在APPLEⅡ机上实现自动定理证明

作者根据锁归结(lock resolution)原理用LISP语言编制了完整的框图和程序,并在APPLE Ⅱ机器上实现了对命题逻辑中的定理的证明。归结原理的证明过程采用了反驳(refutation)的形式,这很像数学中的反证法。为了使一个定理能在计算机上得以证明,首先要把定理的前提和结论都化为逻辑表达式,然后把定     

基于Isabelle定理证明器算法程序的形式化验证

形式化验证对保证软件的正确性和可靠性具有十分重要的意义。定理机械证明是形式化验证的一个重要研究领域,Isabelle系统是一个被广泛运用的定理证明辅助工具。本文在分析Dijkstra最弱前置谓词理论的基础上,根据PAR方法开发的算法程序循环不变式,提出了一种使用Isabelle定理证明器对算法程序进行机械验证的方法。该方法既克服了传统手工验证过程的繁琐性和易错性等缺点,又达到"提高验证效率和保证算法程序高可信"的目标,具有很好的实用价值。     

理论的精确定义

<正> 本章精确地定义我们证明技术的数学理论基础,即我们定理证明器证明定理的理论。这里将要介绍系统引用的公理、定义原理、以及归纳法原理,我们不打算介绍逻辑和等式的常用初等规则,而是假定读者熟悉这些规则。本章把重要的部分都上下加线框起。未框起来的部分就是关于所框素材的启示。我们将按一种“朴素的”集论语言来表述这种启示素材。既可把我们的理论嵌入集论之中,又可从集论中导     

Boyer和Moore获1983年McCarthy奖

两年一度的J.McCarthy奖是为表彰在程序验证领域的杰出工作而设立的。首次奖是根据最近五年在该领域发表的研究工作评定的。美国德克萨斯大学的R.S.Boyer和J S.Moore被评选为首次获奖人。Boyer和Moore从七十年代就合作研究定理证明和程序验证。他们的主要成就是成功地发展了一种可实现高效率的定理证明系统的严谨的逻辑。这种逻辑特别强调了归纳法的使用以表现程序共有的属性。他们的系统是现今最有效的定理证明系统之一。该系统使用启发式来加强其自动定理证明的功能,并允许用户通过人权交换来推导不能由机器完全自动完成的证明。他们还扩充了该系统,使之包括一个FORTRAN验证条件产生程序,使该系统可以处理FORTRAN这     

机器定理证明的反向归约方法

基于代数和递归函数理论，本文定义了代数递归谓词．代数递归谓词是一类广泛的谓词．基于数学归纳法，作者给出了证明代数递归谓词永真性的反向归约方法及相应的算法Ｒｅｄｕｃｔｉｏｎ．由于采用反向归约方式来完成定理证明，从根本上消除了正向组合式定理证明所产生的组合爆炸，因而极大地提高了定理证明的效率．     

具有三种否定的模糊命题逻辑形式系统FLCOM的λ-归结

考虑到模糊逻辑中定理自动证明的重要性以及目前主要研究具有一种否定的模糊逻辑的归结原理,文中对具有三种否定(矛盾否定、对立否定和中介否定)的模糊命题逻辑(FLCOM)的归结原理进行研究.基于FLCOM的一种无穷值语义解释提出λ-可满足的和λ-不可满足的概念.将λ-归结方法引入FLCOM,给出FLCOM的λ-归结演绎定义,讨论FLCOM的λ-归结原理,并证明FLCOM的λ-归结方法的完备性.基于λ-归结方法和已证明的结论给出实例以佐证文中λ-归结方法和结论的正确性和可行性.因此,在FLCOM范围内可判定任一模糊命题公式是否是λ-可满足的或λ-不可满足的.     

多重序集在间歇断言方法验证循环程序中的应用

本文旨在介绍建立在良基集上的多重集合和多重序集。引入这些概念后,可较为简单和直观地取得循环程序的终结函数,这不仅便于循环程序的终结性证明,而且可将结构归纳法建于其上,方便使用间歇断言法验证循环程序的完全正确性。     

基于半扩展规则的定理证明方法

自动定理证明一直是人工智能领域中最重要的问题之一,基于归结的方法是通过推出空子句的方法来判定子句集的可满足性.基于扩展规则的定理证明方法在一定意义上是和归结原理对偶的方法,是通过子句集能否推导出所有极大项组成的子句集来判定可满足性.通过对扩展规则的研究给出了半扩展规则的概念,并提出了基于半扩展规则的定理证明算法SER.然后分析及证明了该算法的正确性、完备性和复杂性.实验结果表明,算法SER的执行效率较基于归结的有向归结算法DR和基于扩展规则算法IER,NER有明显的提高.     

子目标归纳法及其应用示例

子目标归纳法是继各种归纳方法之后出现的又一种验证程序的方法。本文从递归和循环两个方面描述这种方法,给出了作者实践过的用子目标归纳法证明程序正确性的例题,揭示了子目标归纳法与归纳断言法的关系。     

广义归结中的定理的表示

本文为广义归结原理提供了一种方法,该方法能将要证明的定理写成一阶逻辑中较为自然的Skolem标准形式。     

Eliminating duplication with the hyper-linking strategy

The efficiency of almost all theorem proving methods suffers from a phenomenon called duplication of instances of clauses. In this paper, we present a novel technique, called the hyper-linking strategy, to eliminate such duplication. This strategy is complete for the full first-order predicate calculus. We show the effectiveness of this strategy by comparing it with other proving methods. We give empirical evidence that both the Davis-Putnam procedure and the hyper-linking strategy are comparable to each other and better than other common theorem proving strategies on propositional calculus problems. The fact that the Davis-Putnam procedure is faster than resolution and other common methods on propositional problems seems not to be appreciated by a large segment of the theorem proving community. Also, we give empirical evidence that the hyper-linking strategy is better than other common theorem proving methods on near-propositional problems like logic puzzles. We attempt to explain the superior behavior of the hyper-linking strategy and the Davis-Putnam procedure by examining the kinds of duplication that can occur during the search with the different methods. In addition, we show the completeness of the hyper-linking strategy combined with several support strategies.This research was partially supported by NSF under grant CCR-8802282.     

Theorem Proving Based on the Extension Rule

Methods based on resolution have been widely used for theorem proving since it was proposed. This paper presents a new method for theorem proving that uses the inverse of resolution and employs the inclusion-exclusion principle to circumvent the problem of space complexity. We prove its soundness and completeness. The concept of complementary factor is introduced to estimate its complexity. We also show that our method outperforms resolution-based methods in some cases. Thus it is potentially a complementary method to resolution-based methods. This revised version was published online in August 2006 with corrections to the Cover Date.     

使用显式策略进行程序构造

本文提出一种交互式的程序(半)自动综合方法:使用显式策略引导系统进行程序构造。这些策略包括程序员提供的问题求解策略和系统内部的标准策略,如数学归纳法、归结方法和程序变换规则/策略。策略都用高阶函数式元语言TSL/ML统一地描述,它们的施用则通过高阶一致化完成。因此,我们的程序综合方法可以在统一的框架下使用多种软件自动构造技术,且易于自动实现。     

Splitting and reduction heuristics in automatic theorem proving

A theorem proving program has been written in LISP which attempts to speed up automatic theorem proving by the use of heuristics. Some of these heuristics are of a general nature, applicable to the proof of any theorem in mathematics, while others are designed for set theory. Their effect is to break the theorem into parts which are easier to prove. The proof of these parts is then attempted by resolution. Resolution, when used, is relegated to the job it does best, proving relatively easy assertions.     

Automated theorem proving by resolution in non-classical logics

This paper is an overview of a variety of results, all centered around a common theme, namely embedding of non-classical logics into first order logic and resolution theorem proving. We present several classes of non-classical logics, many of which are of great practical relevance in knowledge representation, which can be translated into tractable and relatively simple fragments of classical logic. In this context, we show that refinements of resolution can often be used successfully for automated theorem proving, and in many interesting cases yield optimal decision procedures.     

Term rewriting and beyond — theorem proving in Isabelle

The subject of this paper is theorem proving based on rewriting and induction. Both principles are implemented as tactics within the generic theorem prover Isabelle. Isabelle's higher-order features enable us to go beyond first-order rewriting and express rewriting with conditionals, induction schemata, higher-order functions and program transformers. Applications include the verification and transformation of functional versions of insertion sort and quicksort.     

T-Resolution: Refinements and Model Elimination

T-resolution is a binary rule, proposed by Policriti and Schwartz in 1995 for theorem proving in first-order theories (T-theorem proving) that can be seen – at least at the ground level – as a variant of Stickel's theory resolution. In this paper we consider refinements of this rule as well as the model elimination variant of it. After a general discussion concerning our viewpoint on theorem proving in first-order theories and a brief comparison with theory resolution, the power and generality of T-resolution are emphasized by introducing suitable linear and ordered refinements, uniformly and in strict analogy with the standard resolution approach. Then a model elimination variant of T-resolution is introduced and proved to be sound and complete; some experimental results are also reported. In the last part of the paper we present two applications of T-resolution: to constraint logic programming and to modal logic.     

一种新的基于扩展规则的定理证明算法

基于扩展规则的定理证明方法是一种与归结方法互补的新的定理证明方法,首先通过对扩展规则的深入研究,给出了扩展规则的一个重要性质,设计并实现了该性质的判定算法.此外,从理论上分析及证明了该判定算法的时问和空间复杂性.基于此,提出了一种新的基于扩展规则的定理证明算法NER,将判定子句集可满足性问题转化为一系列文字集合的包含问题,而非计数问题.实验结果表明,算法NER的执行效率较原有扩展规则算法IER和基于归结的有向归结算法DR有明显提高,有些问题可以提高两个数量级.     

基于MCN和MO启发式策略的扩展规则知识编译方法

在基于扩展规则的知识编译算法的基础上提出了2种启发式策略:MCN策略和MO策略.MCN策略和MO策略利用子句集的信息分别选择相应子句和变量,减少扩展规则的使用次数,进而降低知识编译后目标子句集的规模.在此基础上,设计并实现了MCN_KCER,MO_KCER和MCN_MO_KCER算法.实验结果表明:2种启发式策略都可以大幅度减小编译后的子句集规模,同时使用它们的效果更为明显,经过编译后得到的子句集规模是原算法的1/3～1/39,从而大幅度提高之后的在线推理阶段的效率.     

论有穷状态验证方法的局限性

程序有穷状态验证方法是介于程序验证和程序测试之间的一种方法，一方面它如同程序验证一样可以证明某程序具有某些要求的性质，或找出反例证明该程序不具有所要求的性质。另一方面它又不像程序验证那样复杂，要求验证人员具有较高的形式化推理的专业理论和数学水平。但是，现有的有穷状态验证方法有很大的局限性，它要求所论证的性质是有穷自动机所接受的事件序列的集合，或等价地说该性质能表示成为正则表达式。众所周知，有穷自动机所能接受的语言类，按Chomsky字的集合的分类是很小的类。本文讨论了这种局限性，井尝试突破只能使用有穷自动机的限制，提出了一种新的验证方法——有穷路径验证法。在这种方法中，所论证的性质表示可以推广到使用任何一类自动机。作为代价，描写系统的模型限制是无环的。对于有环的描写系统的模型，本文提出了一种称之为“有穷路径测试”的方法。同一般的程序测试一样，用这种方法通过测试不能正面地验证程序的正确性，可是如果通不过测试，则能帮你发现反例，找出程序的错误。与一般的程序测试不同的是这里的测试是相对于模型的路径，而不执行实际的程序。