1,608 research outputs found
Optimization Modulo Theories with Linear Rational Costs
In the contexts of automated reasoning (AR) and formal verification (FV),
important decision problems are effectively encoded into Satisfiability Modulo
Theories (SMT). In the last decade efficient SMT solvers have been developed
for several theories of practical interest (e.g., linear arithmetic, arrays,
bit-vectors). Surprisingly, little work has been done to extend SMT to deal
with optimization problems; in particular, we are not aware of any previous
work on SMT solvers able to produce solutions which minimize cost functions
over arithmetical variables. This is unfortunate, since some problems of
interest require this functionality.
In the work described in this paper we start filling this gap. We present and
discuss two general procedures for leveraging SMT to handle the minimization of
linear rational cost functions, combining SMT with standard minimization
techniques. We have implemented the procedures within the MathSAT SMT solver.
Due to the absence of competitors in the AR, FV and SMT domains, we have
experimentally evaluated our implementation against state-of-the-art tools for
the domain of linear generalized disjunctive programming (LGDP), which is
closest in spirit to our domain, on sets of problems which have been previously
proposed as benchmarks for the latter tools. The results show that our tool is
very competitive with, and often outperforms, these tools on these problems,
clearly demonstrating the potential of the approach.Comment: Submitted on january 2014 to ACM Transactions on Computational Logic,
currently under revision. arXiv admin note: text overlap with arXiv:1202.140
A Decision Procedure for Herbrand Formulas without Skolemization
This paper describes a decision procedure for disjunctions of conjunctions of anti-prenex normal forms of pure first-order logic (FOLDNFs) that do not contain V within the scope of quantifiers. The disjuncts of these FOLDNFs are equivalent to prenex normal forms whose quantifier-free parts are conjunctions of atomic and negated atomic formulae (= Herbrand formulae). In contrast to the usual algorithms for Herbrand formulae, neither skolemization nor unification algorithms with function symbols are applied. Instead, a procedure is described that rests on nothing but equivalence transformations within pure first-order logic (FOL). This procedure involves the application of a calculus for negative normal forms (the NNF-calculus) with
A -||- A & A (= &I) as the sole rule that increases the complexity of given FOLDNFs.
The described algorithm illustrates how, in the case of Herbrand formulae,
decision problems can be solved through a systematic search for proofs that
reduce the number of applications of the rule &I to a minimum in the NNF-calculus. In the case of Herbrand formulae, it is even possible to entirely abstain from applying &I.
Finally, it is shown how the described procedure can be used within an optimized general search for proofs of contradiction and what kind of questions arise for a &I-minimal proof strategy in the case of a general search for proofs of contradiction
Optimization in SMT with LA(Q) Cost Functions
In the contexts of automated reasoning and formal verification, important
decision problems are effectively encoded into Satisfiability Modulo Theories
(SMT). In the last decade efficient SMT solvers have been developed for several
theories of practical interest (e.g., linear arithmetic, arrays, bit-vectors).
Surprisingly, very few work has been done to extend SMT to deal with
optimization problems; in particular, we are not aware of any work on SMT
solvers able to produce solutions which minimize cost functions over
arithmetical variables. This is unfortunate, since some problems of interest
require this functionality.
In this paper we start filling this gap. We present and discuss two general
procedures for leveraging SMT to handle the minimization of LA(Q) cost
functions, combining SMT with standard minimization techniques. We have
implemented the proposed approach within the MathSAT SMT solver. Due to the
lack of competitors in AR and SMT domains, we experimentally evaluated our
implementation against state-of-the-art tools for the domain of linear
generalized disjunctive programming (LGDP), which is closest in spirit to our
domain, on sets of problems which have been previously proposed as benchmarks
for the latter tools. The results show that our tool is very competitive with,
and often outperforms, these tools on these problems, clearly demonstrating the
potential of the approach.Comment: A shorter version is currently under submissio
A Paraconsistent ASP-like Language with Tractable Model Generation
Answer Set Programming (ASP) is nowadays a dominant rule-based knowledge
representation tool. Though existing ASP variants enjoy efficient
implementations, generating an answer set remains intractable. The goal of this
research is to define a new \asp-like rule language, 4SP, with tractable model
generation. The language combines ideas of ASP and a paraconsistent rule
language 4QL. Though 4SP shares the syntax of \asp and for each program all its
answer sets are among 4SP models, the new language differs from ASP in its
logical foundations, the intended methodology of its use and complexity of
computing models.
As we show in the paper, 4QL can be seen as a paraconsistent counterpart of
ASP programs stratified with respect to default negation. Although model
generation of well-supported models for 4QL programs is tractable, dropping
stratification makes both 4QL and ASP intractable. To retain tractability while
allowing non-stratified programs, in 4SP we introduce trial expressions
interlacing programs with hypotheses as to the truth values of default
negations. This allows us to develop a~model generation algorithm with
deterministic polynomial time complexity.
We also show relationships among 4SP, ASP and 4QL
Compositionality for Quantitative Specifications
We provide a framework for compositional and iterative design and
verification of systems with quantitative information, such as rewards, time or
energy. It is based on disjunctive modal transition systems where we allow
actions to bear various types of quantitative information. Throughout the
design process the actions can be further refined and the information made more
precise. We show how to compute the results of standard operations on the
systems, including the quotient (residual), which has not been previously
considered for quantitative non-deterministic systems. Our quantitative
framework has close connections to the modal nu-calculus and is compositional
with respect to general notions of distances between systems and the standard
operations
Horn formula minimization
Horn formulas make up an important subclass of Boolean formulas that exhibits interesting and useful computational properties. They have been widely studied due to the fact that the satisfiability problem for Horn formulas is solvable in linear time. Also resulting from this, Horn formulas play an important role in the field of artificial intelligence. The minimization problem of Horn formulas is to reduce the size of a given Horn formula to find a shortest equivalent representation. Many knowledge bases in propositional expert systems are represented as Horn formulas. Therefore the minimization of Horn formulas can be used to reduce the size of these knowledge bases, thereby increasing the efficiency of queries. The goal of this project is to study the properties of Horn formulas and the minimization of Horn formulas. Topics discussed include The satisfiability problem for Horn formulas. NP-completeness of Horn formula minimization. Subclasses of Horn formulas for which the minimization problem is solvable in polynomial time. Approximation algorithms for Horn formula minimization
- …