# Nathanaël Fijalkow's research while affiliated with University of Bordeaux and other places

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## Publications (76)

This paper studies two-player zero-sum games played on graphs and makes contributions toward the following question: given an objective, how much memory is required to play optimally for that objective? We study regular objectives, where the goal of the first player is that eventually the sequence of colors along the play belongs to some regular la...

We introduce the notion of universal graphs as a tool for constructing algorithms solving games of infinite duration such as parity games and mean payoff games. In the first part we develop the theory of universal graphs, with two goals: showing an equivalence and normalisation result between different recently introduced related models, and constr...

We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several characterisations: polynomially ambiguous weighted automata, copyless cost-register automata, rational formal series, and line...

Do agents know each others’ strategies? In multi-process software construction, each process has access to the processes already constructed; but in typical human-robot interactions, a human may not announce its strategy to the robot (indeed, the human may not even know their own strategy). This question has often been overlooked when modeling and...

We consider the problem of automatically constructing computer programs from input-output examples. We investigate how to augment probabilistic and neural program synthesis methods with new search algorithms, proposing a framework called distribution-based search. Within this framework, we introduce two new search algorithms: Heap Search, an enumer...

Probabilistic automata are an extension of nondeterministic finite automata in which transitions are annotated with probabilities. Despite its simplicity, this model is very expressive and many of the associated algorithmic questions are undecidable. In this work we focus on the emptiness problem (and its variant the value problem), which asks whet...

Linear temporal logic (LTL) is a specification language for finite sequences (called traces) widely used in program verification, motion planning in robotics, process mining, and many other areas. We consider the problem of learning formulas in fragments of LTL without the $$\mathbf {U}$$ U -operator for classifying traces; despite a growing intere...

We study the complexity of representing polynomials by arithmetic
circuits in both the commutative and the non-commutative settings.
Our approach goes through a precise understanding of the more
restricted setting where multiplication is not associative, meaning
that we distinguish (xy)z from x(yz).
Our first and main conceptual result is a charact...

Bertrand et al. introduced a model of parameterised systems, where each agent is represented by a finite state system, and studied the following control problem: for any number of agents, does there exist a controller able to bring all agents to a target state? They showed that the problem is decidable and EXPTIME-complete in the adversarial settin...

We consider the problem of automatically constructing computer programs from input-output examples. We investigate how to augment probabilistic and neural program synthesis methods with new search algorithms, proposing a framework called distribution-based search. Within this framework, we introduce two new search algorithms: Heap Search, an enumer...

Linear temporal logic (LTL) is a specification language for finite sequences (called traces) widely used in program verification, motion planning in robotics, process mining, and many other areas. We consider the problem of learning LTL formulas for classifying traces; despite a growing interest of the research community, existing solutions suffer...

The notion of separating automata was introduced by Bojanczyk and Czerwinski for understanding the first quasipolynomial time algorithm for parity games. In this paper we show that separating automata is a powerful tool for constructing algorithms solving games with combinations of objectives. We construct two new algorithms: the first for disjunct...

We introduce the notion of universal graphs as a tool for constructing algorithms solving games of infinite duration such as parity games and mean payoff games. In the first part we develop the theory of universal graphs, with two goals: showing an equivalence and normalisation result between different recently introduced related models, and constr...

In this paper we initiate the study of the computational complexity of learning linear temporal logic (LTL) formulas from examples. We construct approximation algorithms for fragments of LTL and prove hardness results; in particular we obtain tight bounds for approximation of the fragment containing only the next operator and conjunctions, and prov...

We study alternating automata with qualitative semantics over infinite binary trees: Alternation means that two opposing players construct a decoration of the input tree called a run, and the qualitative semantics says that a run of the automaton is accepting if almost all branches of the run are accepting. In this article, we prove a positive and...

In this paper, we are interested in automata over infinite words and infinite duration games, that we view as general transition systems. We study transformations of systems using a Muller condition into ones using a parity condition, extending Zielonka's construction. We introduce the alternating cycle decomposition transformation, and we prove a...

This paper surveys recent works about the notion of universal graphs. They were introduced in the context of parity games for understanding the recent quasipolynomial time algorithms, but they are defined for arbitrary objectives yielding a new approach for constructing efficient algorithms for solving different classes of games.

Prompt-LTL extends Linear Temporal Logic with a bounded version of the ``eventually'' operator to express temporal requirements such as bounding waiting times. We study assume-guarantee synthesis for prompt-LTL: the goal is to construct a system such that for all environments satisfying a first prompt-LTL formula (the assumption) the system compose...

Learning probabilistic context-free grammars (PCFGs) from strings is a classic problem in computational linguistics since Horning ( 1969 ). Here we present an algorithm based on distributional learning that is a consistent estimator for a large class of PCFGs that satisfy certain natural conditions including being anchored (Stratos et al., 2016 )....

Bertrand et al. introduced a model of parameterised systems, where each agent is represented by a finite state system, and studied the following control problem: for any number of agents, does there exist a controller able to bring all agents to a target state? They showed that the problem is decidable and EXPTIME-complete in the adversarial settin...

We study alternating automata with qualitative semantics over infinite binary trees: alternation means that two opposing players construct a decoration of the input tree called a run, and the qualitative semantics says that a run of the automaton is accepting if almost all branches of the run are accepting. In this paper we prove a positive and a n...

This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp. Motivated by the seminal paper of Rabin from 1963 introducing probabilistic automata, we study the (deterministic) state complexity of pr...

This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp. Motivated by the seminal paper of Rabin from 1963 introducing probabilistic automata, we study the (deterministic) state complexity of pr...

Programming by example is the problem of synthesizing a program from a small set of input / output pairs. Recent works applying machine learning methods to this task show promise, but are typically reliant on generating synthetic examples for training. A particular challenge lies in generating meaningful sets of inputs and outputs, which well-chara...

Bertrand et al. introduced a model of parameterised systems, where each agent is represented by a finite state system, and studied the following control problem: for any number of agents, does there exist a controller able to bring all agents to a target state? They showed that the problem is decidable and EXPTIME-complete in the adversarial settin...

Probabilistic automata are an extension of nondeterministic finite automata in which transitions are annotated with probabilities. Despite its simplicity, this model is very expressive and many algorithmic questions are undecidable. In this work we focus on the emptiness problem (and its variant the value problem), which asks whether a given probab...

The Monniaux Problem in abstract interpretation asks, roughly speaking, whether the following question is decidable: given a program P, a safety (e.g., non-reachability) specification \(\varphi \), and an abstract domain of invariants \(\mathcal {D}\), does there exist an inductive invariant \(\mathcal {I}\) in \(\mathcal {D}\) guaranteeing that pr...

We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several characterisations: polynomially ambiguous weighted automata, copyless cost-register automata, rational formal series, and line...

The Monniaux Problem in abstract interpretation asks, roughly speaking, whether the following question is decidable: given a program $P$, a safety (\emph{e.g.}, non-reachability) specification $\varphi$, and an abstract domain of invariants $\mathcal{D}$, does there exist an inductive invariant $I$ in $\mathcal{D}$ guaranteeing that program $P$ mee...

The \emph{Orbit Problem} consists of determining, given a linear transformation $A$ on $\mathbb{Q}^d$, together with vectors $x$ and $y$, whether the orbit of $x$ under repeated applications of $A$ can ever reach $y$. This problem was famously shown to be decidable by Kannan and Lipton in the 1980s. In this paper, we are concerned with the problem...

Logical characterizations of probabilistic bisimulation and simulation for Labelled Markov Processes were given by Desharnais et al. These results hold for systems defined on analytic state spaces and assume countably many labels in the case of bisimulation and finitely many labels in the case of simulation.
We revisit these results by giving simpl...

We consider the decidability of state-to-state reachability in linear time-invariant control systems over discrete time. We analyse this problem with respect to the allowable control sets, which in general are assumed to be defined by boolean combinations of linear inequalities. Decidability of the version of the reachability problem in which contr...

In this paper, we give a self contained presentation of a recent breakthrough in the theory of infinite duration games: the existence of a quasipolynomial time algorithm for solving parity games. We introduce for this purpose two new notions: good for small games automata and universal graphs.

We study the computational complexity of solving mean payoff games. This class of games can be seen as an extension of parity games, and they have similar complexity status: in both cases solving them is in $\textbf{NP} \cap \textbf{coNP}$ and not known to be in $\textbf{P}$. In a breakthrough result Calude, Jain, Khoussainov, Li, and Stephan const...

This paper is a contribution to the study of parity games and the recent constructions of three quasipolynomial time algorithms for solving them. We revisit a result of Czerwi\'nski, Daviaud, Fijalkow, Jurdzi\'nski, Lazi\'c, and Parys witnessing a quasipolynomial barrier for all three quasipolynomial time algorithms. The argument is that all three...

Program synthesis constructs programs from specifications in an automated way. Strategy Logic (SL) is a powerful and versatile specification language whose goal is to give theoretical foundations for program synthesis in a multi-agent setting. One limitation of Strategy Logic is that it is purely qualitative. For instance it cannot specify quantita...

Several distinct techniques have been proposed to design quasi-polynomial algorithms for solving parity games since the breakthrough result of Calude, Jain, Khoussainov, Li, and Stephan (2017): play summaries, progress measures and register games. We argue that all those techniques can be viewed as instances of the separation approach to solving pa...

This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp. We look at alternating automata as introduced by Chandra, Kozen and Stockmeyer: such machines run independent computations on the word an...

We consider the decidability of state-to-state reachability in linear time-invariant control systems, with control sets defined by boolean combinations of linear inequalities. Decidability of the sub-problem in which control sets are linear subspaces is a fundamental result in control theory. We first show that reachability is undecidable if the se...

The quest for a polynomial time algorithm for solving parity games gained momentum in 2017 when two different quasipolynomial time algorithms were constructed. In this paper, we further analyse the second algorithm due to Jurdzi\'nski and Lazi\'c and called the succinct progress measure algorithm. It was presented as an improvement over a previous...

Semi-Markov processes are Markovian processes in which the firing time of the transitions is modelled by probabilistic distributions over positive reals interpreted as the probability of firing a transition at a certain moment in time. In this paper we consider the trace-based semantics of semi-Markov processes, and investigate the question of how...

The model of probabilistic automata was introduced by Rabin in 1963. Ever since, undecidability results were obtained for this model, showing that although simple, it is very expressive. This paper provides streamlined constructions implying the most important negative results, including the celebrated inapproximability result of Condon and Lipton.

We study two-player games with counters, where the objective of the first player is that the counter values remain bounded. We investigate the existence of a trade-off between the size of the memory and the bound achieved on the counters, which has been conjectured by Colcombet and Loeding. We show that unfortunately this conjecture does not hold:...

We study Monadic Second-Order Logic (MSO) over finite words, extended with (non-uniform arbitrary) monadic predicates. We show that it defines a class of languages that has algebraic, automata-theoretic, and machine-independent characterizations. We consider the regularity question: Given a language in this class, when is it regular? To answer this...

We present Stamina, a tool solving three algorithmic problems in automata theory. First, compute the star height of a regular language, i.e. the minimal number of nested Kleene stars needed for expressing the language with a complement-free regular expression. Second, decide limitedness for regular cost functions. Third, decide whether a probabilis...

We consider the value 1 problem for probabilistic automata over finite words: it asks whether a given probabilistic automaton accepts words with probability arbitrarily close to 1. This problem is known to be undecidable. However, different algorithms have been proposed to partially solve it; it has been recently shown that the Markov Monoid algori...

In this work we build on these models to look at social influence from a strategic perspective. We do so by introducing a new class of games, called games of influence. Specifically, a game of influence is an infinite repeated game with incomplete information in which, at each stage of interaction, an agent can make her opinions visible (public) or...

The notion of online space complexity, introduced by Karp in 1967, quantifies the amount of space required to solve a given problem using an online algorithm, represented by a Turing machine which scans the input exactly once from left to right. In this paper, we study alternating algorithms as introduced by Chandra, Kozen and Stockmeyer in 1976. W...

Given two labelled Markov decision processes (MDPs), the trace-refinement problem asks whether for all strategies of the first MDP there exists a strategy of the second MDP such that the induced labelled Markov chains are trace-equivalent. We show that this problem is decidable in polynomial time if the second MDP is a Markov chain. The algorithm i...

In this paper, we define the online space complexity of languages, as the size of the smallest abstract machine processing words sequentially and able to determine at every point whether the word read so far belongs to the language or not. The first part of this paper motivates this model and provides examples and preliminary results.
One source of...

Given two labelled Markov decision processes (MDPs), the trace-refinement
problem asks whether for all strategies of the first MDP there exists a
strategy of the second MDP such that the induced labelled Markov chains are
trace-equivalent. We show that this problem is decidable in polynomial time if
the second MDP is a Markov chain. The algorithm i...

This thesis is a contribution to the study of quantitative models of automata, and more specifically of automata with counters and probabilistic automata. They have been independently studied for decades, leading to deep theoretical insights of practical value. We investigate here two seemingly unrelated questions, the first about finite-memory det...

We consider probabilistic automata over finite words. Such an automaton defines the language consisting of the set of words accepted with probability greater than a given threshold. We show the existence of a universally non-regular probabilistic automaton, i.e. an automaton such that the language it defines is non-regular for every threshold. As a...

We study two-player games with counters, where the objective of the first player is that the counter values remain bounded. We investigate the existence of a trade-off between the size of the memory and the bound achieved on the counters, which has been conjectured by Colcombet and Löding. We show that unfortunately this conjecture does not hold: t...

We consider the value 1 problem for probabilistic automata over finite words.
This problem is known to be undecidable. However, different algorithms have
been proposed to partially solve it. The aim of this paper is to prove that one
such algorithm, called the Markov Monoid algorithm, is optimal. To this end, we
develop a profinite theory for proba...

We present ACME, a tool implementing algebraic techniques to solve decision problems from automata theory. The core generic algorithm takes as input an automaton and computes its stabilization monoid, which is a generalization of its transition monoid.
Using the stabilization monoid, one can solve many problems: determine whether a B-automaton (whi...

The value 1 problem is a decision problem for probabilistic automata over
finite words: are there words accepted by the automaton with arbitrarily high
probability? Although undecidable, this problem attracted a lot of attention
over the last few years. The aim of this paper is to review and relate the
results pertaining to the value 1 problem. In...

This paper introduces and investigates decision problems for numberless probabilistic automata, i.e. probabilistic automata where the support of each probabilistic transitions is specified, but the exact values of the probabilities are not. A numberless probabilistic automaton can be instantiated into a probabilistic automaton by specifying the exa...

We study two-player zero-sum games over infinite-state graphs with
boundedness conditions. Our first contribution is about the strategy
complexity, i.e the memory required for winning strategies: we prove that over
general infinite-state graphs, memoryless strategies are sufficient for
finitary B\"uchi games, and finite-memory suffices for finitary...

We study two-player games played on finite graphs equipped with costs on
edges and introduce two winning conditions, cost-parity and cost-Streett, which
require bounds on the cost between requests and their responses. Both
conditions generalize the corresponding classical omega-regular conditions and
the corresponding finitary conditions. For parit...

We consider two-player games played on finite graphs equipped with costs on edges and introduce two winning conditions, cost-parity and cost-Streett, which require bounds on the cost between requests and their responses. Both conditions generalize the corresponding classical ω-regular conditions as well as the corresponding finitary conditions. For...

Games on graphs provide a natural model for reactive non-terminating systems.
In such games, the interaction of two players on an arena results in an
infinite path that describes a run of the system. Different settings are used
to model various open systems in computer science, as for instance turn-based
or concurrent moves, and deterministic or st...

This short note aims at proving that the isolation problem is undecidable for
probabilistic automata with only one probabilistic transition. This problem is
known to be undecidable for general probabilistic automata, without restriction
on the number of probabilistic transitions. In this note, we develop a
simulation technique that allows to simula...

The value 1 problem is a decision problem for probabilistic automata over
finite words: given a probabilistic automaton A, are there words accepted by A
with probability arbitrarily close to 1? This problem was proved undecidable
recently. We sharpen this result, showing that the undecidability result holds
even if the probabilistic automata have o...

The class of omega-regular languages provides a robust specification language
in verification. Every omega-regular condition can be decomposed into a safety
part and a liveness part. The liveness part ensures that something good happens
"eventually". Finitary liveness was proposed by Alur and Henzinger as a
stronger formulation of liveness. It requ...

The value 1 problem is a decision problem for probabilistic automata over finite words: given a probabilistic automaton A, are there words accepted by A with probability arbitrarily close to 1? This problem was proved undecidable recently. We sharpen this result, showing that the undecidability holds even if the probabilistic automata have only one...

Games on graphs with omega-regular objectives provide a natural model for reactive systems. In this paper, we consider generalized reachability games: given k subsets of vertices, the objective is to reach one vertex of each subset. We show that solving generalized reachability games is PSPACE-complete in general. In the special case where reachabi...

## Citations

... In the conference version of this paper we explained how to adapt the proof strategy above to obtain the undecidability of the emptiness problem for quadratically ambiguous probabilistic automata, see Fijalkow et al. (2017). We left open whether the undecidability already holds for linearly ambiguous automata. ...

Reference: Probabilistic Automata of Bounded Ambiguity

... It is a classical observation that coefficient sequences of rational series over fields coincide with the sequences arising as solutions to initial value problems for linear difference equationsi.e., linear recurrences -with constant coefficients. As a consequence, unary weighted automata over fields can also be understood as devices realising such sequences; see, for instance, C. Barloy et al. [1] or J. Bell and D. Smertnig [2]. ...

... Section 6 shows that in the important special case of prefixincreasing objectives, we recover the notion of universality studied over finite graphs by Colcombet, Fijalkow, Gawrychowski and Ohlmann [10]. We then advocate in Section 7 for the applicability of our approach by presenting constructions of well-monotonic universal graphs for many different valuations, in some cases extending existing positionality results. ...

... DeepSynth was used for the experiments of the recently published Fijalkow et al. (2022). The main purpose was to evaluate and compare different search algorithms and their integration with neural predictions. ...

... Recently, it has attracted a lot of attention in the recent years due to its application in Explainable AI and Specification Mining. Along with its theoretical analysis [14], there have been a number of efficient algorithms [30,27,7] to solve it. ...

... Let (G, v 0 ) be an initialized two-player zero-sum game with an intersection of homogeneous objectives. Then Problem 4 is -PSPACE-complete for reachability objectives with finite-memory winning strategies for both players [32], -P-complete for Büchi objectives with finite-memory (resp. uniform) winning strategies for player 1 (resp. ...

... In the more recent distribution-based semantics, the outcome of a stochastic process is a sequence of distributions over states [3,19]. This alternative semantics has received some attention recently for theoretical analysis of probabilistic bisimulation [17] and is adequate to describe large populations of agents [14,10] with applications in system biology [19,1]. The behaviour of an agent is modeled as an MDP with some state space Q, and a large population of identical agents is described by a (continuous) distribution d : Q → [0, 1] that gives the fraction d(q) of agents in the population that are in each state q ∈ Q. ...

... This correspondance was then lifted between weighted tree automata on one hand and non-associative arithmetic circuits on the other, where the interpretation of a more general result of Bozapalidis and Louscou-Bozapalidou [BL83] led to a novel characterization of the size of non-associative circuits. Exploiting this characterization we obtained strong new lower bounds both for non-commutative and (set-multilinear) commutative arithmetic circuits [FLO+20]. ...

... Chapters 7 and 8 are (loosely) based on a joint work with Ashwani Anand, Nathanaël Fijalkow, Aliénor Goubault-Larrecq and Jérôme Leroux [AFGL+21] in which we proved similar results by using separating automata. ...

... Moreover, since probabilistic automata are known to be closed under complement (it is easy to define B ( ) = 1 − A ( )) the two threshold problems are equivalent and undecidable [33]. In probabilistic automata the zero isolation problem, due to complementation, is equivalent to the value-1 problem: this is also undecidable [22], but decidable for the special class of leaktight probabilistic automata [17]. The boundedness problem is not interesting for probabilistic automata (since the output is always bounded by 1), but a folklore argument shows that it is undecidable for Q ⩾0 (+, ·) (see Section 3). ...