Semi-Markov decision processes (SMDPs) are continuoustime Markov decision processes where the residence-time on states is governed by generic distributions on the positive real line. In this paper we consider the problem of comparing two SMDPs with respect to their time-dependent behaviour. We propose a hemimetric between processes, which we call simulation distance, measuring the least acceleration factor by which a process needs to speed up its actions in order to behave at least as fast as another process. We show that this distance can be computed in time O(n 2 (f (l) + k) + mn 7), where n is the number of states, m the number of actions, k the number of atomic propositions, and f (l) the complexity of comparing the residence-time between states. The theoretical relevance and applicability of this distance is further argued by showing that (i) it is suitable for compositional reasoning with respect to CSP-like parallel composition and (ii) has a logical characterisation in terms of a simple Markovian logic.
This paper studies the existence of finite equational axiomatisations of the
interleaving parallel composition operator modulo the behavioural equivalences
in van Glabbeek's linear time-branching time spectrum. In the setting of the
process algebra BCCSP over a finite set of actions, we provide finite,
ground-complete axiomatisations for various simulation and (decorated) trace
semantics. We also show that no congruence over BCCSP that includes
bisimilarity and is included in possible futures equivalence has a finite,
ground-complete axiomatisation; this negative result applies to all the nested
trace and nested simulation semantics.
Semi-Markov processes are Markovian processes in which the firing time of the transitions is modelled by probabilistic distributions over positive reals interpreted as the probability of firing a transition at a certain moment in time.In this paper we consider the trace-based semantics of semi-Markov processes, and investigate the question of how to compare two semi-Markov processes with respect to their time-dependent behaviour. To this end, we introduce the relation of being "faster than" between processes and study its algorithmic complexity. Through a connection to probabilistic automata we obtain hardness results showing in particular that this relation is undecidable. However, we present an additive approximation algorithm for a time-bounded variant of the faster-than problem over semi-Markov processes with slow residence-time functions, and a coNP algorithm for the exact faster-than problem over unambiguous semi-Markov processes.
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