A Deductive Database Approach to Automated Geometry Theorem Proving and Discovering

We report our effort to build a geometry deductive database, which can be used to find the fixpoint for a geometric configuration. The system can find all the properties of the configuration that can be deduced using a fixed set of geometric rules. To control the size of the database, we propose the idea of a structured deductive database. Our experiments show that this technique could reduce the size of the database by one hundred times. We propose the data-based search strategy to improve the efficiency of forward chaining. We also make clear progress in the problems of how to select good geometric rules, how to add auxiliary points, and how to construct numerical diagrams as models automatically. The program is tested with 160 nontrivial geometry configurations. For these geometric configurations, the program not only finds most of their well-known properties but also often gives unexpected results, some of which are possibly new. Also, the proofs generated by the program are generally short and totally geometric.     

A New Approach for Automatic Theorem Proving in Real Geometry

We present a new method for proving geometric theorems in the real plane or higher dimension. The method is derived from elimination set ideas for quantifier elimination in linear and quadratic formulas over the reals. In contrast to other approaches, our method can also prove theorems whose complex analogues fail. Moreover, the problem formulation may involve order inequalities. After specification of independent variables, nondegeneracy conditions are generated automatically. Moreover, when trying to prove conjectures that – apart from nondegeneracy conditions – do not hold in the claimed generality, missing premises are found automatically. We demonstrate the applicability of our method to nontrivial examples.     

Theorem Proving Modulo

Deduction modulo is a way to remove computational arguments from proofs by reasoning modulo a congruence on propositions. Such a technique, issued from automated theorem proving, is of general interest because it permits one to separate computations and deductions in a clean way. The first contribution of this paper is to define a sequent calculus modulo that gives a proof-theoretic account of the combination of computations and deductions. The congruence on propositions is handled through rewrite rules and equational axioms. Rewrite rules apply to terms but also directly to atomic propositions. The second contribution is to give a complete proof search method, called extended narrowing and resolution (ENAR), for theorem proving modulo such congruences. The completeness of this method is proved with respect to provability in sequent calculus modulo. An important application is that higher-order logic can be presented as a theory in deduction modulo. Applying the ENAR method to this presentation of higher-order logic subsumes full higher-order resolution. This revised version was published online in August 2006 with corrections to the Cover Date.     

Analogy in Inductive Theorem Proving

In this paper, we investigate analogy-driven proof plan construction in inductive theorem proving. The intention is to produce a plan for a target theorem that is similar to a given source theorem. We identify second-order mappings from the source to the target that preserve induction-specific proof- relevant abstractions dictating whether the source plan can be replayed. We replay the planning decisions taken in the source if the reasons or justifications for these decisions still hold in the target. If the source and target plan differ significantly at some isolated point, additional reformulations are invoked to add, delete, or modify planning steps. These reformulations are not ad hoc but are triggered by peculiarities of the mappings and by failed justifications. Employing analogy on top of the proof planner CLAM has extended the problem-solving horizon of CLAM: With analogy, some theorems could be proved automatically that neither CLAM nor NQTHM could prove automatically.     

A Problem-Reduction Approach to Proving Simulation Between Programs

System correctness often presents itself as the problem of showing that two programs, the "specification" and the "implementation," are in some sense equivalent. Such a concept of equivalence is supplied by Milner's definition of simulation between programs. This paper presents a problem-reduction approach to proving simulation, and describes an interactive system designed for this purpose.     

机械化定理证明研究综述

随着现代社会计算机化程度的提高,与计算机相关的各种系统故障足以造成巨大的经济损失.机械化定理证明能够建立更为严格的正确性,从而奠定系统的高可信性.针对机械化定理证明的逻辑基础和关键技术,详细剖析了一阶逻辑和基于消解的证明技术、自然演绎和类型化的λ演算、3种编程逻辑、基于高阶逻辑的硬件验证技术、程序构造和求精技术之间的联系和发展变迁,其中,3种编程逻辑包括一阶编程逻辑及变体、Floyd-Hoare逻辑和可计算函数逻辑.然后分析、比较了各类主流证明助手的设计特点,阐述了几个具有代表性的证明助手的开发和实现.接下来对它们在数学、编译器验证、操作系统微内核验证、电路设计验证等领域的应用成果进行了细致的分析.最后,对机械化定理证明进行了总结,并提出面临的挑战和未来研究方向.     

Proving Termination by Dependency Pairs and Inductive Theorem Proving

Current techniques and tools for automated termination analysis of term rewrite systems (TRSs) are already very powerful. However, they fail for algorithms whose termination is essentially due to an inductive argument. Therefore, we show how to couple the dependency pair method for termination of TRSs with inductive theorem proving. As confirmed by the implementation of our new approach in the tool AProVE, now TRS termination techniques are also successful on this important class of algorithms.     

An Improvement of Herbrands Theorem and Its Application to Model Generation Theorem Proving

This paper presents an improvement of Herbrand's theorem.We propose a method for specifying a sub- universe of the Herbrand universe of a clause set S for each argument of predicate symbols and function symbols in S. We prove that a clause set S is unsatisfiable if and only if there is a finite unsatisfiable set of ground instances of clauses of S that are derived by only instantiating each variable,which appears as an argument of predicate symbols or function symbols,in S over its corresponding argument's sub-universe of the Herbrand universe of S.Because such sub-universes are usually smaller(sometimes considerably)than the Herbrand universe of S,the number of ground instances may decrease considerably in many cases.We present an algorithm for automatically deriving the sub-universes for arguments in a given clause set,and show the correctness of our improvement.Moreover,we introduce an application of our approach to model generation theorem proving for non-range-restricted problems,show the range-restriction transformation algorithm based on our improvement and provide examples on benchmark problems to demonstrate the power of our approach.     

Automated Theorem Proving in Euler Diagram Systems

Diagrammatic reasoning has the potential to be important in numerous application areas. This paper focuses on the simple, but widely used, Euler diagrams that form the basis of many more expressive logics. We have implemented a diagrammatic theorem prover, called Edith, which has access to four sound and complete sets of reasoning rules for Euler diagrams. Furthermore, for each rule set we develop a sophisticated heuristic to guide the search for a proof. This paper is about understanding how the choice of reasoning rule set affects the time taken to find proofs. Such an understanding will influence reasoning rule design in other logics. Moreover, this work specific to Euler diagrams directly benefits the many logics based on Euler diagrams. We investigate how the time taken to find a proof depends not only on the proof task but also on the reasoning system used. Our evaluation allows us to predict the best choice of reasoning system, given a proof task, in terms of time taken, and we extract a guide for defining reasoning rules for other logics in order to minimize time requirements.     

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.     

Inductive Theorem Proving for Design Specifications

We present a number of new results on inductive theorem provingfor design specifications based on Horn logic with equality.Induction is explicit here because induction orderings aresupposed to be part of the specification. We show how the automatic support for program verification is enhanced if the specification satisfies a bunch of rewrite properties,summarized under the notion of canonicity. The enhancement isdue to inference rules and corresponding strategies whose soundness is implied by the specification's canonicity. The second main result of the paper provides a method for proving canonicity by using the same rules, which are applied in proofs ofconjectures about the specification and the functional-logic programs it contains.     

Machine Learning for First-Order Theorem Proving

We applied two state-of-the-art machine learning techniques to the problem of selecting a good heuristic in a first-order theorem prover. Our aim was to demonstrate that sufficient information is available from simple feature measurements of a conjecture and axioms to determine a good choice of heuristic, and that the choice process can be automatically learned. Selecting from a set of 5 heuristics, the learned results are better than any single heuristic. The same results are also comparable to the prover’s own heuristic selection method, which has access to 82 heuristics including the 5 used by our method, and which required additional human expertise to guide its design. One version of our system is able to decline proof attempts. This achieves a significant reduction in total time required, while at the same time causing only a moderate reduction in the number of theorems proved. To our knowledge no earlier system has had this capability.     

基于Tableau的定理机器证明系统TableauTAP

使用SWI-PROLOG语言在微机上设计实现了基于tableau的定理证明系统TabIeauTAP。该系统可以证明不含等词的经典逻辑公式童耋譬逻辑公式，通过预处理自动生成tableau规则，因此容易对其功能进行扩展。应用该系统对TPTP的400个逻辑问题进行证明，实验结果表明，TableauTAP在时间和空间效率上都是比较高的。     

基于交互式定理证明的并发程序验证工作综述

并发程序与并发系统可以拥有非常高的执行效率和相对串行系统较快的响应速度,在现实中有着非常广泛的应用。但是并发程序与并发系统往往难以保证其实现的正确性,实际应用程序运行中的错误会带来严重的后果。同时,并发程序执行时的不确定性会给其正确性验证带来巨大的困难。在形式化验证方法中,人们可以通过交互式定理证明器严格地对并发程序进行验证。本文对在交互式定理证明中可用于描述并发程序正确性的验证目标进行总结,它们包括霍尔三元组、可线性化、上下文精化和逻辑原子性。交互式定理证明方法中常用程序逻辑对程序进行验证,本文分析了基于并发分离逻辑、依赖保证逻辑、关系霍尔逻辑等理论研究的系列成果与相应形式化方案,并对使用了这些方法的程序验证工具和程序验证成果进行了总结。     

Theorem Proving Techniques for View Deletion in Databases

In this paper, we show how techniques from first-order theorem proving can be used for efficient deductive database updates. The key idea is to transform the given database, together with the update request, into a (disjunctive) logic program and to apply the hyper-tableau calculus (Baumgartner et al. 1996) to solve the original update problem. The resulting algorithm has the following properties: it works goal-directed (i.e. the search is driven by the update request), it is rational in the sense that it satisfies certain rationality postulates stemming from philosophical works on belief dynamics, and, unlike comparable approaches, it is of polynomial space complexity. To obtain soundness and completeness results, the hyper-tableau calculus is slightly modified for minimal model reasoning. Besides a direct proof we give an alternate proof which gives insights into the relation to previous approaches. As a by-product we thereby derive a soundness and completeness result of hyper-tableaux for computing minimal abductive explanations.     

A New Method for the Boolean Ring Based Theorem Proving

A new method for first-order theorem proving based on the Boolean ring approach is proposed. The method is an extension of Hsiang's N-Strategy in two aspects: (1) When the input polynomials are derived from clauses, our method is reduced to a more restricted (but still complete) version of N-Strategy: Only maximal atoms in an N-rule are considered for generating new inferences. (2) When the input polynomials are derived from non-clausal formulas, no new inference rules are needed in our method for ensuring the completeness. Unlike Kapur and Narendran's method which considers every pair of polynomials for superposition, our method restricts the pairs to those one of which consists of an odd number of monomials. The completeness proof of our method with the integration of reduction is also provided and is done by using the technique of semantic trees. The same technique is used to prove the completeness of N-strategy with reduction (using only N-rules and P-rules) for clausal theorem proving, thus it settles a longtime open problem.     

Reducing Higher-Order Theorem Proving to a Sequence of SAT Problems

We describe a complete theorem proving procedure for higher-order logic that uses SAT-solving to do much of the heavy lifting. The theoretical basis for the procedure is a complete, cut-free, ground refutation calculus that incorporates a restriction on instantiations. The refined nature of the calculus makes it conceivable that one can search in the ground calculus itself, obtaining a complete procedure without resorting to meta-variables and a higher-order lifting lemma. Once one commits to searching in a ground calculus, a natural next step is to consider ground formulas as propositional literals and the rules of the calculus as propositional clauses relating the literals. With this view in mind, we describe a theorem proving procedure that primarily generates relevant formulas along with their corresponding propositional clauses. The procedure terminates when the set of propositional clauses is unsatisfiable. We prove soundness and completeness of the procedure. The procedure has been implemented in a new higher-order theorem prover, Satallax, which makes use of the SAT-solver MiniSat. We also describe the implementation and give several examples. Finally, we include experimental results of Satallax on the higher-order part of the TPTP library.     

Applying Light-Weight Theorem Proving to Debugging and Verifying Pointer Programs

We describe a combination of BDDs and superposition theorem proving, called light-weight theorem proving, and its application to the flexible and efficient automation of the reasoning activity required to debug and verify pointer manipulating programs. This class of programs is notoriously challenging to reason about and it is also interesting from a programming point of view since pointers are an important source of bugs. The implementation of our technique (in a system called haRVey) scales up significantly better than state-of-the-art tools such as E (a superposition prover) and Simplify (a prover based on the Nelson and Oppen combination schema of decision procedures which is used in ESC/Java) on a set of proof obligations arising in debugging and verifying C functions manipulating pointers.