Constructing Concise Characteristic Samples for Acceptors of Omega Regular Languages

TL;DR

The algorithms for constructing characteristic sets in polynomial time for the different omega automata require deterministic polynomial time algorithms for equivalence of the respective omega automata, and testing membership of the language of the automaton in the informative classes, which are provided.

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 Büchi, 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 Büchi, coBüchi, 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.

Figures (4)

• Figure 1: Summary of main general results about efficient teachability
• Figure 2: Top: a sample $T$. Middle: The distinguishing prefixes $U$ of words in the sample $T$. Left: Default acceptor of type DBA for sample $T$ (the dead state and the transitions to it are omitted). Right: Default acceptor of type SUBA for sample $T$.
• Figure 3: (a) Graph $G(\mathcal{P})$ with states colored by $\kappa$. (b) The $\textit{minStates}$-forest of $\mathcal{P}$, with $\kappa$-parities of nodes. (c) Canonical forest $\mathcal{F}^*(\mathcal{P})$, with $\kappa$-parities of nodes. (d) Graph $G(\mathcal{P})$ with the canonical coloring $\kappa^*$.
• Figure 4: Example of the construction of $B({\mathcal{M}},F,q)$ with $F = \{q_0,q_1,q_2\}$ and $q = q_0$.

Theorems & Definitions (132)

• Claim 1
• Claim 2
• Lemma 3: Key Property of Characteristic Samples
• proof
• Theorem 3.1: Identification by Enumeration
• proof
• Corollary 4
• Lemma 5
• proof
• Theorem 3.2