   Chapter 12.3, Problem 12ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# How should the proof of property (12.3.1) be modified to prove property (12.3.2)?

To determine

To prove:

The property that k equivalence is an equivalence relation on S, the set of states of A for each k0.

Explanation

Given information:

A is a finite-state automaton and the set of the states of A is denoted by S.

Proof:

Rk is the notation that is used to denote k equivalence of two states.

The properties of reflexivity, symmetricity and transitivity of Rk should be proved for any input string in the accepted language of the finite-state automaton as well as the Rk relation. The condition that should be changed for k equivalence is the length of input string w is less than or equal to k where for each k0.

Suppose s,t and u are three states of S and hereafter the length of input string w is less than or equal k.

Suppose the states s and t are k equivalence states of A. Then these two states send the automaton to a nonaccepting state or an accepting state for any input string w in the set of strings. This property can be denoted by sRkt.

If sRkt for any string input w, the eventual function Nk will be,

Nk(s,w)Nk(t,w)

If Nk(s,w) is a nonaccepting state, then Nk(t,w) is also a nonaccepting state.

By the symmetricity of the above relationship, for input string w ,

Nk(t,w)Nk(s,w)

Hence, sRkt and tRks is equal for input string w.

Therefore, we can conclude that Rk is symmetric

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