8,048 research outputs found
Lazy Model Expansion: Interleaving Grounding with Search
Finding satisfying assignments for the variables involved in a set of
constraints can be cast as a (bounded) model generation problem: search for
(bounded) models of a theory in some logic. The state-of-the-art approach for
bounded model generation for rich knowledge representation languages, like ASP,
FO(.) and Zinc, is ground-and-solve: reduce the theory to a ground or
propositional one and apply a search algorithm to the resulting theory.
An important bottleneck is the blowup of the size of the theory caused by the
reduction phase. Lazily grounding the theory during search is a way to overcome
this bottleneck. We present a theoretical framework and an implementation in
the context of the FO(.) knowledge representation language. Instead of
grounding all parts of a theory, justifications are derived for some parts of
it. Given a partial assignment for the grounded part of the theory and valid
justifications for the formulas of the non-grounded part, the justifications
provide a recipe to construct a complete assignment that satisfies the
non-grounded part. When a justification for a particular formula becomes
invalid during search, a new one is derived; if that fails, the formula is
split in a part to be grounded and a part that can be justified.
The theoretical framework captures existing approaches for tackling the
grounding bottleneck such as lazy clause generation and grounding-on-the-fly,
and presents a generalization of the 2-watched literal scheme. We present an
algorithm for lazy model expansion and integrate it in a model generator for
FO(ID), a language extending first-order logic with inductive definitions. The
algorithm is implemented as part of the state-of-the-art FO(ID) Knowledge-Base
System IDP. Experimental results illustrate the power and generality of the
approach
Extended ASP tableaux and rule redundancy in normal logic programs
We introduce an extended tableau calculus for answer set programming (ASP).
The proof system is based on the ASP tableaux defined in [Gebser&Schaub, ICLP
2006], with an added extension rule. We investigate the power of Extended ASP
Tableaux both theoretically and empirically. We study the relationship of
Extended ASP Tableaux with the Extended Resolution proof system defined by
Tseitin for sets of clauses, and separate Extended ASP Tableaux from ASP
Tableaux by giving a polynomial-length proof for a family of normal logic
programs P_n for which ASP Tableaux has exponential-length minimal proofs with
respect to n. Additionally, Extended ASP Tableaux imply interesting insight
into the effect of program simplification on the lengths of proofs in ASP.
Closely related to Extended ASP Tableaux, we empirically investigate the effect
of redundant rules on the efficiency of ASP solving.
To appear in Theory and Practice of Logic Programming (TPLP).Comment: 27 pages, 5 figures, 1 tabl
Efficient Interpolant Generation in Satisfiability Modulo Theories
The problem of computing Craig Interpolants for propositional (SAT) formulas has recently received a lot of interest, mainly for its applications in formal verification. However, propositional logic is often not expressive enough for representing many interesting verification problems, which can be more naturally addressed in the framework of Satisfiability Modulo Theories, SMT. Although {some} works have addressed the topic of generating interpolants in SMT, the techniques and tools that are currently available have some limitations, and their performance still does not exploit the full power of current state-of-the-art SMT solvers. In this paper we try to close this gap. We present several techniques for interpolant generation in SMT which overcome the limitations of the current generators mentioned above, and which take full advantage of state-of-the-art SMT technology. These novel techniques can lead to substantial performance improvements wrt. the currently available tools. We support our claims with an extensive experimental evaluation of our implementation of the proposed techniques in the MathSAT SMT solver
Efficient Generation of Craig Interpolants in Satisfiability Modulo Theories
The problem of computing Craig Interpolants has recently received a lot of
interest. In this paper, we address the problem of efficient generation of
interpolants for some important fragments of first order logic, which are
amenable for effective decision procedures, called Satisfiability Modulo Theory
solvers.
We make the following contributions.
First, we provide interpolation procedures for several basic theories of
interest: the theories of linear arithmetic over the rationals, difference
logic over rationals and integers, and UTVPI over rationals and integers.
Second, we define a novel approach to interpolate combinations of theories,
that applies to the Delayed Theory Combination approach.
Efficiency is ensured by the fact that the proposed interpolation algorithms
extend state of the art algorithms for Satisfiability Modulo Theories. Our
experimental evaluation shows that the MathSAT SMT solver can produce
interpolants with minor overhead in search, and much more efficiently than
other competitor solvers.Comment: submitted to ACM Transactions on Computational Logic (TOCL
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