Building Decision Procedures in the Calculus of Inductive Constructions

It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an equivalent proposition P' obtained from P thanks to possibly complex calculations. In this paper, we investigate a new version of the calculus of inductive constructions which incorporates arbitrary decision procedures into deduction via the conversion rule of the calculus. The novelty of the problem in the context of the calculus of inductive constructions lies in the fact that the computation mechanism varies along proof-checking: goals are sent to the decision procedure together with the set of user hypotheses available from the current context. Our main result shows that this extension of the calculus of constructions does not compromise its main properties: confluence, subject reduction, strong normalization and consistency are all preserved

Formal verification of concurrent programs

Interactive theorem proving provides a general approach to modeling and verification of both finite-state and infinite-state systems but requires significant human efforts to deal with many tedious proofs. On the other hand, model-checking is limited to some application domain with small finite-state space. A natural thought for this problem is to integrate these two approaches. To keep the consistency of the integration and ensure the correctness of verification, we suggest to use type theory based theorem provers (e.g. Lego) as the platform for the integration and build a model-checker to do parts of the verification automatically. We formalise a verification system of both CCS and an imperative language in the proof development system Lego which can be used to verify both finite-state and infinite-state problems. Then a model-checker, LegoMC, is implemented to generate Lego proof terras for finite-state problems automatically. Therefore people can use Lego to verify a general problem with some of its finite sub-problems verified by LegoMC. On the other hand, this integration extends the power of model-checking to verify more complicated and infinite-state models as well. The development of automatic techniques and the integration of different reasoning methods would directly benefit the verification community. It is expected that further extension and development of this verification environment would be able to handle real life systems. On the other hand, the research gives us some experiences about how to automate proofs in interactive theorem provers and therefore will improve the usability and applicability of the theorem proving technology

Cut elimination for Zermelo set theory

We show how to express intuitionistic Zermelo set theory in deduction modulo (i.e. by replacing its axioms by rewrite rules) in such a way that the corresponding notion of proof enjoys the normalization property. To do so, we first rephrase set theory as a theory of pointed graphs (following a paradigm due to P. Aczel) by interpreting set-theoretic equality as bisimilarity, and show that in this setting, Zermelo's axioms can be decomposed into graph-theoretic primitives that can be turned into rewrite rules. We then show that the theory we obtain in deduction modulo is a conservative extension of (a minor extension of) Zermelo set theory. Finally, we prove the normalization of the intuitionistic fragment of the theory

Programmiersprachen und Rechenkonzepte

Seit 1984 veranstaltet die GI-Fachgruppe "Programmiersprachen und Rechenkonzepte" regelmäßig im Frühjahr einen Workshop im Physikzentrum Bad Honnef. Das Treffen dient in erster Linie dem gegenseitigen Kennenlernen, dem Erfahrungsaustausch, der Diskussion und der Vertiefung gegenseitiger Kontakte. In diesem Forum werden Vorträge und Demonstrationen sowohl bereits abgeschlossener als auch noch laufender Arbeiten vorgestellt, unter anderem (aber nicht ausschließlich) zu Themen wie - Sprachen, Sprachparadigmen - Korrektheit von Entwurf und Implementierung - Werkzeuge - Software-/Hardware-Architekturen - Spezifikation, Entwurf - Validierung, Verifikation - Implementierung, Integration - Sicherheit (Safety und Security) - eingebettete Systeme - hardware-nahe Programmierung. In diesem Technischen Bericht sind einige der präsentierten Arbeiten zusammen gestellt

Transforming OCL to PVS: Using Theorem Proving Support for Analysing Model Constraints

The Unified Modelling Language (UML) is a de facto standard language for describing software systems. UML models are often supplemented with Object Constraint Language (OCL) constraints, to capture detailed properties of components and systems. Sophisticated tools exist for analysing UML models, e.g., to check that well-formedness rules have been satisfied. As well, tools are becoming available to analyse and reason about OCL constraints. Previous work has been done on analysing OCL constraints by translating them to formal languages and then analysing the translated constraints with tools such as theorem provers. This project contributes a transformation from OCL to the specification language of the Prototype Verification System (PVS). PVS can be used to analyse and reason about translated OCL constraints. A particular novelty of this project is that it carries out the transformation of OCL to PVS by using model transformation, as exemplified by the OMG's Model-Driven Architecture. The project implements and automates model transformations from OCL to PVS using the Epsilon Transformation Language (ETL) and tests the results using the Epsilon Comparison Language (ECL )