We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series - the semilinear and the bounded series - both natural generalizations of the semilinear languages, and we study their behaviour under rational operations, morphisms, Hadamard product, and difference. Turning to commutation on sets of words, we then study the notions of centralizer of a language - the largest set commuting with a language -, of root and of primitive root of a set of words. We answer a question raised by Conway more than thirty years ago - asking whether or not the centralizer of any rational language is rational - in the case of periodic, binary, and ternary sets of words, as well as for rational c-codes, the most general results on this problem. We also prove that any code has a unique primitive root and that two codes commute if and only if they have the same primitive root, thus solving two conjectures of Ratoandromanana, 1989. Moreover, we prove that the commutation with an c-code X can be characterized similarly as in free monoids: a language commutes with X if and only if it is a union of powers of the primitive root of X.

The learnability of the class of letter-counts of regular languages (semilinear sets) and other related classes of subsets of N d with respect to the distribution-free learning model of Valiant (PAC-learning model) is characterized. Using the notion of reducibility among learning problems due to Pitt and Warmuth called "prediction preserving reducibility," and a special case thereof, a number of positive and partially negative results are obtained. On the positive side the class of semilinear sets of dimension 1 or 2 is shown to be learnable when the integers are encoded in unary. On the neutral to negative side it is shown that when the integers are encoded in binary the learning problem for semilinear sets as well as a class of subsets of Z d much simpler than semilinear sets is as hard as learning DNF, a central open problem in the field. A number of hardness results for related learning problems are also given. 3 Most of the research reported herein was conducted while the aut...

In the early days of high-speed computers there was a hope that it would be possible to program a computer to translate from one language to another, and this led to an intensive study of language structure. The result has been disappointing in that we are still far from making translations by computer, but that is no cause for

In this paper we discuss the problem of estimating the marking of a Place/Transition net based on event observation. We assume that the net structureisknown while the initial marking is unknown. We define several observability properties and show how they can beproved. In particular we set up a hierarchy considering the possibility that the above properties are satisfied by a net N starting from an initial marking M 0 , by a net N starting from any initial marking M reachable from an initial marking M 0 , or by a net N starting from any marking in N m ,where m is the number of places of the net. 1 Introduction This paper presents a set of analytical tools to determine the observability properties of Petri nets, i.e., algorithms to determine under which conditions it is possible to reconstruct the marking of a Place/Transition net based on eventobservation. Observability is widely studied by automatic control researchers. It is a fundamental property because it allows one to estim...