An Algebraic Approach for the Unsatisfiability of Nonlinear Constraints

Tiwari, Ashish

doi:10.1007/11538363_18

Cited by 23 publications

(14 citation statements)

References 23 publications

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“…Tiwari [30] presents an approach using Gröbner bases and sign conditions on variables to produce unsatisfiability witnesses for nonlinear constraints. The approach depends on appropriate heuristic variable orderings that are formed by successively introducing new variables for polynomial expressions following certain heuristics (which may not terminate).…”

Section: Related Workmentioning

confidence: 99%

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Real World Verification

Platzer

Quesel

Rümmer

2009

Lecture Notes in Computer Science

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Abstract. Scalable handling of real arithmetic is a crucial part of the verification of hybrid systems, mathematical algorithms, and mixed analog/digital circuits. Despite substantial advances in verification technology, complexity issues with classical decision procedures are still a major obstacle for formal verification of real-world applications, e.g., in automotive and avionic industries. To identify strengths and weaknesses, we examine state of the art symbolic techniques and implementations for the universal fragment of real-closed fields: approaches based on quantifier elimination, Gröbner Bases, and semidefinite programming for the Positivstellensatz. Within a uniform context of the verification tool KeYmaera, we compare these approaches qualitatively and quantitatively on verification benchmarks from hybrid systems, textbook algorithms, and on geometric problems. Finally, we introduce a new decision procedure combining Gröbner Bases and semidefinite programming for the real Nullstellensatz that outperforms the individual approaches on an interesting set of problems.

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Section: Related Workmentioning

confidence: 99%

“…Our work and that of Tiwari share the combination of Gröbner bases with witness generation. Yet we follow semidefinite programming for the real Nullstellensatz, whereas [30] uses heuristic generation of polynomial witness expressions. Tiwari uses the Positivstellensatz to prove refutational completeness but not as part of his technique.…”

Section: Related Workmentioning

confidence: 99%

Real World Verification

Platzer

Quesel

Rümmer

2009

Lecture Notes in Computer Science

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“…Tarski [1948] gave a decision procedure for the first-order theory of real-closed fields which has been improved by Cohen [1969] and Hörmander [1983], and by Collins [1975]. Tiwari [2005] has developed a semi-decision procedure for the universal fragment of real closed fields combining the simplex algorithm and Gröbner basis computations. Parrilo [2003] gives an alternative approach.…”

Section: Theory Solversmentioning

confidence: 99%

Automated deduction for verification

Shankar

2009

ACM Comput. Surv.

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Automated deduction uses computation to perform symbolic logical reasoning. It has been a core technology for program verification from the very beginning. Satisfiability solvers for propositional and first-order logic significantly automate the task of deductive program verification. We introduce some of the basic deduction techniques used in software and hardware verification and outline the theoretical and engineering issues in building deductive verification tools. Beyond verification, deduction techniques can also be used to support a variety of applications including planning, program optimization, and program synthesis.

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“…In these cases, the properties of RCF exploited by the CAD procedure have been generalized into the notion of "cellularly decomposable structures" and now bear rich mathematical fruits. [20] program analysis [19], hardware verification [11], hybrid systems [18], and even ongoing large-scale projects in formalised mathematics [10]), simply deciding the satisfiability of boolean combinations of polynomial equations and inequalities over the real numbers is often sufficient. This problem is equivalent to QE for the purely ∃ (dually purely ∀) sentential fragment of the elementary theory of RCF, in which all formulas considered are sentences consisting only of a single block of non-alternating quantifiers.…”

Section: Introductionmentioning

confidence: 99%

Combined Decision Techniques for the Existential Theory of the Reals

Passmore

Jackson

2009

Lecture Notes in Computer Science

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Abstract. Methods for deciding quantifier-free non-linear arithmetical conjectures over R are crucial in the formal verification of many realworld systems and in formalised mathematics. While non-linear (rational function) arithmetic over R is decidable, it is fundamentally infeasible: any general decision method for this problem is worst-case exponential in the dimension (number of variables) of the formula being analysed. This is unfortunate, as many practical applications of real algebraic decision methods require reasoning about high-dimensional conjectures. Despite their inherent infeasibility, a number of different decision methods have been developed, most of which have "sweet spots" -e.g., types of problems for which they perform much better than they do in general. Such "sweet spots" can in many cases be heuristically combined to solve problems that are out of reach of the individual decision methods when used in isolation. RAHD ("Real Algebra in High Dimensions") is a theorem prover that works to combine a collection of real algebraic decision methods in ways that exploit their respective "sweet-spots." We discuss high-level mathematical and design aspects of RAHD and illustrate its use on a number of examples.

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An Algebraic Approach for the Unsatisfiability of Nonlinear Constraints

Cited by 23 publications

References 23 publications

Real World Verification

Real World Verification

Automated deduction for verification

Combined Decision Techniques for the Existential Theory of the Reals

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