The analysis and partial synthesis of truss structures via theorem proving

A use of theorem proving for the analysis and partial synthesis of truss structures is presented. The behavior of a truss structure is modeled as the set union of the behavior of its constituent components where the behavior of each component is modeled by a set of constraints. This component/constraint model is formally represented by a set of axioms using predicate logic. The axioms are then used toanalyze and partiallysynthesize truss structures via theorem proving.Constraint logic programming is identified as a suitable implementation vehicle for the analysis and partial synthesis of truss structures via theorem proving. Several important implications of the formulation for structural design automation, data base integrity, and parallel processing are discussed.     

The automation of proof: a historical and sociological exploration

This article reviews the history of the use of computers to automate mathematical proofs. It identifies three broad strands of work: automatic theorem proving where the aim is to simulate human processes of deduction; automatic theorem proving where any resemblance to how humans deduce is considered to be irrelevant; and interactive theorem proving, where the proof is directly guided by a human being. The first strand has been underpinned by commitment to the goal of artificial intelligence; practitioners of the second strand have been drawn mainly from mathematical logic; and the third strand has been rooted primarily in the verification of computer programs and hardware designs     

Automated verification of refinement laws

Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back’s atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs.     

Formally Analyzing Expected Time Complexity of Algorithms Using Theorem Proving

Probabilistic techniques are widely used in the analysis of algorithms to estimate the computational complexity of algorithms or a computational problem. Traditionally, such analyses are performed using paper-and-pencil proofs and the results are sometimes validated using simulation techniques. These techniques are informal and thus may result in an inaccurate analysis. In this paper, we propose a formal technique for analyzing the expected time complexity of algorithms using higher-order-logic theorem proving. The approach calls for mathematically modeling the algorithm along with its inputs, using indicator random variables, in higher-order logic. This model is then used to formally reason about the expected time complexity of the underlying algorithm in a theorem prover. The paper includes the higher-order-logic formalization of indicator random variables, which are fundamental to the proposed infrastructure. In order to illustrate the practical effectiveness and utilization of the proposed infrastructure, the paper also includes the analysis of algorithms for three well-known problems, i.e., the hat-check problem, the birthday paradox and the hiring problem.     

Combining Symbolic Constraint Solvers on Algebraic Domains

In the context of constraint logic programming and theorem proving, the development of constraint solvers on algebraic domains and their combination is of prime interest. As an example, a constraint solver in finite algebras is presented for a constraint language including for instance equations, disequations and inequations. By extending techniques used for the combination of unification in disjoint equational theories, we show how to combine constraint solvers on different algebraic domains that may share some constant symbols. We illustrate this technique by combining the constraint solver in finite algebras with other unification algorithms, and with another constraint solver on a different finite algebra.     

Heuristic programming: A survey

This survey of heuristic programming starts by considering the nature of an algorithm as an automatic and precise process. A heuristic, by contrast, in an approximate and uncertain process, although algorithms and heuristics should be regarded as lying on a continuum. An examination of various applications of heuristics follows, showing the wide range of their use from game playing and logic theorem proving to shop floor scheduling and natural language programming.     

Refutational theorem proving using term-rewriting systems

In this paper we propose a new approach to theorem proving in first-order logic based on the term-rewriting method. First for propositional calculus, we introduce a canonical term-rewriting system for Boolean algebra. This system enables us to transform the first-order predicate calculus into a form of equational logic, and to develop several complete strategies (both clausal and nonclausal) for first-order theories based on the Knuth-Bendix Completion Procedure. More importantly, our strategies can deal with predicate logic and built-in (equational) theories in a uniform and effective way. We also describe an implementation and comparisons with some other first-order theorem-proving methods.     

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.     

Programming in temporal-nonmonotonic reasoning

Nonmonotonic reasoning has been proposed as an extension to classical first-order logic. Now people are interested in temporal reasoning with nonmonotonic logic [6]. We combine the monotonic logic [7] with a temporal logic to get a more general reasoning language. We discuss a monotonic logic TML which has predicate formulas, temporal formulas and a special modal formula, and give a completeness theorem of it. We use TH() to designate the set of theorems of a temporal-nonmonotonic theory which has the same language with TML. The completeness theorem of the temporal-nonmonotonic logic naturally arises. Like the relationship between predicate logic with a practical logic programming language PROLOG, we propose a useful temporal-nonmonotonic reasoning language TN for the temporal-nonmonotonic logic. As an appendix we supply an algorithm for the programming language TN.     

A note on assumptions about Skolem functions

Skolemization is not an equivalence preserving transformation. For the purposes of refutational theorem proving it is sufficient that skolemization preserves satisfiability and unsatisfiability. Therefore there is sometimes some freedom in interpreting Skolem functions in a particular way. We show that in certain cases it is possible to exploit this freedom for simplifying formulae considerably. Examples for cases where this occurs systematically are the relational translation from modal logics to predicate logic and the relativization of first-order logics with sorts.     

Automatic inductive theorem proving using prolog

Although Prolog is a programming language based on techniques from theorem proving, its use as a base for a theorem prover has not been explored until recently (Stickel, 1984). In this paper, we introduce a Prolog-based deductive theorem proving method for proving theorems in a first-order inductive theory representable in Horn clauses. The method has the following characteristics:

• 1.It automatically partitions the domains over which the variables range into subdomains according to the manner in which the predicate symbols in the theorem are defined.
• 2.For each of the subdomains the prover returns a lemma. If the lemma is true, then the target theorem is true for this subdomain. The lemma could also be an induction hypothesis for the theorem.
• 3.The method does not explicitly use any inductive inference rule. The induction hypothesis, if needed for a certain subdomain, will sometimes be generated from a (limited) forward chaining mechanism in the prover and not from employing any particular inference rule.

In addition to the backward chaining and backtracking facilities of Prolog, our method introduces three new mechanism—skolemization by need, suspended evaluation, and limited forward chaining. These new mechanisms are simple enough to be easily implemented or even incorporated into Prolog. We describe how the theorem prover can be used to prove properties of Prolog programs by showing two simple examples.     

Logic programming and digital circuit analysis

We show how the unique character of logic programming can be exploited for the purpose of specifying and automatically reasoning about electrical circuits. Although propositional logic has long been used for describing the truth functions of combinational circuits, the more powerful predicate calculus on which logic programming is based has seen relatively little use in design automation. Previous researchers have introduced a number of techniques similar to logic programming, but many of the useful consequences of the methodology have not been demonstrated. We describe particular consequences of using this method for writing directly executable specifications of circuits, including the use of quantified variables, verification of hypothetical states, and sequential simulation. We have used these methods to solve problems in gate assignment, specialization of standard definitions, and determination of signal flow.     

Logical foundations for representing object-oriented systems

Abstract

Object-oriented programming languages are designed for computing or simulating the behaviour of interacting objects, but their encapsulated contexts and procedural methods are not well suited to non-procedural techniques in theorem provers, optimizers, and automated design and analysis tools. Logic is the non-procedural system par excellence, but the predicate calculus notation for logic is awkward for representing and reasoning about encapsulated contexts. Conceptual graphs are a graphic system of logic that is better suited to O-O systems. First, they explicitly represent the contexts that are ignored or obscured in predicate calculus. Second, Peirce's rules of inference for reasoning with graphs are explicitly formulated in terms of contexts and the conditions for importing and exporting information from contexts. This article describes the context mechanisms of conceptual graphs, the rules of inference for reasoning with the graphs, and their use as a design language for object-oriented systems.     

Dijkstra-Scholten predicate calculus: concepts and misconceptions

The paper focusses on the logical backgrounds of the Dijkstra-Scholten program development style for correct programs. For proving the correctness of a program (i.e. the fact that the program satisfies its specifications), one often uses a special form of predicate calculus in this style of programming. We call this the Dijkstra-Scholten (DS) predicate calculus, since [DS90] is the first place in which it is described. DS predicate calculus can be conceived of as a logically sound and complete manipulation technique for dealing with logical formulas which also contain programming variables. We relate DS predicate calculus to the classical logical formalism, by contrasting its syntax, derivation rules and semantics to the classical framework. We also comment on two abstractions of DS predicate calculus: the set-theoretical and the algebraic approach. In doing so, we give DS predicate calculus and its abstract variants a firm basis, on a par with the foundations of the well-known first order logic. Such a comparison of DS predicate calculus and classical logic has not yet been sufficiently elaborated before. We conclude our paper with a number of examples showing that the, up to now, unsatisfactory presentation of DS predicate calculus and some of its features (such as the square brackets notation) has led to errors and fallacies in the literature. Received: 22 May 1997 / 5 May 1998     

基于符号模拟和约束逻辑编程的RTL级Verilog谓词抽象方法

利用人工智能最新研究成果--约束逻辑编程对Verilog描述进行谓词抽象,并与目前基于SAT的方法进行了比较.首先通过符号模拟建立Verilog的形式化模型,然后结合要抽象的谓词,将谓词抽象问题转化为约束逻辑编程问题并进行求解.该方法的优点是在计算抽象系统时,不需要像基于SAT的方法那样将字级约束打散成位级约束,求解效率显著提高;提供了一个统一的框架用于描述各种约束.实验结果表明,与基于SAT的抽象技术相比,基于约束逻辑编程的抽象方法的求解速度有显著提高.     

Theorem Proving Guided Development of Formal Assertions in a Resource-Constrained Scheduler for High-Level Synthesis

This paper presents a formal specification and a proof of correctness for the widely-used Force-Directed List Scheduling (FDLS) algorithm for resource-constrained scheduling of data flow graphs in high-level synthesis systems. The proof effort is conducted using a higher-order logic theorem prover. During the proof effort many interesting properties of the FDLS algorithm are discovered. These properties are formally stated and proved in a higher-order logic theorem proving environment. These properties constitute a detailed set of formal assertions and invariants that should hold at various steps in the FDLS algorithm. They are then inserted as programming assertions in the implementation of the FDLS algorithm in a production-strength high-level synthesis system. When turned on, the programming assertions (1) certify whether a specific run of the FDLS algorithm produced correct schedules and, (2) in the event of failure, help discover and isolate programming errors in the FDLS implementation.We present a detailed example and several experiments to demonstrate the effectiveness of these assertions in discovering and isolating errors. Based on this experience, we discuss the role of the formal theorem proving exercise in developing a useful set of assertions for embedding in the scheduler code and argue that in the absence of such a formal proof checking effort, discovering such a useful set of assertions would have been an arduous if not impossible task.     

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.     

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.     

针对教学场景的ZFC集合论Coq形式化

离散数学是计算机类专业的基础课程之一,命题逻辑、一阶逻辑与公理集合论是其重要组成部分.教学实践表明,初学者准确理解语法、语义、推理系统等抽象概念是有一定难度的.近年来,已有一些学者开始在教学中引入交互式定理证明工具,以帮助学生构造形式化证明,更透彻地理解逻辑系统.然而,现有的定理证明器有较高上手门槛,直接使用会增加学生的学习负担.鉴于此,在Coq中开发了针对教学场景的ZFC公理集合论证明器.首先,形式化了一阶逻辑推理系统和ZFC公理集合论;之后,开发了数条自动化推理规则证明策略.学生可以在与教科书风格相同的简洁证明环境中使用自动化证明策略完成定理的形式化证明.该工具被用在了大一新生离散数学课程的教学中,没有定理证明经验的学生使用该工具可以快速完成数学归纳法和皮亚诺算术系统等定理的形式化证明,验证了该工具的实际效果.     

Mechanical proof systems for logic II,consensus programs and their processing

We continue our investigations and study automated theorem proving for reasoning about perception of reasoning agents and their consensus reaching. Using our earlier techniques and those of logic programming we develop the processing techniques for consensus programs.Work partially supported by Polish Goverment grant KBN 2 2051 91 02.Work partially supported by U.S. National Science Foundation grant IRI-9012902.