Non-deterministic weighted automata evaluated over Markov chains

Abstract

We present the first study of non-deterministic weighted automata under probabilistic semantics. In this semantics words are random events, generated by a Markov chain, and functions computed by weighted automata are random variables. We consider the probabilistic questions of computing the expected value and the cumulative distribution for such random variables.The exact answers to the probabilistic questions for non-deterministic automata can be irrational and are uncomputable in general. To overcome this limitation, we propose approximation algorithms for the probabilistic questions, which work in exponential time in the size of the automaton and polynomial time in the size of the Markov chain and the given precision. We apply this result to show that non-deterministic automata can be effectively determinised with respect to the standard deviation metric. ACM Subject ClassificationTheory of computation → Automata over infinite objects; Theory of computation → Quantitative automata Some quantitative-model-checking frameworks [11] are based on the universality problem for non-deterministic automata, which asks whether all words have the value below a given threshold. Unfortunately, the universality problem is undecidable for weighted automata with the sum or the limit average values functions. The distribution question can be considered as a computationally-attractive variant of universality, i.e., we ask whether almost all words have value below some given threshold. We show that if the threshold can be approximated, the distribution question can be computed effectively. Weighted automata have been used to formally study online algorithms [2]. Online algorithms have been modeled by deterministic weighted automata, which make choices based solely on the past, while offline algorithms have been modeled by non-deterministic weighted automata. Relating deterministic and non-deterministic models allowed for formal verification of the worst-case competitiveness ratio of online algorithms. Using the result from our paper, we can extend the analysis from [2] to the average-case competitiveness. Related workThe problem considered in this paper is related to the following areas from the literature. Probabilistic verification of qualitative properties. Probabilistic verification asks for the probability of the set of traces satisfying a given property. For non-weighted automata, it has been extensively studied [33,15,4] and implemented [26,22]. The prevalent approach in this area is to work with deterministic automata, and apply determinisation as needed. To obtain better complexity bounds, the probabilistic verification problem has been directly studied for unambiguous Büchi automata in [5]; the authors explain there the potential pitfalls in the probabilistic analysis of non-deterministic automata.Weighted automata under probabilistic semantics. Probabilistic verification of weighted automata and their extensions has been studied in [12]. All automata considered there are deterministic. (MDPs). MDPs are a classical exte...

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