Calculus for decision systems

The conceptualization of the term system has become highly dependent on the application domain. What a physicist means by the term system might be different than what a sociologist means by the same term. In 1956, Bertalanffy [1] defined a system as a set of units with relationships among them . This and many other definitions of system share the idea of a system as a black box that has parts or elements interacting between each other. This means that at some level of abstraction all systems are similar, what eventually differentiates one system from another is the set of underlining equations which describe how these parts interact within the system. ^ In this dissertation we develop a framework that allows us to characterize systems from an interaction level, i.e., a framework that gives us the capability to capture how/when the elements of the system interact. This framework is a process algebra called Calculus for Decision Systems (CDS). This calculus provides means to create mathematical expressions that capture how the systems interact and react to different stimuli. It also provides the ability to formulate procedures to analyze these interactions and to further derive other interesting insights of the system. ^ After defining the syntax and reduction rules of the CDS, we develop a notion of behavioral equivalence for decision systems. This equivalence, called bisimulation, allows us to compare decision systems from the behavioral standpoint. We apply our results to games in extensive form, some physical systems, and cyber-physical systems. ^ Using the CDS for the study of games in extensive form we were able to define the concept of subgame perfect equilibrium for a two-person game with perfect information. Then, we investigate the behavior of two games played in parallel by one of the players. We also explore different couplings between games, and compare - using bisimulation - the behavior of two games that are the result of two different couplings. The results showed that, with some probability, the behavior of playing a game as first player, or second player, could be irrelevant. ^ Decision systems can be comprised by multiple decision makers. We show that in the case where two decision makers interact, we can use extensive games to represent the conflict resolution. For the case where there are more than two decision makers, we presented how to characterize the interactions between elements within an organizational structure. Organizational structures can be perceived as multiple players interacting in a game. In the context of organizational structures, we use the CDS as an information sharing mechanism to transfer the inputs and outputs from one extensive game to another. We show the suitability of our calculus for the analysis of organizational structures, and point out some potential research extensions for the analysis of organizational structures. ^ The other general area we investigate using the CDS is cyber-physical systems. Cyber-physical systems or CPS is a class of systems that are characterized by a tight relationship between systems (or processes) in the areas of computing, communication and physics. We use the CDS to describe the interaction between elements in some simple mechanical system, as well as a particular case of the generalized railroad crossing (GRC) problem, which is a typical case of CPS. We show two approaches to the solution of the GRC problem. ^ This dissertation does not intend to develop new methods to solve game theoretical problems or equations of motion of a physical system, it aims to be a seminal work towards the creation of a general framework to study systems and equivalence of systems from a formal standpoint, and to increase the applications of formal methods to real-world problems

Resource Contention in Real-time Systems

The divide—and—conquer method is extensively used for system design. For real-time systems the separated components execute concurrently using some common computational infrastructure and this can lead to contention for system resources, such as processors, memory, communication channels, and so on. Unless the resource contention is accommodated, then a system built from the composition of components may not function as expected and the “proven” behaviour of the components can be invalid. To overcome this uncertainty a divide—conquer—and—system-composition method is required. This thesis takes a different approach to many of the existing notations which focus on descriptions of behaviour. The Composite Transition System notation and algebra presented here enables the resource usage of the components to be specified and combined to form a composite system of concurrently executing components. By relating the composite system to the realisable behaviour of the system resources provided by the common infrastructure it becomes possible to determine any violation of the constraints imposed by the system resources. If the composite system model is then constrained by the resource behaviours then it is possible through an extraction operation to determine the modified behaviour of the components that will yield a system free of resource contention. Component specification, concurrent composition, the application of system level constraints and extraction are applied in this thesis to a system encountered in a commercial application. The purpose of this example is to demonstrate contention modelling and the mathematics of the notation, rather than to prove any specific properties of the application. Deployment of the notation to more complex applications will require the development of software tools to compute concurrent composition and extraction, and this is the motivation for the mathematical treatment in this thesis

Jugando con el tiempo : semántica de pruebas para algebras de procesos temporizadas

en el presente trabajo hemos estudiado la semántica de pruebas para alebras de procesos temporizadas. En primer lugar hemos estudiado un álgebra de procesos temporizada básica, se trata de un lenguaje recursivo, secuencial no determinista. Puesto que las semánticas de pruebas son poco manejables, se hace necesario dar una caracterización alternativa de la misma; nosotros hemos dado una caracterizacion que depende unicamente de la semántica operacional de álgebra. A continuación hemos dotado al álgebra de una semántica denotacional, que ha resultado ser completamente abstracta con respecto a la semántica de pruebas. Seguidamente hemos estudiado una semántica axiomática, puesto que conseguimos probar que esta ultima es correcta y completa con respecto a la semántica detonaciones tenemos inmediatamente que tambien será correcta y completa con respecto a la semántica de pruebas. Puesto que todo lo anterior lo habíamos hecho con un álgebra bastante simple, es necesario introducir operadores mas complejos. En concreto hemos estudiado una serie de operadores que aparecen en la mayoria de las alebras de procesos temporizadas: . El operador de paralelo. El operador de ocultamiento, y. El operador de prefijo mediante acción visible con intervalo de tiempo. Por ultimo hemos estudiado un operador de elección tipo ccs, que tiene los problemas típicos con respecto a la congruenci

Stochastic transition systems: bisimulation, logic, and composition

Cyber-physical systems and the Internet of Things raise various challenges concerning the modelling and analysis of large modular systems. Models for such systems typically require uncountable state and action spaces, samplings from continuous distributions, and non-deterministic choices over uncountable many alternatives. In this thesis we fo- cus on a general modelling formalism for stochastic systems called stochastic transition system. We introduce a novel composition operator for stochastic transition systems that is based on couplings of probability measures. Couplings yield a declarative modelling paradigm appropriate for the formalisation of stochastic dependencies that are caused by the interaction of components. Congruence results for our operator with respect to standard notions for simulation and bisimulation are presented for which the challenge is to prove the existence of appropriate couplings. In this context a theory for stochastic transition systems concerning simulation, bisimulation, and trace-distribution relations is developed. We show that under generic Souslin conditions, the simulation preorder is a subset of trace-distribution inclusion and accordingly, bisimulation equivalence is finer than trace-distribution equivalence. We moreover establish characterisations of the simulation preorder and the bisimulation equivalence for a broad subclass of stochastic transition systems in terms of expressive action-based probabilistic logics and show that these characterisations are still maintained by small fragments of these logics, respectively. To treat associated measurability aspects, we rely on methods from descriptive set theory, properties of Souslin sets, as well as prominent measurable-selection principles.:1 Introduction 2 Probability measures on Polish spaces 3 Stochastic transition systems 4 Simulations and trace distributions for Souslin systems 5 Action-based probabilistic temporal logics 6 Parallel composition based on spans and couplings 7 Relations to models from the literature 8 Conclusions 9 Bibliograph