Mechanisation of Mathematics refers to use of computers to generate or check proofs in Mathematics. It involves translation of relevant mathematical theories from one system of logic to another, to render these theories implementable in a computer. This process is termed formalisation of mathematics. Two among the many ways of mechanising are: 1 Generating results using automated theorem provers. 2 Interactive theorem proving in a proof assistant which involves a combination of user intervention and automation. In the first part of this thesis, we reformulate the question of equivalence of two Links in first order logic using braid groups. This is achieved by developing a set of axioms whose canonical model is the braid group on infinite strands B∞. This renders the problem of distinguishing knots and links, amenable to implementation in first order logic based automated theorem provers. We further state and prove results pertaining to models of braid axioms. The second part of the thesis deals with formalising knot Theory in Higher Order Logic using the interactive proof assistant -Isabelle. We formulate equivalence of links in higher order logic. We obtain a construction of Kauffman bracket in the interactive proof assistant called Isabelle proof assistant. We further obtain a machine checked proof of invariance of Kauffman bracket.

Call number: LD2668 .T4 1985 J63 / Master of Science

We explore the possibility of automating <i>NP</i>-hardness reductions. We motivate the problem from an artificial intelligence perspective, then propose the use of second-order existential (<i>SO</i>&#8707;) logic as representation language for decision problems. Building upon the theoretical framework of J. Antonio Medina, we explore the possibility of implementing seven syntactic operators. Each operator transforms <i>SO</i>&#8707; sentences in a way that preserves <i>NP</i>-completeness. We subsequently propose a program which implements these operators. We discuss a number of theoretical and practical barriers to this task. We prove that determining whether two <i>SO</i>&#8707; sentences are equivalent is as hard as GRAPH ISOMORPHISM, and prove that determining whether an arbitrary <i>SO</i>&#8707; sentence represents an <i>NP</i>-complete problem is undecidable.

Le travail présenté dans cette thèse est une contribution aux méthodes formelles de spécification et de vérification. Les spécifications formelles sont utilisées pour décrire un logiciel, ou plus généralement un système, d'une manière mathématique sans ambiguïté. Des techniques de vérification formelle sont définies sur la base de ces spécifications afin d'assurer l'exactitude d'un système donné. Cependant, les méthodes formelles ne sont souvent pas pratiques et facile à utiliser dans des systèmes réels. L'une des raisons est que de nombreux formalismes de spécification ne sont pas assez riches pour couvrir à la fois les exigences orientées données et orientées comportement. Certains langages de spécification ont été proposés pour couvrir ce genre d'exigences. Le langage Circus se distingue parmi ces langues par une syntaxe et une sémantique riche et complètement intégrées.L'objectif de cette thèse est de fournir un cadre formel pour la spécification et la vérification de systèmes complexes. Les spécifications sont écrites en Circus et la vérification est effectuée soit par des tests ou par des preuves de théorèmes. Des environnements similaires de spécification et de vérification ont déjà été proposés dans la littérature. Une spécificité de notre approche est de combiner des preuves de théorème avec la génération de test. En outre, la plupart des méthodes de génération de tests sont basés sur une caractérisation syntaxique des langages étudiés. Notre environnement est différent car il est basé sur la sémantique dénotationnelle et opérationnelle de Circus. L'assistant de preuves Isabelle/HOL constitue la plateforme formelle au-dessus de laquelle nous avons construit notre environnement de spécification et de vérification.La première contribution principale de notre travail est l'environnement formel de spécification et de preuve Isabelle/Circus, basé sur la sémantique dénotationnelle de Circus. Sur la base d’Isabelle/HOL nous avons fourni une intégration vérifiée d’UTP, la base de la sémantique de Circus. Cette intégration est utilisée pour formaliser la sémantique dénotationnelle du langage Circus. L'environnement Isabelle/Circus associe à cette sémantique des outils de parsing qui aident à écrire des spécifications Circus. Le support de preuve d’Isabelle/HOL peut être utilisé directement pour raisonner sur ces spécifications grâce à la représentation superficielle de la sémantique (shallow embedding). Nous présentons une application de l'environnement à des preuves de raffinement sur des processus Circus (impliquant à la fois des données et des aspects comportementaux).La deuxième contribution est l'environnement de test CirTA construit au-dessus d’Isabelle/Circus. Cet environnement fournit deux tactiques de génération de tests symboliques qui permettent la vérification de deux notions de raffinement: l'inclusion des traces et la réduction de blocages. L'environnement est basé sur une formalisation symbolique de la sémantique opérationnelle de Circus avec Isabelle/Circus. Plusieurs définitions symboliques et tactiques de génération de test sont définies dans le cadre de CirTA. L'infrastructure formelle permet de représenter explicitement les théories de test ainsi que les hypothèses de sélection de test. Des techniques de preuve et de calculs symboliques sont la base des tactiques de génération de test. L'environnement de génération de test a été utilisé dans une étude de cas pour tester un système existant de contrôle de message. Une spécification du système est écrite en Circus, et est utilisé pour générer des tests pour les deux relations de conformité définies pour Circus. Les tests sont ensuite compilés sous forme de méthodes de test JUnit qui sont ensuite exécutées sur une implémentation Java du système étudié. / The work presented in this thesis is a contribution to formal specification and verification methods. Formal specifications are used to describe a software, or more generally a system, in a mathematical unambiguous way. Formal verification techniques are defined on the basis of these specifications to ensure the correctness of the resulting system. However, formal methods are often not convenient and easy to use in real system developments. One of the reasons is that many specification formalisms are not rich enough to cover both data-oriented and behavioral requirements. Some specification languages were proposed to cover this kind of requirements. The Circus language distinguishes itself among these languages by a rich syntax and a fully integrated semantics.The aim of this thesis is to provide a formal environment for specifying and verifying complex systems. Specifications are written in Circus and verification is performed either by testing or by theorem proving. Similar specifications and verification environment have already been proposed. A specificity of our approach is to combine supports for proofs and test generation. Moreover, most test generation methods are based on a syntactic characterization of the studied languages. Our proposed environment is different since it is based on the denotational and operational semantics of Circus. The Isabelle/HOL theorem prover is the formal platform on top of which we built our specification and verification environment.The first main contribution of our work is the Isabelle/Circus specification and proof environment based on the denotational semantics of Circus. On top of Isabelle/HOL we provide a machine-checked shallow embedding of UTP, the semantics basis of Circus. This embedding is used to formalize the denotational semantics of the Circus language. The Isabelle/Circus environment associates to this semantics some parsing facilities that help writing Circus specifications. The proof support of Isabelle/HOL can be used directly to reason on these specifications thanks to the shallow embedding of the semantics. We present an application of the environment to refinement proofs on Circus processes (involving both data and behavioral aspects). The second main contribution is the CirTA testing framework build on top of Isabelle/Circus. The framework provides two symbolic test generation tactics that allow checking two notions of refinement: traces inclusion and deadlocks reduction. The framework is based on a shallow symbolic formalization of the operational semantics of Circus using Isabelle/Circus. Several symbolic definition and test generation tactics are defined in the CirTA framework. The formal infrastructure allows us to represent explicitly test theories as well as test selection hypothesis. Proof techniques and symbolic computations are the basis of test generation tactics. The test generation environment was used for a case study to test an existing message monitoring system. A specification of the system is written in Circus, and used to generate tests following the defined conformance relations. The tests are then compiled in forms of JUnit test methods and executed against a Java implementation of the monitoring system.This thesis is a step towards, on one hand, the development of sophisticated testing tools making use of proof techniques and, on the other hand, the integration of testing and proving within formally verified software developments.

Embedded software today is pervasive: they can be found everywhere, from coffee makers and medical devices, to cars and aircraft. Embedded software today is also open and connected to the Internet, exposing them to external attacks that can cause its Control-Flow Integrity (CFI) to be violated. Control-Flow Integrity is an important safety property of software, which ensures that the behavior of the software is not inadvertently changed. The violation of CFI in software can cause unintended behaviors, and can even lead to catastrophic incidents in safety-critical systems. This dissertation develops a two-part approach for CFI: (i) prescribing source-code safetychecks, that prevent the root-causes of CFI, that programmers can insert themselves, and (ii) formally proving CFI for the machine-code of programs with source-code safety-checks. First, our prescribed safety-checks, when applied, prevent the root-causes of CFI, thereby enabling software to recover from CFI violations in a customizable way. In addition, our prescribed safety-checks are visible to programmers, empowering them to ensure that the behavior of their software is not inadvertently changed by the prescribed safety-checks. However, programmer-inserted safety-checks may be incomplete. Thus, current techniques for proving CFI, which assume that safety-checks are complete, may not work. Second, this dissertation develops a logic approach that automates formal proofs of CFI for the machine-code of software containing both source-code CFI safety-checks and system calls. We extend an existing trustworthy Hoare logic with new proof rules, proof tactics, and a novel proof-search algorithm, which exploit the principle of local reasoning for safety properties to automatically generate CFI proofs for the machine-code of programs compiled with our prescribed source-code safety-checks. To the best of our knowledge, our approach to CFI is the first to combine programmer-visible source-code enforcement mechanisms for CFI–enabling programmers to customize them and observe that their software is not inadvertently changed–with machine-code proofs of CFI that can be automated, and that does not require a trusted or verified compiler to ensure its proven properties hold in machine-code. We evaluate our CFI approach on realistic embedded software. We evaluate our approach on the MiBench and WCET benchmarks, implementations of common file utilities, and programs interfacing with hardware inputs and outputs on the Raspberry Pi single-board-computer. The variety of our target programs, and our ability to support useful features such as file and hardware inputs and outputs, demonstrate the wide applicability of our approach.

As verification efforts using interactive theorem proving grow, we are in need of certified algorithms in computer algebra to tackle problems over the real numbers. This is important because uncertified procedures can drastically increase the size of the trust base and under- mine the overall confidence established by interactive theorem provers, which usually rely on a small kernel to ensure the soundness of derived results. This thesis describes an ongoing effort using the Isabelle theorem prover to certify the cylindrical algebraic decomposition (CAD) algorithm, which has been widely implemented to solve non-linear problems in various engineering and mathematical fields. Because of the sophistication of this algorithm, people are in doubt of the correctness of its implementation when deploying it to safety-critical verification projects, and such doubts motivate this thesis. In particular, this thesis proposes a library of real algebraic numbers, whose distinguishing features include a modular architecture and a sign determination algorithm requiring only rational arithmetic. With this library, an Isabelle tactic based on univariate CAD has been built in a certificate-based way: external, untrusted code delivers solutions in the form of certificates that are checked within Isabelle. To lay the foundation for the multivariate case, I have formalised various analytical results including Cauchy's residue theorem and the bivariate case of the projection theorem of CAD. During this process, I have also built a tactic to evaluate winding numbers through Cauchy indices and verified procedures to count complex roots in some domains. The formalisation effort in this thesis can be considered as the first step towards a certified computer algebra system inside a theorem prover, so that various engineering projections and mathematical calculations can be carried out in a high-confidence framework.

Thesis. 1977. M.S.--Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Bibliography: leaves 157-158. / M.S.

Abstract not available

Systems code is almost universally written in the C programming language or a variant. C has a very low level of type and memory abstraction and formal reasoning about C systems code requires a memory model that is able to capture the semantics of C pointers and types. At the same time, proof-based verification demands abstraction, in particular from the aliasing and frame problems. In this thesis, we study the mechanisation of a series of models, from semantic to separation logic, for achieving this abstraction when performing interactive theorem-prover based verification of C systems code in higher- order logic. We do not commit common oversimplifications, but correctly deal with C's model of programming language values and the heap, while developing the ability to reason abstractly and efficiently. We validate our work by demonstrating that the models are applicable to real, security- and safety-critical code by formally verifying the memory allocator of the L4 microkernel. All formalisations and proofs have been developed and machine-checked in the Isabelle/HOL theorem prover.

Outils Génériques de Modélisation et de Démonstration pour la Formalisation des Mathématiques en Théorie des Types. Application à la Théorie des Catégories.