# Re: Left and Right recursive non-terminals

## bart@time.cirl.uoregon.edu (Barton C. Massey)15 Dec 1996 16:19:01 -0500

From comp.compilers

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 From: bart@time.cirl.uoregon.edu (Barton C. Massey) Newsgroups: comp.compilers Date: 15 Dec 1996 16:19:01 -0500 Organization: University of Oregon References: 96-12-089 Keywords: parse, theory

<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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