Trace Expressiveness of Timed and Probabilistic Automata

Automata expressiveness is an essential feature in understanding which of the formalisms available should be chosen for modelling a particular problem. Probabilistic and stochastic automata are suitable for modelling systems exhibiting probabilistic behaviour and their expressiveness has been studied relative to non-probabilistic transition systems and Markov chains. In this paper, we consider previous formalisms of Timed, Probabilistic and Stochastic Timed Automata, we present our new model of Timed Automata with Polynomial Delay, we introduce a measure of expressiveness for automata we call trace expressiveness and we characterize the expressiveness of these models relative to each other under this new measure. Timed and probabilistic automata may be used for modelling systems that exhibit timed and probabilistic behaviour as diverse as safety-critical navigation, web security, algorithmic trading, search-engine optimization, communication protocol design and hardware failure prediction. It is important to understand the expressive power of these models and incorporate them within an expressiveness hierarchy, as properties of given machines may be deduced upwards and downwards the hierarchy. There exists, for example, a trade-off between the expressiveness of a given model of computation and the tractability/decidability of its Model Checking problem. This paper provides a unifying expressiveness framework for the aforementioned models. We also use our model of Timed Automata with Polynomial Delay, a restriction of Stochastic Timed Automata to transitions characterized by polynomials. Lehmann and Rabin in [20] show that distributed systems of probabilistic processors are essentially more powerful than distributed systems of deterministic processors. Of interest to us, due to the concepts and machinery employed, is also Thin and Thick Timed Regular Languages by Basset and Asarin [2] in which information-theoretic arguments are applied for characterizing trajectories of timed automata. In [1] it is developed the theory of timed automata to model the behavior of real-time, safety-critical systems over time. The definition provides a way to include timing information within state-transition machines using real-valued variables called clocks. The paper [9] gives an introduction to probabilistic automata, describing how distributed systems with discrete probabilities can be modeled and analyzed by means of this model and extending the basic techniques of analysis of non-probabilistic automata to probabilistic systems. Probabilistic Automata were introduced in [21] and are also treated in [22] and in [23], [24], in [11], [12] and [10]. Probabilistic timed automata are timed automata extended with discrete probability distributions, and can be used to model timed randomized protocols or fault-tolerant systems. They were introduced in [7] and are also treated in [19] and in [13]. The authors use probabilistic decision protocols to model realtime models exhibiting probabilistic behavior. Stochastic Timed Automata are treated in [3], [18], [15] and [14], [6]. These are timed automata equipped on each edge with probability distributions depending on time. A mathematical treatment of concurrent timed and probabilistic systems is given in the monograph [8]. The theory of Stochastic Processes is given in [5]. There exists the hierarchy of formal languages introduced in [17] which induces expressiveness results on certain type of automata. As expected, with a very expressive model of computation come various undecidable features. In [9] it is shown that probabilistic automata subsume non-probabilistic automata, Markov chains and Markov decision processes. In [16] we introduced two measures of expressiveness on runs, isomorphic and homomorphic expressiveness and observed a clear separation between all formalisms considered when applying isomorphic expressiveness and a separation between stochastic and non-stochastic automata when considering homomorphic expressiveness. In the present paper we consider traces which may roughly be equated with the “external” behaviour of a machine; we observe a similar separation between stochastic and non-stochastic models. We achieve this by assigning weights to edges of non-probabilistic machines and using the edge probabilities for the probabilistic machines and defining a notion of isomorphism between traces; this induces a measure on the traces of automata and different formalisms can then be compared subject to their collection of traces being isomorphic. We proceed by explaining the notation used and introduce finite-state automata, we introduce adapted versions of timed automata, probabilistic automata, probabilistic timed automata and stochastic timed automata and our model of timed automata with polynomial delay. In the third section we outline our expressiveness results from [16]. The fourth section introduces new results concerning trace expressiveness.