Jean Mairesse

Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture

Given an Iterated Function System (IFS) of topical maps verifying some conditions, we prove that the asymptotic height optimization problems are equivalent to finding the extrema of a continuous functional, the average height, on some compact space of measures. We give general results to determine these extrema, and then apply them to two concrete problems. First, we give a new proof of the theorem that the densest heaps of two Tetris pieces are sturmian. Second, we construct an explicit counterexample to the Lagarias-Wang finiteness conjecture.

Deciding unambiguity and sequentiality from a finitely ambiguous max-plus automaton

Finite automata with weights in the max-plus semiring are considered. The main result is: it is decidable whether a series that is recognized by a finitely ambiguous max-plus automaton is unambiguous, or is sequential. Furthermore, the proof is constructive. A collection of examples is given to illustrate the hierarchy of max-plus series with respect to ambiguity.

Idempotency: Task resource models and (max, +) automata

We show that a typical class of timed concurrent systems can be modeled as automata with multiplicities in the (max,+) semiring. This representation can be seen as a timed extension of the logical modeling in terms of trace monoids. We briefly discuss the applications of this algebraic modeling to performance evaluation.

Deficiency Zero Petri Nets and Product Form

Consider a Markovian Petri net with race policy. The marking process has a “product form” stationary distribution if the probability of viewing a given marking can be decomposed as the product over places of terms depending only on the local marking. First we observe that the Deficiency Zero Theorem of Feinberg, developed for chemical reaction networks, provides a structural and simple sufficient condition for the existence of a product form. In view of this, we study the classical subclass of free-choice nets. Roughly, we show that the only Petri nets of this class which have a product form are the state machines, which can alternatively be viewed as Jackson networks.

Around probabilistic cellular automata

We survey probabilistic cellular automata with approaches coming from combinatorics, statistical physics, and theoretical computer science, each bringing a different viewpoint. Some of the questions studied are specific to a domain, and some others are shared, most notably the ergodicity problem.

ON THE FINITENESS PROBLEM FOR AUTOMATON (SEMI)GROUPS

This paper addresses a decision problem highlighted by Grigorchuk, Nekrashevich, and Sushchanskii, namely the finiteness problem for automaton (semi)groups. For semigroups, we give an effective sufficient but not necessary condition for finiteness and, for groups, an effective necessary but not sufficient condition. The efficiency of the new criteria is demonstrated by testing all Mealy automata with small stateset and alphabet. Finally, for groups, we provide a necessary and sufficient condition that does not directly lead to a decision procedure.

Growth series for Artin groups of dihedral type

We consider the Artin groups of dihedral type I2(k) defined by the presentation Ak = 〈a,b | prod(a,b;k) = prod(b,a;k)〉 where prod(s,t;k) = ststs …, with k terms in the product on the right-hand side. We prove that the spherical growth series and the geodesic growth series of Ak with respect to the Artin generators {a,b,a-1, b-1} are rational. We provide explicit formulas for the series.

Asymptotic results on infinite tandem queueing networks

Abstract. We consider an infinite tandem queueing network consisting of ·/GI/1/∞ stations with i.i.d. service times. We investigate the asymptotic behavior of t(n, k), the inter-arrival times between customers n and n + 1 at station k, and that of w(n, k), the waiting time of customer n at station k. We establish a duality property by which w(n, k) and the “idle times”y(n, k) play symmetrical roles. This duality structure, interesting by itself, is also instrumental in proving some of the ergodic results. We consider two versions of the model: the quadrant and the half-plane. In the quadrant version, the sequences of boundary conditions {w(0,k), k∈ℕ} and {t(n, 0), n∈ℕ}, are given. In the half-plane version, the sequence {t(n, 0), n∈ℕ} is given. Under appropriate assumptions on the boundary conditions and on the services, we obtain ergodic results for both versions of the model. For the quadrant version, we prove the existence of temporally ergodic evolutions and of spatially ergodic ones. Furthermore, the process {t(n, k), n∈ℕ} converges weakly with k to a limiting distribution, which is invariant for the queueing operator. In the more difficult half plane problem, the aim is to obtain evolutions which are both temporally and spatially ergodic. We prove that 1/n∑k=1nw(0, k) converges almost surely and in L1 to a finite constant. This constitutes a first step in trying to prove that {t(n,k), n∈ℤ} converges weakly with k to an invariant limiting distribution.

Randomly growing braid on three strands and the manta ray.

Consider the braid group B3 = and the nearest neighbor random walk defined by a probability \nu with support {a,b,a^-1,b^-1}. The rate of escape of the walk is explicitely expressed in function of the unique solution of a set of eight polynomial equations of degree three over eight indeterminates. We also explicitely describe the harmonic measure of the induced random walk on B3 quotiented by its center. The method and results apply, mutatis mutandis, to nearest neighbor random walks on dihedral Artin groups.

Blocking a transition in a free choice net and what it tells about its throughput

In a live and bounded free choice Petri net, pick a non-conflicting transition. Then there exists a unique reachable marking in which no transition is enabled except the selected one. For a routed live and bounded free choice net, this property is true for any transition of the net. Consider now a live and bounded stochastic routed free choice net, and assume that the routings and the firing times are independent and identically distributed. Using the above results, we prove the existence of asymptotic firing throughputs for all transitions in the net. Furthermore, the vector of the throughputs at the different transitions is explicitly computable up to a multiplicative constant.