It is proved that there exist context-free languages L1, L2, L3, and words w1, w2 such that it is recursively unsolvable to determine for an arbitrary word w(1 ∼ 4) whether w1 \ L1 ∩ w \ L1 is the empty set (a finite set, a regular set, a context-free language, respectively), (5 ∼ 8) whether w \ L2 is the empty set (a finite set, a regular set, a context-free language, respectively), (9) whether w \ L2 is a regular set, (10) whether w3 \ L3 ∪| w \ L3, and (11) whether w3 \ L3 = w \ L3.
Let M be an on-line recognizer of a set X of words. Suppose that M is given an input word wu. Let δ(w) denote the configuration of M immediately before it reads the first letter of u. M is said to be nonredundant if δ(w) = δ(w′) for all words w, w′ such that w \ X = wm ′ \ X. From unsolvable problem (11) above it follows that there exists a context-free language that has no nonredundant on-line recognizer.
