Computation in Real Closed Infinitesimal and Transcendental Extensions of the Rationals

Moura, Leonardo de; Passmore, Grant Olney

doi:10.1007/978-3-642-38574-2_12

Cited by 24 publications

(16 citation statements)

References 24 publications

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“…Sturm sequences and several speed enhancements such as the use of pseudo division are implemented in the SMT solver Z3 [31]. That implementation was the inspiration for the work presented in this paper.…”

Section: Related Workmentioning

confidence: 99%

“…Such bounds are commonly referred to as Cauchy bounds, and can help reduce any unbounded interval to a bounded interval on which the polynomial is nonnegative if and only if it is nonnegative on the original unbounded interval. While there are many possible definitions of such bounds, the formalization presented in this paper uses Knuth's bound as in [31]. The reader is referred to the development reals in the NASA PVS library, where the precise definition of the function Knuth poly root strict bound, which specifies Knuth's bound.…”

Section: Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Formally-Verified Decision Procedures for Univariate Polynomial Computation Based on Sturm’s and Tarski’s Theorems

2015

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Sturm's theorem is a well-known result in real algebraic geometry that provides a function that computes the number of roots of a univariate polynomial in a semi-open interval, not counting multiplicity. A generalization of Sturm's theorem is known as Tarski's theorem, which provides a linear relationship between functions known as Tarski queries and cardinalities of certain sets. The linear system that results from this relationship is in fact invertible and can be used to explicitly count the number of roots of a univariate polynomial on a set defined by a system of polynomial relations. This paper presents a formalization of these results in the PVS theorem prover, including formal proofs of Sturm's and Tarski's theorems. These theorems are at the basis of two decision procedures, which are implemented as computable functions in PVS. The first, based on Sturm's theorem, determines satisfiability of a single polynomial relation over an interval. The second, based on Tarski's theorem, determines the satisfiability of a system of polynomial relations over the real line. The soundness and completeness properties of these decision procedures are formally verified in PVS. The procedures and their correctness properties enable the implementation of PVS strategies for automatically proving existential and universal statements on polynomial systems. Since the decision procedures are formally verified in PVS, the soundness of the strategies depends solely on the internal logic of PVS rather than on an external oracle.

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Section: Related Workmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

Formally-Verified Decision Procedures for Univariate Polynomial Computation Based on Sturm’s and Tarski’s Theorems

2015

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“…While some work has been done in rigorous global optimization that formally verifies nonlinear functions including semialgebraic and transcendental functions (Domes, 2009;Domes and Neumaier, 2014), the most commonly-used solver software generates relaxations and cutting planes via floating point arithmetic and then uses floating point-based LP and NLP solvers for finding underestimators and heuristic solutions. Numerical instability can be at least partially mitigated using validated interval arithmetic (Brönnimann et al, 2003(Brönnimann et al, , 2006de Moura and Passmore, 2013) for FBBT but, especially for badlyscaled optimization problems, combining diverent solving strategies may induce numerical trouble because of the variance in tolerances between solvers (including clash between different sub-solvers of a single meta-solver).…”

Section: Numerical Issuesmentioning

confidence: 99%

Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO

Boukouvala

Misener

Floudas

2016

European Journal of Operational Research

199

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This manuscript reviews recent advances in deterministic global optimization for Mixed-Integer Nonlinear Programming (MINLP), as well as Constrained DerivativeFree Optimization (CDFO). This work provides a comprehensive and detailed literature review in terms of significant theoretical contributions, algorithmic developments, software implementations and applications for both MINLP and CDFO. Both research areas have experienced rapid growth, with a common aim to solve a wide range of real-world problems. We show their individual prerequisites, formulations and applicability, but also point out possible points of interaction in problems which contain hybrid characteristics. Finally, an inclusive and complete test suite is provided for both MINLP and CDFO algorithms, which is useful for future benchmarking.

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“…In order to address the ever-present concern that numerical errors can result in incorrect answers in these methods, it has been proposed to relax constraints and consider δ-complete decision procedures [17,18]. As opposed to numerically-driven approaches, recently symbolic CAD-based techniques have been successfully integrated in a model-constructing DPLL(T )-style procedure [19,20], and several libraries and toolboxes have been made publicly available for the development of symbolically-driven solvers [21,22].…”

Section: Introductionmentioning

confidence: 99%

Minimal-Model-Guided Approaches to Solving Polynomial Constraints and Extensions

Larraz

Oliveras

Rodríguez-Carbonell

et al. 2014

Lecture Notes in Computer Science

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Abstract. In this paper we present new methods for deciding the satisfiability of formulas involving integer polynomial constraints. In previous work we proposed to solve SMT(NIA) problems by reducing them to SMT(LIA): non-linear monomials are linearized by abstracting them with fresh variables and by performing case splitting on integer variables with finite domain. When variables do not have finite domains, artificial ones can be introduced by imposing a lower and an upper bound, and made iteratively larger until a solution is found (or the procedure times out). For the approach to be practical, unsatisfiable cores are used to guide which domains have to be relaxed (i.e., enlarged) from one iteration to the following one. However, it is not clear then how large they have to be made, which is critical. Here we propose to guide the domain relaxation step by analyzing minimal models produced by the SMT(LIA) solver. Namely, we consider two different cost functions: the number of violated artificial domain bounds, and the distance with respect to the artificial domains. We compare these approaches with other techniques on benchmarks coming from constraint-based program analysis and show the potential of the method. Finally, we describe how one of these minimalmodel-guided techniques can be smoothly adapted to deal with the extension Max-SMT of SMT(NIA) and then applied to program termination proving.

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Computation in Real Closed Infinitesimal and Transcendental Extensions of the Rationals

Cited by 24 publications

References 24 publications

Formally-Verified Decision Procedures for Univariate Polynomial Computation Based on Sturm’s and Tarski’s Theorems

Formally-Verified Decision Procedures for Univariate Polynomial Computation Based on Sturm’s and Tarski’s Theorems

Global optimization advances in Mixed-Integer Nonlinear Programming, MINLP, and Constrained Derivative-Free Optimization, CDFO

Minimal-Model-Guided Approaches to Solving Polynomial Constraints and Extensions

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