Summary
Not every unambiguous regular grammar can be parsed by a finite state machine, even if a lookahead facility is added to the machine's capabilities. Those which can be parsed with a fixed lookahead of k are said to be FL(k). If such a grammar has n non-terminals, it never needs more than (n(n−1)/2) + 1 lookahead, and there exist grammars which do require this much. An algorithm is presented for determining whether a grammar is fixed lookahead parsable, and if so, for finding the minimum lookahead needed. The algorithm sets up and solves a set of O(n2) equations using O(n4) steps. Two parsing methods for FL(k) grammars are discussed. One uses a large precomputed parsing table, and operates in real time. The second parses an input string in time proportional to its length, while using approximately 3n storage locations.
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Ostrand, T.J., Paull, M.C. & Weyuker, E.J. Parsing regular grammars with finite lookahead. Acta Informatica 16, 125–138 (1981). https://doi.org/10.1007/BF00261256
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DOI: https://doi.org/10.1007/BF00261256