How to prove for string of triplets that it is a Regular Language?

Asked Jul 07 '17 at 19:25

Active Jul 07 '17 at 19:25

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Let Σ₂ = {0, 1}, and define Σ = Σ₂³. Informally, Σ* is the set of triples of the form (a, b, c) where a, b, c are single binary digits. Consider a string s ∈ Σ* : it is a sequence of such triples. Have to “verify” binary addition of numbers in the first two coordinates by checking that it is equal to the third. Let A be the language of such triples such that the concatenation of the first coordinates, as a number, and the concatenation of the second coordinates, as a number, sum to be equal to the third.

For example, if w = (0, 1, 1)(1, 1, 1)(0, 0, 1)(1, 0, 1), this is encoding 0101₂ + 1100₂ = 1111₂, which is false; therefore, w ∉ A. How to prove A is Regular ?

Only thing I could think of is somehow has to prove for Reversal is Regular and then this could be proved, but no further idea.

asked Jul 07 '17 at 19:25

Desperado

Simulate addition with carry. Two states should suffice. – Yuval Filmus Jul 07 '17 at 19:33
Not getting your point ! Simulate addition with carry. NFA for addition - two states would be enough fine. But missing somethings I guess. – Desperado Jul 07 '17 at 19:52
Actually I was thinking of a DFA. – Yuval Filmus Jul 07 '17 at 19:53
yeah DFA is fine. But not able to understand . – Desperado Jul 07 '17 at 19:56
It's easier to read the bits LSB to MSB. – Yuval Filmus Jul 07 '17 at 19:57
Okay lets simplify . So i will make a DFA for the addition of Binary Numbers but I will use LSB to MSB configuration. So far good. I was also thinking this way , By reversal I meant the same thing. So if I could do that does it mean that original language in Question is Regular as the reversal is regular ? – Desperado Jul 07 '17 at 20:00
The class of regular languages is closed under reversal. Hopefully you learned that in class. – Yuval Filmus Jul 07 '17 at 20:00
Yes learned that in class. Do i need to take some special care for triplet sequence in my case ( the question ). w = (0, 1, 1)(1, 1, 1)(0, 0, 1)(1, 0, 1) ? – Desperado Jul 07 '17 at 20:07
They're nothing special. Just an alphabet. Perhaps it helps if you thought of them as ABCDEFGH instead of triplets. – Yuval Filmus Jul 07 '17 at 20:08

How to prove for string of triplets that it is a Regular Language?

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