Experimenting on Solving Nonlinear Integer Arithmetic with Incremental Linearization

Cimatti, Alessandro; Griggio, Alberto; Irfan, Ahmed; Roveri, Marco; Sebastiani, Roberto

doi:10.1007/978-3-319-94144-8_23

Cited by 16 publications

(10 citation statements)

References 15 publications

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“…Z3 and CVC4 use various strategies and techniques for quantifier instantiation including E-matching [18], and enumerative [24] and conflictbased [27] instantiation. For non-linear integer arithmetic, CVC4 uses an approach based on incremental linearization [7,6,26]. Vampire is a superposition-based theorem prover for first-order logic based on the AVATAR framework [31], which has been extended also to support some theories including integer arithmetic [23].…”

Section: Case Studiesmentioning

confidence: 99%

Towards Bit-Width-Independent Proofs in SMT Solvers

Niemetz

Preiner

Reynolds

et al. 2019

Lecture Notes in Computer Science

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Many SMT solvers implement efficient SAT-based procedures for solving fixed-size bit-vector formulas. These approaches, however, cannot be used directly to reason about bit-vectors of symbolic bit-width. To address this shortcoming, we propose a translation from bit-vector formulas with parametric bit-width to formulas in a logic supported by SMT solvers that includes non-linear integer arithmetic, uninterpreted functions, and universal quantification. While this logic is undecidable, this approach can still solve many formulas by capitalizing on advances in SMT solving for non-linear arithmetic and universally quantified formulas. We provide several case studies in which we have applied this approach with promising results, including the bit-width independent verification of invertibility conditions, compiler optimizations, and bit-vector rewrites. Symbol SMT-LIB Syntax

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Section: Case Studiesmentioning

confidence: 99%

Towards Bit-Width-Independent Proofs in SMT Solvers

Niemetz

Preiner

Reynolds

et al. 2019

Lecture Notes in Computer Science

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“…Given a polynomial template B(c, x) for B(x), where c is the coefficients of the monomials to be decided in B(c, x), we substitute B(c, x) for B(x) occurring in the conditions (5)- (7) to obtain three polynomial identities in R[x] with linear polynomials in R[c, λ] as their coefficients, where λ is a vector composed of all the λ α , λ β , λ γ occurring in (5)- (7). Since (5)- (7) are identities, then all the coefficients of the corresponding monomials on both sides of the identities must be identical. By collecting the corresponding coefficients of the monomials on both sides of the identities and let them equal respectively, we obtain a system S of linear equations and inequalities on c, λ.…”

Section: Robust Barrier Certificate By Linear Programmingmentioning

confidence: 99%

“…This theorem provides us with a solution to solve barrier certificate by linear programming. Given a polynomial template B(c, x) for B(x), where c is the coefficients of the monomials to be decided in B(c, x), we substitute B(c, x) for B(x) occurring in the conditions ( 5)-( 7) to obtain three polynomial identities in R[x] with linear polynomials in R[c, λ] as their coefficients, where λ is a vector composed of all the λ α , λ β , λ γ occurring in ( 5)- (7). Since ( 5)-( 7) are identities, then all the coefficients of the corresponding monomials on both sides of the identities must be identical.…”

Section: Robust Barrier Certificate By Linear Programmingmentioning

confidence: 99%

“…Only recently, tools such as CLRT [9,10], Flow* [6], MathSAT SMT solver [8,7], HySAT/iSAT [12], dReach [25], C2E2 [11] and CORA [1], have made some progresses in verifying nonlinear continuous and hybrid models. Some of these tools [7,25,12] are based on decision procedures that overcome the theoretical limits in nonlinear theories over the reals. The main idea is to encode the reachability problem for nonlinear systems as first-order logic formulas over the real numbers.…”

Section: Introductionmentioning

confidence: 99%

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Piecewise Robust Barrier Tubes for Nonlinear Hybrid Systems with Uncertainty

Kong

Bartocci

Jiang

et al. 2019

Lecture Notes in Computer Science

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Piecewise Barrier Tubes (PBT) is a new technique for flowpipe overapproximation for nonlinear systems with polynomial dynamics, which leverages a combination of barrier certificates. PBT has advantages over traditional time-step based methods in dealing with those nonlinear dynamical systems in which there is a large difference in speed between trajectories, producing an overapproximation that is time independent. However, the existing approach for PBT is not efficient due to the application of interval methods for enclosure-box computation, and it can only deal with continuous dynamical systems without uncertainty. In this paper, we extend the approach with the ability to handle both continuous and hybrid dynamical systems with uncertainty that can reside in parameters and/or noise. We also improve the efficiency of the method significantly, by avoiding the use of interval-based methods for the enclosure-box computation without loosing soundness. We have developed a C++ prototype implementing the proposed approach and we evaluate it on several benchmarks. The experiments show that our approach is more efficient and precise than other methods in the literature. arXiv:1907.11514v1 [eess.SY]

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“…The basic algorithm of the tool relies on a translation from bit-vectors to non-linear integer arithmetic with uninterpreted functions, followed by a CEGAR loop [7] that lazily instantiates bit-vector axioms over the translation. The idea of using integer reasoning for bit-vector solving is not new (see, e.g., [1,4]), however, it is worth revisiting due to recent improvements in solvers for non-linear integer arithmetic [5,8,10]. We expect this solver to perform better on benchmarks that involve arithmetic bit-vector constraints and large bit-widths because the encoding of arithmetic constraints is straightforward and independent of bit-width, as opposed to the encoding of bit-wise constraints which is less natural and hindered by larger bit-widths.…”

mentioning

confidence: 99%

lazybvtoint at the SMT Competition 2020

Zohar,

Irfan,

Mann

et al. 2021

Preprint

Self Cite

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lazybv2int is a new prototype SMT-solver, that will participate in the incremental and non-incremental tracks of the QF BV logic.Overview. lazybv2int is a prototype SMT-solver for the theory of fixed-width bit-vectors and uninterpreted functions. This is the first year it will compete in SMT-COMP. It will participate in the incremental and non-incremental QF BV tracks. The basic algorithm of the tool relies on a translation from bit-vectors to non-linear integer arithmetic with uninterpreted functions, followed by a CEGAR loop [7] that lazily instantiates bit-vector axioms over the translation. The idea of using integer reasoning for bit-vector solving is not new (see, e.g., [1,4]), however, it is worth revisiting due to recent improvements in solvers for non-linear integer arithmetic [5,8,10]. We expect this solver to perform better on benchmarks that involve arithmetic bit-vector constraints and large bit-widths because the encoding of arithmetic constraints is straightforward and independent of bit-width, as opposed to the encoding of bit-wise constraints which is less natural and hindered by larger bit-widths. The tool is open-source and is available at https://github.com/yoni206/lazybv2int.

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Experimenting on Solving Nonlinear Integer Arithmetic with Incremental Linearization

Cited by 16 publications

References 15 publications

Towards Bit-Width-Independent Proofs in SMT Solvers

Towards Bit-Width-Independent Proofs in SMT Solvers

Piecewise Robust Barrier Tubes for Nonlinear Hybrid Systems with Uncertainty

lazybvtoint at the SMT Competition 2020

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