Infinite Structures in Timed Systems

Sammanfattning: Real time systems distinguish themselves by explicitly stating timing constraints in the system specification. This requires specific methods and tools in system design to ensure such constraints. We focus on one of the methods applied in the validation phase, namely formal verification. This method automatically establishes correctness of the system model with mathematical rigor. In order to apply mechanical procedures to determine whether the system satisfies the requirements, we first have to model the validated part of the system in a mathematical form. This thesis deals with one such formalism - timed automata - and investigates different types of infinite state structures arising in the verification procedures related to this formalism. There are two different views which open the door for introduction of such structures.First, we turn outwards and extend timed automata with additional infinite data structures - unbounded queues. These queues serve different purposes. In one case, the queues contain computation tasks and, together with a timed automaton, model a real-time system with tasks. The problem of interest in this setting is schedulability analysis. We investigate the decidability boundary in presence of various features such as preemption, variable computation times of tasks, and communication between the timed automaton and the task queue. In the other case, we use queues for asynchronous communication between timed automata running synchronously in parallel. These queues store messages issued by one automaton and waiting to be read by another automaton. Such situations occur among other cases in real-time control systems where several concurrently running tasks communicate via buffers. We study the decidability border for reachability analysis depending on various communication topologies of these systems.Secondly, we turn inwards and study a peculiar feature of timed automata which allows them to enforce behaviors where time distances between events monotonically grow while being bounded by some integer. This feature can be characterized by unbounded counters recording the number of such enforced increases. When we switch from the dense time semantics used for modeling to an implementation with a fixed clock rate (sampled semantics), only behaviors which correspond to a bounded usage of these counters are preserved. We describe operation of these counters as a new type of a counter automaton and prove that one can effectively check whether the counters are used in a bounded way. As a result, it is possible to check for a given timed automaton whether there is an implementation with a fixed sampling rate which preserves all qualitative behaviors.