Publications

Zélus is a new programming language for modeling systems that mix discrete logical time and continuous time behaviors. From a user's perspective, its main originality is to extend an existing -like synchronous language with Ordinary Differential Equations (ODEs). The extension is conservative: any synchronous program expressed as data-flow equations and hierarchical automata can be composed arbitrarily with ODEs in the same source code.

A dedicated type system and causality analysis ensure that all discrete changes are aligned with zero-crossing events so that no side effects or discontinuities occur during integration. Programs are statically scheduled and translated into sequential code which, by construction, runs in bounded time and space. Compilation is effected by source-to-source translation into a small synchronous subset which is processed by a standard synchronous compiler architecture. The resulting code is paired with an off-the-shelf numeric solver.

This experiment show that it is possible to build a modeler for explicit hybrid systems à la Simulink/Stateflow on top of an existing synchronous language, using it both as a semantic basis and as a target for code generation.

Efficiently distributing synchronous programs is a challenging and long-standing subject. This paper introduces the use of futures in a Lustre-like language, giving the programmer control over the expression of parallelism. In the synchronous model where computations are considered instantaneous, futures increase expressiveness by decoupling the beginning from the end of a computation.

Through a number of examples, we show how to desynchronize long computations and implement parallel patterns such as fork-join, pipelining and data parallelism. The proposed extension preserves the main static properties of the base language, including static resource bounds and the absence of deadlock, livelock and races. Moreover, we prove that adding or removing futures preserves the underlying synchronous semantics.

The generation of efficient sequential code for synchronous data-flow languages raises two intertwined issues: control and memory optimization. While the former has been extensively studied, for instance in the compilation of Lustre and SIGNAL, the latter has been only addressed in a restricted manner. Yet, memory optimization becomes a pressing issue when arrays are added to such languages.

This article presents a two-levels solution to the memory optimization problem. It combines a compile-time optimization algorithm, reminiscent of register allocation, paired with language annotations on the source given by the designer. Annotations express in-place modifications and control where allocation is performed. Moreover, they allow external functions performing in-place modifications to be imported safely. Soundness of annotations is guaranteed by a semilinear type system and additional scheduling constraints. A key feature is that annotations for well-typed programs do not change the semantics of the language: removing them may lead to a less efficient code but with the very same semantics.

The method has been implemented in a new compiler for a Lustre-like synchronous language extended with hierarchical automata and arrays. Experiments show that the proposed approach removes most of the unnecessary array copies, resulting in faster code that uses less memory.

Hybrid system modelers have become a corner stone of complex embedded system development. Embedded systems include not only control components and software, but also physical devices. In this area, Simulink is a de facto standard design framework, and Modelica a new player. However, such tools raise several issues related to the lack of reproducibility of simulations (sensitivity to simulation parameters and to the choice of a simulation engine).

In this paper we propose using techniques from non-standard analysis to define a semantic domain for hybrid systems. Non-standard analysis is an extension of classical analysis in which infinitesimal (the ε and η in the celebrated generic sentence for allεthere existsη... of college maths) can be manipulated as first class citizens. This approach allows us to define both a denotational semantics, a constructive semantics, and a Kahn Process Network semantics for hybrid systems, thus establishing simulation engines on a sound but flexible mathematical foundation. These semantics offer a clear distinction between the concerns of the numerical analyst (solving differential equations) and those of the computer scientist (generating execution schemes).

We also discuss a number of practical and fundamental issues in hybrid system modelers that give rise to non reproducibility of results, nondeterminism, and undesirable side effects. Of particular importance are cascaded mode changes (also called “zero-crossings” in the context of hybrid systems modelers).

Lucy-n is a data-flow programming language similar to Lustre extended with a buffer operator. It is based on the n-synchronous model which was initially introduced for programming multimedia streaming applications. In this article, we show that Lucy-n is also applicable to model Latency Insensitive Designs (LID). In order to model latency introduced by wires, we add a delay operator. Thanks to this new operator, a LID can be described by a Lucy-n program. Then, the Lucy-n compiler automatically provides static schedules for computation nodes and buffer sizes needed in shell wrappers.

Hybrid modeling tools such as Simulink have evolved from simulation platforms into development platforms on which simulation, testing, formal verification and code generation are performed. It is thus critical to place them on a firm semantical basis where it can be proven that the results of simulation, compilation and verification are mutually consistent. Synchronous languages have addressed these issues but only for discrete systems. They cannot be used to model hybrid systems with both efficiency and precision.

Following the approach of Benveniste et al., we present the design of a hybrid modeler built from a synchronous language and an off-the-shelf numerical solver. The main novelty is a language which includes control structures, such as hierarchical automata, for both continuous and discrete contexts. These constructs can be arbitrarily mixed with data-flow and ordinary differential equations (ODEs). A type system is required to statically ensure that all discrete state changes are aligned with zero-crossing events and that the function passed to the numerical solver is free of side-effects during integration. We show that well-typed programs can be compiled through a source-to-source translation into synchronous code which is then translated into sequential code using an existing synchronous language compiler.

Based on the presented material, a compiler for a new synchronous language with hybrid features has been developed. We demonstrate its effectiveness on some examples.

Hybrid modelers such as Simulink have become corner stones of embedded systems development. They allow both discrete controllers and their continuous environments to be expressed in a single language. Despite the availability of such tools, there remain a number of issues related to the lack of reproducibility of simulations and to the separation of the continuous part, which has to be exercised by a numerical solver, from the discrete part, which must be guaranteed not to evolve during a step.

Starting from a minimal, yet full-featured, Lustre-like synchronous language, this paper presents a conservative extension where data-flow equations can be mixed with ordinary differential equations (ODEs) with possible reset. A type system is proposed to statically distinguish discrete computations from continuous ones and to ensure that signals are used in their proper domains. We propose a semantics based on non-standard analysis which gives a synchronous interpretation to the whole language, clarifies the discrete/continuous interaction and the treatment of zero-crossings, and also allows the correctness of the type system to be established.

The extended data-flow language is realized through a source-to-source transformation into a synchronous subset, which can then be compiled using existing tools into routines that are both efficient and bounded in their use of memory. These routines are orchestrated with a single off-the-shelf numerical solver using a simple but precise algorithm which treats causally-related cascades of zero-crossings. We have validated the viability of the approach through experiments with the SUNDIALS library.

Hybrid systems modelers have become the corner stone of embedded system development, with Simulink a de facto standard and Modelica a new player. Such tools still raise a number of issues that, we believe, require more fundamental understanding.

In this paper we propose using non standard analysis as a semantic domain for hybrid systems - non standard analysis is an extension of classical analysis in which infinitesimals (the ε and η in the celebrated generic sentence for allε. there existsη... in college maths) can be manipulated as first class citizens. This allows us to provide a denotational semantics and a constructive semantics for hybrid systems, thus establishing simulation engines on a firm mathematical basis. In passing, we cleanly separate the job of the numerical analyst (solving differential equations) from that of the computer scientist (generating execution schemes).

Synchronous functional languages such as Lustre or Lucid Synchrone define a restricted class of Kahn Process Networks which can be executed with no buffer. Every expression is associated to a clock indicating the instants when a value is present. A dedicated type system, the clock calculus, checks that the actual clock of a stream equals its expected clock and thus does not need to be buffered. The n-synchrony relaxes synchrony by allowing the communication through bounded buffers whose size is computed at compile-time. It is obtained by extending the clock calculus with a subtyping rule which defines buffering points.

This paper presents the first implementation of the n-synchronous model inside a Lustre-like language called LucyN. The language extends Lustre with an explicit "buffer" construct whose size is automatically computed during the clock calculus. This clock calculus is defined as an inference type system and is parametrized by the clock language and the algorithm used to solve subtyping constraints. We detail here one algorithm based on the abstraction of clocks, an idea originally introduced in [16]. The paper presents a simpler, yet more precise, clock abstraction for which the main algebraic properties have been proved in Coq. Finally, we illustrate the language on various examples including a video application.

This paper addresses the question of producing modular sequential imperative code from synchronous data-flow networks. Precisely, given a system with several input and output flows, how to decompose it into a minimal number of classes executed atomically and statically scheduled without restricting possible feedback loops between input and output? Though this question has been identified by Raymond in the early years of LUSTRE, it has almost been left aside until the recent work of Lublinerman, Szegedy and Tripakis. The problem is proven to be intractable, in the sense that it belongs to the family of optimization problems where the corresponding decision problem - there exists a solution with size c - is NP-complete. Then, the authors derive an iterative algorithm looking for solutions for c = 1, 2,... where each step is encoded as a satisfiability (SAT) problem.

Despite the apparent intractability of the problem, our experience is that real programs do not exhibit such a complexity. Based on earlier work by Raymond, the current paper presents a new encoding of the problem in terms of input/output relations. This encoding simplifies the problem, in the sense that it rejects some solutions, while keeping all the optimal ones. It allows, in polynomial time, (1) to identify nodes for which several schedules are feasible and thus are possible sources of combinatorial explosion; (2) to obtain solutions which in some cases are already optimal; (3) otherwise, to get a non trivial lower bound for c to start an iterative combinatorial search. The method has been validated on several industrial examples.

This paper addresses the question of producing modular sequential imperative code from synchronous data-flow networks. Precisely, given a system with several input and output flows, how to decompose it into a minimal number of classes executed atomically and statically scheduled without restricting possible feedback loops between input and output? Though this question has been identified by Raymond in the early years of Lustre, it has almost been left aside until the recent work of Lublinerman, Szegedy and Tripakis. The problem is proven to be intractable, in the sense that it belongs to the family of optimization problems where the corresponding decision problem - there exists a solution with size c - is NP-complete. Then, the authors derive an iterative algorithm looking for solutions for c = 1,2,... where each step is encoded as a SAT problem.

Despite the apparent intractability of the problem, our experience is that real programs do not exhibit such a complexity. Based on earlier work by Raymond, this paper presents a new symbolic encoding of the problem in terms of input/output relations. This encoding simplifies the problem, in the sense that it rejects solutions, while keeping all the optimal ones. It allows, in polynomial time, (1) to identify nodes for which several schedules are feasible and thus are possible sources of combinatorial explosion; (2) to obtain solutions which in some cases are already optimal; (3) otherwise, to get a non trivial lower bound for c to start an iterative combinatorial search.

The solution applies to a large class of block-diagram formalisms based on atomic computations and a delay operator, ranging from synchronous languages such as Lustre or SCADE to modeling tools such as Simulink.

This paper addresses the problem of designing and implementing complex control systems for real-time embedded software. Typical applications involve different control laws corresponding to different phases or modes, e.g., take-off, full flight and landing in a fly-by-wire control system. On one hand, existing methods such as the combination of Simulink/Stateflow provide powerful but unsafe mechanisms by means of imperative updates of shared variables. On the other hand, synchronous languages and tools such as Esterel or Scade/Lustre are too restrictive and forbid to fully separate the specification of modes from their actual instantiation with a particular control automaton.

In this paper, we introduce a conservative extension of a synchronous data-flow language close to Lustre, in order to be able to define systems with modes in a more modular way, while insuring the absence of data-races. We show that such a system can be viewed as an object where modes are methods acting on a shared memory. The object is associated to a scheduling policy which specifies the ways methods can be called to build a valid synchronous reaction. We show that the verification of the proper use of an object reduces to a type inference problem using row types introduced by Wand, Rémy and Vouillon. We define the semantics of the extended synchronous language and the type system. The proposed extension has been implemented and we illustrate its use through several examples.

Synchronous data-flow languages such as Lustre manage infinite sequences or streams as basic values. Each stream is associated to a clock which defines the instants where the current value of the stream is present. This clock is a type information and a dedicated type system - the so-called clock-calculus - statically rejects programs which cannot be executed synchronously. In existing synchronous languages, it amounts at asking whether two streams have the same clocks and thus relies on clock equality only. Recent works have shown the interest of introducing some relaxed notion of synchrony, where two streams can be composed as soon as they can be synchronized through the introduction of a finite buffer (as done in the SDF model of Edward Lee). This technically consists in replacing typing by sub-typing. The present paper introduces a simple way to achieve this relaxed model through the use of clock envelopes. These clock envelopes are set of concrete clocks which are not necessarily periodic. This allows to model various features in real-time embedded software such as bounded jitter as found in video-systems, execution time of real-time processes and scheduling resources or the communication through buffers. We present the algebra of clock envelopes and its main theoretical properties.

We address the design of distributed systems with synchronous dataflow programming languages. As modular design entails handling both architecture and functional modularity, our first contribution is to extend an existing synchronous dataflow programming language with primitives allowing the description of a distributed architecture and the localization of some expressions onto some processors. We also present a distributed semantics to formalize the distributed execution of synchronous programs. Our second contribution is to provide a type system, in order to infer the localization of non-annotated values by means of type inference and to ensure, at compilation time, the consistency of the distribution. Our third contribution is to provide a type-directed projection operation to obtain automatically, from a centralized typed program, the local program to be executed by each computing resource. The type system as well as the automatic distribution mechanism has been fully implemented in the compiler of an existing synchronous data-flow programming language.

The compilation of synchronous block diagrams into sequential imperative code has been addressed in the early eighties and can now be considered as folklore. However, modular code generation, though largely used in existing compilers and particularly in industrial ones, has never been precisely described or entirely formalized. Such a formalization is now fundamental in the long-term goal to develop a mathematically certified compiler for a synchronous language as well as in simplifying existing implementations.

This article presents in full detail the modular compilation of synchronous block diagrams into sequential code. We consider a first-order functional language reminiscent of Lustre, which it extends with a general n-ary merge operator, a reset construct, and a richer notion of clocks. The clocks are used to express activation of computations in the program and are specifically taken into account during the compilation process to produce efficient imperative code. We introduce a generic machine-based intermediate language to represent transition functions, and we present a concise clock-directed translation from the source to this intermediate language. We address the target code generation phase by describing a translation from the intermediate language to Java and C.

Lucid Synchrone is a programming language dedicated to the design of reactive systems. It is based on the synchronous model of Lustre which it extends with features usually found in functional languages such as higher-order or constructed data-types. The language is equipped with several static analysis, all expressed as special type-systems and used to ensure the absence of certain run-time errors on the final application. It provides, in particular, automatic type and clock inference and statically detects initialization issues or dead-locks. Finally, the language offers both data-flow and automata-based programming inside a unique framework.

This paper presents a programming experiment of complex network routing protocols for mobile ad hoc networks within the reactive language ReactiveML.

Mobile ad hoc networks are highly dynamic networks characterized by the absence of physical infrastructure. In such networks, nodes are able to move, evolve concurrently and synchronize continuously with their neighbors. Due to mobility, connections in the network can change dynamically and nodes can be added or removed at any time. All these characteristics - concurrency with many communications and the need of complex data-structure - combined to our routing protocol specifications make the use of standard simulation tools (e.g., NS-2, OPNET) inadequate. Moreover network protocols appear to be very hard to program efficiently in conventional programming languages.

In this paper, we show that the synchronous reactive model as introduced in the pioneering work of Boussinot matters for programming such systems. This model provides adequate programming constructs - namely synchronous parallel composition, broadcast communication and dynamic creation - which allow a natural implementation of the hard part of the simulation. This thesis is supported by two concrete examples: the first example is a routing protocol in mobile ad hoc networks and the simulation focuses only on the network layer. The second one is a routing protocol for sensors networks and the wholes layers are faithfully simulated (hardware, MAC and network layers). More importantly, the physical environment (e.g., clouds) has also been integrated into the simulation using the tool Lucky.

The implementation has been done in ReactiveML, an embedding of the reactive model inside a statically typed, strict functional language. ReactiveML provides reactive programming constructs together with most of the features of Ocaml. Moreover, it provides an efficient execution scheme for reactive constructs which made the simulation of real-size examples (with several thousand of nodes) feasible.

Synchronous data-flow languages such as SCADE/Lustre manage infinite sequences or streams as primitive values making them naturally adapted to the description of data-dominated systems. Their conservative extension with means to define control-structures or modes have been a long-term research topic and several solutions have emerged.

In this paper, we pursue this effort and generalize existing solutions by providing two constructs: a general form of state machines called parameterized state machines, and valued signals, as can be found in Esterel. Parameterized state machines greatly reduce the reliance on error-prone mechanisms such as shared memory in automaton-based programming. Signals provide a new way of programming with multi-rate data in synchronous data-flow languages. Together, they allow for a much more direct and natural programming of systems that combine data-flow and state-machines.

The proposed extension is fully implemented in the new Lucid Synchrone compiler.

The design of high-performance stream-processing systems is a fast growing domain, driven by markets such like high-end TV, gaming, 3D animation and medical imaging. It is also a surprisingly demanding task, with respect to the algorithmic and conceptual simplicity of streaming applications. It needs the close cooperation between numerical analysts, parallel programming experts, real-time control experts and computer architects, and incurs a very high level of quality insurance and optimization.

In search for improved productivity, we propose a programming model and language dedicated to high-performance stream processing. This language builds on the synchronous programming model and on domain knowledge - the periodic evolution of streams - to allow correct-by-construction properties to be proven by the compiler. These properties include resource requirements and delays between input and output streams. Automating this task avoids tedious and error-prone engineering, due to the combinatorics of the composition of filters with multiple data rates and formats. Correctness of the implementation is also difficult to assess with traditional (asynchronous, simulation-based) approaches. This language is thus provided with a relaxed notion of synchronous composition, called n-synchrony: two processes are n-synchronous if they can communicate in the ordinary (0-)synchronous model with a FIFO buffer of size n.

Technically, we extend a core synchronous data-flow language with a notion of periodic clocks, and design a relaxed clock calculus (a type system for clocks) to allow non strictly synchronous processes to be composed or correlated. This relaxation is associated with two sub-typing rules in the clock calculus. Delay, buffer insertion and control code for these buffers are automatically inferred from the clock types through a systematic transformation into a standard synchronous program. We formally define the semantics of the language and prove the soundness and completeness of its clock calculus and synchronization transformation. Finally, the language is compared with existing formalisms.

Ce chapitre présente Lucid Synchrone, un langage dédié à la programmation de systèmes réactifs. Il est fondé sur le modèle synchrone de Lustre qu'il étend avec des caractéristiques présentes dans les langages fonctionnels tels que l'ordre supérieur ou l'inférence des types. Il offre un mécanisme de synthèse automatique des horloges et permet de décrire, dans un cadre unifié, une programmation flot de données et une programmation par automates.

Ce chapitre est une présentation du langage, destinée à la fois au programmeur cherchant à se familiariser avec celui-ci, mais aussi au chercheur s'intéressant aux systèmes réactifs et aux techniques qui peuvent être appliquées au niveau d'un langage, pour faciliter les activités de spécification, de programmation et de vérification.

This paper presents an extension of a synchronous data-flow language such as Lustre with imperative features expressed in terms of powerful state machine à la SyncChart. This extension is fully conservative in the sense that all the programs from the basic language still make sense in the extended language and their semantics is preserved.

From a syntactical point of view this extension consists in hierarchical state machines that may carry at each hierarchy level a bunch of equations. This proposition is an alternative to the joint use of Simulink and Stateflow but improves it by allowing a fine grain mix of both styles.

The central idea of the paper is to base this extension on the use of clocks, translating imperative constructs into well clocked data-flow programs from the basic language. This clock directed approach is an easy way to define a semantics for the extension, it is light to implement in an existing compiler and experiments show that the generated code compete favorably with ad-hoc techniques. The proposed extension has been implemented in the ReLuC compiler of Scade/Lustre and in the Lucid Synchrone compiler.

We present ReactiveML, a programming language dedicated to the implementation of complex reactive systems as found in graphical user interfaces, video games or simulation problems. The language is based on the reactive model introduced by Boussinot. This model combines the so-called synchronous model found in Esterel which provides instantaneous communication and parallel composition with classical features found in asynchronous models like dynamic creation of processes.

The language comes as a conservative extension of an existing call-by-value ML language and it provides additional constructs for describing the temporal part of a system. The language receives a behavioral semantics à la Esterel and a transition semantics describing precisely the interaction between ML values and reactive constructs. It is statically typed through a Milner type inference system and programs are compiled into regular ML programs. The language has been used for programming several complex simulation problems (e.g., routing protocols in mobile ad-hoc networks).

Nous présentons ReactiveML, un langage dédié à la programmation de systèmes réactifs complexes tels que les interfaces graphiques, les jeux vidéo ou les problèmes de simulation. Le langage est basé sur le modèle réactif synchrone introduit dans les années 90 par Frédéric Boussinot. Ce modèle combine les principes du modèle synchrone avec la possibilité de créer dynamiquement des processus.

ReactiveML est une extension conservative d'un langage ML avec appel par valeur. Il ajoute des constructions supplémentaires pour décrire les comportements temporels des systèmes. Nous illustrons l'expressivité du langage sur des exemples et donnons une sémantique formelle du noyau du langage. La sémantique exprime les interactions entre les expressions ML et les constructions réactives. Le langage est statiquement typé avec un système d'inférence de type à la Milner.

This paper presents a programming experiment of a complex network routing protocol for mobile ad hoc networks within the ReactiveML language.

Mobile ad hoc networks are highly dynamic networks characterized by the absence of physical infrastructure. In such networks, nodes are able to move, evolve concurrently and synchronize continuously with their neighbors. Due to mobility, connections in the network can change dynamically and nodes can be added or removed at any time. All these characteristics - concurrency with many communications and the need of complex data-structure - combined to our routing protocol specifications make the use of standard simulation tools (e.g., NS, OPNET) inadequate. Moreover network protocols appear to be very hard to program efficiently in conventional programming languages.

In this paper, we show that the synchronous reactive model, as introduced in the pioneering work of Boussinot, matters for programming such systems. This model provides adequate programming constructs - namely synchronous parallel composition, broadcast communication and dynamic creation - which allow a natural implementation of the hard part of the simulation.

The implementation has been done in ReactiveML, an embedding of the reactive model inside a statically typed, strict functional language. ReactiveML provides reactive programming constructs together with most of the features of Ocaml. Moreover, it provides an efficient execution scheme for reactive constructs which made the simulation of real-size examples feasible. Experimental results show that the ReactiveML implementation is two orders of magnitude faster than the original C version; it was able to simulate more than 1000 nodes where the original C version failed (after 200 nodes) and compares favorably with the version programmed in NAB.

The paper introduces a higher-order synchronous data-flow language in which communication channels may themselves transport programs. This provides a mean to dynamically reconfigure data-flow processes. The language comes as a natural and strict extension of both Lustre and Lucid Synchrone. This extension is conservative, in the sense that a first-order restriction of the language can receive the same semantics. We illustrate the expressivity of the language with some examples, before giving the formal semantics of the underlying calculus. The language is equipped with a polymorphic type system allowing types to be automatically inferred and a clock calculus rejecting programs for which synchronous execution cannot be statically guaranteed. To our knowledge, this is the first higher-order synchronous data-flow language where stream functions are first class citizens.

Clocks in synchronous data-flow languages are the natural way to define several time scales in reactive systems. They play a fundamental role during the specification of the system and are largely used in the compilation process to generate efficient sequential code. Based on the formulation of clocks as dependent types, the paper presents a simpler clock calculus reminiscent to ML type systems with first order abstract types à la Laufer & Odersky. Not only this system provides clock inference, it shares efficient implementations of ML type systems and appears to be expressive enough for many real applications.

One of the appreciated features of the synchronous data-flow approach is that a program defines a perfectly deterministic behavior. But the use of the delay primitive leads to undefined values at the first cycle; thus a data-flow program is really deterministic only if it can be shown that such undefined values do not affect the behavior of the system.

This paper presents an initialization analysis that guarantees the deterministic behavior of programs. This property being undecidable in general, the paper proposes a safe approximation of the property, precise enough for most data-flow programs. This analysis is a one-bit analysis - expressions are either initialized or uninitialized - and is defined as an inference type system with sub-typing constraints. This analysis has been implemented in the Lucid Synchrone compiler and in a new Scade-Lustre prototype compiler. It gives very good results in practice.

We express reactive programs in Coq using data-flow synchronous operators. Following Lucid Synchrone approach, synchronous static constraints are here expressed using dependent types. Hence, our analysis of synchrony is here directly performed by Coq typechecker.

This article presents a causality analysis for a synchronous stream language with higher-order functions. This analysis takes the shape of a type system with rows. Rows were originally designed to add extensible records to the ML type system (Didier Rémy, Mitchell Wand). We also restate briefly the coiterative semantics for synchronous streams (Paul Caspi, Marc Pouzet), and prove the correctness of our analysis with respect to this semantics.

This paper presents an extension of a synchronous data-flow language providing full functionality with a modular reset operator. This operator can be considered as a basic primitive for describing dynamic reconfigurations in a purely data-flow framework. The extension proposed here is conservative with respect to the fundamental properties of the initial language: reactivity (i.e, execution in bounded memory and time) and referential transparency are kept. The reset operator is thus compatible with higher-order. This is obtained by extending the clock calculus of the initial language and providing a compilation method. We illustrate the use of this operator by describing an automatic encoding of Mode-automata. All the experiments presented in the paper has been done with Lucid Synchrone, an ML extension of Lustre.