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Given the regular languages L1 and L2, prove that the following language is also regular:

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I know that U,*,. between 2 regular languages provides a regular language. But, the following regular language is infinite so I don't really have an idea how to 'look' at it properly.

would love to get some assistance.

Rouki
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  • I'm having trouble deciphering the language definition, as it uses comprehension and quantification in a nonstandard way (normally you can't say for all 1 and for all 2, and quantifiers don't normally occur before the vertical bar). – ibid Nov 19 '13 at 09:27
  • the : 'For all 1' refers to the 2i + 1 . and the 'For all 2' refers to the 2i. (Odd and Even) – Rouki Nov 19 '13 at 12:11
  • I suppse you're trying to say that in both cases we are quantifying over i, subject to the constraint given? Abuse of notation goes only so far – it has to be understandable :) – ibid Nov 20 '13 at 13:11
  • I think I now understand what SuperStar is trying to do. Do you understand what it does? – Understanding is the first step toward a proof. – ibid Nov 20 '13 at 13:12
  • Also, is this homework? – ibid Nov 20 '13 at 13:13
  • No. I think that I do understand abit what it does but ive no idea how to start proving it. – Rouki Nov 20 '13 at 15:19

2 Answers2

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Superstar consists of concatenations of words such that

  • words on odd positions are taken from L1
  • words on even positions are taken from L2

L1 and L2 are regular languages Then L1L2 is also a regular language, say L' := L1L2 Now L' = { w1w2 | w1 taken from L1 and w2 taken from L2 } and L' is regular

S -> epsilon | L'S | L1

is also a regular language, where epsilon is the zero-length empty word.

S will generate L'L'L'……L' and an optional L1 at the end, where L' = L1L2.
David Angyal
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Your language is (L1L2)* U (L1L2)*L1.

J.-E. Pin
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