How to show that given language is unambiguous

Question

Given following grammar:

$$ \begin{align} S \rightarrow &A1B \\ A \rightarrow & 0A \mid \varepsilon \\ B \rightarrow & 0B \mid 1B \mid \varepsilon \\ \end{align} $$

How can I show that this grammar is unambiguous? I need to find a grammar for the same language that is ambiguous, and demonstrate it.

I know if I was asked to prove that the language is ambigious then I should find two different parse trees for same string, but I don't know what to do.

score 4 · Accepted Answer · answered Dec 20 '12 at 01:29

4

To show a grammar is unambiguous you have to argue that for each string in the language there is only one derivation tree.

In this particular case you can observe that $A$ only generates $0$'s, so the $1$ generated by the start symbol $S$ must be the first $1$ in the string.

Any grammar can be made ambiguous by adding chain productions like $S\to S$.

answered Dec 20 '12 at 01:29

Hendrik Jan

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so if i search an unambiguity over a language i should check if there exists any chain production whatever it is ? and by the way thanks for your reply. – quartaela Dec 20 '12 at 12:38
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For ambiguity you try to fiund one string with at least two parse trees (derivation trees). A chain production has the form $A\to A$. If such a production exists and $A$ occurs in the tree, you can find another tree by just adding $A\to A$. This will not change the word generated. And, no, it does not suffice to check for these productions. There might also be more general causes for ambiguity. – Hendrik Jan Dec 20 '12 at 13:32

score 1 · Answer 2 · edited Dec 30 '12 at 18:31

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This grammar is equivalent with $$ \begin{align} S \rightarrow &0A1B\mid 1B \\ A \rightarrow & 0A \mid \varepsilon \\ B \rightarrow & 0B \mid 1B \mid \varepsilon \\ \end{align} $$ and so like a simple grammar we can show that this grammar is not ambiguous. Of course this grammar is not simple.

edited Dec 30 '12 at 18:31

A.Schulz

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answered Dec 21 '12 at 20:57

M.K. Dadsetani

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What exactly is a »simple gramar«? Do you mean »regular grammar«? – fuz Dec 23 '12 at 12:06
a grammar G is said to be simple, if all productions are of the form $$ A \rightarrow ax $$ where $A\in V, a\in T,x\in V^\ast$, and any pair $(A,a)$ occurs at most once in productions. – M.K. Dadsetani Dec 24 '12 at 10:07
Are the $\varepsilon$ productions allowed in simple grammars? – Hendrik Jan Dec 29 '12 at 00:42

How to show that given language is unambiguous

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