Synthesis of Data Word Transducers

In reactive synthesis, the goal is to automatically generate an implementation from a specification of the reactive and non-terminating input/output behaviours of a system. Specifications are usually modelled as logical formulae or automata over infinite sequences of signals ($\omega$-words), while implementations are represented as transducers. In the classical setting, the set of signals is assumed to be finite. In this paper, we consider data $\omega$-words instead, i.e., words over an infinite alphabet. In this context, we study specifications and implementations respectively given as automata and transducers extended with a finite set of registers. We consider different instances, depending on whether the specification is nondeterministic, universal or deterministic, and depending on whether the number of registers of the implementation is given or not. In the unbounded setting, we show undecidability for both universal and nondeterministic specifications, while decidability is recovered in the deterministic case. In the bounded setting, undecidability still holds for nondeterministic specifications, but can be recovered by disallowing tests over input data. The generic technique we use to show the latter result allows us to reprove some known result, namely decidability of bounded synthesis for universal specifications

Computability of Data-Word Transductions over Different Data Domains

In this paper, we investigate the problem of synthesizing computable functions of infinite words over an infinite alphabet (data $\omega$-words). The notion of computability is defined through Turing machines with infinite inputs which can produce the corresponding infinite outputs in the limit. We use non-deterministic transducers equipped with registers, an extension of register automata with outputs, to describe specifications. Being non-deterministic, such transducers may not define functions but more generally relations of data $\omega$-words. In order to increase the expressive power of these machines, we even allow guessing of arbitrary data values when updating their registers. For functions over data $\omega$-words, we identify a sufficient condition (the possibility of determining the next letter to be outputted, which we call next letter problem) under which computability (resp. uniform computability) and continuity (resp. uniform continuity) coincide. We focus on two kinds of data domains: first, the general setting of oligomorphic data, which encompasses any data domain with equality, as well as the setting of rational numbers with linear order; and second, the set of natural numbers equipped with linear order. For both settings, we prove that functionality, i.e. determining whether the relation recognized by the transducer is actually a function, is decidable. We also show that the so-called next letter problem is decidable, yielding equivalence between (uniform) continuity and (uniform) computability. Last, we provide characterizations of (uniform) continuity, which allow us to prove that these notions, and thus also (uniform) computability, are decidable. We even show that all these decision problems are PSpace-complete for (N,<) and for a large class of oligomorphic data domains, including for instance (Q,<).Comment: Extended version of arxiv:2002.0820

Synthèse Automatique de Systèmes avec Données

A reactive system is a system that continuously interacts with its environment. The environment provides an input signal, to which the system reacts with an output signal, and so on ad infinitum. In reactive synthesis, the goal is to automatically generate an implementation from a specification of the reactive and non-terminating input/output behaviours of a system. In the classical setting, the set of signals is assumed to be finite. however, this assumption is not realistic to model systems which process sequences of signals accompanied with data from a possibly infinite set (e.g. a client id, a sensor value, etc.), which need to be stored in memory and compared against each other.The goal of this thesis is to lift the theory of reactive system synthesis over words on a finite alphabet to data words. The data domain consists in an infinite set whose structure is given by predicates and constants enriched with labels from a finite alphabet. In this context, specifications and implementations are respectively given as automata and transducers extended with a finite set of registers that they use to store data values. To determine the transition to take, they compare the input data with the content of the registers using the predicates of the domain.In a first part, we consider both the bounded and unbounded synthesis problem; the former additionally asks for a bound on the number of registers of the implementation, along with the specification. We do so for different instances, depending on whether the specification is a nondeterministic, universal (a.k.a. co-non-deterministic) or deterministic automaton, for various domains.While the bounded synthesis problem is undecidable for non-deterministic specifications, we provide a generic approach consisting in a reduction to the finite alphabet case, that is done through automata-theoretic constructions. This allows to reprove decidability of bounded synthesis for universal specifications over (ℕ,=), and to obtain new ones, such as the case of a dense order, or the ability of data guessing, all with a 2-ExpTime complexity.We then move to the unbounded synthesis problem, which is undecidable for specifications given by non-deterministic and universal automata, but decidable and ExpTime-complete for deterministic ones over (ℕ,=) and (ℚ,<). We also exhibit a decidable subclass in the case of (ℕ,<), namely one-sided specifications.In a second part, we lift the reactivity assumption, considering the richer class of implementations that are allowed to wait for additional input before reacting, again over data words. Specifications are modelled as non-deterministic asynchronous transducers, that output a (possibly empty) word when they read an input data. Already in the finite alphabet case, their synthesis problem is undecidable.A way to circumvent the difficulty is to focus on functional specifications, for which any input sequence admits at most one acceptable output. Targeting programs computed by input-deterministic transducers is again undecidable, so we shift the focus to deciding whether a specification is computable, in the sense of the classical extension of Turing-computability to infinite inputs. We relate this notion with that of continuity for the Cantor distance, which yields a decidable characterisation of computability for functional specifications given by asynchronous register transducers over (ℕ,=) and for the superseding class of oligomorphic data domains, that also encompasses (ℚ,<). The study concludes with the case of (ℕ,<), that is again decidable. Overall, we get PSpace-completeness for the problems of deciding computability and refined notions, as well as functionality.Les systèmes réactifs sont caractérisés par une interaction constante avec leur environnement : celui-ci fournit un signal d’entrée, auquel le système répond par un signal de sortie, et ainsi de suite à l’infini. L’objectif de la synthèse réactive est de générer automatiquement l’implémentation d’un tel système à partir de la spécification de son comportement. Classiquement, l’ensemble des signaux est supposé fini. Cependant, ce cadre échoue à modéliser des systèmes qui traitent des signaux accompagnés de données issues d’un ensemble potentiellement infini (un identifiant unique, la valeur d’un capteur, etc.), qui doivent être stockées et comparées entre elles.L’objectif de cette thèse est d’étendre la théorie de la synthèse réactive sur les mots à alphabet fini au cas des mots de données. Le domaine de données consiste en un ensemble infini, dont la structure est définie par des prédicats et des constantes, enrichi par un ensemble fini de signaux. Les spécifications et les implémentations sont alors respectivement représentées par des automates et des transducteurs à registres, qu’ils utilisent pour stocker les données. Pour déterminer la transition à prendre, ils comparent la donnée d’entrée au contenu de leurs registres à l’aide des prédicats du domaine.Dans une première partie, nous considérons les problèmes de la synthèse bornée et non-bornée. Dans le premier cas, l’algorithme prend en entrée une borne sur le nombre de registres de l’implémentation, en plus de la spécification à implémenter. Nous considérons plusieurs instances, selon que la spécification est un automate non-déterministe, universel (ou co-non-déterministe), ou encore déterministe, pour plusieurs domaines de données.Tandis que le problème de la synthèse bornée est indécidable pour les spécifications non-déterministes, nous élaborons une approche générique qui permet de le réduire au cas d’un alphabet fini. Celle-ci permet de redémontrer la décidabilité de la synthèse bornée à partir d’automates universels sur (ℕ,=) et d’étendre le résultat à (ℚ,<), y compris en autorisant l’automate à deviner des données, tout cela en 2-ExpTime.Quant à la synthèse non bornée, elle est indécidable pour les spécifications données par des automates non-déterministes ou universels, mais décidable et ExpTime-complète pour les automates déterministes sur (ℕ,=) et (ℚ,<). Nous exhibons également une sous-classe décidable dans le cas de (ℕ,<), à savoir les spécifications unilatérales.Dans une seconde partie, nous examinons comment étendre au cas non-réactif, où l’implémentation est autorisée à attendre d’obtenir plus d’information avant de sélectionner son signal de sortie, toujours dans le cadre des mots de données. Les spécifications sont modélisées par des transducteurs non-déterministes asynchrones, qui produisent un mot (possiblement vide) à chaque fois qu’ils lisent une entrée. Déjà dans le cas fini, un tel problème est indécidable pour cette classe de spécifications.Une manière de contourner la difficulté est de traiter le cas des spécifications fonctionnelles, pour lesquelles chaque suite infinie d’entrées admet au plus une suite de sorties. Pour les implémentations données par des transducteurs déterministes sur l’entrée, le problème est indécidable, aussi nous intéressons-nous au problème de la calculabilité au sens de Turing, classiquement étendue au cas des mots infinis. Nous lions cette notion à celle de continuité pour la distance de Cantor, ce qui nous fournit une caractérisation de la calculabilité qui est décidable pour les fonctions définies par des transducteurs non-déterministes asynchrones sur (ℕ,=) et pour la classe des domaines oligomorphes, qui englobe (ℕ,=) et (ℚ,<). L’étude se conclut par le cas de (ℕ,<), également décidable. Pour ces trois domaines, les problème de calculabilité et ses déclinaisons, ainsi que la fonctionnalité, sont décidables en espace polynomial (PSpace)

The Complexity of Transducer Synthesis from Multi-Sequential Specifications

International audienceThe transducer synthesis problem on finite words asks, given a specification S ⊆ I × O, where I and O are sets of finite words, whether there exists an implementation f : I → O which (1) fulfils the specification, i.e., (i, f (i)) ∈ S for all i ∈ I, and (2) can be defined by some input-deterministic (aka sequential) transducer T f. If such an implementation f exists, the procedure should also output T f. The realisability problem is the corresponding decision problem. For specifications given by synchronous transducers (which read and write alternately one symbol), this is the finite variant of the classical synthesis problem on ω-words, solved by Büchi and Landweber in 1969, and the realisability problem is known to be ExpTime-c in both finite and ω-word settings. For specifications given by asynchronous transducers (which can write a batch of symbols, or none, in a single step), the realisability problem is known to be undecidable. We consider here the class of multi-sequential specifications, defined as finite unions of sequential transducers over possibly incomparable domains. We provide optimal decision procedures for the realisability problem in both the synchronous and asynchronous setting, showing that it is PSpace-c. Moreover, whenever the specification is realisable, we expose the construction of a sequential transducer that realises it and has a size that is doubly exponential, which we prove to be optimal. Acknowledgements We warmly thank the anonymous reviewers for their helpful comments and Christof Löding for pointing us to some related references

Synthesis of Data Word Transducers

International audienceIn reactive synthesis, the goal is to automatically generate an implementation from a specification of the reactive and non-terminating input/output behaviours of a system. Specifications are usually modelled as logical formulae or automata over infinite sequences of signals (ω-words), while implementations are represented as transducers. In the classical setting, the set of signals is assumed to be finite. In this paper, we consider data ωwords instead, i.e., words over an infinite alphabet. In this context, we study specifications and implementations respectively given as automata and transducers extended with a finite set of registers. We consider different instances, depending on whether the specification is nondeterministic, universal or deterministic, and depending on whether the number of registers of the implementation is given or not. In the unbounded setting, we show undecidability for both universal and nondeterministic specifications, while decidability is recovered in the deterministic case. In the bounded setting, undecidability still holds for nondeterministic specifications, but can be recovered by disallowing tests over input data. The generic technique we use to show the latter result allows us to reprove some known result, namely decidability of bounded synthesis for universal specifications

A Synthesis Tool for Optimal Monitors in a Branching-Time Setting

International audienceMonitorability is a characteristic that delineates between the properties that can be runtime verified by a monitor and those that cannot. Existing notions of monitorability for branching-time specifications are quite restrictive, limiting the set of monitorable properties to a small logical fragment. A recent study has enlarged the set of monitorable branching-time properties by weakening the requirements expected of the monitors effecting the verification: it defines a novel notion of optimal monitor that carries out the maximum number of detections that can be effected for any property, thereby turning a branching-time property into a monitorable one. The study also outlines a method for obtaining a unique optimal monitor from any branching-time property but falls short of providing an automation for this procedure. In this paper, we present a prototype tool that generates monitorable properties for branchingtime properties expressed in a variant of the modal μ-calculus, based on this procedure. We also assess the performance of the prototype tool by evaluating its performance against several specifications