Explanation Given information: A is a finite-state automaton and the set of the states of A is denoted by S . Proof: R k is the notation that is used to denote k − equivalence of two states. The properties of reflexivity, symmetricity and transitivity of R k should be proved for any input string in the accepted language of the finite-state automaton as well as the R k 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 k ≥ 0 . 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 s R k t . If s R k t for any string input w , the eventual function N k will be, N k ( s , w ) ⇔ N k ( t , w ) If N k ( s , w ) is a nonaccepting state, then N k ( t , w ) is also a nonaccepting state. By the symmetricity of the above relationship, for input string w , N k ( t , w ) ⇔ N k ( s , w ) Hence, s R k t and t R k s is equal for input string w . Therefore, we can conclude that R k is symmetric

5th Edition

ISBN: 9780357097717

Author: EPP

Publisher: CENGAGE L

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Chapter 12.3, Problem 12ES

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To prove:

The property that k− equivalence is an equivalence relation on S, the set of states of A for each k≥0.

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Chapter 12 Solutions

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To prove: The property that k − equivalence is an equivalence relation on S , the set of states of A for each k ≥ 0 .

Solution Summary: The author explains the properties of reflexivity, symmetry and transitivity of R_k, which should be proved for any input string in the accepted language of the finite-state automaton.

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To prove:

The property that k− equivalence is an equivalence relation on S, the set of states of A for each k≥0.

Chapter 12 Solutions

To prove: The property that k − equivalence is an equivalence relation on S , the set of states of A for each k ≥ 0 .

Solution Summary: The author explains the properties of reflexivity, symmetry and transitivity of R_k, which should be proved for any input string in the accepted language of the finite-state automaton.

Concept explainers

Videos

To prove: The property that k− equivalence is an equivalence relation on S, the set of states of A for each k≥0.

Chapter 12 Solutions

To prove:

The property that k− equivalence is an equivalence relation on S, the set of states of A for each k≥0.