Floating-Point Verification Using Theorem Proving

Harrison, John

doi:10.1007/11757283_8

Cited by 46 publications

(24 citation statements)

References 41 publications

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“…An ultimate way to verify floating-point programs is to give a formal proof of their correctness. To achieve this goal, there exist several formalizations of the floating-point standard in proof assistants [38,26]. Boldo et al [4] formalized a non-trivial floating-point program for solving a wave equation.…”

Section: Related Workmentioning

confidence: 99%

Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions

Solovyev

Jacobsen

Rakamarić

et al. 2015

FM 2015: Formal Methods

123

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Rigorous estimation of maximum floating-point round-off errors is an important capability central to many formal verification tools. Unfortunately, available techniques for this task often provide overestimates. Also, there are no available rigorous approaches that handle transcendental functions. We have developed a new approach called Symbolic Taylor Expansions that avoids this difficulty, and implemented a new tool called FPTaylor embodying this approach. Key to our approach is the use of rigorous global optimization, instead of the more familiar interval arithmetic, affine arithmetic, and/or SMT solvers. In addition to providing far tighter upper bounds of round-off error in a vast majority of cases, FPTaylor also emits analysis certificates in the form of HOL Light proofs. We release FPTaylor along with our benchmarks for evaluation. Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor ExpansionsAlexey Solovyev, Charles Jacobsen, Zvonimir Rakamarić, and Ganesh Gopalakrishnan School of Computing, University of Utah, Salt Lake City, UT 84112, USA {monad,charlesj,zvonimir,ganesh}@cs.utah.edu Abstract. Rigorous estimation of maximum floating-point round-off errors is an important capability central to many formal verification tools. Unfortunately, available techniques for this task often provide overestimates. Also, there are no available rigorous approaches that handle transcendental functions. We have developed a new approach called Symbolic Taylor Expansions that avoids this difficulty, and implemented a new tool called FPTaylor embodying this approach. Key to our approach is the use of rigorous global optimization, instead of the more familiar interval arithmetic, affine arithmetic, and/or SMT solvers. In addition to providing far tighter upper bounds of round-off error in a vast majority of cases, FPTaylor also emits analysis certificates in the form of HOL Light proofs. We release FPTaylor along with our benchmarks for evaluation.

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Section: Related Workmentioning

confidence: 99%

Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions

Solovyev

Jacobsen

Rakamarić

et al. 2015

FM 2015: Formal Methods

123

View full text Add to dashboard Cite

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“…For the purposes of this paper, we include in the definition of formal methods all forms of formal analysis including static code analysis, abstract interpretation, model-checking, and theorem proving. Formal methods have had V&V successes previously in communities such as computer hardware and software security [1] [2]. However, these techniques have made few inroads into the safety-critical software arena.…”

Section: Introductionmentioning

confidence: 99%

Study on the Barriers to the Industrial Adoption of Formal Methods

Davis

Clark

Cofer

et al. 2013

Formal Methods for Industrial Critical Systems

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Abstract.The authors conducted an informal survey of contractors, customers, and certification authorities in the United States aerospace domain to identify barriers to the adoption of formal methods and suggested mitigations for those barriers. We surveyed 31 individuals from the following nine organizations: United States Army, Boeing, FAA, Galois, Honeywell, Lockheed Martin, NASA, Rockwell Collins, and Wind River. The top three barrier categories were education, tools, and the industrial environment (i.e., non-technical barriers with respect to personnel changes, contracts, and schedules). The top three mitigation categories were education, improving tool integration, and creating and disseminating evidence of the benefits of formal analysis. Strategies to accelerate adoption of formal methods include making formal methods a part of the undergraduate software engineering curriculum, hosting courses in formal methods for working engineers, funding the integration of tools, funding improvements to tool interfaces, and promoting/requiring the use of formal methods on future contracts.

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“…A similar approach is taken by [4] which generate verification conditions that are discharged by various theorem provers. Harisson has also done significant work on proving floating-point programs in the HOL Light theorem prover [30]. Our approach makes a different compromise on the precision vs. automation tradeoff, by being less precise, but automatic.…”

Section: Related Workmentioning

confidence: 98%

Sound compilation of reals

Darulová

Kunčak

2014

Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages

114

110

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Writing accurate numerical software is hard because of many sources of unavoidable uncertainties, including finite numerical precision of implementations. We present a programming model where the user writes a program in a real-valued implementation and specification language that explicitly includes different types of uncertainties. We then present a compilation algorithm that generates a finite-precision implementation that is guaranteed to meet the desired precision with respect to real numbers. Our compilation performs a number of verification steps for different candidate precisions. It generates verification conditions that treat all sources of uncertainties in a unified way and encode reasoning about finiteprecision roundoff errors into reasoning about real numbers. Such verification conditions can be used as a standardized format for verifying the precision and the correctness of numerical programs. Due to their non-linear nature, precise reasoning about these verification conditions remains difficult and cannot be handled using state-of-the art SMT solvers alone. We therefore propose a new procedure that combines exact SMT solving over reals with approximate and sound affine and interval arithmetic. We show that this approach overcomes scalability limitations of SMT solvers while providing improved precision over affine and interval arithmetic. Our implementation gives promising results on several numerical models, including dynamical systems, transcendental functions, and controller implementations.

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Floating-Point Verification Using Theorem Proving

Abstract: Abstract. This chapter describes our work on formal verification of floating-point algorithms using the HOL Light theorem prover.

Cited by 46 publications

References 41 publications

Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions

Rigorous Estimation of Floating-Point Round-off Errors with Symbolic Taylor Expansions

Study on the Barriers to the Industrial Adoption of Formal Methods

Sound compilation of reals

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