Abstract

The aim of the Platform-Independent Approach to Formal Specification and Verification of Standard Mathematical Functions project is the development of an incremental combined approach to specification and verification of standard mathematical functions like sqrt, cos, sin, etc. The term “platform-independence” means that we attempt to design a relatively simple axiomatization of computer arithmetic in terms of real arithmetic (i.e., the arithmetic of the field ℝ of real numbers), but do not specify either the base of the computer arithmetic or the format of the representation of numbers. The incrementality means that we start with the most straightforward specification of the simplest algorithm in real numbers and finish with a realistic specification and a verification of the algorithm in computer arithmetic. We call our approach combined because we start with consideration of a “basic” case, the manual (pen-and-paper) verification of the algorithm in real numbers, then use this verification as proof-outlines for the manual verification of the algorithm in computer arithmetic, and finish with a computer-aided validation of the manual proofs with a proof-assistant system to avoid appeals to “obviousness” that are common in human-carried proofs. In the paper, we apply our platform-independent incremental combined approach to specification and verification of the standard mathematical square root function. By now, the computer-aided validation of the developed algorithms has been carried out only partially to prove, using the ACL2 system, the correctness (consistency) of our fixed-point arithmetic and the existence of a look-up table with the initial approximations of the square roots for fixed-point numbers.

中文翻译：

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独立于平台的规范和标准数学平方根函数的验证

摘要

平台无关的形式规范和标准数学函数验证的项目的目的是开发一种增量组合方法来规范和验证标准数学函数（例如sqrt，cos，sin等）。术语“平台无关”意味着我们试图根据实数算法（即实数字段the的算术）设计相对简单的计算机算术公理化，但没有指定计算机算术的基础或形式的表示形式数字。增量性意味着我们从最简单的算法的实数最简单的说明开始，到最后以现实的说明和计算机算法对算法的验证结束。之所以将我们的方法称为“组合方法”，是因为我们从考虑“基本”情况开始，即以实数对算法进行手动（笔纸验证），然后将此验证用作对算法进行手动验证的证明大纲。计算机算术，并通过带有证明辅助系统的手动证明的计算机辅助验证来完成，以避免对人类携带的证明中常见的“显而易见性”产生吸引力。在本文中，我们将独立于平台的增量组合方法应用于标准数学平方根函数的规范和验证。到目前为止，对开发的算法的计算机辅助验证仅通过ACL2系统进行了部分验证，