4,594 research outputs found
A Reduction from Unbounded Linear Mixed Arithmetic Problems into Bounded Problems
We present a combination of the Mixed-Echelon-Hermite transformation and the
Double-Bounded Reduction for systems of linear mixed arithmetic that preserve
satisfiability and can be computed in polynomial time. Together, the two
transformations turn any system of linear mixed constraints into a bounded
system, i.e., a system for which termination can be achieved easily. Existing
approaches for linear mixed arithmetic, e.g., branch-and-bound and cuts from
proofs, only explore a finite search space after application of our two
transformations. Instead of generating a priori bounds for the variables, e.g.,
as suggested by Papadimitriou, unbounded variables are eliminated through the
two transformations. The transformations orient themselves on the structure of
an input system instead of computing a priori (over-)approximations out of the
available constants. Experiments provide further evidence to the efficiency of
the transformations in practice. We also present a polynomial method for
converting certificates of (un)satisfiability from the transformed to the
original system
A Reduction from Unbounded Linear Mixed Arithmetic Problems into Bounded Problems
International audienceWe present a combination of the Mixed-Echelon-Hermite transformation and the Double-Bounded Reduction for systems of linear mixed arithmetic that preserve satisfiability and can be computed in polynomial time. Together, the two transformations turn any system of linear mixed constraints into a bounded system, i.e., a system for which termination can be achieved easily. Existing approaches for linear mixed arithmetic, e.g., branch-and-bound and cuts from proofs, only explore a finite search space after application of our two transformations. Instead of generating a priori bounds for the variables, e.g., as suggested by Papadimitriou, unbounded variables are eliminated through the two transformations. The transformations orient themselves on the structure of an input system instead of computing a priori (over- )approximations out of the available constants. Experiments provide further evidence to the efficiency of the transformations in practice. We also present a polynomial method for converting certificates of (un)satisfiability from the transformed to the original system
Decision procedures for linear arithmetic
In this thesis, we present new decision procedures for linear arithmetic in the context of SMT solvers and theorem provers: 1) CutSat++, a calculus for linear integer arithmetic that combines techniques from SAT solving and quantifier elimination in order to be sound, terminating, and complete. 2) The largest cube test and the unit cube test, two sound (although incomplete) tests that find integer and mixed solutions in polynomial time. The tests are especially efficient on absolutely unbounded constraint systems, which are difficult to handle for many other decision procedures. 3) Techniques for the investigation of equalities implied by a constraint system. Moreover, we present several applications for these techniques. 4) The Double-Bounded reduction and the Mixed-Echelon-Hermite transformation, two transformations that reduce any constraint system in polynomial time to an equisatisfiable constraint system that is bounded. The transformations are beneficial because they turn branch-and-bound into a complete and efficient decision procedure for unbounded constraint systems. We have implemented the above decision procedures (except for Cut- Sat++) as part of our linear arithmetic theory solver SPASS-IQ and as part of our CDCL(LA) solver SPASS-SATT. We also present various benchmark evaluations that confirm the practical efficiency of our new decision procedures.In dieser Arbeit prĂ€sentieren wir neue Entscheidungsprozeduren fĂŒr lineare Arithmetik im Kontext von SMT-Solvern und Theorembeweisern: 1) CutSat++, ein korrekter und vollstĂ€ndiger KalkĂŒl fĂŒr ganzzahlige lineare Arithmetik, der Techniken zur Entscheidung von Aussagenlogik mit Techniken aus der Quantorenelimination vereint. 2) Der GröĂte-WĂŒrfeltest und der EinheitswĂŒrfeltest, zwei korrekte (wenn auch unvollstĂ€ndige) Tests, die in polynomieller Zeit (gemischt-)ganzzahlige Lösungen finden. Die Tests sind besonders effizient auf vollstĂ€ndig unbegrenzten Systemen, welche fĂŒr viele andere Entscheidungsprozeduren schwer sind. 3) Techniken zur Ermittlung von Gleichungen, die von einem linearen Ungleichungssystem impliziert werden. Des Weiteren prĂ€sentieren wir mehrere Anwendungsmöglichkeiten fĂŒr diese Techniken. 4) Die Beidseitig-Begrenzte-Reduktion und die Gemischte-Echelon-Hermitesche- Transformation, die ein Ungleichungssystem in polynomieller Zeit auf ein erfĂŒllbarkeitsĂ€quivalentes System reduzieren, das begrenzt ist. Vereint verwandeln die Transformationen Branch-and-Bound in eine vollstĂ€ndige und effiziente Entscheidungsprozedur fĂŒr unbeschrĂ€nkte Ungleichungssysteme. Wir haben diese Techniken (ausgenommen CutSat++) in SPASS-IQ (unserem theory solver fĂŒr lineare Arithmetik) und in SPASS-SATT (unserem CDCL(LA) solver) implementiert. Basierend darauf prĂ€sentieren wir Benchmark-Evaluationen, die die Effizienz unserer Entscheidungsprozeduren bestĂ€tigen
Complexity of short Presburger arithmetic
We study complexity of short sentences in Presburger arithmetic (Short-PA).
Here by "short" we mean sentences with a bounded number of variables,
quantifiers, inequalities and Boolean operations; the input consists only of
the integers involved in the inequalities. We prove that assuming Kannan's
partition can be found in polynomial time, the satisfiability of Short-PA
sentences can be decided in polynomial time. Furthermore, under the same
assumption, we show that the numbers of satisfying assignments of short
Presburger sentences can also be computed in polynomial time
Decidability of Univariate Real Algebra with Predicates for Rational and Integer Powers
We prove decidability of univariate real algebra extended with predicates for
rational and integer powers, i.e., and . Our decision procedure combines computation over real algebraic
cells with the rational root theorem and witness construction via algebraic
number density arguments.Comment: To appear in CADE-25: 25th International Conference on Automated
Deduction, 2015. Proceedings to be published by Springer-Verla
A Survey of Symbolic Execution Techniques
Many security and software testing applications require checking whether
certain properties of a program hold for any possible usage scenario. For
instance, a tool for identifying software vulnerabilities may need to rule out
the existence of any backdoor to bypass a program's authentication. One
approach would be to test the program using different, possibly random inputs.
As the backdoor may only be hit for very specific program workloads, automated
exploration of the space of possible inputs is of the essence. Symbolic
execution provides an elegant solution to the problem, by systematically
exploring many possible execution paths at the same time without necessarily
requiring concrete inputs. Rather than taking on fully specified input values,
the technique abstractly represents them as symbols, resorting to constraint
solvers to construct actual instances that would cause property violations.
Symbolic execution has been incubated in dozens of tools developed over the
last four decades, leading to major practical breakthroughs in a number of
prominent software reliability applications. The goal of this survey is to
provide an overview of the main ideas, challenges, and solutions developed in
the area, distilling them for a broad audience.
The present survey has been accepted for publication at ACM Computing
Surveys. If you are considering citing this survey, we would appreciate if you
could use the following BibTeX entry: http://goo.gl/Hf5FvcComment: This is the authors pre-print copy. If you are considering citing
this survey, we would appreciate if you could use the following BibTeX entry:
http://goo.gl/Hf5Fv
On the Path-Width of Integer Linear Programming
We consider the feasibility problem of integer linear programming (ILP). We
show that solutions of any ILP instance can be naturally represented by an
FO-definable class of graphs. For each solution there may be many graphs
representing it. However, one of these graphs is of path-width at most 2n,
where n is the number of variables in the instance. Since FO is decidable on
graphs of bounded path- width, we obtain an alternative decidability result for
ILP. The technique we use underlines a common principle to prove decidability
which has previously been employed for automata with auxiliary storage. We also
show how this new result links to automata theory and program verification.Comment: In Proceedings GandALF 2014, arXiv:1408.556
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