Linear Context-Free Languages Are in NC

This paper concerns linear context-free languages (LIN). We prove that LIN ⊆ NC (under UE∗−uniformity reduction). We introduce a new normal form for context-free grammars, called Dyck normal form. Using this new normal form we prove that for each context-free language L there exist an integer K and a homomorphism φ such that L = φ(D′ K), where D′ K ⊆ DK , and DK is the one-sided Dyck language over K letters. Based on these results we prove that each linear context-free language can be recognized in O(log n) time and space by an indexing alternating Turing machine (ATM). Since the class of languages recognizable by an indexing ATM in logarithmic time equals the UE∗ -uniform NC 1 class, result proved in [15], we obtain that LIN ⊆ NC, and consequently, LIN ⊆ L (where L is the class of languages recognizable by a deterministic Turing machine in logarithmic space). On the other hand, according to [17], each language in LIN belongs to L if and only if L = NL (where NL is the class of languages recognizable by a nondeterministic Turing machine in logarithmic space). Hence, besides the inclusion of linear context-free languages in NC, problem left open in [12], we also resolve the longstanding open question L ? = NL in the favor of L = NL.