The Hot List Strategy

Experimentation strongly suggests that, for attacking deep questions and hard problems with the assistance of an automated reasoning program, the more effective paradigms rely on the retention of deduced information. A significant obstacle ordinarily presented by such a paradigm is the deduction and retention of one or more needed conclusions whose complexity sharply delays their consideration. To mitigate the severity of the cited obstacle, I formulated and feature in this article the hot list strategy. The hot list strategy asks the researcher to choose, usually from among the input statements characterizing the problem under study, one or more statements that are conjectured to play a key role for assignment completion. The chosen statements – conjectured to merit revisiting, again and again – are placed in an input list of statements, called the hot list. When an automated reasoning program has decided to retain a new conclusion C – before any other statement is chosen to initiate conclusion drawing – the presence of a nonempty hot list (with an appropriate assignment of the input parameter known as heat) causes each inference rule in use to be applied to C together with the appropriate number of members of the hot list. Members of the hot list are used to complete applications of inference rules and not to initiate applications. The use of the hot list strategy thus enables an automated reasoning program to briefly consider a newly retained conclusion whose complexity would otherwise prevent its use for perhaps many CPU-hours. To give evidence of the value of the strategy, I focus on four contexts: (1) dramatically reducing the CPU time required to reach a desired goal, (2) finding a proof of a theorem that had previously resisted all but the more inventive automated attempts, (3) discovering a proof that is more elegant than previously known, and (4) answering a question that had steadfastly eluded researchers relying on an automated reasoning program. I also discuss a related strategy, the dynamic hot list strategy (formulated by my colleague W. McCune), that enables the program during a run to augment the contents of the hot list. In the Appendix, I give useful input files and interesting proofs. Because of frequent requests to do so, I include challenge problems to consider, commentary on my approach to experimentation and research, and suggestions to guide one in the use of McCunes automated reasoning program OTTER.     

The power of combining resonance with heat

In this article, I present experimental evidence of the value of combining two strategies each of which has proved powerful in various contexts. The resonance strategy gives preference (for directing a program's reasoning) to equations or formulas that have the same shape (ignoring variables) as one of the patterns supplied by the researcher to be used as a resonator. The hot list strategy rearranges the order in which conclusions are drawn, the rearranging caused by immediately visiting and, depending on the value of the heat parameter, even immediately revisiting a set of input statements chosen by the researcher; the chosen statements are used to complete applications of inference rules rather than to initiate them. Combining these two strategies often enables an automated reasoning program to attack deep questions and hard problems with far more effectiveness than using either alone. The use of this combination in the context of cursory proof checking produced most unexpected and satisfying results, as I show here. I present the material (including commentary) in the spirit of excerpts from an experimenter's notebook, thus meeting the frequent request to illustrate how a researcher can make wise choices from among the numerous options offered by McCune's automated reasoning program OTTER. I include challenges and topics for research and, to aid the researcher, in the Appendix a sample input file and a number of intriguing proofs.This work was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Local proofs for global safety properties

This paper explores locality in proofs of global safety properties of concurrent programs. Model checking on the full state space is often infeasible due to state explosion. A local proof, in contrast, is a collection of per-process invariants, which together imply the desired global safety property. Local proofs can be more compact than global proofs, but local reasoning is also inherently incomplete. In this paper, we present an algorithm for safety verification that combines local reasoning with gradual refinement. The algorithm gradually exposes facts about the internal state of components, until either a local proof or a real error is discovered. The refinement mechanism ensures completeness. Experiments show that local reasoning can have significantly better performance over the traditional reachability computation. Moreover, for some parameterized protocols, a local proof can be used as the basis of a correctness proof over all instances.     

OTTER and the Moufang identity problem

This article provides additional evidence of the value of using an automated reasoning program as a research assistant. Featured is the use of Bill McCune's program OTTER to find proofs of theorems taken from the study of Moufang loops, but not just any proofs. Specifically, the proofs satisfy the property of purity. In particular, when given, say, four equivalent identities (which is the case in this article), one is asked to prove the second identity from the first, the third from the second, the fourth from the third, and the first from the fourth. If the proof that 1 implies 2 does not rely on 3 or 4, then by definition the proof is pure with respect to 3 and 4, or simply the proof is pure. If for the four identities one finds four pure proofs showing that 1 implies 2, 2 implies 3, 3 implies 4, and 4 implies 1, then by definition one has found a circle of pure proofs. By finding the needed twelve pure proofs, this article shows that there does exist a circle of pure proofs for the four equivalent identities for Moufang loops and for all orderings of the identities; however, for much of this article, the emphasis is on the first three identities. In addition — in part to promote the use of automated reasoning programs and to answer questions concerning the choice of options — featured here is the methodology that was employed and a discussion of some of the obstacles, some of which are subtle. The approach relies on paramodulation (which generalizes equality substitution), on demodulation, and — so crucial for attacking deep questions and hard problems — on various strategies, most important of which are the hot list strategy, the set of support strategy, and McCune's ratio strategy. To permit verification of the results presented here, extension of them, and application of the methodology to other unrelated fields, a sample input file and four proofs (relevant to a circle of pure proofs for the four identities) are included. Research topics and challenges are offered at the close of this article.This work was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Missing Proofs Found

For close to a century, despite the efforts of fine minds that include Hilbert and Ackermann, Tarski and Bernays, ukasiewicz, and Rose and Rosser, various proofs of a number of significant theorems have remained missing – at least not reported in the literature – amply demonstrating the depth of the corresponding problems. The types of such missing proofs are indeed diverse. For one example, a result may be guaranteed provable because of being valid, and yet no proof has been found. For a second example, a theorem may have been proved via metaargument, but the desired axiomatic proof based solely on the use of a given inference rule may have eluded the experts. For a third example, a theorem may have been announced by a master, but no proof was supplied. The finding of missing proofs of the cited types, as well as of other types, is the focus of this article. The means to finding such proofs rests with heavy use of McCune's automated reasoning program OTTER, reliance on a variety of powerful strategies this program offers, and employment of diverse methodologies. Here we present some of our successes and, because it may prove useful for circuit design and program synthesis as well as in the context of mathematics and logic, detail our approach to finding missing proofs. Well-defined and unmet challenges are included.     

Round-optimal zero-knowledge proofs of knowledge for NP

It is well known that all the known black-box zero-knowledge proofs of knowledge for NP are nonconstant-round.Whether there exit constant-round black-box zero-knowledge proofs of knowledge for all NP languages under certain standard assumptions is an open problem.This paper focuses on the problem and gives a positive answer by presenting two constructions of constant-round(black-box) zero-knowledge proofs of knowledge for the HC(hamiltonian cycle) problem.By the recent result of Katz,our second construction which relies on the existence of claw-free functions has optimal round complexity(5-round) assuming the polynomial hierarchy does not collapse.     

Automating the Search for Elegant Proofs

The research reported in this article was spawned by a colleague's request to find an elegant proof (of a theorem from Boolean algebra) to replace his proof consisting of 816 deduced steps. The request was met by finding a proof consisting of 100 deduced steps. The methodology used to obtain the far shorter proof is presented in detail through a sequence of experiments. Although clearly not an algorithm, the methodology is sufficiently general to enable its use for seeking elegant proofs regardless of the domain of study. In addition to (usually) being more elegant, shorter proofs often provide the needed path to constructing a more efficient circuit, a more effective algorithm, and the like. The methodology relies heavily on the assistance of McCune's automated reasoning program OTTER. Of the aspects of proof elegance, the main focus here is on proof length, with brief attention paid to the type of term present, the number of variables required, and the complexity of deduced steps. The methodology is iterative, relying heavily on the use of three strategies: the resonance strategy, the hot list strategy, and McCune's ratio strategy. These strategies, as well as other features on which the methodology relies, do exhibit tendencies that can be exploited in the search for shorter proofs and for other objectives. To provide some insight regarding the value of the methodology, I discuss its successful application to other problems from Boolean algebra and to problems from lattice theory. Research suggestions and challenges are also offered.     

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.     

NP问题的最优轮复杂性知识的零知识证明

NP问题已有的知识的(黑箱)零知识证明都是非常数轮的,因此,在标准的复杂性假设下,NP问题是否存在常数轮的(黑箱)知识的零知识证明是一个有意义的问题.本文对该问题进行了研究,在一定的假设下给出了HC问题的两个常数轮知识的零知识证明系统.根据Katz最近的研究结果,在多项式分层不坍塌的条件下,本文基于claw-free陷门置换给出的HC问题的5轮知识的零知识证明系统具有最优的轮复杂性.     

复杂表达式推理在计算机自动推理系统中的研究

计算机自动推理和几何定理机器证明已经取得令人瞩目的成果,自动推理的软件也出现了很多。完善解决表达式的推理是数学定理机械化证明必须达到的目的,但是关于表达式的推理还缺乏研究,因为表达式的推理不同于一般信息的推理,它没有固定的格式,信息表述复杂。本文在之前研究工作的基础上,通过对表达式在推理时的特征分析,提出一种表达式推理的方法,是向这个方向的一个尝试。在该方法中,通过适当的替换,将表达式化为空,从而实现了表达式的简单处理,并在文中列举几个实例进行分析。实践证明,利用这种方法,可对大多数的结论为表达式的几何命题给出可读证明过程。     

Automated generation of readable proofs with geometric invariants

We present a set of rules based on full-angles as the basis of automated geometry theorem proving. We extend the idea of eliminating variables and points to the idea of eliminating lines. We also discuss how to combine the forward chaining and backward chaining to achieve higher efficiency. The prover based on the full-angle method has been used to produce short and elegant proofs for more than one hundred difficult geometry theorems. The proofs of many of those theorems produced by our previous area method are relatively long.This work was supported in part by the NSF Grants CCR-9117870, CCR-9420857 and the Chinese NSF.     

A Milestone Reached and a Secret Revealed

In this special issue of the Journal of Automated Reasoning, this article sets the stage for the succeeding articles, all of which focus on finding proofs of theorems of formal logic and on the various methodologies that were employed. The proofs that are offered mark an important milestone for automated reasoning and for logic, for each of them is indeed new. One key question this article answers is why an automated reasoning program was able to find proofs that had eluded some of the finest mathematicians and logicians for many, many decades.     

Constant-round zero-knowledge proofs of knowledge with strict polynomial-time extractors for NP

Barak and Lindell showed that there exist constant-round zero-knowledge arguments of knowledge with strict polynomial-time extractors.This leaves the open problem of whether it is possible to obtain an analogous result regarding constant-round zero-knowledge proofs of knowledge for NP.This paper focuses on this problem and gives a positive answer by presenting a construction of constant-round zero-knowledge proofs of knowledge with strict polynomial-time extractors for NP.     

Complexity of Positivstellensatz proofs for the knapsack

A lower bound is established on degrees of Positivstellensatz calculus refutations (over a real field) introduced in (Grigoriev & Vorobjov 2001; Grigoriev 2001) for the knapsack problem. The bound depends on the values of coefficients of an instance of the knapsack problem: for certain values the lower bound is linear and for certain values the upper bound is constant, while in the polynomial calculus the degree is always linear (regardless of the values of coefficients) (Impagliazzo et al. 1999). This shows that the Positivstellensatz calculus can be strictly stronger than the polynomial calculus from the point of view of the complexity of the proofs. Received: February 9, 2000.     

数据可恢复性的零知识证明

可恢复性证明是一种外包存储服务中的数据完整性检测技术.本文致力于交互式证明系统标准模型下的可恢复性证明协议构造方法研究,提出了首个能够防止证明者欺骗和验证数据泄露的交互式可恢复性证明协议,并通过构造多项式时间的可回卷知识抽取器,给出了基于计算Diffie-Hellman假设下的协议完备性、稳固性,以及零知识性证明.特别是,所提方案的验证过程只要求低的、固定大小的负载,达到了最小化通信复杂性的目的.     

Writing programs that construct proofs

When we learn mathematics, we learn more than definitions and theorems. We learn techniques of proof. In this paper, we describe a particular way to express these techniques and incorporate them into formal theories and into computer systems used to build such theories. We illustrate the methods as they were applied in the -PRL system, essentially using the ML programming language from Edinburgh LCF [23] as the formalised metalanguage. We report our experience with such an approach emphasizing the ideas that go beyond the LCF work, such as transformation tactics and special purpose reasoners. We also show how the validity of tactics can be guaranteed. The introduction places the work in historical context and the conclusion briefly describes plans to carry the methods further. The majority of the paper presents the -PRL approach in detail.Department of Computer Science Technical Report TR84-645. This research supported in part by the National Science Foundation under grant MCS-81-04018.     

基于层次因果诊断模型的搜索策略研究

文章针对定性推理QSIM算法存在组合爆炸的问题,根据水产专家的多年总结的实践经验与知识,对鱼病的数据、信息进行了分析,将领域专家的知识作为约束条件,提出了基于层次因果的诊断模型的搜索策略,减少了推理分支,提高了效率。