Ce mémoire étudie le comportement des particules dont la position est maximale au temps t dans la marche aléatoire branchante et le mouvement brownien branchant sur R, pour des valeurs de t grandes. Plus exactement, on regarde le comportement du maximum d’une marche aléatoire branchante dans un environnement inhomogène en temps, au sens où la loi des accroissements varie en fonction du temps. On compare avec des modèles connus ou simplifiés, en particulier le modèle i.i.d., où l’on observe des marches aléatoires indépendantes et le modèle de la marche aléatoire homogène. On s’intéresse par la suite aux corrélations entre les particules maximales d’un mouvement brownien branchant. Plus précisément, on étudie le temps de branchement entre deux particules maximales. Finalement, on applique les méthodes et les résultats des premiers chapitres afin d’étudier les corrélations dans un mouvement brownien branchant dans un environnement inhomogène. Le résultat principal du mémoire stipule qu’il y a existence de temps de branchement au centre de l’intervalle [0, t] dans le mouvement brownien branchant inhomogène, ce qui n’est pas le cas pour le mouvement brownien branchant standard. On présentera également certaines simulations numériques afin de corroborer les résultats numériques et pour établir des hypothèses pour une recherche future. / This thesis studies the behavior of particles that are maximal at time t in branching random walk and branching Brownian motion on R, for large values of t. Precisely, we look at the behavior of the maximum in a branching random walk in a time-inhomogeneous environment, where the law of the increments varies with respect to time. We compare with known or simplified models such as the model where random walks are taken to be i.i.d. and the branching random walk in a time-homogeneous environment model. We then take a look at the correlations between maximal particles in a branching brownian motion. Specifically, we look at the branching time between those maximal particles. Finally, we apply results and methods from the first chapters to study those same correlations in branching Brownian motion in a inhomogeneous environment. The thesis’ main result establishes existence of branching time at the center of the interval [0, t] for the branching Brownian motion in a inhomogeneous environment, which is not the case for standard branching brownian motion.We also present results of simulations that agree with theoretical results and help establishing new hypotheses for future research.

Ce mémoire étudie le comportement des particules dont la position est maximale au temps t dans la marche aléatoire branchante et le mouvement brownien branchant sur R, pour des valeurs de t grandes. Plus exactement, on regarde le comportement du maximum d’une marche aléatoire branchante dans un environnement inhomogène en temps, au sens où la loi des accroissements varie en fonction du temps. On compare avec des modèles connus ou simplifiés, en particulier le modèle i.i.d., où l’on observe des marches aléatoires indépendantes et le modèle de la marche aléatoire homogène. On s’intéresse par la suite aux corrélations entre les particules maximales d’un mouvement brownien branchant. Plus précisément, on étudie le temps de branchement entre deux particules maximales. Finalement, on applique les méthodes et les résultats des premiers chapitres afin d’étudier les corrélations dans un mouvement brownien branchant dans un environnement inhomogène. Le résultat principal du mémoire stipule qu’il y a existence de temps de branchement au centre de l’intervalle [0, t] dans le mouvement brownien branchant inhomogène, ce qui n’est pas le cas pour le mouvement brownien branchant standard. On présentera également certaines simulations numériques afin de corroborer les résultats numériques et pour établir des hypothèses pour une recherche future. / This thesis studies the behavior of particles that are maximal at time t in branching random walk and branching Brownian motion on R, for large values of t. Precisely, we look at the behavior of the maximum in a branching random walk in a time-inhomogeneous environment, where the law of the increments varies with respect to time. We compare with known or simplified models such as the model where random walks are taken to be i.i.d. and the branching random walk in a time-homogeneous environment model. We then take a look at the correlations between maximal particles in a branching brownian motion. Specifically, we look at the branching time between those maximal particles. Finally, we apply results and methods from the first chapters to study those same correlations in branching Brownian motion in a inhomogeneous environment. The thesis’ main result establishes existence of branching time at the center of the interval [0, t] for the branching Brownian motion in a inhomogeneous environment, which is not the case for standard branching brownian motion.We also present results of simulations that agree with theoretical results and help establishing new hypotheses for future research.

Information and communication systems enter an increasing number of areas of daily lives. Our reliance and dependence on the functioning of such systems is rapidly growing together with the costs and the impact of system failures. At the same time the complexity of hardware and software systems extends to new limits as modern hardware architectures become more and more parallel, dynamic and heterogenous. These trends demand for a closer integration of formal methods and system engineering to show the correctness of complex systems within the design phase of large projects. The goal of this thesis is to introduce a formal holistic approach for modeling, analysis and synthesis of parallel systems that potentially addresses complex system behavior at any layer of the hardware/software stack. Due to the complexity of modern hardware and software systems, we aim to have a hierarchical modeling framework that allows to specify the behavior of a parallel system at various levels of abstraction and that facilitates designing complex systems in an iterative refinement procedure, in which more detailed behavior is added successively to the system description. In this context, the major challenge is to provide modeling formalisms that are expressive enough to address all of the above issues and are at the same time amenable to the application of formal methods for proving that the system behavior conforms to its specification. In particular, we are interested in specification formalisms that allow to apply formal verification techniques such that the underlying model checking problems are still decidable within reasonable time and space bounds. The presented work relies on an exogenous modeling approach that allows a clear separation of coordination and computation and provides an operational semantic model where formal methods such as model checking are well suited and applicable. The channel-based exogenous coordination language Reo is used as modeling formalism as it supports hierarchical modeling in an iterative top-down refinement procedure. It facilitates reusability, exchangeability, and heterogeneity of components and forms the basis to apply formal verification methods. At the same time Reo has a clear formal semantics based on automata, which serve as foundation to apply formal methods such as model checking. In this thesis new modeling languages are presented that allow specifying complex systems in terms of Reo and automata models which yield the basis for a holistic approach on modeling, verification and synthesis of parallel systems. The second main contribution of this thesis are tailored branching-time and alternating time temporal logics as well as corresponding model checking algorithms. The thesis includes results on the theoretical complexity of the underlying model checking problems as well as practical results. For the latter the presented approach has been implemented in the symbolic verification tool set Vereofy. The implementation within Vereofy and evaluation of the branching-time and alternating-time model checker is the third main contribution of this thesis.

Information and communication systems enter an increasing number of areas of daily lives. Our reliance and dependence on the functioning of such systems is rapidly growing together with the costs and the impact of system failures. At the same time the complexity of hardware and software systems extends to new limits as modern hardware architectures become more and more parallel, dynamic and heterogenous. These trends demand for a closer integration of formal methods and system engineering to show the correctness of complex systems within the design phase of large projects. The goal of this thesis is to introduce a formal holistic approach for modeling, analysis and synthesis of parallel systems that potentially addresses complex system behavior at any layer of the hardware/software stack. Due to the complexity of modern hardware and software systems, we aim to have a hierarchical modeling framework that allows to specify the behavior of a parallel system at various levels of abstraction and that facilitates designing complex systems in an iterative refinement procedure, in which more detailed behavior is added successively to the system description. In this context, the major challenge is to provide modeling formalisms that are expressive enough to address all of the above issues and are at the same time amenable to the application of formal methods for proving that the system behavior conforms to its specification. In particular, we are interested in specification formalisms that allow to apply formal verification techniques such that the underlying model checking problems are still decidable within reasonable time and space bounds. The presented work relies on an exogenous modeling approach that allows a clear separation of coordination and computation and provides an operational semantic model where formal methods such as model checking are well suited and applicable. The channel-based exogenous coordination language Reo is used as modeling formalism as it supports hierarchical modeling in an iterative top-down refinement procedure. It facilitates reusability, exchangeability, and heterogeneity of components and forms the basis to apply formal verification methods. At the same time Reo has a clear formal semantics based on automata, which serve as foundation to apply formal methods such as model checking. In this thesis new modeling languages are presented that allow specifying complex systems in terms of Reo and automata models which yield the basis for a holistic approach on modeling, verification and synthesis of parallel systems. The second main contribution of this thesis are tailored branching-time and alternating time temporal logics as well as corresponding model checking algorithms. The thesis includes results on the theoretical complexity of the underlying model checking problems as well as practical results. For the latter the presented approach has been implemented in the symbolic verification tool set Vereofy. The implementation within Vereofy and evaluation of the branching-time and alternating-time model checker is the third main contribution of this thesis.

Es wird die Axiomatisierbarkeit einer Klasse von temporalen Prädikatenlogiken über verzweigenden Strukturen gezeigt. Entscheidbarkeitsresultate folgen für diverse Fragmente dieser Logiken. Anwendungen werden diskutiert.

Es wird die Axiomatisierbarkeit einer Klasse von temporalen Prädikatenlogiken über verzweigenden Strukturen gezeigt. Entscheidbarkeitsresultate folgen für diverse Fragmente dieser Logiken. Anwendungen werden diskutiert.