A Randomized Algorithm for the Capacity of Finite-State Channels

Inspired by ideas from the field of stochastic approximation, we propose a ran- domized algorithm to compute the capacity of a finite-state channel with a Markovian input. When the mutual information rate of the channel is concave with respect to the chosen parameterization, the proposed algorithm proves to be convergent to the ca- pacity of the channel almost surely with the derived convergence rate. We also discuss the convergence behavior of the algorithm without the concavity assumption.published_or_final_versio

The Degree of a Finite Set of Words

We generalize the notions of the degree and composition from uniquely decipherable codes to arbitrary finite sets of words. We prove that if X = Y?Z is a composition of finite sets of words with Y complete, then d(X) = d(Y) ? d(Z), where d(T) is the degree of T. We also show that a finite set is synchronizing if and only if its degree equals one. This is done by considering, for an arbitrary finite set X of words, the transition monoid of an automaton recognizing X^* with multiplicities. We prove a number of results for such monoids, which generalize corresponding results for unambiguous monoids of relations

A Unifying View of Loosely Time-Triggered Architectures

Cyber-Physical Systems require distributed architectures to support safety critical real-time control. Hermann Kopetz' Time-Triggered Architecture (TTA) has been proposed as both an architecture and a comprehensive paradigm for systems architecture, for such systems. TTA offers the programmer a logical discrete time compliant with synchronous programming, together with timing bounds. A clock synchronization protocol is required, unless the local clocks used themselves provide the recquired accuracy. To relax the strict requirements on synchronization imposed by TTA, Loosely Time-Triggered Architectures (LTTA) have been proposed. In LTTA, computation and communication units are all triggered by autonomous, unsynchronized, clocks. Communication media act as shared memories between writers and readers and communication is non blocking. This is at the price of communication artifacts (such as duplication or loss of data), which must be compensated for by using some "LTTA protocol". In this paper we pursue our previous work by providing a unified presentation of the two variants of LTTA (token- and time-based), with simplified analyses. We compare these two variants regarding performance and robustness and we provide ways to combine them. This report was prepared for a lecture in Gérard Berry's seminar series at the Collège de France, March 5, 2014; it is a corrected version of a paper, which appeared at Emsoft'2010. It is dedicated to our close friend Paul Caspi who died in April 2012.Les infrastructures de calcul distribuées pour le contrôle des systèmes embarqués critiques requièrent des propriétés particulières destinées à préserver les caractéristiques attendues du contrôleur. Les architectures TTA (Time-Triggered Architectures) ont été proposées par Hermann Kopetz, à la fois comme une architecture de calcul et comme une méthodologie de conception des systèmes. TTA offre au programmeur un temps logique conforme à celui de la programmation synchrone, avec en outre un contrôle strict du temps. Il requiert un protocole de synchronisation entre les horloges du système réparti. Pour affaiblir ces hypothèses, les architectures LTTA (Loosely Time-Triggered Architectures) ont été proposées récemment. Dans LTTA, les calculs et les communications sont rythmées par des horloges locales, non synchronisées. Les supports de communication se comportent comme des mémoires partagées. La communication est donc non-bloquante. Ce type de communiccation crée évidemment des artefacts à combattre par un protocole dit "LTTA". Dans cet article nous présentons une approche unifiée des deux techniques connues pour ce type de protocole, reposant sur l'usage, soit de jetons, soit du temps. On compare ces deux variantes et on étudie leur performance. Le présent rapport est une version corrigée d'un article paru à Emsoft'2010. Il est dédié à notre très cher ami Paul Caspi, décédé en Avril 2012

Sparse {Fourier Transform} by Traversing {Cooley-Tukey FFT} Computation Graphs

Computing the dominant Fourier coefficients of a vector is a common task in many fields, such as signal processing, learning theory, and computational complexity. In the Sparse Fast Fourier Transform (Sparse FFT) problem, one is given oracle access to a $d$-dimensional vector $x$ of size $N$, and is asked to compute the best $k$-term approximation of its Discrete Fourier Transform, quickly and using few samples of the input vector $x$. While the sample complexity of this problem is quite well understood, all previous approaches either suffer from an exponential dependence of runtime on the dimension $d$ or can only tolerate a trivial amount of noise. This is in sharp contrast with the classical FFT algorithm of Cooley and Tukey, which is stable and completely insensitive to the dimension of the input vector: its runtime is $O(N\log N)$ in any dimension $d$. In this work, we introduce a new high-dimensional Sparse FFT toolkit and use it to obtain new algorithms, both on the exact, as well as in the case of bounded $\ell_2$ noise. This toolkit includes i) a new strategy for exploring a pruned FFT computation tree that reduces the cost of filtering, ii) new structural properties of adaptive aliasing filters recently introduced by Kapralov, Velingker and Zandieh'SODA'19, and iii) a novel lazy estimation argument, suited to reducing the cost of estimation in FFT tree-traversal approaches. Our robust algorithm can be viewed as a highly optimized sparse, stable extension of the Cooley-Tukey FFT algorithm. Finally, we explain the barriers we have faced by proving a conditional quadratic lower bound on the running time of the well-studied non-equispaced Fourier transform problem. This resolves a natural and frequently asked question in computational Fourier transforms. Lastly, we provide a preliminary experimental evaluation comparing the runtime of our algorithm to FFTW and SFFT 2.0

Traversing the FFT Computation Tree for Dimension-Independent Sparse Fourier Transforms

We consider the well-studied Sparse Fourier transform problem, where one aims to quickly recover an approximately Fourier $k$-sparse vector $\widehat{x} \in \mathbb{C}^{n^d}$ from observing its time domain representation $x$. In the exact $k$-sparse case the best known dimension-independent algorithm runs in near cubic time in $k$ and it is unclear whether a faster algorithm like in low dimensions is possible. Beyond that, all known approaches either suffer from an exponential dependence on the dimension $d$ or can only tolerate a trivial amount of noise. This is in sharp contrast with the classical FFT of Cooley and Tukey, which is stable and completely insensitive to the dimension of the input vector: its runtime is $O(N\log N)$ in any dimension $d$ for $N=n^d$. Our work aims to address the above issues. First, we provide a translation/reduction of the exactly $k$-sparse FT problem to a concrete tree exploration task which asks to recover $k$ leaves in a full binary tree under certain exploration rules. Subsequently, we provide (a) an almost quadratic in $k$ time algorithm for this task, and (b) evidence that a strongly subquadratic time for Sparse FT via this approach is likely impossible. We achieve the latter by proving a conditional quadratic time lower bound on sparse polynomial multipoint evaluation (the classical non-equispaced sparse FT) which is a core routine in the aforementioned translation. Thus, our results combined can be viewed as an almost complete understanding of this approach, which is the only known approach that yields sublinear time dimension-independent Sparse FT algorithms. Subsequently, we provide a robustification of our algorithm, yielding a robust cubic time algorithm under bounded $\ell_2$ noise. This requires proving new structural properties of the recently introduced adaptive aliasing filters combined with a variety of new techniques and ideas

Compositional Contract Abstraction for System Design

Contract-based design has been recently proposed as a framework for concurrent system design in the context of complex supplier chains, where sub-system design can be sub-contracted to suppliers while guaranteeing correct system integration. A unifying meta-theory of contracts was proposed in [Benveniste et al. 2012], which subsumes known frameworks such as interface theories, modal interfaces, and Assume/Guarantee contracts. This report proposes, for this meta-theory of contracts, a generic abstraction technique allowing to prove contract properties based on their abstractions. More precisely, we show how to lift abstractions, from components to contracts, in a systematic way. In doing so, fundamental relations such as being a correct implementation or a valid environment, refining, can be checked on abstractions. Our abstraction technique is fully compositional with respect to contract conjunction. Compositionality of abstraction with respect to contract composition is only partially achieved. We believe that the results we obtain are the best achievable ones and we explain the obstructions we see against improving them. Our abstraction technique complements observers, proposed as a testing technique adapted to contracts in [6]. The latter allow disproving properties, whereas abstraction allows proving them. Key-words: system design, component based design, contract, interface, abstraction, abstract interpretation.La conception par contrats a été proposée récemment comme une approche formelle pour la conception de systèmes permettant le développement parallèle de sysèmes dans un contexte de chaine complexe de sous-traitants. Les théories d'interfaces, les interfaces modales et les contrats hypothèse/garantie, sont autant de formalismes en ce sens. L'article collectif [Benveniste et al. 2012] a proposé une "méta-théorie" des contrats, unifiant les formalismes précédents. Le présent rapport développe, pour cette méta-théorie des contrats, une technique systématique d'abstraction. Les propriétés fondamentales des contrats (relation d'implémentation, d'environnement, de raffinement) peuvent être prouvées sur les abstractions. L'abstraction proposée offre de bonnes propriétés de compositionnalité, même si toutes les propriétés souhaitables ne sont pas valides. Cette technique d'abstraction complète celle des observateurs, qui permettent d'invalider des propriétés de contrats par une approche de type test. Mots-clés : conception des systèmes, composant, contrat, interface, abstraction, interprétation abstraite

A Unifying View of Loosely Time-Triggered Architectures

Cyber-Physical Systems require distributed architectures to support safety critical real-time control. Hermann Kopetz' Time-Triggered Architecture (TTA) has been proposed as both an architecture and a comprehensive paradigm for systems architecture, for such systems. TTA offers the programmer a logical discrete time compliant with synchronous programming, together with timing bounds. A clock synchronization protocol is required, unless the local clocks used themselves provide the recquired accuracy. To relax the strict requirements on synchronization imposed by TTA, Loosely Time-Triggered Architectures (LTTA) have been proposed. In LTTA, computation and communication units are all triggered by autonomous, unsynchronized, clocks. Communication media act as shared memories between writers and readers and communication is non blocking. This is at the price of communication artifacts (such as duplication or loss of data), which must be compensated for by using some "LTTA protocol". In this paper we pursue our previous work by providing a unified presentation of the two variants of LTTA (token- and time-based), with simplified analyses. We compare these two variants regarding performance and robustness and we provide ways to combine them. This report was prepared for a lecture in Gérard Berry's seminar series at the Collège de France, March 5, 2014; it is a corrected version of a paper, which appeared at Emsoft'2010. It is dedicated to our close friend Paul Caspi who died in April 2012.Les infrastructures de calcul distribuées pour le contrôle des systèmes embarqués critiques requièrent des propriétés particulières destinées à préserver les caractéristiques attendues du contrôleur. Les architectures TTA (Time-Triggered Architectures) ont été proposées par Hermann Kopetz, à la fois comme une architecture de calcul et comme une méthodologie de conception des systèmes. TTA offre au programmeur un temps logique conforme à celui de la programmation synchrone, avec en outre un contrôle strict du temps. Il requiert un protocole de synchronisation entre les horloges du système réparti. Pour affaiblir ces hypothèses, les architectures LTTA (Loosely Time-Triggered Architectures) ont été proposées récemment. Dans LTTA, les calculs et les communications sont rythmées par des horloges locales, non synchronisées. Les supports de communication se comportent comme des mémoires partagées. La communication est donc non-bloquante. Ce type de communiccation crée évidemment des artefacts à combattre par un protocole dit "LTTA". Dans cet article nous présentons une approche unifiée des deux techniques connues pour ce type de protocole, reposant sur l'usage, soit de jetons, soit du temps. On compare ces deux variantes et on étudie leur performance. Le présent rapport est une version corrigée d'un article paru à Emsoft'2010. Il est dédié à notre très cher ami Paul Caspi, décédé en Avril 2012