Regular Language to Regular Grammar

Question

I think the below language is Regular-
L = {xy | n_a(x) = n_b(y) where  x,y $\in$ (a,b)* }

Doubt : Since if we consider any string in given language is split in such a way so that we satisfy the required condition. like
(abbbaa)(bbaba), (bbbbb)(a)  etc.
(Note - brackets are just for understanding purpose). Can some one write the regular grammar for this language?

Comments

it will be (a+b)*...any string can be partitioned in two parts such that  it satisfies above language..

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Yeah it is (a+b)*

and its grammar will be

S-> aS / bS / epsilon

but how we can show that where to split?

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i all first thought how DFA will know what is pattern for a given language because it has to see in partitions which require counting, which is not possible with a DFA....than intituinally i observe that every string can be partitioned in two parts to follow above conditions, hence it is (a+b)*

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We don't need to show the range of partitions for each string. If a language is given, we only need to check what all strings can be accepted by its corresponding machine. 
DFA for this language will accept everything, So this is a regular language because we can have a dfa for it.

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Dfa accept all the string of Language but also reject all the string which not belong to Language

Answer 1

Is it regular or CFL

I think it is CFL because Finite Automata can't compare .

Comments

If we take -aaaaa

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in 'aaaaa'..

take x= epsilon and y="aaaaa"...

na(x)=nb(y)=0

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I got it x and y also may be epsilon

Thanks conceptual questions

Thanks for clearing doubt