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By Ivanyi A. (ed.)

Ivanyi A. (ed.) Algorithms of informatics, vol.1.. foundations (2007)(ISBN 9638759615)

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N}. 1. 20. 21 to which a regular expression is associated by Method 1. 21. k We can prove by induction that sets Rij can be described by regular expressik ons. Indeed, if k = −1, then for all i and j languages Rij are nite, so they can be expressed by regular expressions representing exactly these languages. Moreover, if k−1 for all i and j language Rij can be expressed by regular expression, then language k Rij can be expressed also by regular expression, which can be corresponding constk−1 k−1 k−1 k−1 ructed from regular expressions representing languages Rij , Rik , Rkk and Rkj k respectively, using the above formula for Rij .

This may be extended also to sets Λ(S) = Λ(q), ∀S ⊆ Q . q∈S Clearly, for all q ∈ Q and S ⊆ Q both Λ(q) and Λ(S) may be computed. Suppose in the sequel that these are given. The following algorithm determine the transitions using the transition function δ , which is dened in line 5. If |Q| = n and |Σ| = m,, then lines 26 show that the running time in worst case is O(n2 m). 15 Consider the FA with ε-moves in Fig. 11. The corresponding transition table is: δ 0 1 ε q0 q1 ∅ {q1 } {q0 } ∅ {q1 } {q2 } q2 ∅ {q2 } ∅ Apply algorithm Eliminate-Epsilon-Moves.

Marking pair {p, q} means putting a star in the cell corresponding to row p and column q (or row q and column p). 2. Finite automata and regular languages 47 First we mark pairs {q2 , q0 }, {q2 , q1 }, {q2 , q3 }, {q2 , q4 } and {q2 , q5 } (because q2 is the single nal state). Then consider all unmarked pairs and examine them as the algorithm requires. Let us begin with pair {q0 , q1 }. Associate with it pairs {elem δ(q0 , 0) , elem δ(q1 , 0) }, {elem δ(q0 , 1) , elem δ(q1 , 1) }, that is {q1 , q4 }, {q4 , q2 }.

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