A Completeness Theorem for Probabilistic Regular Expressions

TL;DR

This paper introduces Probabilistic Regular Expressions (PRE), a probabilistic extension of regular expressions, and proves a completeness theorem for an inference system that determines probabilistic language equivalence.

Contribution

It presents the first formal inference system for probabilistic regular expressions and establishes its completeness, extending Kleene Algebra to probabilistic languages.

Findings

Inference system for PRE is complete.

Probabilistic language equivalence can be reasoned about systematically.

Extends Kleene Algebra to probabilistic settings.

Abstract

We introduce Probabilistic Regular Expressions (PRE), a probabilistic analogue of regular expressions denoting probabilistic languages in which every word is assigned a probability of being generated. We present and prove the completeness of an inference system for reasoning about probabilistic language equivalence of PRE based on Salomaa's axiomatisation of Kleene Algebra.