Is the word problem of CFLs in NC? Consider the membership problem for a context-free language. An instance of this problem can be described as pair $(G,w)$, where $G$ is a context-free grammar and $w$ is a string. 

Lets say I have a PRAM and I want it to decide in $\log^k n$ time and $n^c$ space complexity whether $w$ is in $L(G)$ ($c$ and $k$ are both constants)&nbsp;&ndash; that is, the complexity stays within Nick's class. 

Is there such an algorithm? If so what would be an idea?
The problem is not in NC unless P=NC. Jones and Laaser showed in "Complete Problems for Deterministic Polynomial Time" that the problem (therein called CFMEMBER) is P-complete.
Interestingly, Ullman and Van Gelder showed in "Parallel Complexity of of logical query programs" that the same problem where the input CFG has no epsilon productions is in NC (Corollary 7.2).