173 research outputs found
Propositional Encoding of Constraints over Tree-Shaped Data
We present a functional programming language for specifying constraints over
tree-shaped data. The language allows for Haskell-like algebraic data types and
pattern matching. Our constraint compiler CO4 translates these programs into
satisfiability problems in propositional logic. We present an application from
the area of automated analysis of (non-)termination of rewrite systems
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
FO(FD): Extending classical logic with rule-based fixpoint definitions
We introduce fixpoint definitions, a rule-based reformulation of fixpoint
constructs. The logic FO(FD), an extension of classical logic with fixpoint
definitions, is defined. We illustrate the relation between FO(FD) and FO(ID),
which is developed as an integration of two knowledge representation paradigms.
The satisfiability problem for FO(FD) is investigated by first reducing FO(FD)
to difference logic and then using solvers for difference logic. These
reductions are evaluated in the computation of models for FO(FD) theories
representing fairness conditions and we provide potential applications of
FO(FD).Comment: Presented at ICLP 2010. 16 pages, 1 figur
BigraphER: rewriting and analysis engine for bigraphs
BigraphER is a suite of open-source tools providing an effi-
cient implementation of rewriting, simulation, and visualisation for bigraphs,
a universal formalism for modelling interacting systems that
evolve in time and space and first introduced by Milner. BigraphER consists
of an OCaml library that provides programming interfaces for the
manipulation of bigraphs, their constituents and reaction rules, and a
command-line tool capable of simulating Bigraphical Reactive Systems
(BRSs) and computing their transition systems. Other features are native
support for both bigraphs and bigraphs with sharing, stochastic reaction
rules, rule priorities, instantiation maps, parameterised controls, predicate
checking, graphical output and integration with the probabilistic
model checker PRISM
Counterexample Guided Abstraction Refinement Algorithm for Propositional Circumscription
Circumscription is a representative example of a nonmonotonic reasoning
inference technique. Circumscription has often been studied for first order
theories, but its propositional version has also been the subject of extensive
research, having been shown equivalent to extended closed world assumption
(ECWA). Moreover, entailment in propositional circumscription is a well-known
example of a decision problem in the second level of the polynomial hierarchy.
This paper proposes a new Boolean Satisfiability (SAT)-based algorithm for
entailment in propositional circumscription that explores the relationship of
propositional circumscription to minimal models. The new algorithm is inspired
by ideas commonly used in SAT-based model checking, namely counterexample
guided abstraction refinement. In addition, the new algorithm is refined to
compute the theory closure for generalized close world assumption (GCWA).
Experimental results show that the new algorithm can solve problem instances
that other solutions are unable to solve
Resolution over Linear Equations and Multilinear Proofs
We develop and study the complexity of propositional proof systems of varying
strength extending resolution by allowing it to operate with disjunctions of
linear equations instead of clauses. We demonstrate polynomial-size refutations
for hard tautologies like the pigeonhole principle, Tseitin graph tautologies
and the clique-coloring tautologies in these proof systems. Using the
(monotone) interpolation by a communication game technique we establish an
exponential-size lower bound on refutations in a certain, considerably strong,
fragment of resolution over linear equations, as well as a general polynomial
upper bound on (non-monotone) interpolants in this fragment.
We then apply these results to extend and improve previous results on
multilinear proofs (over fields of characteristic 0), as studied in
[RazTzameret06]. Specifically, we show the following:
1. Proofs operating with depth-3 multilinear formulas polynomially simulate a
certain, considerably strong, fragment of resolution over linear equations.
2. Proofs operating with depth-3 multilinear formulas admit polynomial-size
refutations of the pigeonhole principle and Tseitin graph tautologies. The
former improve over a previous result that established small multilinear proofs
only for the \emph{functional} pigeonhole principle. The latter are different
than previous proofs, and apply to multilinear proofs of Tseitin mod p graph
tautologies over any field of characteristic 0.
We conclude by connecting resolution over linear equations with extensions of
the cutting planes proof system.Comment: 44 page
Effective problem solving using SAT solvers
In this article we demonstrate how to solve a variety of problems and puzzles
using the built-in SAT solver of the computer algebra system Maple. Once the
problems have been encoded into Boolean logic, solutions can be found (or shown
to not exist) automatically, without the need to implement any search
algorithm. In particular, we describe how to solve the -queens problem, how
to generate and solve Sudoku puzzles, how to solve logic puzzles like the
Einstein riddle, how to solve the 15-puzzle, how to solve the maximum clique
problem, and finding Graeco-Latin squares.Comment: To appear in Proceedings of the Maple Conference 201
Solving and Verifying the Boolean Pythagorean Triples Problem via Cube-and-Conquer
We solved a long-outstanding open problem in Ramsey theory, using SAT solving
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