Interpretation Frameworks Patrick Cousot LIENS, Ecole Normale Superieure 45, rue d’Ulm 75230 Paris cedex 05 (France) cousot@dmi.ens.fr Radhia Cousot LIX, Ecole Polytechnique 91128 Palaiseau cedex (France) radhia@polytechnique.fr

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Abstract Interpretation Frameworks

This paper is about better engineering of cyber-physical systems (CPSs) through better models. Deterministic models have historically proven extremely useful and arguably form the kingpin of the industrial revolution and the digital and information technology revolutions. Key deterministic models that have proven successful include differential equations, synchronous digital logic and single-threaded imperative programs. Cyber-physical systems, however, combine these models in such a way that determinism is not preserved. Two projects show that deterministic CPS models with faithful physical realizations are possible and practical. The first project is PRET, which shows that the timing precision of synchronous digital logic can be practically made available at the software level of abstraction. The second project is Ptides (programming temporally-integrated distributed embedded systems), which shows that deterministic models for distributed cyber-physical systems have practical faithful realizations. These projects are existence proofs that deterministic CPS models are possible and practical.

The past, present and future of cyber-physical systems: a focus on models.

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Real-Time Systems

Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary meta-theory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic meta-theory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are self-justifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic first-order logic. It is also established that, in many cases, the natural deduction systems induce well-known intuitionistic modal logics, previously given by Hilbert-style axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their susceptibility to proof-theoretic techniques. Strong normalization (and confluence) results are proved for all of the systems. Normalization is then used to establish the completeness of cut-free sequent calculi for all of the systems, and decidability for some of the systems. Lastly, techniques developed throughout the thesis are used to establish that those intuitionistic modal logics proved decidable also satisfy the finite model property. For the logics considered, decidability and the finite model property presented open problems.

The proof theory and semantics of intuitionistic modal logic

We reconsider the foundations of modal logic, following Martin-Lof's methodology of distinguishing judgments from propositions. We give constructive meaning explanations for necessity and possibility, which yields a simple and uniform system of natural deduction for intuitionistic modal logic that does not exhibit anomalies found in other proposals. We also give a new presentation of lax logic and find that the lax modality is already expressible using possibility and necessity. Through a computational interpretation of proofs in modal logic we further obtain a new formulation of Moggi's monadic metalanguage.

A judgmental reconstruction of modal logic

We investigate a peculiar intuitionistic modal logic, called Propositional Lax Logic (PLL), which has promising applications to the formal verification of computer hardware. The logic has emerged from an attempt to express correctness up to behavioural constraints?a central notion in hardware verification?as a logical modality. As a modal logic it is special since it features a single modal operator ? that has a flavour both of possibility and of necessity. In the paper we provide the motivation for PLL and present several technical results. We investigate some of its proof-theoretic properties, presenting a cut-elimination theorem for a standard Gentzen-style sequent presentation of the logic. We go on to define a new class of fallible two-frame Kripke models for PLL. These models are unusual since they feature worlds with inconsistent information; furthermore, the only frame condition imposed is that the ?-frame be a subrelation of the ?-frame. We give a natural translation of these models into Goldblatt's J-space models of PLL. Our completeness theorem for these models yields a Godel-style embedding of PLL into a classical bimodal theory of type (S4,S4) and underpins a simple proof of the finite model property. We proceed to prove soundness and completeness of several theories for specialized classes of models. We conclude with a brief exploration of two concrete and rather natural types of model from hardware verification for which the modality ? models correctness up to timing constraints. We obtain decidability of ?-free fragment of the logic of the first type of model, which coincides with the stable form of Maksimova's intermediate logicL?.

Propositional Lax Logic

We consider two systems of constructive modal logic which are computationally motivated. Their modalities admit several computational interpretations and are used to capture intensional features such as notions of computation, constraints, concurrency, etc. Both systems have so far been studied mainly from type-theoretic and category-theoretic perspectives, but Kripke models for similar systems were studied independently. Here we bring these threads together and prove duality results which show how to relate Kripke models to algebraic models and these in turn to the appropriate categorical models for these logics.

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Categorical and Kripke Semantics for Constructive S4 Modal Logic

The semantics of Statecharts macro steps, as introduced by Pnueli and Shalev [1991], lacks compositionality. This article first analyzes the compositionality problem and traces it back to the invalidity of the Law of the Excluded Middle. It then characterizes the semantics via a particular class of linear intuitionistic Kripke models. This yields, for the first time in the literature, a simple fully abstract semantics that interprets Pnueli and Shalev's concept of failure naturally. The results not only give insight into the semantic subtleties of Statecharts, but also provide a basis for an implementation, for developing algebraic theories for macro steps, and for comparing different Statecharts variants.

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The intuitionism behind Statecharts steps

Mosel is a new tool-set for the analysis and verification in Monadic Second-order Logic. In this paper we concentrate on the system's design: Mosel is a tool-set to include a flexible set of decision procedures for several theories of the logic complemented by a variety of support components for input format translations, visualization, and interfaces to other logics and tools. The main distinguishing features of Mosel are its layered approach to the logic, based on a formal semantics for a minimal subset, its modular design, and its integration in a heterogeneous analysis and verification environment.

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MOSEL: A FLexible Toolset for Monadic Second-Order Logic

We present a new visual language, SCCharts, designed for specifying safety-critical reactive systems. SCCharts use a statechart notation and provide determinate concurrency based on a synchronous model of computation (MoC), without restrictions common to previous synchronous MoCs. Specifically, we lift earlier limitations on sequential accesses to shared variables, by leveraging the sequentially constructive MoC. The semantics and key features of SCCharts are defined by a very small set of elements, the Core SCCharts, consisting of state machines plus fork/join concurrency. We also present a compilation chain that allows efficient synthesis of software and hardware.

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SCCharts: sequentially constructive statecharts for safety-critical applications: HW/SW-synthesis for a conservative extension of synchronous statecharts

Michael Mendler

Papers

Propositional Lax Logic

Categorical and Kripke Semantics for Constructive S4 Modal Logic

The intuitionism behind Statecharts steps

MOSEL: A FLexible Toolset for Monadic Second-Order Logic

SCCharts: sequentially constructive statecharts for safety-critical applications: HW/SW-synthesis for a conservative extension of synchronous statecharts