Les systèmes à canaux non fiables ont été introduits à l'origine comme un modèle de communication. Ils ont donné naissance à une classe de complexité restée mal comprise pendant longtemps. Dans cette thèse, nous étudions et comblons certaines des plus importantes lacunes dans la connaissance de cette classe. Nous fournissons entre autres des bornes inférieure et supérieure qui se rejoignent pour la complexité en temps. Puis nous proposons un nouvel outil de preuve : le Problème de Sous Mot de Post (PEP). C'est un problème simple, inspiré du Problème de Correspondance de Post, et complet pour cette classe de complexité. Nous étudions ensuite PEP et ses variantes, ainsi que les langages de solutions de PEP sur lesquels nous avons fourni des résultats de complexité et des outils de preuve tels que des lemmes de pompage. / Lossy channel systems were originally introduced to model communication protocols. It gave birth to a complexity class wich remained scarcely undersood for a long time. In this thesis we study some of the most important gaps. In particular, we bring matching upper and lower bounds for the time complexity. Then we describe a new proof tool : the Post Embedding Problem (PEP) which is a simple problem, closely related to the Post Correspondence Problem, and complete for this complexity class. Finally, we study PEP, its variants and the languages of solutions of PEP on which we provide complexity results and proof tools like pumping lemmas.

Difference equations are the discrete analogs to differential equations. While the independent variable of differential equations normally is a continuous time variable, t, that of a difference equation is a discrete time variable, n, which measures time in intervals. A feature of difference equations not shared by differential equations is that they can be characterized as recursive functions. Examples of their use include modeling population changes from one season to another, modeling the spread of disease, modeling various business phenomena, discrete simulations applications, or giving rise to the phenomena chaos. The key is that they are discrete, recursive relations. Systems of difference equations are similar in structure to systems of differential equations. Systems of first-order linear difference equations are of the form x(n + 1) = Ax(n) , and systems of first-order linear differential equations are of the form x(t) = Ax(t). In each case A is a 2x2 matrix and x(n +1), x(n), x(t), and x(t) are all vectors of length 2. The methods used in analyzing systems of difference equations are similar to those used in differential equations.Solutions of scalar, second-order linear difference equations are similar to those of scalar, second-order differential equations, but with one major difference: the composition of their general solutions. When the eigenvalues of A, λ1 and λ2, are real and distinct, general solutions of differential equations are of the form x(t) = c1eλ1t +c2eλ2t, while general solutions of difference equations are of form x(n) = 1λn1 + c2λn2. So, on the one hand, while the methods used in examining systems of difference equations are similar to those used for systems of differential equations; on the other hand, their general solutions can exhibit significantly different behavior.Chapter 1 will cover systems of first-order and second-order linear difference equations that are autonomous (all coefficients are constant). Chapter 2 will apply that theory to the local stability analysis of systems of nonlinear difference equations. Finally, Chapter 3 will give some example of the types of models to which systems of difference equations can be applied.

Lossy channel systems were originally introduced to model communication protocols. It gave birth to a complexity class wich remained scarcely undersood for a long time. In this thesis we study some of the most important gaps. In particular, we bring matching upper and lower bounds for the time complexity. Then we describe a new proof tool : the Post Embedding Problem (PEP) which is a simple problem, closely related to the Post Correspondence Problem, and complete for this complexity class. Finally, we study PEP, its variants and the languages of solutions of PEP on which we provide complexity results and proof tools like pumping lemmas.