Re: Left and Right recursive non-terminals

<cowboy@vnet.ibm.com> wrote:
> I'm sure I read somewhere that a non-terminal that is both left and right
> recursive renders the grammar ambiguous (I think for any k).


A reachable non-terminal that is both left and right recursive and
which derives any nonempty string of terminals renders a grammar
ambiguous, yes. I don't know a reference, but the proof is pretty
easy...


Consider a such nonterminal A, with production
    A -> A b A
where b is any string of terminals and nonterminals (need a
Greek keyboard :-).


Now, for there to be any finite-length strings recognizable by A, it
must be that it is possible to derive the empty sentence e from A.
Assume further that there is a nonempty sentence s such that s is
derivable from b (this is informal, but you get the idea).


Consider parsing the sentence s s s with A. We can parse this
as
    A -> A b A -> (A b A) b A -> ((A b A) b A) b A -*-> ((s) s) s
or as
    A -> A b A -> (A b A) b A -> (A b A) b (A b A) -*-> (s) s (s)


In other words, we can have at least two parse trees for the
sentence s s s


    s s
      \ / \
        s s s
          \
            s


Finally, we note that by the definition of A, A must recognize the
sentence s s s , therefore A is ambiguous.


Hope this helps, (and that I didn't just do your homework for you :-)!


Bart Massey
bart@cs.uoregon.edu
--

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