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