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There are m variable in a grammar. The number of productions after removal of unit productions in the worst case is ,(Assume there are no null productions)

(a) O(m)
(b) O($m^2$)

(c) O($k^m$)

(d) O($2^m$)


My attempt:

He asked about number of total production remains in the minimized grammar , so , if the there is no null production , it is advantage , as you know when we convert a grammar without null production , generally , removal of all production introduces new production in the resultant grammar (it can be upto set of all subset of given max RHS number of variable , so exponetial ) .

But , in given grammar , you don't have null production , so you are removing only unit production (i.e. A--->B type productions ) , resultant grammar maximum 'm' production , where m is number of variable . example :- S---->A A---->B B---->a/b/c resultant grammar , S---->a/b/c So , it is O(m) time .

Can you explain in formal way, please?

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  • $\begingroup$ Hint: there is more than one correct answer. (What is $k$?) $\endgroup$ – Raphael Oct 2 '15 at 19:59
  • $\begingroup$ $k$ is constant $\endgroup$ – ً ً Oct 3 '15 at 6:02
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    $\begingroup$ Then all answers are correct (if $k\geq 2$). $\endgroup$ – Raphael Oct 3 '15 at 10:44
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Hint: Consider the grammar $$S \to \overbrace{X\cdots X}^{\text{$n$ times}} \\ X\to a|b$$ What happens when you remove unit productions?

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  • $\begingroup$ Resultant grammar will be S--->a|b|..|a|b....2n-times , so total number of productions are O(2n) , minimized grammar is S--->a|b . rt , sir ? $\endgroup$ – ً ً Sep 3 '15 at 6:09
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    $\begingroup$ No, the minimized grammar is rather different from $S\to a|b$. $\endgroup$ – Yuval Filmus Sep 3 '15 at 6:23
  • $\begingroup$ If I am consider given grammar as S---->XXXXX....n-times, and X---->a|b , then minimized grammar should be O($2^n$) productions . Sir, Again I have two queris (1) Is the production S----->XXXXX... unit production ? (2) If , there was X---->a|b|....m-times , then the minimized productions O($m^n$) ? Am I rt ? $\endgroup$ – ً ً Sep 3 '15 at 6:51

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