Perspectives for proof unwinding by programming languages techniques

In this chapter, we propose some future directions of work, potentially beneficial to Mathematics and its foundations, based on the recent import of methodology from the theory of programming languages into proof theory. This scientific essay, written for the audience of proof theorists as well as the working mathematician, is not a survey of the field, but rather a personal view of the author who hopes that it may inspire future and fellow researchers

Unwinding biological systems

Unwinding conditions have been fruitfully exploited in Information Flow Security to define persistent security properties. In this paper we investigate their meaning and possible uses in the analysis of biological systems. In particular, we elaborate on the notion of robustness and propose some instances of unwinding over the process algebra Bio-PEPA and over hybrid automata. We exploit such instances to analyse two case-studies: Neurospora crassa circadian system and Influenza kinetics models

Secrecy for Mobile Implementations of Security Protocols

Mobile code technology offers interesting possibilities to the practitioner, but also raises strong concerns about security. One aspect of security is secrecy, the preservation of confidential information. This thesis investigates the modelling, specification and verification of secrecy in mobile applications which access and transmit confidential information through a possibly compromised medium (e.g. the Internet). These applications can be expected to communicate secret information using a security protocol, a mechanism to guarantee that the transmitted data does not reach unauthorized entities. The central idea is therefore to relate the secrecy properties of the application to those of the protocol it implements, through the definition of a ``confidential protocol implementation'' relation. The argument takes an indirect form, showing that a confidential implementation transmits secret data only in the ways indicated by the protocol. We define the implementation relation using labelled transition semantics, bisimulations and relabelling functions. To justify its technical definition, we relate this property to a notion of noninterference for nondeterministic systems derived from Cohen's definition of Selective Independency. We also provide simple and local conditions that greatly simplify its verification, and report on our experiments on an architecture showing how the proposed formulations could be used in practice to enforce secrecy of mobile code

The Quantum Monadology

The modern theory of functional programming languages uses monads for encoding computational side-effects and side-contexts, beyond bare-bone program logic. Even though quantum computing is intrinsically side-effectful (as in quantum measurement) and context-dependent (as on mixed ancillary states), little of this monadic paradigm has previously been brought to bear on quantum programming languages. Here we systematically analyze the (co)monads on categories of parameterized module spectra which are induced by Grothendieck's "motivic yoga of operations" -- for the present purpose specialized to HC-modules and further to set-indexed complex vector spaces. Interpreting an indexed vector space as a collection of alternative possible quantum state spaces parameterized by quantum measurement results, as familiar from Proto-Quipper-semantics, we find that these (co)monads provide a comprehensive natural language for functional quantum programming with classical control and with "dynamic lifting" of quantum measurement results back into classical contexts. We close by indicating a domain-specific quantum programming language (QS) expressing these monadic quantum effects in transparent do-notation, embeddable into the recently constructed Linear Homotopy Type Theory (LHoTT) which interprets into parameterized module spectra. Once embedded into LHoTT, this should make for formally verifiable universal quantum programming with linear quantum types, classical control, dynamic lifting, and notably also with topological effects.Comment: 120 pages, various figure

Causality-based verification

Program verification is one of the central research topics in computer science since its inception – we can consider the field to be initiated as early as in 1949, with Alan Turing’s pioneering paper “Checking a Large Routine.” Yet, we are still far from the dream of automatically proving every computer program correct. Two aspects make this problem particularly challenging: concurrent program execution on parallel processors, and large, or even infinite, state spaces of data-manipulating programs. Nowadays, with concurrency entering everywhere, from smartphones to aircrafts, proving the correctness of infinite-state concurrent programs becomes increasingly more important: we do want to be sure that the program that controls the airplane we are flying in is correct. In this thesis we propose a new approach to the verification of infinitestate concurrent programs. We call it causality-based, because it captures in an automatic proof system the “cause-effect” reasoning principles, which are often used informally in manual correctness proofs. While traditionally automatic methods are based on the state space exploration, our method is based on a new concurrency model, called concurrent traces, which are the abstractions of the history of a concurrent program to some key events and the relationships between them. Causality-based proof rules relate concurrent traces with each other, by formally tracking what are the necessary consequences (the “effects”) from a particular analysis situation (the “cause”). The full correctness proof is then a composition of such primitive proof steps. We study the syntactic and language-based properties of concurrent traces, and characterize the complexity of such operations as emptiness checking and language inclusion. Regarding the program correctness, we develop proof systems for the broad classes of safety and liveness properties, and provide algorithms for the automatic construction of correctness proofs. We demonstrate that for practically relevant classes of programs, such as multi-threaded programs with binary semaphores, the constructed proofs are of polynomial size, and can be also checked in polynomial time. The methods of the thesis have been implemented in Arctor, the first scalable termination prover for concurrent programs, which is able to handle programs with hundreds of non-trivial threads.Die Programmverifikation ist seit den Anfängen der Informatik eines ihrer zentralen Forschungsfelder. Als Beginn dieser Forschungsrichtung kann bereits das Jahr 1949 betrachtet werden, in dem Alan Turings bahnbrechende Arbeit “Checking a Large Routine” erschien. Der Traum, die Korrektheit von Programmen stets automatisch beweisen zu können, ist aber auch heute noch weit davon entfernt, Realität zu sein. Es gibt zwei Aspekte, die dieses Problem zu einer solch großen Herausforderung machen: die nebenläufige Ausführung von Programmen auf Parallelrechnern, und die großen, oder sogar unendlichen, Zustandsräume von datenverarbeitenden Programmen. Nebenläufige Programme werden in immer mehr Anwendungsbereichen, von Handys bis zur Luftfahrt, eingesetzt. Automatische Korrektheitsbeweise werden daher immer wichtiger: wenn wir mit dem Flugzeug reisen, möchten wir sicher sein, dass das Programm, das das Flugzeug steuert, auch tatsachlich korrekt ist. In dieser Arbeit schlagen wir einen neuen Ansatz für die Verifikation von nebenläufigen Programmen mit unendlichem Zustandsraum vor. Wir nennen den Ansatz “kausalitätsbasiert”, weil er im Rahmen eines automatischen Beweissystems die “Ursache-Wirkung”-Beziehungen erfasst, die sonst eher informell in manuellen Korrektheitsbeweisen benutzt werden. Anders als traditionelle automatische Methoden, die den Zustandsraums explorieren, baut unser Ansatz auf einem neuen nebenläufigen Berechnungsmodell, dem der “nebenläufigen Spuren”, auf. Eine nebenläufige Spur ist eine Abstraktion der Vergangenheit eines nebenläufigen Programms im Hinblick auf bestimmte Schlüsselereignisse und die Beziehungen zwischen diesen Ereignissen. Kausalitätsbasierte Beweis-regeln setzen nebenläufige Spuren zueinander in Bezug, indem die Konsequenzen (die “Wirkungen”) einer bestimmten analytischen Situation (der “Ursache”) auf eine formale Art und Weise verfolgt werden. Der vollständige Korrektheitsbeweis setzt sich dann aus solchen einfachen Beweisschritten zusammen. Wir untersuchen die syntaktischen und sprachtheoretischen Eigenschaften von nebenläufigen Spuren, und charakterisieren die Komplexität von Operationen wie den Tests auf leere Sprache und Sprachinklusion. Wir entwickeln Beweissysteme zum Nachweis der Programmkorrektheit für die allgemeinen Klassen der Sicherheits- und Lebendigkeitseigenschaften, und stellen Algorithmen vor, die solche Beweise automatisch konstruieren. Für aus praktischer Sicht relevante Klassen von Programmen, wie Multi-Thread Programme mit binären Semaphoren, zeigen wir, dass die konstruierten Beweise polynomiell groß sind und auch in polynomieller Zeit geprüft werden können. Die in der Arbeit vorgestellten Methoden wurden im Verifikationswerkzeug Arctor implementiert. Arctor is der erste skalierbare Terminierungsbeweiser für nebenläufige Programme. Arctor kann Programme mit Hunderten nicht-trivialer Threads verarbeiten