Recent Posts

March 12, 2019
Learning probabilistic automata using nonnegative matrix factorisation
Nathanaël FijalkowWe discuss an approach to learning probabilistic automata based on nonnegative matrix factorisation.We use the notations from this post about weighted automata and the Hankel matrix.It may be useful to have read this post about learning weighted ...

February 10, 2019
Mean payoff games and universal graphs
Nathanaël FijalkowThis post discusses using separating automata for constructing algorithms for solving mean payoff games.The goal of this post is to present the main results of this paper,meaning upper and lower bounds on separating automata for mean payoff games....

February 09, 2019
Separating automata and universal graphs
Nathanaël FijalkowThis post discusses the equivalence between separating automata and universal graphs beyond parity games.The goal of this post is to present again the result of this paper, which is a joint work with Thomas Colcombet, but this time in more general...

October 19, 2018
Minimising weighted tree automata and contextfree grammars
Nathanaël FijalkowWe discuss an extension of Fliess' theorem for minimising weighted tree automata.This post is somehow a followup of this one, but it can be read independently.Minimising weighted tree automataWe consider tree formal series (here over the reals), ...

October 12, 2018
Parity games and universal graphs
Nathanaël FijalkowThis post introduces the notion of universal graphs and their applications to parity games.The goal of this post is to prove the main result of this paper,meaning that separating automata have at least quasipolynomial size.This quasipolynomial bar...

October 11, 2018
The fundamental theorem of parity games
Nathanaël FijalkowThis post proves a very important theorem about parity games, showing the interplay with (universal) trees.The goal of this post is to prove the following result.This is the key technical point of a recent joint work with Thomas Colcombet proving ...

October 10, 2018
Separation for parity games
Nathanaël FijalkowThis post discusses a framework for reducing parity games to safety games by the construction of an automaton.There are three quasipolynomial time algorithms for parity games, namely the statistics games of Calude, Jain, Khoussainov, Li, and Steph...

September 04, 2018
A robust class of sequences
Corentin Barloy and Nathanaël FijalkowWe introduce a subclass of linear recurrence sequences (LRS).We show that this class is robust by giving several characterisations.The results presented here are mostly due to Corentin Barloy, who took a summer internship in LaBRI, Bordeaux, under...

August 15, 2018
Boundedness for (min,plus)automata
Nathanaël FijalkowWe prove that the boundedness problem for (min,plus)automata is decidable.One of the main application of this result is that the starheigh problem of regular expressions reduces to (a slight extension of) it.The main point is to present a very u...

August 10, 2018
Value iteration for parity games
Nathanaël FijalkowThis post presents a generic value iteration algorithm for parity gamesparametrised by universal trees. As special cases this extends the small progress measure of Jurdziński and the succinct progress measure of Jurdziński and Lazić...

August 08, 2018
The universality problem for automata with bounded ambiguity
Ritam Raha and Nathanaël FijalkowWe prove that the universality problem for automata with fixed ambiguity is decidable in polynomial time.Given an automaton $\A$ on alphabet $A$ recognising the language $L(\A)$, the universality problem asks whetherThe universality problem is PSP...

August 03, 2018
Universal trees
Nathanaël FijalkowThis post introduces the notion of universal trees and poses an open problem:what is the exact size of the smallest universal tree?The motivation for studying universal trees comes from parity games. This post does not explain this connection at a...

August 03, 2018
Generalized reachability games
Nathanaël FijalkowThis post is about generalized reachability games. The main purpose is to pose an open problem:what is the complexity of solving 2generalized reachability games?This post is about this paper. This is the first research paper I have worked on (wit...

August 02, 2018
Positional determinacy for parity games, a forward approach
Nathanaël FijalkowThis post gives a proof of positional determinacy for parity games.One may recognise it in the works of Emerson and Jutla (91) using modal mucalculus, and also more explicitely in the works of Kupferman and Vardi (98).We fix some notations. Consi...

August 02, 2018
Positional determinacy for parity games, a backward approach by Muller and Schupp
Nathanaël FijalkowThis post revisits Muller and Schupp's backward approach to prove the positionality of parity games.The argument is simple and beautiful; credits go to Thomas Colcombet for this presentation.We discussed the technical details of the construction i...

April 13, 2018
Angluin's style learning for weighted automata
Nathanaël FijalkowWe show that weighted automata over the reals can be learned efficiently in Angluin's supervised scenario.This post uses all the notations of the previous post,and the result presented here was proved in this paper.The scenario is Angluinâ€™s style ...

April 12, 2018
Fliess' theorem for minimising weighted automata
Nathanaël FijalkowWe state and prove Fliess' theorem, which says that the minimal automaton recognising a given formal series has size the rank of its Hankel matrix.A formal series (here over the reals) is a function .The Hankel matrix of is a biinfinite matrix ...

April 02, 2018
The fundamental theorem of statistical learning
Nathanaël FijalkowWe state and prove the fundamental theorem of statistical learning, which says that a class of functions is learnableif and only if it has finite VC dimension.This post uses all the notations of the previous post. Theorem: If $H$ has infinit...

March 15, 2018
VC dimension, Rademacher complexity, and growth function
Nathanaël FijalkowWe define the VC dimension, the Rademacher complexity, and the growth function of a class of functions.Let $X$ be the set of inputs. The general learning question we consider is the following: we want to learn a target function through a number o...