Kleene's algorithm

In theoretical computer science, in particular in formal language theory, Kleene's algorithm transforms a given nondeterministic finite automaton (NFA) into a regular expression. Together with other conversion algorithms, it establishes the equivalence of several description formats for regular languages. Alternative presentations of the same method include the "elimination method" attributed to Brzozowski and McCluskey, the algorithm of McNaughton and Yamada, and the use of Arden's lemma. According to Gross and Yellen (2004), the algorithm can be traced back to Kleene (1956). A presentation of the algorithm in the case of deterministic finite automata (DFAs) is given in Hopcroft and Ullman (1979). The presentation of the algorithm for NFAs below follows Gross and Yellen (2004). Given a nondeterministic finite automaton M = (Q, Σ, δ, q0, F), with Q = { q0,...,qn } its set of states, the algorithm computes the sets R of all strings that take M from state qi to qj without going through any state numbered higher than k. Here, "going through a state" means entering and leaving it, so both i and j may be higher than k, but no intermediate state may. Each set R is represented by a regular expression; the algorithm computes them step by step for k = -1, 0, ..., n. Since there is no state numbered higher than n, the regular expression R represents the set of all strings that take M from its start state q0 to qj. If F = { q1,...,qf } is the set of accept states, the regular expression R | ... | R represents the language accepted by M. The initial regular expressions, for k = -1, are computed as follows for i≠j: R = a1 | ... | am where qj ∈ δ(qi,a1), ..., qj ∈ δ(qi,am) and as follows for i=j: R = a1 | ... | am | ε where qi ∈ δ(qi,a1), ..., qi ∈ δ(qi,am) In other words, R mentions all letters that label a transition from i to j, and we also include ε in the case where i=j.