A Simplex-Based Extension of Fourier-Motzkin for Solving Linear Integer Arithmetic

Bobot, François; Conchon, Sylvain; Contejean, Evelyne; Iguernelala, Mohamed; Mahboubi, Assia; Mebsout, Alain; Melquiond, Guillaume

doi:10.1007/978-3-642-31365-3_8

Cited by 13 publications

(22 citation statements)

References 10 publications

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“…We also compared our test with the ctrl-ergo solver, which includes a subroutine that is essentially the dual to our largest cube test [4]. As expected, both approaches are comparable for infinite lattice width polyhedra.…”

Section: Methodsmentioning

confidence: 75%

“…Moreover, let the variables x 2 , x 3 , x 4 , and x 5 be tightly bounded such that x 2 = 1, x 3 = 0, x 4 = 1, and x 5 = 3. If we now replace the tightly non-basic variables, in the definitions of x 6 and x 7 we get that both of their normalized representations are 2x 1 and we have actively used the tight bounds x 2 = 1, x 4 = 1, x 5 = 3 to compute this normalization.…”

Section: Incrementality Explanations and Justificationsmentioning

confidence: 99%

“…This problem has been investigated in different research areas, e.g., in optimization via (mixed) integer linear programming (MILP) [19] and in constraint solving via satisfiability modulo theories (SMT) [4,6,11,16]. For commercial MILP implementations, it is standard to integrate preprocessing techniques, heuristics, and specialized tests [19].…”

Section: Introductionmentioning

confidence: 99%

“…Also, Bobot et al [4] discuss relations between hypercubes and polyhedra including infinite lattice width and positive linear combinations between inequalities. Our largest cube test can also detect these relations because it is, with some minor changes, the dual of the linear optimization problem of Bobot et al In contrast to the linear optimization problem of Bobot et al, our tests are closer to the original polyhedron and, therefore, easier to construct.…”

Section: Introductionmentioning

confidence: 99%

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New techniques for linear arithmetic: cubes and equalities

Bromberger

Weidenbach

2017

Form Methods Syst Des

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We present several new techniques for linear arithmetic constraint solving. They are all based on the linear cube transformation, a method presented here, which allows us to efficiently determine whether a system of linear arithmetic constraints contains a hypercube of a given edge length. Our first findings based on this transformation are two sound tests that find integer solutions for linear arithmetic constraints. While many complete methods search along the problem surface for a solution, these tests use cubes to explore the interior of the problems. The tests are especially efficient for constraints with a large number of integer solutions, e.g., those with infinite lattice width. Inside the SMT-LIB benchmarks, we have found almost one thousand problem instances with infinite lattice width. Experimental results confirm that our tests are superior on these instances compared to several state-ofthe-art SMT solvers. We also discovered that the linear cube transformation can be used to investigate the equalities implied by a system of linear arithmetic constraints. For this purpose, we developed a method that computes a basis for all implied equalities, i.e., a finite representation of all equalities implied by the linear arithmetic constraints. The equality basis has several applications. For instance, it allows us to verify whether a system of linear arithmetic constraints implies a given equality. This is valuable in the context of NelsonOppen style combinations of theories.

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Section: Methodsmentioning

confidence: 75%

Section: Incrementality Explanations and Justificationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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New techniques for linear arithmetic: cubes and equalities

Bromberger

Weidenbach

2017

Form Methods Syst Des

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“…A simplex-based algorithm [3] then finds the optimal bounds by linear combination of these. Improved bounds may then lead to another round of saturation by application of the instantiated axioms.…”

Section: Gappa As a Matching Algorithmmentioning

confidence: 99%

Built-in Treatment of an Axiomatic Floating-Point Theory for SMT Solvers

Conchon¹,

Melquiond²,

Roux³

et al.

EPiC Series in Computing

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The treatment of the axiomatic theory of floating-point numbers is out of reach of current SMT solvers, especially when it comes to automatic reasoning on approximation errors. In this paper, we describe a dedicated procedure for such a theory, which provides an interface akin to the instantiation mechanism of an SMT solver. This procedure is based on the approach of the Gappa tool: it performs saturation of consequences of the axioms, in order to refine bounds on expressions. In addition to the original approach, bounds are further refined by a constraint solver for linear arithmetic. Combined with the natural support for equalities provided by SMT solvers, our approach improves the treatment of goals coming from deductive verification of numerical programs. We have implemented it in the Alt-Ergo SMT solver.

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