Formal languages and their relation to automata
Formal languages and their relation to automata
The reduction of two-way automata to one-way automata
IBM Journal of Research and Development
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Hopcroft and Ullman (problem 3.10 [1]) pose the amusing question of whether the "first third" of a regular language L, FIRST-THIRD(L) = {x| x is a prefix of a member of L of length 3|x|}, is necessarily regular. To see that it is, we can adapt of 1-way deterministic finite-state acceptor (an FA, for short) for L to get a 2-way non-deterministic finite-state acceptor with endmarkers for FIRST-THIRD(L). This acceptor behaves like the FA on x until it reaches the right endmarker, and then it uses another pass over x at half speed to behave like the FA on some nondeterministically chosen continuation of length 2|x|. That such an acceptor accepts a regular language follows from an argument similar to that of Shepherdson for deterministic acceptors [2].