Strategic Issues, Problems and Challenges in Inductive Theorem Proving

(Automated) Inductive Theorem Proving (ITP) is a challenging field in automated reasoning and theorem proving. Typically, (Automated) Theorem Proving (TP) refers to methods, techniques and tools for automatically proving general (most often first-order) theorems. Nowadays, the field of TP has reached a certain degree of maturity and powerful TP systems are widely available and used. The situation with ITP is strikingly different, in the sense that proving inductive theorems in an essentially automatic way still is a very challenging task, even for the most advanced existing ITP systems. Both in general TP and in ITP, strategies for guiding the proof search process are of fundamental importance, in automated as well as in interactive or mixed settings. In the paper we will analyze and discuss the most important strategic and proof search issues in ITP, compare ITP with TP, and argue why ITP is in a sense much more challenging. More generally, we will systematically isolate, investigate and classify the main problems and challenges in ITP w.r.t. automation, on different levels and from different points of views. Finally, based on this analysis we will present some theses about the state of the art in the field, possible criteria for what could be considered as substantial progress, and promising lines of research for the future, towards (more) automated ITP.     

Unification Algorithms for Eliminating and Introducing Quantifiers in Natural Deduction Automated Theorem Proving

A natural deduction system was adapted from Gentzen system. It enables valid wffs to be deduced in a very natural way. One need not transform a formula into other normal forms. Robinsons unification algorithm is used to handle clausal formulas. Algorithms for eliminating and introducing quantifiers without Skolemization are presented, and unification theorems for them are proved. A natural deduction automated theorem prover based on the algorithms was implemented. The rules for quantifiers are controlled by the algorithms. The Andrews challenge and the halting problem were proved by the system.     

Automated reasoning in geometry theorem proving with Prolog

This paper describes automated reasoning in a PROLOG Euclidean geometry theorem-prover. It brings into focus general topics in automated reasoning and the ability of Prolog in coping with them.     

一个用于一阶逻辑自动定理证明的新算法

本文提出一个用于一阶逻辑(FOPC)自动定理证明的并行算法,它基于分治的思想,把原问题子句集S划分成两个独立的子句集S1和S2,并通过并行地证明S_1和S_2的不可满足性。本文首先讨论了子句集的划分问题,引入了导出子句集及划分因子的概念;然后,在此基础上,提出了FOPC定理证明的并行算法;最后,给出了算法的有效性和完备性证明。文中还讨论了子句集的化简及算法性能评价等问题。     

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.     

Automated proofs of equality problems in Overbeek's competition

This paper reports the detailed computer proofs of nine equality problems in Overbeek's competition obtained by Herky, a completion-based theorem prover developed by the author. Advanced techniques implemented in Herky made it a high-performance theorem prover for equational reasoning. Herky is able to prove the first nine of the ten equality problems in the competition (the tenth is an open problem). These equality problems are likely to serve as good exercises for theorem provers based on different approaches, and the proofs of these problems may help people to solve them using their own theorem provers.     

本体推理在几何定理机器证明中的应用

文中阐述了平面几何定理机器证明的基本原理及方法,针对几何定理机器证明过程中可读证明的产生,及推理信息快速增长的问题,提出了一种基于本体推理的几何定理机器证明方法。通过具体案例,描述了以Protégér软件为工具,基于WordNet重用的领域本体半自动构建方法,构建几何本体模型的过程,并结合Prolog规则进行双向推理。结果表明将本体引入几何定理机器证明是可行的,且本体推理脱离了代数形式,使得推理过程更接近自然语言的描述,同时推理效率更高。     

Case studies of Z-module reasoning: Proving benchmark theorems from ring theory

A new method, called Z-module reasoning, was formulated for proving and discovering theorems from ring theory. In a case study, the ZMR system designed to implement this method was used to prove the benchmark x ³ ring theorem from associative ring theory. The system proved the theorem quite efficiently. The system was then used to prove the x ⁴ ring theorem from associative ring theory. Again, a proof was produced easily. These proofs, together with the successes in proving other difficult theorems from ring theory suggest that the Z-module reasoning method is useful for solving a class of problems relying on equality reasoning. This paper illustrates the Z-module reasoning method, and analyzes the computer proof of the x ³ ring theorem.This reasearch was supported in part by the Applied Mathematical Sciences subprogram of the office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38.     

Automated production of traditional proofs in solid geometry

This paper presents a method of producing readable proofs for theorems in solid geometry. The method is for a class of constructive geometry statements about straight lines, planes, circles, and spheres. The key idea of the method is to eliminate points from the conclusion of a geometric statement using several (fixed) high-level basic propositions about the signed volumes of tetrahedrons and Pythagorean differences of triangles. We have implemented the algorithm, and more than 80 examples from solid geometry have been used to test the program. Our program is efficient and the proofs produced by it are generally short and readable.The work reported here was supported in part by the NSF Grant CCR-9117870 and Chinese National Science Foundation.On leave from Institute of Systems Sciences, Academia Sinica, Beijing 100080, P.R. China.     

应用本体和AllegroGraph实现几何定理证明

由于传统的定理机器证明方法是基于规则的,使得定理证明出现几何信息增长迅猛,推理和计算效率低以及过程可读性差等问题。针对以上情况,提出了基于本体和AllegroGraph的几何定理证明方法。该方法通过本体构建几何定理命题模型,然后采用Prolog规则描述语言对几何定理性质进行描述,同时通过分析本体模型和规则描述的对应关系,提出定理规则半自动生成方法。最后以AllegroGraph（ AG）图形数据库的推理机制为基础,完成几何定理证明。实验结果表明,将本体和AllegroGraph推理机应用于几何定理证明领域可以摆脱以往几何定理证明代数化问题,几何证明过程容易理解,同时合理地控制了信息的增长,支持定理可持续证明。     

Automated Theorem Proving in Temporal Logic:T-Resolution

This paper presentes a novel resolution method,T-resolution,based on the first order temporal logic.The primary claim of this method is its soundness and completeness.For this purpose,we construct the corresponding semantic trees and extend Herbrand‘s Theorem.     

基于消点法的几何自动推理系统实现

为了实现几何自动推理的可读性证明,并提高推理效率,介绍了一个基于消点法的可构造性几何命题自动推理系统的设计与实现。该系统提供作图的方式接受用户的几何命题前提条件的输入,可以对初等几何中的大部分可构造性几何问题进行自动证明和求解,并生成可读的证明步骤,大大方便了初高等几何教育和相关研究者的需要。     

Gazing: An approach to the problem of definition and lemma use

A major outstanding problem in automated theorem proving research is determining the appropriate use of definitions and previously proved theorems within a proof. Presenting the theorem prover with only the formulae that are necessary for the proof might be viewed as ‘cheating’ and requires that the user prove the theorem ahead of time. In real world applications of theorem proving, this is almost certain to be infeasible. On the other hand providing the prover with all formulae that might be relevant rapidly swamps the prover with unnecessary information. A technique for the selective retrieval of formulae based on features of those formulae and the conjecture at hand is required to solve this problem. In this paper I describe an abstraction-based technique which addresses this problem. Implicit hypotheses such as definitions, axioms and previously proved theorems are stored in a database which may be accessed by a heuristic rule of inference calledgazing. Before accessing this database the gazing rule plans the use of these formulae in a hierarchy of abstraction spaces. When the planning phase is complete, the system can use the indicated formulae with some confidence that they are relevant to the proof. Since the technique is abstraction-based, no guarantee that the plant will be eventually applicable or successful can be made. However, because the plans are built by considering increasing amounts of detail, the number of ways in which the application of a plan can fail is limited. Plan failures may be ‘patched’ in a uniform way.     

The evidence algorithm and problems of representation and processing of mathematical computer knowledge

A series of research works on the realization of the “evidence algorithm” program initiated by V. M. Glushkov in the sixties is described. The works are planned in the context of a new understanding of the theorem-proving problem and according to modern trends in information technologies and computer science. Translated from Kibemetika i Sistemnyi Analiz, No. 6, pp. 9–17, November–December, 1999.     

SGARP中符号计算模块的实现及其应用

可持续发展的几何自动推理平台(sustainable geometry automated reasoning platform, SGARP)支持用户按需添加或修改几何定理机器证明所涉及的几何对象、谓词、定理和规则,以发展多种多样基于规则的机器自动推理或人机交互推理方法.为进一步提高SGARP的推理能力和扩展其适用范围,提出一种在SGARP中实现符号计算功能的快捷方法,并成功添加了质点法和解析法推理模块.质点法可证明希尔伯特交点类几何命题,解析法能用于辅助证明各种类型有一定难度的几何定理,如著名的Thebault定理.对这两种方法用基于Web的机器证明测试用的几何问题库(thousands of geometric problems for geometric theorem provers, TGTP)中180道几何题进行评估,均在合理时间内给出令人满意的可读机器证明,表明升级后的SGARP能更好地满足用户学习与发展几何机器推理的需求.     

An overview of automated reasoning and related fields

This article provides an overview of automated reasoning and of the various fields for which it is relevant. It takes the form of a collection of articles, each covering some field and each written by an expert in that field. A field is introduced, its elements reviewed, the current state of the art given, the basic problems discussed, and the various goals listed. Although individually the goals of each field present a wide spectrum, collectively the fields share the interest of automating the process known as reasoning.     

关于鞅超收敛定理与遗忘因子最小二乘算法的收敛性分析

鞅超收敛定理是研究随机时变系统辨识算法有界收敛性的一个有效数学工具,它是鞅收益是在随机时变系统中的推广。文「１」用它证明了遗忘因子最小二乘算法参数估计误差的有界收敛性,但是文「１」假设系统的理各态遍历的,且协方差阵是用它的数学期望代替的,所得到的结果是近似的。而本文精确地给出了协方差阵的上下界,改进了文「１」的结果。     

Statement and solution of some problems on a mathematical safe

The problem on a mathematical safe is formulated and studied in terms of graph theory. The cases of simplest digraphs such as paths, contours, and doubly connected components are analyzed. A number of statements on the existence of solutions to these problems are proved. The results obtained are extended to the case of corresponding nondirected graphs. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 3–14, May–June 2006.