How to determine whether a language is regular or not using Myhill-Nerode theorem?
I already have got this notes from my lectures, so I don't want a theoretical answer something that repeats the same thing:
Here are the steps to prove that a language is not regular using the Myhill-Nerode theorem:
1) Define the relation L on the language L as follows:
For any two strings x and y in *, x L y if and only if, for any string z in *, either all of xz and yz are in L, or neither of them are in L.
2) Construct an infinite set of pairwise distinguishable strings {w1, w2, w3, ...} such that for any i and j, if i j, then w i and w j are not equivalent under the relation L.
To show that the set of strings is pairwise distinguishable, you need to show that for any two distinct strings w i and w j , there exists a string z such that either w i z is in L and w j z is not in L, or w j z is in L and w i z is not in L. This can be done by constructing such a string z for each pair of strings.
3) Show that there are an infinite number of equivalence classes of strings under L, which implies that the language L is not regular.
If you can successfully perform these steps and show that there are an infinite number of equivalence classes, you can conclude that the language L is not regular. Language is regular if and only if there are a finite number of equivalence classes of strings under a certain relation.
I want an explanation with some examples (regular language and non-regular language) to get an understanding of this.
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How to determine whether a language is regular or not using Myhill-Ner.docx
1. How to determine whether a language is regular or not using Myhill-Nerode theorem?
I already have got this notes from my lectures, so I don't want a theoretical answer something
that repeats the same thing:
Here are the steps to prove that a language is not regular using the Myhill-Nerode theorem:
1) Define the relation L on the language L as follows:
For any two strings x and y in *, x L y if and only if, for any string z in *, either all of xz and yz
are in L, or neither of them are in L.
2) Construct an infinite set of pairwise distinguishable strings {w1, w2, w3, ...} such that for any
i and j, if i j, then w i and w j are not equivalent under the relation L.
To show that the set of strings is pairwise distinguishable, you need to show that for any two
distinct strings w i and w j , there exists a string z such that either w i z is in L and w j z is not in L,
or w j z is in L and w i z is not in L. This can be done by constructing such a string z for each pair
of strings.
3) Show that there are an infinite number of equivalence classes of strings under L, which
implies that the language L is not regular.
If you can successfully perform these steps and show that there are an infinite number of
equivalence classes, you can conclude that the language L is not regular. Language is regular if
and only if there are a finite number of equivalence classes of strings under a certain relation.
I want an explanation with some examples (regular language and non-regular language) to get an
understanding of this.
