Learning from Positive and Negative Examples: New Proof for Binary Alphabets

TL;DR

This paper revisits the NP-completeness of learning deterministic finite automata from positive and negative examples over a binary alphabet, correcting previous misconceptions and providing a new proof for DFAs.

Contribution

The authors identify issues in prior proofs based on Mealy machines and present a new, accurate construction demonstrating NP-hardness for learning binary DFA from examples.

Findings

Gold's original proof was based on Mealy machines, not DFAs.

The paper provides a corrected construction for binary DFA.

It establishes NP-hardness of learning DFA with positive and negative examples.

Abstract

One of the most fundamental problems in computational learning theory is the the problem of learning a finite automaton $A$ consistent with a finite set $P$ of positive examples and with a finite set $N$ of negative examples. By consistency, we mean that $A$ accepts all strings in $P$ and rejects all strings in $N$. It is well known that this problem is NP-complete. In the literature, it is stated that this NP-hardness holds even in the case of a binary alphabet. As a standard reference for this theorem, the work of Gold from 1978 is either cited or adapted. But as a crucial detail, the work of Gold actually considered Mealy machines and not deterministic finite state automata (DFAs) as they are considered nowadays. As Mealy automata are equipped with an output function, they can be more compact than DFAs which accept the same language. We show that the adaptions of Gold's construction for Mealy machines stated in the literature have some issues and give a new construction for DFAs with a binary alphabet ourselves.