Re: LR(k)

[Paulo Barreto <BARRETO%VELAHF@ECCSA.TR.UNISYS.COM>]

Myself (Paulo):
>Knuth's theorem states that *all* deterministic context-free languages
>have an LR(1) grammar.

John:
>I'm not familiar with the theorem. Surely not all languages are LALR(1)

See Knuth, D.E. [1965] "On the translation of languages from left to
right", Information and Control 8(6), p.607-639. (I could fax it to
you on request). I remember to have seen this proof also in Aho, A.V.
and Ullman, J.D. [1972] "The Theory of Parsing, Translation and
Compiling", Prentice-Hall (I don't remember the pages). With some
(correction: much) effort, I could also find my class notes with the
proof (if it was an exercise, I don't guarantee it's correct :-)

John:
>Surely not all languages are LALR(1)

I wouldn't be so sure about that :-)

There's a stronger result: any deterministic context-free language with
the prefix property (i.e. no string of the language is a prefix of any
other string of the same language) has an LR(0) grammar.

The subset of Lojban where all utterances finish in an explicit FA'O
presents the prefix property, hence it has an LR(0) grammar. If the
preparser supplies FA'O when it is ellipsized, full Lojban can be
modelled on an LR(0) grammar. And YACC needs an "end-of-file token",
anyway.

I think this line of reasoning can be followed to prove that all
deterministic context-free languages *are* LALR(1). However, I wish to
point out that this is not a panacea. The LR(0) grammar assured by this
theorem may reflect a syntax structure that is entirely different from
the current grammar.

>From a practical point of view, the "hacked" YACC grammar is more or
less fine (some more comments would make it easier to read). But we
assert that Lojban is unambiguous and that it can be reasonably well
described by a context-free grammar; hence, it *should* be possible to
show an LR(1) grammar for it, and perhaps an LR(k) grammar that
keeps the overall phrase structure of the YACC grammar. Even a less
tricky YACC grammar would be interesting in itself.

Finally, I don't see why Lojbab's afraid of trying to find such a
grammar, as its equivalence (in the sense of generating the same
language) to the current YACC grammar should be a stated in the form
of a *theorem*, not a gratuitous assertion. (I don't think the proof
would be longer than that of Fermat's last theorem, or the four-color
theorem for planar graphs :-)

co'o mi'e paulos.

    Paulo S. L. M. Barreto  --  Software Analyst  --  Unisys Brazil
    Standard disclaimer applies ("I do not speak for Unisys", etc.)
                       e'osai ko sarji la lojban.