Optimal Test Sets for Context-Free Languages

TL;DR

This paper introduces the first polynomial-time algorithm to compute minimal cubic-sized test sets for all context-free languages, bridging the gap between known lower and upper bounds.

Contribution

It provides a novel cubic-size test set algorithm for context-free grammars, resolving a longstanding theoretical gap.

Findings

Cubic-sized test sets exist for all context-free grammars.

The algorithm computes minimal test sets efficiently.

This work closes the gap between lower and upper bounds on test set sizes.

Abstract

A test set for a formal language (set of strings) L is a subset T of L such that for any two string homomorphisms f and g defined on L, if the restrictions of f and g on T are identical functions, then f and g are identical on the entire L. Previously, it was shown that there are context-free grammars for which smallest test sets are cubic in the size of the grammar, which gives a lower bound on tests set size. Existing upper bounds were higher degree polynomials; we here give the first algorithm to compute test sets of cubic size for all context-free grammars, settling the gap between the upper and lower bound.