ETPS: A System to Help Students Write Formal Proofs

ETPS (Educational Theorem Proving System) is a program that logic students can use to write formal proofs in first-order logic or higher-order logic. It enables students to concentrate on the essential logical problems involved in proving theorems, and it automatically checks the proofs.     

支持索引式的PPTL定理证明器的实现

定理证明是目前主流的形式化验证方法,拥有强大的抽象和逻辑表达能力,且不存在状态空间爆炸问题,可用于有穷和无穷状态系统,但其不能完全自动化,并且要求用户掌握较强的数学知识.含索引式的命题投影时序逻辑(PPTL)是一种具有完全正则表达能力,并且包含LTL的时序逻辑,具有较强的建模和性质描述能力.目前,一个可靠完备的含索引式的PPTL公理系统已被构建,然而基于该公理系统的定理证明尚未得到良好工具的支持,存在证明自动化程度较低以及证明冗长易错的问题.鉴于此,首先设计了支持索引式的PPTL定理证明器的实现框架,包括公理系统的形式化与交互式定理证明;然后,在Coq中形式化定义了含索引式的PPTL公式、公理与推理规则,完成了框架中公理系统的实现;最后,通过两个实例的交互式证明验证了该定理证明器的可用性.     

Mechanically verifying concurrent programs with the Boyer-Mooreprover

A proof system suitable for the mechanical verification of concurrent programs is described. This proof system is based on Unity, and may be used to specify and verify both safety and liveness properties. However, it is defined with respect to an operational semantics of the transition system model of concurrency. Proof rules are simply theorems of this operational semantics. This methodology makes a clear distinction between the theorems in the proof system and the logical inference rules and syntax which define the underlying logic. Since this proof system essentially encodes Unity in another sound logic, and this encoding has been mechanically verified, this encoding proves the soundness of this formalization of Unity. This proof system has been mechanically verified by the Boyer-Moore prover. This proof system has been used to mechanically verify the correctness of a distributed algorithm that computes the minimum node value in a tree     

Composition of Semantic Web services using Linear Logic theorem proving

This paper introduces a method for automatic composition of Semantic Web services using Linear Logic (LL) theorem proving. The method uses a Semantic Web service language (DAML-S) for external presentation of Web services, while, internally, the services are presented by extralogical axioms and proofs in LL. LL, as a resource conscious logic, enables us to capture the concurrent features of Web services formally (including parameters, states and non-functional attributes). We use a process calculus to present the process model of the composite service. The process calculus is attached to the LL inference rules in the style of type theory. Thus, the process model for a composite service can be generated directly from the complete proof. We introduce a set of subtyping rules that defines a valid dataflow for composite services. The subtyping rules that are used for semantic reasoning are presented with LL inference figures. We propose a system architecture where the DAML-S Translator, LL Theorem Prover and Semantic Reasoner can operate together. This architecture has been implemented in Java.     

Focusing and polarization in linear,intuitionistic, and classical logics

A focused proof system provides a normal form to cut-free proofs in which the application of invertible and non-invertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cut-free proofs. Within intuitionistic and classical logics, there are various different proof systems in the literature that exhibit focusing behavior. These focused proof systems have been applied to both the proof search and the proof normalization approaches to computation. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other intuitionistic proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system LKF for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard’s LC and LU proof systems.     

Natural deduction as higher-order resolution

An interactive theorem prover, Isabelle, is under development. In lcf, each inference rule is represented by one function for forwards proof and another (a tactic) for backwards proof. In Isabelle, each inference rule is represented by a Horn clause. Resolution gives both forwards and backwards proof, supporting a large class of logics. Isabelle has been used to prove theorems in Martin-Löf's constructive type theory. Quantifiers pose several difficulties: substitution, bound variables, Skolemization. Isabelle's representation of logical syntax is the typed λ-calculus, requiring higher-order unification. It may have potential for logic programming. Depth-first subgoaling along inference rules constitutes a higher-order PROLOG.     

Coq Formalization of ZFC Set Theory for Teaching Scenarios

Discrete mathematics is a foundation course for computer-related majors, and propositional logic, first-order logic, and the axiomatic set theory are important parts of this course. Teaching practice shows that beginners find it difficult to accurately understand abstract concepts, such as syntax, semantics, and reasoning system. In recent years, some scholars have begun introducing interactive theorem provers into teaching to help students construct formal proofs so that they can understand logic systems more thoroughly. However, directly employing the existing theorem provers will increase students'' learning burden since these tools have a high threshold for getting started with them. To address this problem, we develop a prover for the Zermelo-Fraenkel set theory with the axiom of Choice (ZFC) in Coq for teaching scenarios. Specifically, the first-order logical reasoning system and the axiomatic set theory ZFC are formalized, and several automated proof tactics specific to reasoning rules are then developed. Students can utilize these automated proof tactics to construct formal proofs of theorems in a textbook-style concise proving environment. This tool has been introduced into the teaching of the course of discrete mathematics for freshmen. Students with no prior theorem-proving experience can quickly construct formal proofs of theorems including mathematical induction and Peano arithmetic with this tool, which verifies the practical effectiveness of this tool.     

Proof rules for fault tolerant distributed programs

Proving the properties of a program which must execute on a distributed system whose nodes may fail is a complex task. Such proofs must take into account the effects of hardware failures at all possible points in the execution of individual processes. The difficulty in accomplishing this is compounded by the need to cater also for the simultaneous failure of two or more processing nodes. In this paper, we consider programs written in a version of Hoare's CSP and define a set of axioms and inference rules by which proofs can be constructed in three steps: proving the properties of each process when its communicants are prone to failure, establishing the effects of failure of each process, and combining these two steps to determing the fault tolerant properties of the whole program. The proof system is thus compositional, in the sense that proofs can be constructed in a modular way.     

Rule-based induction

Induction is indispensable for reasoning about parameterized logical models, as they arise in the context of hardware and software verification. Sophisticated heuristics for mechanizing induction proofs exist and have been successfully implemented in theorem provers for first-order logic. This article is a presentation of a complementary induction method that makes extensive use of the expressiveness of higher-order logic and higher-order unification facilities ofLAMBDA. Rather than hard-coded routines, rules are used that incarnate induction schemes in a concise way. By means of instantiatable meta-variables, these rules are more general and preserve the flexibility to gradually specify an induction scheme throughout a proof process, as more and more information is available, rather than fully determining the induction scheme right at the beginning. Safe inference mechanisms and general-purpose proof automation utilities in the proof checkerLAMBDA are well-suited for an interactive induction assistant that integrates automatic induction heuristics and at they same time accepts human guidance.     

A Proof Search Specification of the π-Calculus

We present a meta-logic that contains a new quantifier (for encoding “generic judgments”) and inference rules for reasoning within fixed points of a given specification. We then specify the operational semantics and bisimulation relations for the finite π-calculus within this meta-logic. Since we restrict to the finite case, the ability of the meta-logic to reason within fixed points becomes a powerful and complete tool since simple proof search can compute this one fixed point. The quantifier helps with the delicate issues surrounding the scope of variables within π-calculus expressions and their executions (proofs). We shall illustrate several merits of the logical specifications we write: they are natural and declarative; they contain no side conditions concerning names of variables while maintaining a completely formal treatment of such variables; differences between late and open bisimulation relations are easy to see declaratively; and proof search involving the application of inference rules, unification, and backtracking can provide complete proof systems for both one-step transitions and for bisimulation.     

Automatic Derivation of the Irrationality of e

As part of a project on automatic generation of proofs involving both logic and computation, we have automatically generated a proof of the irrationality of e. The proof involves inequalities, bounds on infinite series, type distinctions (between real numbers and natural numbers), a subproof by mathematical induction, and significant mathematical steps, including correct simplification of expressions involving factorials and summing an infinite geometrical series. Metavariables are instantiated by inference rules embodying mathematical knowledge, rather than only by unification. The proof is generated completely automatically, without any interactive component.     

On rules of abduction

We present a logical theory of abduction based on the idea of recognizing explanation, or abduction, as a separate reasoning activity. We describe a formalism for writing rules of abduction; furthermore, we define a validity criterion for such rules. The criterion is based on the concept of invariants. This idea allows us to link abduction with induction and deduction. We believe that the three types of inference rules can best be understood in terms of symmetry, i.e. types of relations they preserve, namely: explainability, falsifiability and truth. We also formulate a model theory of abduction and link it with a proof theory. We discuss a variety of rules of abduction and argue that logical forms of abduction do not have to be restricted to the reversemodus ponens. These rules are used to describe such tasks as word-sense disambiguation and anaphora resolution in natural language processing, as well as abduction-based diagnosis.     

The construction of a fuzzy inference network by extension of the rule inference network

Fuzzy logic can bring about inappropriate inferences as a result of ignoring some information in the reasoning process. Neural networks are powerful tools for pattern processing, but are not appropriate for the logical reasoning needed to model human knowledge. The use of a neural logic network derived from a modified neural network, however, makes logical reasoning possible. In this paper, we construct a fuzzy inference network by extending the rule–inference network based on an existing neural logic network. The propagation rule used in the existing rule–inference network is modified and applied. In order to determine the belief value of a proposition pertaining to the execution part of the fuzzy rules in a fuzzy inference network, the nodes connected to the proposition to be inferenced should be searched for. The search costs are compared and evaluated through application of sequential and priority searches for all the connected nodes.     

Proof by consistency

Advances of the past decade in methods and computer programs for showing consistency of proof systems based on first-order equations have made it feasible, in some settings, to use proof by consistency as an alternative to conventional rules of inference. Musser described the method applied to proof of properties of inductively defined objects. Refinements of this inductionless induction method were discussed by Kapur, Goguen, Huet and Hullot, Huet and Oppen, Lankford, Dershowitz, Paul, and more recently by Jouannaud and Kounalis as well as by Kapur, Narendran and Zhang. This paper gives a very general account of proof by consistency and inductionless induction, and shows how previous results can be derived simply from the general theory. New results include a theorem giving characterizations of an unambiguity property that is key to applicability of proof by consistency, and a theorem similar to the Birkhoff's Completeness Theorem for equational proof systems, but concerning inductive proof.     

An Introduction to IN CAPS System

INCAPS，a subsystem of XYZ system,is an INteractive Computer-Assisted Proving System,The primary targets to develop it range from proving temporal logic formal theorem to verifying XYZ/SE program‘s correctness which are supported respectively by the mechanized logics-FOTL logic and Hoare-like proof system.This paper discusses five main topics concerning INCAPS system:the rules,implementation,tactics,forward proof and backward proof.It also gives several typical examples for demonstration of INCAPS‘ working principle.The achievement to data in that we have now accomplished successfully the verification of the hierarchical specification of AB protocol and the correctness of XYZ/SE program.     

Program Development Schemata as Derived Rules

We show how the formalization and application of schemata for program development can be reduced to the formalization and application of derived rules of inference. We formalize and derive schemata as rules in theories that axiomatize program data and programs themselves. We reduce schema-based program development to ordinary theorem proving, where higher-order unification is used to apply rules. Conceptually, our formalization is simple and unifies divergent views of schemata, program synthesis, and program transformation. Practically, our formalization yields a simple methodology for carrying out development using existing logical frameworks; we illustrate this in the domain of logic program synthesis and transformation using the Isabelle logical framework.     

On Automating Diagrammatic Proofs of Arithmetic Arguments

Theorems in automated theorem proving are usually proved by formal logical proofs. However, there is a subset of problems which humans can prove by the use of geometric operations on diagrams, so called diagrammatic proofs. Insight is often more clearly perceived in these proofs than in the corresponding algebraic proofs; they capture an intuitive notion of truthfulness that humans find easy to see and understand. We are investigating and automating such diagrammatic reasoning about mathematical theorems. Concrete, rather than general diagrams are used to prove particular concrete instances of the universally quantified theorem. The diagrammatic proof is captured by the use of geometric operations on the diagram. These operations are the inference steps of the proof. An abstracted schematic proof of the universally quantified theorem is induced from these proof instances. The constructive -rule provides the mathematical basis for this step from schematic proofs to theoremhood. In this way we avoid the difficulty of treating a general case in a diagram. One method of confirming that the abstraction of the schematic proof from the proof instances is sound is proving the correctness of schematic proofs in the meta-theory of diagrams. These ideas have been implemented in the system, called Diamond, which is presented here.     

An approach to automatic deductive synthesis of functional programs

The work deals with automatic deductive synthesis of functional programs. Formal specification of a program is taken as a mathematical existence theorem and while proving it, we derive a program and simultaneously prove that this program corresponds to given specification. Several problems have to be resolved for automatic synthesis: the choice of synthesis rules that allows us to derive the basic constructions of a functional program, order of rule application and choice of a particular induction rule. The method proposed here is based on the deductive tableau method. The basic method gives rules for functional program construction. To determine the proof strategy we use some external heuristics, including rippling. And for the induction hypothesis formation the combination of rippling and the deductive tableau method became very useful. The proposed techniques are implemented in the system ALISA (Automatic Lisp Synthesizer) and used for automatic synthesis of several functions in the Lisp language. The work has been partially supported by RFBR grant 05-01-00948a.