By Luke Y.L.

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These states are merged into a class numbered 1. The radix sort of the four states of height 2 gives the sequence (10, 11, 2, 6), so 10, 11 are grouped into a class 2 and 2, 6 are grouped into a class 3. 2 6 10 11 : : : : 0a1b1 0a1b1 0a1b0 0a1b0 (c) Signatures of states of height 2. ν 0 1 2 3 4 5 6 7 8 9 10 11 3 1 0 3 1 0 2 2 (d) The corresponding states of the minimal automaton. The states of height 3 all give singleton classes, because the signatures are diﬀerent. This is already clear because they have distinct lengths.

AutomataProduct(A, B) 1 C ← NewAutomaton() 2 InitialC ← InitialA 3 TerminalC ← TerminalB 4 Merge(TerminalA , InitialB ) 5 return C Version June 23, 2004 38 Algorithms on Words AutomatonStar(A) 1 B ← NewAutomaton() 2 Next1(InitialB ) ← (ε, InitialA ) 3 Next2(InitialB ) ← (ε, TerminalB ) 4 Next1(TerminalA ) ← (ε, InitialA ) 5 Next1(TerminalA ) ← (ε, TerminalB ) 6 return C The practical implementation of these algorithms on a regular expression is postponed to the next section. 26. It has 21 states and 27 edges.

0 1 2 3 4 5 6 7 8 9 10 11 1 0 1 0 (b) The corresponding states of the minimal automaton. The states of height 1 have the signatures given below. Observe that in a signature, the next state appearing in an edge is replaced by its class. This can be done because the algorithm works by increasing height. These states are merged into a class numbered 1. The radix sort of the four states of height 2 gives the sequence (10, 11, 2, 6), so 10, 11 are grouped into a class 2 and 2, 6 are grouped into a class 3.