On Regularity of Unary Probabilistic Automata

The quantitative verification of Probabilistic Automata (PA) is undecidable in general. Unary PA are a simpler model where the choice of action is fixed. Still, the quantitative verification problem is open and known to be as hard as Skolem\u27s problem, a problem on linear recurrence sequences, whose decidability is open for at least 40 years. In this paper, we approach this problem by studying the languages generated by unary PAs (as defined below), whose regularity would entail the decidability of quantitative verification. Given an initial distribution, we represent the trajectory of a unary PA over time as an infinite word over a finite alphabet, where the n-th letter represents a probability range after n steps. We extend this to a language of trajectories (a set of words), one trajectory for each initial distribution from a (possibly infinite) set. We show that if the eigenvalues of the transition matrix associated with the unary PA are all distinct positive real numbers, then the language is effectively regular. Further, we show that this result is at the boundary of regularity, as non-regular languages can be generated when the restrictions are even slightly relaxed. The regular representation of the language allows us to reason about more general properties, e.g., robustness of a regular property in a neighbourhood around a given distribution

Behavioural Preorders on Stochastic Systems - Logical, Topological, and Computational Aspects

Computer systems can be found everywhere: in space, in our homes, in our cars, in our pockets, and sometimes even in our own bodies. For concerns of safety, economy, and convenience, it is important that such systems work correctly. However, it is a notoriously difficult task to ensure that the software running on computers behaves correctly. One approach to ease this task is that of model checking, where a model of the system is made using some mathematical formalism. Requirements expressed in a formal language can then be verified against the model in order to give guarantees that the model satisfies the requirements. For many computer systems, time is an important factor. As such, we need our formalisms and requirement languages to be able to incorporate real time. We therefore develop formalisms and algorithms that allow us to compare and express properties about real-time systems. We first introduce a logical formalism for reasoning about upper and lower bounds on time, and study the properties of this formalism, including axiomatisation and algorithms for checking when a formula is satisfied. We then consider the question of when a system is faster than another system. We show that this is a difficult question which can not be answered in general, but we identify special cases where this question can be answered. We also show that under this notion of faster-than, a local increase in speed may lead to a global decrease in speed, and we take step towards avoiding this. Finally, we consider how to compare the real-time behaviour of systems not just qualitatively, but also quantitatively. Thus, we are interested in knowing how much one system is faster or slower than another system. This is done by introducing a distance between systems. We show how to compute this distance and that it behaves well with respect to certain properties.Comment: PhD dissertation from Aalborg Universit