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I am trying to find the context free grammar for this equation

L = {a ᶦ b ʲ c ᵏ |  i = j or i = k, i, j, k ≥ 0}

I have been able to work out the grammar for when a and b have equal exponent such as

S->ɛ
S->XY
S->ab
S->aXb
Y->cY | ɛ

but I am having troubles when I try to do when a and c have the same exponent, as the b now is on the way. Can someone give me some good hints or corrections?

giacomo venturelli
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1 Answers1

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S → ab S → aXb means, exactly, that S is an ab surrounded by equal length strings of as and bs.

The ab is completely orthogonal:

  • S → x S → a S b

    x, axb, aaxbb, aaaxbbb, aaaaxbbbb, …

  • S → centre S → a S b

    x, acentreb, aacentrebb, aaacentrebbb, aaaacentrebbbb, …

  • S → ε S → a S b

    ε, ab, aabb, aaabbb, aaaabbbb, …

It works just the same if the thing in the middle is a non-terminal. Or a sequence of terminals and non-terminals. This is one of the ways you can compose grammars.

rici
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