1,527 research outputs found
Combining Lists with Non-Stably Infinite Theories
http://www.springerlink.comIn program verification one has often to reason about lists over elements of a given nature. Thus, it becomes important to be able to combine the theory of lists with a generic theory modeling the elements. This combination can be achieved using the Nelson-Oppen method only if is stably infinite. The goal of this paper is to relax the stable-infiniteness requirement. More specifically, we provide a new method that is able to combine the theory of lists with any theory of the elements, regardless of whether is stably infinite or not. The crux of our combination method is to guess an arrangement over a set of variables that is larger than the one considered by Nelson and Oppen. Furthermore, our results entail that it is also possible to combine with the more general theory of lists with a length function
New results on rewrite-based satisfiability procedures
Program analysis and verification require decision procedures to reason on
theories of data structures. Many problems can be reduced to the satisfiability
of sets of ground literals in theory T. If a sound and complete inference
system for first-order logic is guaranteed to terminate on T-satisfiability
problems, any theorem-proving strategy with that system and a fair search plan
is a T-satisfiability procedure. We prove termination of a rewrite-based
first-order engine on the theories of records, integer offsets, integer offsets
modulo and lists. We give a modularity theorem stating sufficient conditions
for termination on a combinations of theories, given termination on each. The
above theories, as well as others, satisfy these conditions. We introduce
several sets of benchmarks on these theories and their combinations, including
both parametric synthetic benchmarks to test scalability, and real-world
problems to test performances on huge sets of literals. We compare the
rewrite-based theorem prover E with the validity checkers CVC and CVC Lite.
Contrary to the folklore that a general-purpose prover cannot compete with
reasoners with built-in theories, the experiments are overall favorable to the
theorem prover, showing that not only the rewriting approach is elegant and
conceptually simple, but has important practical implications.Comment: To appear in the ACM Transactions on Computational Logic, 49 page
ILP Modulo Data
The vast quantity of data generated and captured every day has led to a
pressing need for tools and processes to organize, analyze and interrelate this
data. Automated reasoning and optimization tools with inherent support for data
could enable advancements in a variety of contexts, from data-backed decision
making to data-intensive scientific research. To this end, we introduce a
decidable logic aimed at database analysis. Our logic extends quantifier-free
Linear Integer Arithmetic with operators from Relational Algebra, like
selection and cross product. We provide a scalable decision procedure that is
based on the BC(T) architecture for ILP Modulo Theories. Our decision procedure
makes use of database techniques. We also experimentally evaluate our approach,
and discuss potential applications.Comment: FMCAD 2014 final version plus proof
A Rewriting Approach to the Combination of Data Structures with Bridging Theories
International audienceWe introduce a combination method Ă la Nelson-Oppen to solve the satisfiability problem modulo a non-disjoint union of theories connected with bridging functions. The combination method is particularly useful to handle verification conditions involving functions defined over inductive data structures. We investigate the problem of determining the data structure theories for which this combination method is sound and complete. Our completeness proof is based on a rewriting approach where the bridging function is defined as a term rewrite system, and the data structure theory is given by a basic congruence relation. Our contribution is to introduce a class of data structure theories that are combinable with a disjoint target theory via an inductively defined bridging function. This class includes the theory of equality, the theory of absolutely free data structures, and all the theories in between. Hence, our non-disjoint combination method applies to many classical data structure theories admitting a rewrite-based satisfiability procedure
Combination of convex theories: Modularity, deduction completeness, and explanation
AbstractDecision procedures are key components of theorem provers and constraint satisfaction systems. Their modular combination is of prime interest for building efficient systems, but their effective use is often limited by poor interface capabilities, when such procedures only provide a simple âsat/unsatâ answer. In this paper, we develop a framework to design cooperation schemas between such procedures while maintaining modularity of their interfaces. First, we use the framework to specify and prove the correctness of classic combination schemas by NelsonâOppen and Shostak. Second, we introduce the concept of deduction complete satisfiability procedures, we show how to build them for large classes of theories, then we provide a schema to modularly combine them. Third, we consider the problem of modularly constructing explanations for combinations by re-using available proof-producing procedures for the component theories
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
On the Convexity of a Fragment of Pure Set Theory with Applications within a Nelson-Oppen Framework
In Proceedings GandALF 2021, arXiv:2109.0779
Delayed theory combination vs. Nelson-Oppen for satisfiability modulo theories: a comparative analysis
Most state-of-the-art approaches for Satisfiability Modulo Theories rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory . Often is the combination of two (or more) simpler theories , s.t. the specific must be combined. Up to a few years ago, the standard approach to was to integrate the SAT solver with one combined , obtained from two distinct by means of evolutions of Nelson and Oppen's (NO) combination procedure, in which the deduce and exchange interface equalities. Nowadays many state-of-the-art SMT solvers use evolutions of a more recent procedure called Delayed Theory Combination (DTC), in which each interacts directly and only with the SAT solver, in such a way that part or all of the (possibly very expensive) reasoning effort on interface equalities is delegated to the SAT solver itself. In this paper we present a comparative analysis of DTC vs. NO for . On the one hand, we explain the advantages of DTC in exploiting the power of modern SAT solvers to reduce the search. On the other hand, we show that the extra amount of Boolean search required to the SAT solver can be controlled. In fact, we prove two novel theoretical results, for both convex and non-convex theories and for different deduction capabilities of the , which relate the amount of extra Boolean search required to the SAT solver by DTC with the number of deductions and case-splits required to the by NO in order to perform the same tasks: (i) under the same hypotheses of deduction capabilities of the required by NO, DTC causes no extra Boolean search; (ii) using with limited or no deduction capabilities, the extra Boolean search required can be reduced down to a negligible amount by controlling the quality of the -conflict sets returned by the ${\mathcal{T}_i}{\text {-}}solvers
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