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1

Compositionality for Tightly Coupled Systems: A New Application of the Propositions-as-Types Interpretation

Stehr M-O.

Fundamenta Informaticae

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2008

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Vol. 82, nr 4

311-340

EN

The design of complex software systems fundamentally relies on the understanding of abstract components and their interactions. Although compositional techniques are being successfully employed in practice, the use of such techniques is often rather informal and intuitive, and typically a justification for correct behaviour of the composed system exists but is not expressed explicitly. In this paper, we show what can be gained from treating such justifications as first-class citizens. The fairly general setting for this paper is a formal development of a UNITY-style temporal logic for labeled transition systems in the calculus of inductive constructions which has been conducted using the Coq proof assistant in a formally rigorous way. Our development not only subsumes the original UNITY approach to program verification and the more recent approach of New UNITY, but goes beyond it in several essential aspects, such as the generality of the program/system model (arbitrary labeled transition systems instead of UNITY programs), the notion of fairness (weak group fairness instead of unconditional fairness), and the issue of compositionality (not only for safety but also for liveness assertions). The last aspect, which we feel is crucial in the foundations for software engineering, is subject of this paper. We present a general proof rule for compositional verification of liveness assertions in tightly coupled systems. It relies on a notion of compositional proofs, which in turn is closely related to classical work on interference-free proofs for parallel programs. The formulation of this new proof rule and the verification of its soundness does not only exploit the strong inductive reasoning capabilities of the calculus of inductive constructions, but it also uses the propositions-as-types interpretation and the associated proofs-as-objects interpretation in an essential way.

2

The Open Calculus of Constructions (Part I) : An Equational Type Theory with Dependent Types for Programming, Specification, and Interactive Theorem Proving

Stehr M-O.

Fundamenta Informaticae

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2005

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Vol. 68, nr 1/2

131--174

EN

The open calculus of constructions integrates key features of Martin-Löf's type theory, the calculus of constructions, membership equational logic, and rewriting logic into a single uniform language. The two key ingredients are dependent function types and conditional rewriting modulo equational theories. We explore the open calculus of constructions as a uniform framework for programming, specification and interactive verification in an equational higher-order style. By having equational logic and rewriting logic as executable sublogics we preserve the advantages of a first-order semantic and logical framework and we provide a foundation for a broad spectrum of applications ranging from what could be called executable mathematics, involving symbolic computations and logical proofs, to software and system engineering applications, involving symbolic execution and analysis of nondeterministic and concurrent systems.

3

Revisiting the Algebra of Petri Net Processes under the Collective Token Philosophy

Coja-Oghlan A., Stehr M-O.

Fundamenta Informaticae

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2003

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Vol. 54, nr 2,3

151-164

EN

Revisiting the view of ``Petri nets as monoids'' suggested by Meseguer and Montanari, we give a direct proof of the well-known result that the class of Best/Devillers processes, which represents the behavior of Petri nets under the collective token semantics, has a sound and complete axiomatization in terms of symmetric monoidal categories. Using membership equational logic for the axiomatization, we prove the result by an explicit construction of a natural isomorphism between suitable functors. Our interest in the collective token semantics is motivated by earlier work on the use of rewriting logic as a uniform framework for different Petri net classes, especially including high-level Petri nets, where individuality of tokens can be already expressed at the system level.