# Non-deterministic weighted automata evaluated over Markov chains

## TL;DR

This paper introduces a probabilistic semantics for non-deterministic weighted automata over Markov chains, providing approximation algorithms for key probabilistic questions and demonstrating effective determinisation under certain metrics.

## Contribution

It is the first to study non-deterministic weighted automata with probabilistic semantics and offers approximation methods for their probabilistic analysis.

## Key findings

- Approximation algorithms for expected value and distribution of automata outputs.
- Automata can be effectively determinised with respect to the standard deviation metric.
- Probabilistic questions are uncomputable in general but approximable in exponential time.

## 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.