Constructing Deterministic Parity Automata from Positive and Negative Examples

TL;DR

This paper introduces a polynomial time algorithm to construct deterministic parity automata from positive and negative examples, effectively learning $ ext{omega}$-regular languages with potential minimal automata, and identifies parameters that optimize the number of examples needed.

Contribution

The paper presents a complete polynomial time algorithm for constructing DPAs from examples and defines the precise DPA based on right congruences, with bounds on example complexity.

Findings

Algorithm is complete for $ ext{omega}$-regular languages.

Precise DPA can be exponential or minimal depending on language structure.

Number of examples needed can be polynomial with fixed parameters.

Abstract

We present a polynomial time algorithm that constructs a deterministic parity automaton (DPA) from a given set of positive and negative ultimately periodic example words. We show that this algorithm is complete for the class of $\omega$-regular languages, that is, it can learn a DPA for each regular $\omega$-language. For use in the algorithm, we give a definition of a DPA, that we call the precise DPA of a language, and show that it can be constructed from the syntactic family of right congruences for that language (introduced by Maler and Staiger in 1997). Depending on the structure of the language, the precise DPA can be of exponential size compared to a minimal DPA, but it can also be a minimal DPA. The upper bound that we obtain on the number of examples required for our algorithm to find a DPA for $L$ is therefore exponential in the size of a minimal DPA, in general. However we identify two parameters of regular $\omega$-languages such that fixing these parameters makes the bound polynomial.