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I have a BNF defined as follow:

 -> 0  -> 1  ->  

I think this grammar is not ambiguous, but the solution was 'yes'. Any idea?
The book solution is odd to me, because I think it is the same as the one above.

-> 0|1| 

Thanks,
Chan

  • 0
    You give two times the same grammar. An unambiguous grammar for the same language is $S \rightarrow 0S \mid 1S \mid 0 \mid 1$, for instance.2011-02-17

4 Answers 4

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A general technique to check for ambiguity is this. Transform the grammar in an equation system:

$S(x) = S(x)^2 + 2x$

Solving for $S(x)$ yields $S(x) = \frac{1}{2}(1 - \sqrt{1 - 8x})$ which is the ordinary generating function enumerating the grammar's left-derivation trees.

The generated language is $\{0,1\}^+$ which has the generating function $\frac{1}{1-2x} - 1$. This is obviously different from the above, therefore there are more left-derivation trees (for words of length $n$) than words (of length $n$).

This technique goes back to Chomsky and Schützenberger, 1963 (PDF).

  • 1
    Yea, few have. Which is funny given how old it is. My advisor uses generating functions heavily and happily, so he teaches us such things. Note, though, that this approach can be arbitrarily hard to execute (solving the equation system and comparing the resulting functions). Computer algebra systems help a lot.2011-02-17
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Consider the following:

$S \to S S \to S S S \to 1 S S \to 1 1 S \to 1 1 1$

$S \to S S \to 1 S \to 1 S S \to 1 1 S\to 1 1 1$

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    That is possible with ambiguous grammars, even in more ways than one.2011-02-18
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Consider that string = 000

1) S = S S = (S S) S = (0 0) 0 = 000

2) S = S S = S ( S S) = 0 (0 0 ) = 000

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    If I read your non-standard notation correctly, 1) is a left-derivation and 2) a right-derivation. Hence, this is not a witness for the grammar being ambiguous.2014-04-25
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You may be write LHS(left hand side) of your BNF and RHS(right hand side) like above cmt to solve that. After have LHS and RHS of your BNF, we will compare two of these, if there are no different between of these => Your BNF not ambigous and vice versa. Good luck :D