Greedy Grammar Induction with Indirect Negative Evidence
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This paper proposes a non-lexicalized grammar-induction procedure that separates two tests: recognition of the observed finite presentation, and rejection of short preterminal strings generated by a hypothesis but unsupported by the evidence. The central object is the rule-coverage bound \(\ell^*(G)\): the maximum, over rules in \(G\), of the length of the shortest preterminal string whose derivation uses that rule. This bound induces the comparison universe \(\Sigma_{\mathrm{pre}}^{\le \ell^*(G)}\), where unsupported generated strings serve as indirect evidence against overgenerating hypotheses. We give a greedy search algorithm over rule sets and prove a conditional weak-recovery theorem: under explicit reachability conditions and sufficient saturation of the presentation, the exact learner reaches a grammar weakly equivalent to the unknown target. The complexity analysis is slice-wise: for each fixed incrementality radius \(k\), the search explores polynomially many rule-set extensions in the finite rule universe. Across 31 benchmark runs spanning Dyck-\(k\) languages \((1\le k\le4)\), palindromes, \(a^n b^n\), English-like recursive fragments, and an inherently ambiguous union language, grammar-level analysis establishes weak equivalence between every returned grammar and its target.