the compactness theorem and semi-decidability of HoCHC
for semi-decidable background theories, and we prove that HoCHC satisfies a
canonical model property. Moreover a variant of the well-known translation from
higher-order to 1st-order logic is shown to be sound and complete for HoCHC in
standard semantics. We illustrate how to transfer decidability results for
(fragments of) 1st-order logic modulo theories to our higher-order setting,
using as example the Bernays-Schonfinkel-Ramsey fragment of HoCHC modulo a
restricted form of Linear Integer Arithmetic
Tractable Query Answering and Optimization for Extensions of Weakly-Sticky Datalog+-
We consider a semantic class, weakly-chase-sticky (WChS), and a syntactic
subclass, jointly-weakly-sticky (JWS), of Datalog+- programs. Both extend that
of weakly-sticky (WS) programs, which appear in our applications to data
quality. For WChS programs we propose a practical, polynomial-time query
answering algorithm (QAA). We establish that the two classes are closed under
magic-sets rewritings. As a consequence, QAA can be applied to the optimized
programs. QAA takes as inputs the program (including the query) and semantic
information about the "finiteness" of predicate positions. For the syntactic
subclasses JWS and WS of WChS, this additional information is computable.Comment: To appear in Proc. Alberto Mendelzon WS on Foundations of Data
Management (AMW15
Quadratic Word Equations with Length Constraints, Counter Systems, and Presburger Arithmetic with Divisibility
Word equations are a crucial element in the theoretical foundation of
constraint solving over strings, which have received a lot of attention in
recent years. A word equation relates two words over string variables and
constants. Its solution amounts to a function mapping variables to constant
strings that equate the left and right hand sides of the equation. While the
problem of solving word equations is decidable, the decidability of the problem
of solving a word equation with a length constraint (i.e., a constraint
relating the lengths of words in the word equation) has remained a
long-standing open problem. In this paper, we focus on the subclass of
quadratic word equations, i.e., in which each variable occurs at most twice. We
first show that the length abstractions of solutions to quadratic word
equations are in general not Presburger-definable. We then describe a class of
counter systems with Presburger transition relations which capture the length
abstraction of a quadratic word equation with regular constraints. We provide
an encoding of the effect of a simple loop of the counter systems in the theory
of existential Presburger Arithmetic with divisibility (PAD). Since PAD is
decidable, we get a decision procedure for quadratic words equations with
length constraints for which the associated counter system is \emph{flat}
(i.e., all nodes belong to at most one cycle). We show a decidability result
(in fact, also an NP algorithm with a PAD oracle) for a recently proposed
NP-complete fragment of word equations called regular-oriented word equations,
together with length constraints. Decidability holds when the constraints are
additionally extended with regular constraints with a 1-weak control structure.Comment: 18 page
Revisiting Chase Termination for Existential Rules and their Extension to Nonmonotonic Negation
Existential rules have been proposed for representing ontological knowledge,
specifically in the context of Ontology- Based Data Access. Entailment with
existential rules is undecidable. We focus in this paper on conditions that
ensure the termination of a breadth-first forward chaining algorithm known as
the chase. Several variants of the chase have been proposed. In the first part
of this paper, we propose a new tool that allows to extend existing acyclicity
conditions ensuring chase termination, while keeping good complexity
properties. In the second part, we study the extension to existential rules
with nonmonotonic negation under stable model semantics, discuss the relevancy
of the chase variants for these rules and further extend acyclicity results
obtained in the positive case.Comment: This paper appears in the Proceedings of the 15th International
Workshop on Non-Monotonic Reasoning (NMR 2014
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