It usually works: The temporal logic of stochastic systems

Abstract

In this paper the branching time logic pCTL* is defined.pCTL* expresses quantitative bounds on the probabilities of correct behavior; it can be interpreted over discrete Markov processes. A bisimulation relation is defined on finite Markov processes, and shown to be sound and complete with respect to pCTL*. We extend the universe of models to generalized Markov processes in order to support notions of refinement, abstraction, and parametrization. Model checking pCTL* over generalized Markov processes is shown to be elementary by a reduction to RCF. We conclude by describing practical and theoretical avenues for further work.

A Logic of Probability with Decidable Model-Checking

“…In the following, we sketch the proof of Theorem 7 (2). In the proof we use 'equivalent' for 'equivalent over the probabilistic structure generated by ðM, s 0 Þ)'.…”

“…The first applications of temporal logic to probabilistic systems were considered while studying which temporal properties are satisfied with probability 1 by systems modeled as finite Markov chains [17]. Later, references [12,2] introduced pCTL and pCTL* logics that can express quantitative bounds on the probability of system evolutions. This approach is surveyed, e.g., in [11,4,19].…”

Group-by-Group Probabilistic Bisimilarities and Their Logical Characterizations

“…A first work in this direction was [4]; it showed that bisimilarity and stuttering bisimilarity are, respectively, in full agreement with the logical equivalences induced by CTL* and by CTL* without the next-time operator when interpreted over Kripke structures (state-labeled transition systems) [5]. In a subsequent work, it was shown that the equivalence induced by the probabilistic temporal logic PCTL*, interpreted over probabilistic Kripke structures, coincides with probabilistic bisimilarity [1]. A more recent work is [25], which introduces new probabilistic bisimilarities that are in full agreement with the logical equivalences induced by PCTL, PCTL*, and their variants without the next-time operator interpreted over nondeterministic and probabilistic Kripke structures [3].…”

“…Finally, we denote by PML ∃,I and PML ∀,I two further variants generalizing the previous two logics, in which the probability value p is replaced by a prob- 1] are such that p 1 ≤ p 2 -and the resulting diamond operator is interpreted as follows:…”

“…For instance, the two states s 1 and s 2 in Fig. 2, which are not related by ∼ PB,gbg,= as can be seen by considering the PML ∃,I formula offer [0.5,0.5] head [1,1] true, cannot be distinguished by any PML ∃,≥ formula.…”