# Recent Posts

• April 06, 2019

### A polynomial time algorithm for the equivalence problem of weighted automata

Nathanaël Fijalkow

We present an algebraic algorithm solving the equivalence problem for weighted automata (in particular probabilistic automata) in polynomial time.A formal series (here over the reals) is a function .For recognising formal series we use weighted au...

• March 26, 2019

### Weighted automata and matrix factorisations

Nathanaël Fijalkow

We discuss extensions of Fliess' theorem which says that the smallest weighted automaton for a function is exactly the rank of its Hankel matrix.The goal is to extend this theorem to subclasses of weighted automata such as probabilistic automata.M...

• March 26, 2019

### PAC-learning and compression schemes

Nathanaël Fijalkow

We state and prove the equivalence between PAC-learnability, finite VC dimension, and the existence of compression schemes.Many thanks to Borja Balle, Pascale Gourdeau, and Pierre Ohlmann, for discussions on compression schemes and for the joint e...

• March 25, 2019

### Strong and weak separating automata

Nathanaël Fijalkow

This post discusses the difference between strong and weak separating automata.The goal of this post is to discuss the notion of separating automata, which already appear in many previous posts, see this post and this post for parity games,this po...

• February 10, 2019

### Mean payoff games and universal graphs

Nathanaël Fijalkow

This 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 Fijalkow

This 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 context-free grammars

Nathanaël Fijalkow

We discuss an extension of Fliess' theorem for minimising weighted tree automata.This post is somehow a follow-up 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 Fijalkow

This 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 Fijalkow

This 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 Fijalkow

This 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 Fijalkow

We 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 Fijalkow

We prove that the boundedness problem for (min,plus)-automata is decidable.One of the main application of this result is that the star-heigh 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 Fijalkow

This 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 Fijalkow

We 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 Fijalkow

This 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 Fijalkow

This post is about generalized reachability games. The main purpose is to pose an open problem:what is the complexity of solving 2-generalized 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 Fijalkow

This post gives a proof of positional determinacy for parity games.One may recognise it in the works of Emerson and Jutla (91) using modal mu-calculus, 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 Fijalkow

This 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 Fijalkow

We 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 Angluin’s style ...

• April 12, 2018

### Fliess' theorem for minimising weighted automata

Nathanaël Fijalkow

We 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 bi-infinite matrix ...