Constraint linear-time temporal logic (CLTL) is an extension of LTL that is interpreted on sequences of valuations of variables over an infinite domain. The atomic formulas are interpreted as constraints on the valuations. The atomic formulas can constrain valuations over a range of positions along a sequence, with the range being bounded by a parameter depending on the formula. The satisfiability and model checking problems for CLTL have been studied by Demri and D'Souza. We consider the realizability problem for CLTL. The set of variables is partitioned into two parts, with each part controlled by a player. Players take turns to choose valuations for their variables, generating a sequence of valuations. The winning condition is specified by a CLTL formula -- the first player wins if the sequence of valuations satisfies the specified formula. We study the decidability of checking whether the first player has a winning strategy in the realizability game for a given CLTL formula. We prove that it is decidable in the case where the domain satisfies the completion property, a property introduced by Balbiani and Condotta in the context of satisfiability. We prove that it is undecidable over $(\mathbb{Z},<,=)$, the domain of integers with order and equality. We prove that over $(\mathbb{Z},<,=)$, it is decidable if the atomic constraints in the formula can only constrain the current valuations of variables belonging to the second player, but there are no such restrictions for the variables belonging to the first player. We call this single-sided games.

International audience

Weighted Timed Games (WTG for short) are the most widely used model to describe controller synthesis problems involving real-time issues. Unfortunately, they are notoriously difficult, and undecidable in general. As a consequence, one-clock WTGs have attracted a lot of attention, especially because they are known to be decidable when only non-negative weights are allowed. However, when arbitrary weights are considered, despite several recent works, their decidability status was still unknown. In this paper, we solve this problem positively and show that the value function can be computed in exponential time (if weights are encoded in unary).

Weighted timed games are two-player zero-sum games played in a timed automaton equipped with integer weights. We consider optimal reachability objectives, in which one of the players, that we call Min, wants to reach a target location while minimising the cumulated weight. While knowing if Min has a strategy to guarantee a value lower than a given threshold is known to be undecidable (with two or more clocks), several conditions, one of them being divergence, have been given to recover decidability. In such weighted timed games (like in untimed weighted games in the presence of negative weights), Min may need finite memory to play (close to) optimally. This is thus tempting to try to emulate this finite memory with other strategic capabilities. In this work, we allow the players to use stochastic decisions, both in the choice of transitions and of timing delays. We give a definition of the expected value in weighted timed games. We then show that, in divergent weighted timed games as well as in (untimed) weighted games (that we call shortest-path games in the following), the stochastic value is indeed equal to the classical (deterministic) value, thus proving that Min can guarantee the same value while only using stochastic choices, and no memory.

Weighted automata (WA) are an extension of finite automata that define functions from words to values in a given semiring. An alternative deterministic model, called Cost Register Automata (CRA), was introduced by Alur et al. It enriches deterministic finite automata with a finite number of registers, which store values, updated at each transition using the operations of the semiring. It is known that CRA with register updates defined by linear maps have the same expressiveness as WA. Previous works have studied the register minimization problem: given a function computable by a WA and an integer k, is it possible to realize it using a CRA with at most k registers? In this paper, we solve this problem for CRA over a field with linear register updates, using the notion of linear hull, an algebraic invariant of WA introduced recently by Bell and Smertnig. We then generalise the approach to solve a more challenging problem, that consists in minimizing simultaneously the number of states and that of registers. In addition, we also lift our results to the setting of CRA with affine updates. Last, while the linear hull was recently shown to be computable by Bell and Smertnig, no complexity bounds were given. To fill this gap, we provide two new algorithms to compute invariants of WA. This allows us to show that the register (resp. state-register) minimization problem can be solved in 2-ExpTime (resp. in NExpTime).

International audience

Weighted timed games are zero-sum games played by two players on a timed automaton equipped with weights, where one player wants to minimise the cumulative weight while reaching a target. Used in a reactive synthesis perspective, this quantitative extension of timed games allows one to measure the quality of controllers in real-time systems. Weighted timed games are notoriously difficult and quickly undecidable, even when restricted to non-negative weights. For non-negative weights, the largest class that can be analysed has been introduced by Bouyer, Jaziri and Markey in 2015. Though the value problem is undecidable, the authors show how to approximate the value by considering regions with a refined granularity. In this work, we extend this class to incorporate negative weights, allowing one to model energy for instance, and prove that the value can still be approximated, with the same complexity. A small restriction also allows us to obtain a class of decidable weighted timed games with negative weights and an arbitrary number of clocks. In addition, we show that a symbolic algorithm, relying on the paradigm of value iteration, can be used as an approximation/computation schema over these classes. We also consider the special case of untimed weighted games, where the same fragments are solvable in polynomial time: this contrasts with the pseudo-polynomial complexity, known so far, for weighted games without restrictions.

Weighted Timed Games (WTG for short) are the most widely used model to describe controller synthesis problems involving real-time issues. Unfortunately, they are notoriously difficult, and undecidable in general. As a consequence, one-clock WTG has attracted a lot of attention, especially because they are known to be decidable when only non-negative weights are allowed. However, when arbitrary weights are considered, despite several recent works, their decidability status was still unknown. In this paper, we solve this problem positively and show that the value function can be computed in exponential time (if weights are encoded in unary).

The Kleene theorem establishes a fundamental link between automata and expressions over the free monoid. Numerous generalisations of this result exist in the literature; on one hand, lifting this result to a weighted setting has been widely studied. On the other hand, beyond the free monoid, different monoids can be considered: for instance, two-way automata, and even tree-walking automata, can be described by expressions using the free inverse monoid. In the present work, we aim at combining both research directions and consider weighted extensions of automata and expressions over a class of monoids that we call pre-rational, generalising both the free inverse monoid and graded monoids. The presence of idempotent elements in these pre-rational monoids leads in the weighted setting to consider infinite sums. To handle such sums, we will have to restrict ourselves to rationally additive semirings. Our main result is thus a generalisation of the Kleene theorem for pre-rational monoids and rationally additive semirings. As a corollary, we obtain a class of expressions equivalent to weighted two-way automata, as well as one for tree-walking automata.

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 $(\mathbb{N},<)$ and for a large class of oligomorphic data domains, including for instance $(\mathbb{Q},<)$.

Abstract In this paper, we propose a novel framework for the synthesis of robust and optimal energy-aware controllers. The framework is based on energy timed automata, allowing for easy expression of timing constraints and variable energy rates. We prove decidability of the energy-constrained infinite-run problem in settings with both certainty and uncertainty of the energy rates. We also consider the optimization problem of identifying the minimal upper bound that will permit existence of energy-constrained infinite runs. Our algorithms are based on quantifier elimination for linear real arithmetic. Using Mathematica and Mjollnir, we illustrate our framework through a real industrial example of a hydraulic oil pump. Compared with previous approaches our method is completely automated and provides improved results.

Copyless streaming string transducers (copyless SST) have been introduced by R. Alur and P. Černý in 2010 as a one-way deterministic automata model to define transductions of finite strings. Copyless SST extend deterministic finite state automata with a set of variables in which to store intermediate output strings, and those variables can be combined and updated all along the run, in a linear manner, i.e., no variable content can be copied on transitions. It is known that copyless SST capture exactly the class of MSO-definable string-to-string transductions, and are as expressive as deterministic two-way transducers. They enjoy good algorithmic properties. Most notably, they have decidable equivalence problem (in PSpace). On the other hand, HDT0L systems have been introduced for a while, the most prominent result being the decidability of the equivalence problem. In this paper, we propose a semantics of HDT0L systems in terms of transductions, and use it to study the class of deterministic copyful SST. Our contributions are as follows: (i)HDT0L systems and total deterministic copyful SST have the same expressive power, (ii)the equivalence problem for deterministic copyful SST and the equivalence problem for HDT0L systems are inter-reducible, in quadratic time. As a consequence, equivalence of deterministic SST is decidable, (iii)the functionality of non-deterministic copyful SST is decidable, (iv)determining whether a non-deterministic copyful SST can be transformed into an equivalent non-deterministic copyless SST is decidable in polynomial time.

Shortest-path games are two-player zero-sum games played on a graph equipped with integer weights. One player, that we call Min, wants to reach a target set of states while minimising the total weight, and the other one has an antagonistic objective. This combination of a qualitative reachability objective and a quantitative total-payoff objective is one of the simplest settings where Min needs memory (pseudo-polynomial in the weights) to play optimally. In this article, we aim at studying a tradeoff allowing Min to play at random, but using no memory. We show that Min can achieve the same optimal value in both cases. In particular, we compute a randomised memoryless ε-optimal strategy when it exists, where probabilities are parametrised by ε. We also show that for some games, no optimal randomised strategies exist. We then characterise, and decide in polynomial time, the class of games admitting an optimal randomised memoryless strategy.

Abstract In this paper, we investigate the problem of synthesizing computable functions of infinite words over an infinite alphabet (data $$\omega $$ ω -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 specify functions. Such transducers may not define functions but more generally relations of data $$\omega $$ ω -words, and we show that it is PSpace-complete to test whether a given transducer defines a function. Then, given a function defined by some register transducer, we show that it is decidable (and again, PSpace-c) whether such function is computable. As for the known finite alphabet case, we show that computability and continuity coincide for functions defined by register transducers, and show how to decide continuity. We also define a subclass for which those problems are PTime.

We solve in a purely symbolic way the robust controller synthesis problem in timed automata with Büchi acceptance conditions. The goal of the controller is to play according to an accepting lasso of the automaton, while resisting to timing perturbations chosen by a competing environment. The problem was previously shown to be PSPACE -complete using regions-based techniques, but we provide a first tool solving the problem using zones only, thus more resilient to state-space explosion problem. The key ingredient is the introduction of branching constraint graphs allowing to decide in polynomial time whether a given lasso is robust, and even compute the largest admissible perturbation if it is. We also make an original use of constraint graphs in this context in order to test the inclusion of timed reachability relations, crucial for the termination criterion of our algorithm. Our techniques are illustrated using a case study on the regulation of a train network.

Transducers extend finite state automata with outputs, and describe transformations from strings to strings. Sequential transducers, which have a deterministic behaviour regarding their input, are of particular interest. However, unlike finite-state automata, not every transducer can be made sequential. The seminal work of Choffrut allows to characterise, amongst the functional one-way transducers, the ones that admit an equivalent sequential transducer. In this work, we extend the results of Choffrut to the class of transducers that produce their output string by adding simultaneously, at each transition, a string on the left and a string on the right of the string produced so far. We call them the string-to-context transducers. We obtain a multiple characterisation of the functional string-to-context transducers admitting an equivalent sequential one, based on a Lipschitz property of the function realised by the transducer, and on a pattern (a new twinning property). Last, we prove that given a string-to-context transducer, determining whether there exists an equivalent sequential one is in coNP.

We consider the problem of evaluating in streaming (i.e., in a single left-to-right pass) a nested word transduction with a limited amount of memory. A transduction T is said to be height bounded memory (HBM) if it can be evaluated with a memory that depends only on the size of T and on the height of the input word. We show that it is decidable in coNPTime for a nested word transduction defined by a visibly pushdown transducer (VPT), if it is HBM. In this case, the required amount of memory may depend exponentially on the height of the word. We exhibit a sufficient, decidable condition for a VPT to be evaluated with a memory that depends quadratically on the height of the word. This condition defines a class of transductions that strictly contains all determinizable VPTs.

Weighted timed games are zero-sum games played by two players on a timed automaton equipped with weights, where one player wants to minimise the accumulated weight while reaching a target. Weighted timed games are notoriously difficult and quickly undecidable, even when restricted to non-negative weights. For non-negative weights, the largest class that can be analysed has been introduced by Bouyer, Jaziri and Markey in 2015. Though the value problem is undecidable, the authors show how to approximate the value by considering regions with a refined granularity. In this work, we extend this class to incorporate negative weights, allowing one to model energy for instance, and prove that the value can still be approximated, with the same complexity. In addition, we show that a symbolic algorithm, relying on the paradigm of value iteration, can be used as an approximation schema on this class.

Automata-logic connections are pillars of the theory of regular languages. Such connections are harder to obtain for transducers, but important results have been obtained recently for word-to-word transformations, showing that the three following models are equivalent: deterministic two-way transducers, monadic second-order (MSO) transducers, and deterministic one-way automata equipped with a finite number of registers. Nested words are words with a nesting structure, allowing to model unranked trees as their depth-first-search linearisations. In this paper, we consider transformations from nested words to words, allowing in particular to produce unranked trees if output words have a nesting structure. The model of visibly pushdown transducers allows to describe such transformations, and we propose a simple deterministic extension of this model with two-way moves that has the following properties: i) it is a simple computational model, that naturally has a good evaluation complexity; ii) it is expressive: it subsumes nested word-to-word MSO transducers, and the exact expressiveness of MSO transducers is recovered using a simple syntactic restriction; iii) it has good algorithmic/closure properties: the model is closed under composition with a unambiguous one-way letter-to-letter transducer which gives closure under regular look-around, and has a decidable equivalence problem.

Weighted automata (WA) extend finite-state automata by associating with transitions weights from a semiring S, defining functions from words to S. Recently, cost register automata (CRA) have been introduced as an alternative model to describe any function realised by a WA by means of a deterministic machine. Unambiguous WA over a monoid (M, ⊗) can equivalently be described by cost register automata whose registers take their values in M, and are updated by operations of the form x: = y ⊗ c, with c ∈ M. This class is denoted by CRA⊗c(M).

Robustness of timed systems aims at studying whether infinitesimal perturbations in clock values can result in new discrete behaviors. A model is robust if the set of discrete behaviors is preserved under arbitrarily small (but positive) perturbations. We tackle this problem for time Petri nets (TP Ns, for short) by considering the model of parametric guard enlargement which allows time-intervals constraining the firing of transitions in TPNs to be enlarged by a (positive) parameter. We show that TPNs are not robust in general and checking if they are robust with respect to standard properties (such as boundedness, safety) is undecidable. We then extend the marking class timed automaton construction for TPNs to a parametric setting, and prove that it is compatible with guard enlargements. We apply this result to the (undecidable) class of TPNs which are robustly bounded (i.e., whose finite set of reachable markings remains finite under infinitesimal perturbations): we provide two decidable robustly bounded subclasses, and show that one can effectively build a timed automaton which is timed bisimilar even in presence of perturbations. This allows us to apply existing results for timed automata to these TPNs and show further robustness properties.

Any two-way finite state automaton is equivalent to some one-way finite state automaton. This well-known result, shown by Rabin and Scott and independently by Shepherdson, states that two-way finite state automata (even non-deterministic) characterize the class of regular languages. It is also known that this result does not extend to finite string transductions: (deterministic) two-way finite state transducers strictly extend the expressive power of (functional) one-way transducers. In particular deterministic two-way transducers capture exactly the class of MSO-transductions of finite strings. In this paper, we address the following definability problem: given a function defined by a two-way finite state transducer, is it definable by a one-way finite state transducer? By extending Rabin and Scott's proof to transductions, we show that this problem is decidable. Our procedure builds a one-way transducer, which is equivalent to the two-way transducer, whenever one exists.

This paper presents the Monotone-Pruning algorithm (MP) for computing the minimal coverability set of Petri nets. The original Karp and Miller algorithm (K&M) unfolds the reachability graph of a Petri net and uses acceleration on branches to ensu

We consider the problem of evaluating in streaming (i.e. in a single left-to-right pass) a nested word transduction with a limited amount of memory. A transduction T is said to be height bounded memory (HBM) if it can be evaluated with a memory that depends only on the size of T and on the height of the input word. We show that it is decidable in coNPTime for a nested word transduction defined by a visibly pushdown transducer (VPT), if it is HBM. In this case, the required amount of memory may depend exponentially on the height of the word. We exhibit a sufficient, decidable condition for a VPT to be evaluated with a memory that depends quadratically on the height of the word. This condition defines a class of transductions that strictly contains all determinizable VPTs.

In this work, we study decision problems related to timed automata with silent transitions (TA $_{ϵ}$ ) which strictly extend the expressiveness of timed automata (TA). We first answer negatively a central question raised by the introduction of silent transitions: can we decide whether the language recognized by a TA $_{ϵ}$ can be recognized by some TA? Then we establish in the framework of TA $_{ϵ}$ some old open conjectures that O. Finkel has recently solved for TA. His proofs follow a generic scheme which relies on the fact that only a finite number of configurations can be reached by a TA while reading a timed word. This property does not hold for TA $_{ϵ}$ , the proofs in the framework of TA $_{ϵ}$ thus require more elaborated arguments. We establish undecidability of complementability, minimization of the number of clocks, and closure under shuffle. We also show these results in the framework of infinite timed languages.