Proof strategies in linear logic

Linear logic, introduced by J.-Y. Girard, is a refinement of classical logic providing means for controlling the allocation of resources. It has aroused considerable interest from both proof theorists and computer scientists. In this paper we investigate methods for automated theorem proving in propositional linear logic. Both the bottom-up (tableaux) and top-down (resolution) proof strategies are analyzed. Various modifications of sequent rules and efficient search strategies are presented along with the experiments performed with the implemented theorem provers.     

出具证明编译器中线性整数命题证明的自动生成

近年来,出具证明编译器作为构建高可信软件的重要途径,逐渐成为编译器理论和形式化验证的研究热点.在其理论框架中,编译器需要借助自动定理证明技术,自动地证明验证条件并生成机器可检查的证明项,因此好的自动定理证明器对出具证明编译器至关重要.本文基于Simplex算法在出具证明编译器的框架内设计并实现了一个支持线性整数命题求解的自动定理证明器,并且提出一套证明项构造方法,将其应用于自动定理证明器中可生成Coq可检查的证明.     

Using Resolution for Testing Modal Satisfiability and Building Models

This paper presents a translation-based resolution decision procedure for the multimodal logic K _(m)(,,) defined over families of relations closed under intersection, union, and converse. The relations may satisfy certain additional frame properties. Different from previous resolution decision procedures that are based on ordering refinements, our procedure is based on a selection refinement, the derivations of which correspond to derivations of tableaux or sequent proof systems. This procedure has the advantage that it can be used both as a satisfiability checker and as a model builder. We show that tableaux and sequent-style proof systems can be polynomially simulated with our procedure. Furthermore, the finite model property follows for a number of extended modal logics.     

A Mechanized Proof Environment for the Convenient Computations Proof Method

A mechanized verification environment made up of theories over the deductive mechanized theorem prover PVS is presented, which allows taking advantage of the convenient computations method. This method reduces the conceptual difficulty of proving a given property for all the possible computations of a system by separating two different concerns: (1) proving that special convenient computations satisfy the property, and (2) proving that every computation is related to a convenient one by a relation which preserves the property. The approach is especially appropriate for applications in which the first concern is trivial once the second has been shown, e.g., where the specification itself is that every computation reduces to a convenient one. Two examples are the serializability of transactions in distributed databases, and sequential consistency of distributed shared memories. To reduce the repetition of effort, a clear separation is made between infrastructural theories to be supplied as a proof environment PVS library to users, and the specification and proof of particular examples. The provided infrastructure formally defines the method in its most general way. It also defines a computation model and a reduction relation—the equivalence of computations that differ only in the order of finitely many independent operations. One way to prove that this relation holds between every computation and some convenient one involves the definition of a measure function from computations into a well-founded set. Two possible default measures, which can be applied in many cases, are also defined in the infrastructure, along with useful lemmas that assist in their usage. We show how the proof environment can be used, by a step-by-step explanation of an application example.     

Decidability by Resolution for Propositional Modal Logics

The paper shows that satisfiability in a range of popular propositional modal systems can be decided by ordinary resolution procedures. This follows from a general result that resolution combined with condensing, and possibly some additional form of normalization, is a decision procedure for the satisfiability problem in certain so-called path logics. Path logics arise from normal propositional modal logics by the optimized functional translation method. The decision result provides an alternative method of proving decidability for modal logics, as well as closely related systems of artificial intelligence. This alone is not interesting. A more far-reaching consequence of the result has practical value, namely, many standard first-order theorem provers that are based on resolution are suitable for facilitating modal reasoning.     

基于函数式语义的循环和递归程序结构通用证明技术

各类安全攸关系统的可靠运行离不开软件程序的正确执行.程序的演绎验证技术为程序执行的正确性提供高度保障.程序语言种类繁多,且用途覆盖高可靠性场景的新式语言不断涌现,难以为每种语言设计支撑其程序验证任务的整套逻辑规则,并证明其相对于形式语义的可靠性和完备性.语言无关的程序验证技术提供以程序语言的语义为参数的验证过程及其可靠性结果.对每种程序语言,提供其形式语义后可直接获得面向该语言的程序验证过程.提出一种面向大步操作语义的语言无关演绎验证技术,其核心是对不同语言中循环、递归等可导致无界行为的语法结构进行可靠推理的通用方法.特别地,借助大步操作语义的一种函数式形式化提供表达程序中子结构所执行计算的能力,从而允许借助辅助信息对子结构进行推理.证明所提出验证技术的可靠性和相对完备性,通过命令式、函数式语言中的程序验证实例初步评估了该技术的有效性,并在Coq辅助证明工具中形式化了所有理论结果和验证实例,为基于辅助证明工具实现面向大步语义的语言无关程序验证工具提供了基础.     

A Formal Proof of Sylow's Theorem

The theorem of Sylow is proved in Isabelle HOL. We follow the proof by Wielandt that is more general than the original and uses a nontrivial combinatorial identity. The mathematical proof is explained in some detail, leading on to the mechanization of group theory and the necessary combinatorics in Isabelle. We present the mechanization of the proof in detail, giving reference to theorems contained in an appendix. Some weak points of the experiment with respect to a natural treatment of abstract algebraic reasoning give rise to a discussion of the use of module systems to represent abstract algebra in theorem provers. Drawing from that, we present tentative ideas for further research into a section concept for Isabelle.     

Dialogue Systems as Proof Editors

This paper shows how a dialogue system for information-seekingdialogues can be implemented in a type-theory-based syntax editor,originally developed for editing mathematical proofs.The implementation gives a simple logical metatheory tosuch dialogue systems and also suggests new functions forthem, e.g., a local undo operation. The method developed provides alogically based declarative way of implementing simple dialoguesystems that is easy to port to new domains.     

Automated reasoning about elementary point-set topology

In this paper we present first-order formulas for basic point-set topology, in an attempt to extend the mathematical range available for exploration with automated theorem-proving programs. We present topology definitions and sample lemmas both in first-order logic and in clausal form. We then illustrate some of the difficulties of these sample lemmas through a proof of a basic lemma in five parts.This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38, and by the Division of Educational Programs, Argonne National Laboratory.     

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.     

Automated deduction in von Neumann-Bernays-Gödel set theory

I present a new clausal version of NGB set theory, and compare my version with that first given by Boyer et al. [4]. A complete set of reductions for Boolean rings is given, derived from those of Hsiang [7]. I list over 400 theorems proved semiautomatically in elementary set theory, and supply the proofs of several of these, including Cantor's theorem. I present a semiautomated proof that the composition of homomorphisms is a homomorphism, thus solving a challenge problem given in [4]. Using the clauses and heuristics presented, there is no apparent obstacle to the semiautomated development of set theory through considerably more difficult theorems.     

Computer proofs in Group Theory

This paper reports the results of an experiment in using the Boyer-Moore Theorem Prover in seeking computer-checked proofs in Group Theory. Starting from the axioms for groups and elementary list and number theory, the theorem prover was led to the proofs of two basic theorems in Elementary Group Theory by a sequence of lemmas suggested by the author. The computer proofs illustrate the effective use of the Boyer-Moore Theorem Prover in Group Theory.     

Position Paper: Thoughts on Programming with Proof Assistants

Today the reigning opinion about computer proof assistants based on constructive logic (even from some of the developers of these tools!) is that, while they are very helpful for doing math, they are an absurdly heavy-weight solution to use for practical programming. Yet the Curry-Howard isomorphism foundation of proof assistants like Coq [Yves Bertot and Pierre Castéran. Interactive Theorem Proving and Program Development. Coq'Art: The Calculus of Inductive Constructions. Texts in Theoretical Computer Science. Springer Verlag, 2004] gives them clear interpretations as programming environments.My purpose in this position paper is to make the general claim that Coq is already quite useful today for non-trivial certified programming tasks, as well as to highlight some reasons why you might want to consider using it as a base for your next project in dependently-typed programming.     

Efficient Algorithms to Detect and Restore Minimality, an Extension of the Regular Restriction of Resolution

A given binary resolution proof, represented as a binary tree, is said to be minimal if the resolutions cannot be reordered to generate an irregular proof. Minimality extends Tseitin"s regularity restriction and still retains completeness. A linear-time algorithm is introduced to decide whether a given proof is minimal. This algorithm can be used by a deduction system that avoids redundancy by retaining only minimal proofs and thus lessens its reliance on subsumption, a more general but more expensive technique.Any irregular binary resolution tree is made strictly smaller by an operation called Surgery, which runs in time linear in the size of the tree. After surgery the result proved by the new tree is nonstrictly more general than the original result and has fewer violations of the regular restriction. Furthermore, any nonminimal tree can be made irregular in linear time by an operation called Splay. Thus a combination of splaying and surgery efficiently reduces a nonminimal tree to a minimal one.Finally, a close correspondence between clause trees, recently introduced by the authors, and binary resolution trees is established. In that sense this work provides the first linear-time algorithms that detect minimality and perform surgery on clause trees.     

A Skeptic's Approach to Combining HOL and Maple

We contrast theorem provers and computer algebra systems, pointing out the advantages and disadvantages of each, and suggest a simple way to achieve a synthesis of some of the best features of both. Our method is based on the systematic separation of search for a solution and checking the solution, using a physical connection between systems. We describe the separation of proof search and checking in some detail, relating it to proof planning and to the complexity class NP, and discuss different ways of exploiting a physical link between systems. Finally, the method is illustrated by some concrete examples of computer algebra results proved formally in the HOL theorem prover with the aid of Maple.     

A note on answer extraction in resolution-based systems

An algorithm for answer extraction in resolution-based systems is presented. The algorithm, an extension of the Luckham-Nilsson algorithm, accomplishes the answer extraction in a single traversal of the proof graph without requiring the appending of tautologies to clauses from the negation of the theorem. It enables realization of a significant savings in time and space resources. In addition, an extension to the algorithm is described by which some types of multiple answers to Wh and counting type questions can be obtained.This work was supported by funds from National Aeronautics and Space Administration, Grant NGR21-002-270, Supplement 5, to the University of Maryland.     

Resolution theorem proving in reified modal logics

This paper is concerned with the application of the resolution theorem proving method to reified logics. The logical systems treated include the branching temporal logics and logics of belief based on K and its extensions. Two important problems concerning the application of the resolution rule to reified systems are identified. The first is the redundancy in the representation of truth functional relationships and the second is the axiomatic reasoning about modal structure. Both cause an unnecessary expansion in the search space. We present solutions to both problems which allow the axioms defining the reified logic to be eliminated from the database during theorem proving hence reducing the search space while retaining completeness. We describe three theorem proving methods which embody our solutions and support our analysis with empirical results.Much of the research reported in this paper was supported by DTI IED SERC grant No. GR/F 35968, and was carried out whilst Han Reichgelt was at the University of Nottingham.