By Ivanyi A. (ed.)
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Additional resources for Algorithms of informatics, vol. 1
E. for each q ∈ Q, u ∈ Σ∗ and v ∈ W ∗ the triplet (q, u, v) can be a configuration. If u = a1 a2 . . ak and v = x1 x2 . . xm , then the pushdown automaton can change its configuration in two ways: • (q, a1 a2 . . ak , x1 x2 . . xm−1 xm ) =⇒ (p, a2 a3 . . ak , x1 , x2 . . xm−1 w), if q, (a1 , xm /w), p ∈ E • (q, a1 a2 . . ak , x1 x2 . . xm ) =⇒ (p, a1 a2 . . ak , x1 , x2 . . xm−1 w), if q, (ε, xm /w), p ∈ E. ∗ The reflexive and transitive closure of the relation =⇒ will be denoted by =⇒.
For example in the case of Fig. 20 let these variable X, Y, Z, U corresponding to the states q0 , q1 , q2 , q3 . The corresponding equation are X=ε Y = X1 Z = X0 + Y 0 + Z0 U = Y 1 + Z1. If an equation is of the form X = Xα + β, where α, β are arbitrary words not containing X, then it is easy to see by a simple substitution that X = βα∗ is a solution of the equation. Because these equations are linear, all of them can be written in the form X = Xα + β or X = Xα, where α do not contain any variable.
Define regular operations on NFA: union, product and iteration. The result will be an FA with ε-moves. Operation will be given also by diagrams. An NFA is given as in Fig. 13(a). Initial states are represented by a circle with an arrow, final states by a double circle. Let A1 = (Q1 , Σ1 , E1 , I1 , F1 ) and A2 = (Q2 , Σ2 , E2 , I2 , F2 ) be NFA. The result of any operation is a FA with ε-moves A = (Q, Σ, E, I, F ). Suppose that Q1 ∩Q2 = ∅ always. If not, we can rename the elements of any set of states.