Lower Bounds and Reductions Between Grammar Problems

Abstract

AESTRA~ r Many parsing techniques are parameterlzed by the amount of context allowed in making parsmg decisions (e g the LR(k), LL(k), and SLR(k) hierarchies). The best-known algorithms for testing membership in these classes depend exponentially on k when the test is performed determinlsttcally, and polynomlally on k when the test IS done nondetermmistically This paper shows that the deterministic complexity of any uniform algorithm for, say, LR(k) testing is a polynommi function of k if and only if P = NP. We then provide a lower bound on the growth rate with respect to k As to the relationships between the various classes, it is shown that the time needed for testing the LL(k) property ~s a lower bound on the time needed for testing membership in any subset of the LR(k) grammars that includes all the LL(k) grammars These results suggest that the complexity of LL(I) testing is a lower bound on the complexity of testing membership m most useful classes of grammars Finally, some relative decision problems are mvesUgated, that is, problems of testing whether a given member of some class is also a member of some other specified class A polynomial time algorithm ts given for determining for an LR(k) grammar whether there exists a k' for which the grammar is LL(k'). Slmdar results are shown for the SLR(k) and strong LL(k) hierarchies