Probabilistic Rewrite Theories Nirman Kumar, Koushik Sen, Jos´ e Meseguer, Gul Agha Department of Computer Science, University of Illinois at Urbana Champaign. {nkumar5,ksen,meseguer,agha}@cs.uiuc.edu Abstract We propose probabilistic rewrite theories as a general semantic framework supporting high- level specification of probabilistic systems that can be massively concurrent. We give the defini- tion and semantics of probabilistic rewrite theories and discuss the mappings between different classes of theories and models. We then define the semantics of probabilistic temporal formulae for a given probabilistic rewrite theory. We explain how real-time probabilistic systems whose time is discrete can be expressed as probabilistic rewrite theories without any extension. Finally we give our design ideas for PMaude, an implementation of probabilistic rewrite theories on top of Maude 2.0. We shall report a running prototype of PMaude in the final version of the paper. 1 Introduction We propose a natural extension of rewrite theories, called probabilistic rewrite theories, as a general high-level formalism to specify probabilistic systems of the kind used in performance and reliability modelling, and in distributed randomized algorithms. We are particularly interested in specifying next-generation large networks of embedded hybrid systems. Such systems are distributed, highly asynchronous, have essential real-time features, and have unreliable communication media and faults. The behavior of these systems can be modelled probabilistically by replacing nondeterminism due to unreliability and faults by probabilities. There has been considerable research on models, logics, and model checking for probabilistic systems. Work in this area includes, among others, probabilistic process algebra approaches such as [12, 11, 25], probabilistic Petri nets [18, 19], the hybrid I/O automata by Lynch [15], probabilistic automata by Segala and Lynch [27], and probabilistic nondeterministic systems by Bianco and de Alfaro [8]. Theories related to timed automata were developed by Alur and Dill [3], and Alur and Dill [26] presents a survey of real-time logics, including RTL and MTL. Marta Kwiatkowska and others combined these works to develop the theory of probabilistic timed automata and tools related to verification of such systems [14]. The logics underlying such tools include Computational Stochastic Logic (CSL), Probabilistic Computational Tree Logic (PCTL) and Probabilistic Timed Computation Tree Logic (PTCTL). The PRISM tool [13] of Kwiatowska et al. has been developed to model check systems based on these models for the above mentioned logics. It implements efficient symbolic model checking techniques as proposed in [5]. The situation with models at present is that, either they build in some existing model of concurrency, such as a variant of CCS, Petri nets, or I/O automata, or they are low-level automaton like models such as Markov chains or probabilistic nondeterministic systems (PNSs). In this context, 1