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MultiProver Verification of FloatingPoint Programs ⋆
"... Abstract. In the context of deductive program verification, supporting floatingpoint computations is tricky. We propose an expressive language to formally specify behavioral properties of such programs. We give a firstorder axiomatization of floatingpoint operations which allows to reduce verifica ..."
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Cited by 28 (5 self)
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Abstract. In the context of deductive program verification, supporting floatingpoint computations is tricky. We propose an expressive language to formally specify behavioral properties of such programs. We give a firstorder axiomatization of floatingpoint operations which allows to reduce verification to checking the validity of logic formulas, in a suitable form for a large class of provers including SMT solvers and interactive proof assistants. Experiments using the FramaC platform for static analysis of C code are presented. 1
Behavioral Properties of FloatingPoint Programs ⋆
"... Abstract. We propose an expressive language to specify formally behavioral properties of programs involving floatingpoint computations. We present a deductive verification technique, which allows to prove formally that a given program meets its specifications, using either SMTclass automatic theor ..."
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Cited by 3 (3 self)
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Abstract. We propose an expressive language to specify formally behavioral properties of programs involving floatingpoint computations. We present a deductive verification technique, which allows to prove formally that a given program meets its specifications, using either SMTclass automatic theorem provers or general interactive proof assistants. Experiments using the FramaC platform for static analysis of C code are presented. 1
Elimination of Square Roots and Divisions by Partial Inlining
"... Computing accurately with real numbers is always a challenge. This is particularly true in critical embedded systems since memory issues do not allow the use of dynamic data structures. This constraint imposes a finite representations of the real numbers, provoking uncertainties and rounding error ..."
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Computing accurately with real numbers is always a challenge. This is particularly true in critical embedded systems since memory issues do not allow the use of dynamic data structures. This constraint imposes a finite representations of the real numbers, provoking uncertainties and rounding errors that might modify the actual behavior of a program from its ideal one. This article presents a solution to this problem with a program transformation that eliminates square roots and divisions in straight line programs without nested function calls. These two operations are the source of infinite sequences of digits in numerical representations, thus, eliminating these operations allows to compute exactly using for example a fixedpoint number representation with a sufficient number of bits. In order to avoid an explosion of the size of the produced code this transformation relies on a particular antiunification to realize a partial inlining of the variable and function definitions. This transformation targeting code for aeronautics certified in PVS, we want to prove the semantics preservation in this proof assistant. Thus we use both an OCaml implementation and the subtyping features of PVS to ensure the correctness of the transformation by defining a proofproducing (certifying) program transformation, providing a specific semantics preservation lemma for every definition in the transformed program.
An Automatable Formal Semantics for IEEE754 FloatingPoint Arithmetic
"... Abstract—Automated reasoning tools often provide little or no support to reason accurately and efficiently about floatingpoint arithmetic. As a consequence, software verification systems that use these tools are unable to reason reliably about programs containing floatingpoint calculations or may ..."
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Abstract—Automated reasoning tools often provide little or no support to reason accurately and efficiently about floatingpoint arithmetic. As a consequence, software verification systems that use these tools are unable to reason reliably about programs containing floatingpoint calculations or may give unsound results. These deficiencies are in stark contrast to the increasing awareness that the improper use of floatingpoint arithmetic in programs can lead to unintuitive and harmful defects in software. To promote coordinated efforts towards building efficient and accurate floatingpoint reasoning engines, this paper presents a formalization of the IEEE754 standard for floatingpoint arithmetic as a theory in manysorted firstorder logic. Benefits include a standardized syntax and unambiguous semantics, allowing tool interoperability and sharing of benchmarks, and providing a basis for automated, formal analysis of programs that process floatingpoint data. I.
ProjectTeam Proval Proof of programs
"... c t i v it y e p o r t 2009 Table of contents ..."
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Rigorous Estimation of FloatingPoint Roundoff Errors with Symbolic Taylor Expansions
, 2015
"... Rigorous estimation of maximum floatingpoint roundoff 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. W ..."
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Rigorous estimation of maximum floatingpoint roundoff 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 roundoff 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.
Certified, Efficient and Sharp Univariate Taylor Models in COQ
, 2013
"... Abstract—We present a formalisation, within the COQ proof assistant, of univariate Taylor models. This formalisation being executable, we get a generic library whose correctness has been formally proved and with which one can effectively compute rigorous and sharp approximations of univariate functi ..."
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Abstract—We present a formalisation, within the COQ proof assistant, of univariate Taylor models. This formalisation being executable, we get a generic library whose correctness has been formally proved and with which one can effectively compute rigorous and sharp approximations of univariate functions composed of usual functions such as 1/x, √ x, e x, sin x among others. In this paper, we present the key parts of the formalisation and we evaluate the quality of our certified library on a set of examples. I. Introduction and Motivations Polynomial approximations are a practical way to represent realvalued functions. In fact, on processors, where the only primitive arithmetic operations are +, −, and ×, they are the only effective way to compute realvalued
Formal Proofs for Nonlinear Optimization
"... We present a formally verified global optimization framework. Given a semialgebraic or transcendental function f and a compact semialgebraic domain K, we use the nonlinear maxplus template approximation algorithm to provide a certified lower bound of f over K. This method allows to bound in a modul ..."
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We present a formally verified global optimization framework. Given a semialgebraic or transcendental function f and a compact semialgebraic domain K, we use the nonlinear maxplus template approximation algorithm to provide a certified lower bound of f over K. This method allows to bound in a modular way some of the constituents of f by suprema of quadratic forms with a well chosen curvature. Thus, we reduce the initial goal to a hierarchy of semialgebraic optimization problems, solved by sums of squares relaxations. Our implementation tool interleaves semialgebraic approximations with sums of squares witnesses to form certificates. It is interfaced with Coq and thus benefits from the trusted arithmetic available inside the proof assistant. This feature is used to produce, from the certificates, both valid underestimators and lower bounds for each approximated constituent. The application range for such a tool is widespread; for instance Hales ’ proof of Kepler’s conjecture yields thousands of multivariate transcendental inequalities. We illustrate the performance of our formal framework on some of these inequalities as well as on examples from the global optimization literature.
To cite this version:
, 2013
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.