Two-sided disjunctions and the formal semi-supervised criterion of Blum–Balcan: a counting lemma, two learners, and what remains open

Blum and Balcan (Open Problems in Efficient Semi-Supervised PAC Learning, COLT 2007) asked for an efficient semi-supervised algorithm for two-sided disjunctions whose number of labelled examples is polynomial in log|CD,χ(ε)|, 1/ε and 1/δ, for every distribution consistent with the compatibility notion. Taking the class to be all pairs of monotone disjunctions, including the empty disjunction, and using one ε for labelled and unlabelled error as in the formal definition, we prove a hitting-set lemma: log2|CD,χ(ε)| ≥ ⌊εn⌋ for every such distribution. Consequently, classical supervised elimination already meets the stated criterion. A second counting lemma sharpens the label bound and gives a stopping rule from unlabelled data. Since no unlabelled data is needed to classify, the formal criterion is satisfied degenerately; this does not show that unlabelled data helps. Without the empty disjunction the hitting-set lemma fails, and only the second bound survives with an extra ln n. For the cover-based strengthening we give a family with ε-cover of size 4 on which elimination-type rules need Θ(n log min(n,1/ε)) labels; that version remains open. Files: manuscript (PDF), LaTeX source, and a single Python 3 verification script (standard library only). Contact: [email protected]