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Verifying the accuracy of polynomial approximations in HOL
 Theorem Proving in Higher Order Logics: 10th International Conference, TPHOLs’97
, 1997
"... . Many modern algorithms for the transcendental functions rely on a large table of precomputed values together with a loworder polynomial to interpolate between them. In verifying such an algorithm, one is faced with the problem of bounding the error in this polynomial approximation. The most s ..."
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. Many modern algorithms for the transcendental functions rely on a large table of precomputed values together with a loworder polynomial to interpolate between them. In verifying such an algorithm, one is faced with the problem of bounding the error in this polynomial approximation. The most straightforward methods are based on numerical approximations, and are not prima facie reducible to a formal HOL proof. We discuss a technique for proving such results formally in HOL, via the formalization of a number of results in polynomial theory, e.g. squarefree decomposition and Sturm's theorem, and the use of a computer algebra system to compute results that are then checked in HOL. We demonstrate our method by tackling an example from the literature. 1 Introduction Many algorithms for the transcendental functions such as exp, sin and ln in floating point arithmetic are based on table lookup. Suppose that a transcendental function f(x) is to be calculated. Values of f(a i ) are...
Assisted verification of elementary functions
, 2005
"... The implementation of a correctly rounded or interval elementary function needs to be proven carefully in the very last details. The proof requires a tight bound on the overall error of the implementation with respect to the mathematical function. Such work is function specific, concerns tens of lin ..."
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The implementation of a correctly rounded or interval elementary function needs to be proven carefully in the very last details. The proof requires a tight bound on the overall error of the implementation with respect to the mathematical function. Such work is function specific, concerns tens of lines of code for each function, and will usually be broken by the smallest change to the code (e.g. for maintenance or optimization purpose). Therefore, it is very tedious and errorprone if done by hand. This article discusses the use of the Gappa proof assistant in this context. Gappa has two main advantages over previous approaches: Its input format is very close to the actual C code to validate, and it automates error evaluation and propagation using interval arithmetic. Besides, it can be used to incrementally prove complex mathematical properties pertaining to the C code. Yet it does not require any specific knowledge about automatic theorem proving, and thus is accessible to a wider community. Moreover, Gappa may generate a formal proof of the results that can be checked independently by a lowerlevel proof assistant like Coq, hence providing an even higher confidence in the certification of the numerical code.
A Meta Linear Logical Framework
 In 4th International Workshop on Logical Frameworks and MetaLanguages (LFM’04
, 2003
"... Over the years, logical framework research has produced various type theories designed primarily for the representation of deductive systems. Reasoning about these representations requires expressive special purpose meta logics, that are in general not part of the logical framework. ..."
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Over the years, logical framework research has produced various type theories designed primarily for the representation of deductive systems. Reasoning about these representations requires expressive special purpose meta logics, that are in general not part of the logical framework.
Software techniques for perfect elementary functions in floatingpoint interval arithmetic
 IN REAL NUMBERS AND COMPUTERS
, 2006
"... ..."
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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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
Formal Reasoning about Expectation Properties for Continuous Random Variables
"... Abstract. Expectation (average) properties of continuous random variables are widely used to judge performance characteristics in engineering and physical sciences. This paper presents an infrastructure that can be used to formally reason about expectation properties of most of the continuous random ..."
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Abstract. Expectation (average) properties of continuous random variables are widely used to judge performance characteristics in engineering and physical sciences. This paper presents an infrastructure that can be used to formally reason about expectation properties of most of the continuous random variables in a theorem prover. Starting from the relatively complex higherorderlogic definition of expectation, based on Lebesgue integration, we formally verify key expectation properties that allow us to reason about expectation of a continuous random variable in terms of simple arithmetic operations. In order to illustrate the practical effectiveness and utilization of our approach, we also present the formal verification of expectation properties of the commonly used continuous random variables: Uniform, Triangular and Exponential. 1
Provably faithful evaluation of polynomials
 In Proceedings of the 21st Annual ACM Symposium on Applied Computing
, 2006
"... We provide sufficient conditions that formally guarantee that the floatingpoint computation of a polynomial evaluation is faithful. To this end, we develop a formalization of floatingpoint numbers and rounding modes in the Program Verification System (PVS). Our work is based on a wellknown formali ..."
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We provide sufficient conditions that formally guarantee that the floatingpoint computation of a polynomial evaluation is faithful. To this end, we develop a formalization of floatingpoint numbers and rounding modes in the Program Verification System (PVS). Our work is based on a wellknown formalization of floatingpoint arithmetic in the proof assistant Coq, where polynomial evaluation has been already studied. However, thanks to the powerful proof automation provided by PVS, the sufficient conditions proposed in our work are more general than the original ones.
A Formal Model of IEEE Floating Point Arithmetic
, 2013
"... This development provides a formal model of IEEE754 floatingpoint arithmetic. This formalization, including formal specification of the standard and proofs of important properties of floatingpoint arithmetic, forms the foundation for verifying programs with floatingpoint computation. There is als ..."
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This development provides a formal model of IEEE754 floatingpoint arithmetic. This formalization, including formal specification of the standard and proofs of important properties of floatingpoint arithmetic, forms the foundation for verifying programs with floatingpoint computation. There is also a code generation setup for floats so that we can execute programs using this formalization in functional programming languages. The definitions of the IEEE standard in Isabelle is ported from HOL Light [1].
Formal Analysis and Verification of an OFDM Modem Design using HOL
"... Abstract — In this paper we formally specify and verify an implementation of the IEEE802.11a standard physical layer based OFDM (Orthogonal Frequency Division Multiplexing) modem using the HOL (Higher Order Logic) theorem prover. The versatile expressive power of HOL helped model the original design ..."
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Abstract — In this paper we formally specify and verify an implementation of the IEEE802.11a standard physical layer based OFDM (Orthogonal Frequency Division Multiplexing) modem using the HOL (Higher Order Logic) theorem prover. The versatile expressive power of HOL helped model the original design at all abstraction levels starting from a floatingpoint model to the fixedpoint design and then synthesized and implemented in FPGA technology. The paper also investigates the rounding error accumulated during ideal real to floatingpoint and fixedpoint transitions at the algorithmic level. I.