Constructing Concise Characteristic Samples for Acceptors of Omega Regular Languages

TL;DR

This paper investigates the size and construction of characteristic samples for omega automata, providing polynomial-time algorithms for certain deterministic classes but showing non-polynomial bounds for nondeterministic types.

Contribution

It establishes polynomial-time methods for constructing characteristic samples for deterministic omega automata isomorphic to their right congruence automata, and analyzes limitations for nondeterministic automata.

Findings

Non-deterministic omega automata lack polynomial-sized characteristic samples.

Polynomial-time algorithms are provided for deterministic omega automata.

Constructing characteristic samples depends on automata equivalence and membership testing algorithms.

Abstract

A characteristic sample for a language $L$ and a learning algorithm $\textbf{L}$ is a finite sample of words $T_L$ labeled by their membership in $L$ such that for any sample $T \supseteq T_L$ consistent with $L$, on input $T$ the learning algorithm $\textbf{L}$ returns a hypothesis equivalent to $L$. Which omega automata have characteristic sets of polynomial size, and can these sets be constructed in polynomial time? We address these questions here. In brief, non-deterministic omega automata of any of the common types, in particular B\"uchi, do not have characteristic samples of polynomial size. For deterministic omega automata that are isomorphic to their right congruence automata, the fully informative languages, polynomial time algorithms for constructing characteristic samples and learning from them are given. The algorithms for constructing characteristic sets in polynomial time for the different omega automata (of types B\"uchi, coB\"uchi, parity, Rabin, Street, or Muller), require deterministic polynomial time algorithms for (1) equivalence of the respective omega automata, and (2) testing membership of the language of the automaton in the informative classes, which we provide.