188 research outputs found
Pushing the envelope of Optimization Modulo Theories with Linear-Arithmetic Cost Functions
In the last decade we have witnessed an impressive progress in the
expressiveness and efficiency of Satisfiability Modulo Theories (SMT) solving
techniques. This has brought previously-intractable problems at the reach of
state-of-the-art SMT solvers, in particular in the domain of SW and HW
verification. Many SMT-encodable problems of interest, however, require also
the capability of finding models that are optimal wrt. some cost functions. In
previous work, namely "Optimization Modulo Theory with Linear Rational Cost
Functions -- OMT(LAR U T )", we have leveraged SMT solving to handle the
minimization of cost functions on linear arithmetic over the rationals, by
means of a combination of SMT and LP minimization techniques. In this paper we
push the envelope of our OMT approach along three directions: first, we extend
it to work also with linear arithmetic on the mixed integer/rational domain, by
means of a combination of SMT, LP and ILP minimization techniques; second, we
develop a multi-objective version of OMT, so that to handle many cost functions
simultaneously; third, we develop an incremental version of OMT, so that to
exploit the incrementality of some OMT-encodable problems. An empirical
evaluation performed on OMT-encoded verification problems demonstrates the
usefulness and efficiency of these extensions.Comment: A slightly-shorter version of this paper is published at TACAS 2015
conferenc
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
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
Understanding The Impact of Solver Choice in Model-Based Test Generation
Background: In model-based test generation, SMT solvers explore the state-space of the model in search of violations of specified properties. If the solver finds that a predicate can be violated, it produces a partial test specification demonstrating the violation.Aims: The choice of solvers is important, as each may produce differing counterexamples. We aim to understand how solver choice impacts the effectiveness of generated test suites at finding faults.Method: We have performed experiments examining the impact of solver choice across multiple dimensions, examining the ability to attain goal satisfaction and fault detection when satisfaction is achieved---varying the source of test goals, data types of model input, and test oracle.Results: The results of our experiment show that solvers vary in their ability to produce counterexamples, and---for models where all solvers achieve goal satisfaction---in the resulting fault detection of the generated test suites. The choice of solver has an impact on the resulting test suite, regardless of the oracle, model structure, or source of testing goals.Conclusions: The results of this study identify factors that impact fault-detection effectiveness, and advice that could improve future approaches to model-based test generation
A Mixed Real and Floating-Point Solver
Reasoning about mixed real and floating-point constraints is essential for developing accurate analysis tools for floating-point pro- grams. This paper presents FPRoCK, a prototype tool for solving mixed real and floating-point formulas. FPRoCK transforms a mixed formula into an equisatisfiable one over the reals. This formula is then solved using an off-the-shelf SMT solver. FPRoCK is also integrated with the PRECiSA static analyzer, which computes a sound estimation of the round-off error of a floating-point program. It is used to detect infeasible computational paths, thereby improving the accuracy of PRECiSA
Generating and Searching Families of FFT Algorithms
A fundamental question of longstanding theoretical interest is to prove the
lowest exact count of real additions and multiplications required to compute a
power-of-two discrete Fourier transform (DFT). For 35 years the split-radix
algorithm held the record by requiring just 4n log n - 6n + 8 arithmetic
operations on real numbers for a size-n DFT, and was widely believed to be the
best possible. Recent work by Van Buskirk et al. demonstrated improvements to
the split-radix operation count by using multiplier coefficients or "twiddle
factors" that are not n-th roots of unity for a size-n DFT. This paper presents
a Boolean Satisfiability-based proof of the lowest operation count for certain
classes of DFT algorithms. First, we present a novel way to choose new yet
valid twiddle factors for the nodes in flowgraphs generated by common
power-of-two fast Fourier transform algorithms, FFTs. With this new technique,
we can generate a large family of FFTs realizable by a fixed flowgraph. This
solution space of FFTs is cast as a Boolean Satisfiability problem, and a
modern Satisfiability Modulo Theory solver is applied to search for FFTs
requiring the fewest arithmetic operations. Surprisingly, we find that there
are FFTs requiring fewer operations than the split-radix even when all twiddle
factors are n-th roots of unity.Comment: Preprint submitted on March 28, 2011, to the Journal on
Satisfiability, Boolean Modeling and Computatio
Symbolic execution of verification languages and floating-point code
The focus of this thesis is a program analysis technique named symbolic
execution. We present three main contributions to this field.
First, an investigation into comparing several state-of-the-art program
analysis tools at the level of an intermediate verification language over a
large set of benchmarks, and improvements to the state-of-the-art of symbolic
execution for this language. This is explored via a new tool, Symbooglix, that
operates on the Boogie intermediate verification language.
Second, an investigation into performing symbolic execution of floating-point
programs via a standardised theory of floating-point arithmetic that is
supported by several existing constraint solvers. This is investigated via two
independent extensions of the KLEE symbolic execution engine to support
reasoning about floating-point operations (with one tool developed by the
thesis author).
Third, an investigation into the use of coverage-guided fuzzing as a means for
solving constraints over finite data types, inspired by the difficulties
associated with solving floating-point constraints. The associated prototype
tool, JFS, which builds on the LibFuzzer project, can at present be applied to
a wide range of SMT queries over bit-vector and floating-point variables, and
shows promise on floating-point constraints.Open Acces
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