Operator precedence in Lojban mathematics: we need your help again!

Yet another of those appeals to the Lojbani masses (or the net-aware masses,
anyhow) about a fine point in Lojban mathematical expression (mekso).
This one involves how Lojban manages the precedence of its operators.
"Precedence" is the property of an operator that decides whether it is to
be applied before or after other operators.  For example, in normal
mathematics

1)	3 + 4 x 5

means 23, because multiplication has precedence over addition.  If addition
had precedence, the result would be 35.

The Lojban parser does not assign precedence to the various built-in
and ad hoc operators: all of them are equal, and grouping is left to right,
as in most parts of the language.  So the straight translation of Example 1
is:

2)	ci su'i vo pi'i mu

and this does mean 35, because of left-grouping.  This result violates the
general design goal of making mekso "spoken mathematics", which requires
matching the conventions of mathematics as much as possible.

Let me quote from the current draft of my paper on the subject:

The first solution is to ignore the problem.  People will say "li ci
su'i vo pi'i mu" and mean 23 by it, because the notion that multiplication
takes precedence over addition is too deeply ingrained to be eradicated
by Lojban parsing, which totally ignores semantics.  This convention
essentially allows semantics to dominate syntax in this one area.

(Why not hard-wire the precedences into the grammar, as is done in computer
programming languages?  Essentially because there are too many operators,
known and unknown, with levels of precedence that vary according to usage.
The programming language 'C' has 13 levels of precedence, and its list
of operators is not even extensible.  For Lojban this approach is just not
practical.  In addition, hard-wired precedence could not be overridden in
mathematical systems such as spreadsheets where the conventions are
different.)

The second solution is to use explicit means to specify the precedence
of operators.  This approach is fully general, but clumsy.  [It involves
the use of the metalinguistic declaration "ti'o", which a smart parser
could use to change an operator's precedence.   The current parser cannot
interpret such declarations.]

The third solution is simple but not very general.  It builds on a
mechanism used elsewhere in the language: the short-scope cmavo "bo".
When an operator is prefixed with "bo", it becomes automatically of
higher precedence than other operators not so prefixed.  Thus,

3)	li ci su'i vo bo pi'i mu du li reci
	the-number three plus four-times-five equals the-number two-three
	3 + 4 x 5 = 23

is a truthful Lojban bridi.

In addition, of course, Lojban has the mathematical parentheses "vei" and
"ve'o", which can be used just like their written equivalents "(" and ")"
to group expressions in any way desired.

### end of excerpt ###

In reviewing this paper, Colin Fine has proposed that we explicitly
implement a fixed priority of operators within the grammar so as to get
the simple cases right:

> [P]recedence.  I find this very unsatisfactory as stated. Any
> of the solutions will do, but the uncertainty is intolerable. In your
> quadratic [formula] example later, you use the 'conventional' interpretation.

The quadratic formula is the most complex mekso example in the paper, and it
was full of subtle precedence errors.  Thanks to Colin and to Mark Shoulson
for pointing these out and helping me work out the correct answer.

> There are two possible problems in this interpretation: the opposing
> pull of syntactic and semantic structure and the problem of operator
> precedence.  [...]
> The precedence problem is also superable. I suggest the following:
> 1) define the order of precedence for VUhU (don't try and give them
> precedence values, that's an irrelevant nuisance).
> 2) define a means for setting/overriding precedence, as suggested
> elsewhere in the paper.
> 3) define all nonce and constructed operators to have the same default
> precedence, probably coming below all the defined ones, but that will
> need a bit of thought. In any case, in a defined place in the VUhU
> hierarchy.
> 4) operators with the same precedence associate to the left.

A discussion of this idea with Bob and Nora produced the following suggestion:
Currently there are basically only two rules in the YACC grammar for handling
infix mekso of the kind discussed here:  one for ordinary operators, and one
for operators with prefixed "bo".  If we split each rule into some number
of sub-rules (say ten), and divide the VUhU selma'o (which contains the
built-in mekso operators) into ten sub-selma'o, then we can allow the part
of the parser that assigns cmavo to their selma'o, which is currently just
a big lookup table, to be a little smarter.

In the absence of other information, the parser would place all mekso operators
into a single sub-selma'o, say "VUhU1".  All would therefore have the same
precedence.  Through appropriate parsing options, however, a different table
would be installed which could assign the VUhUs to different sub-selma'o,
giving them in effect different precedences.  For example, "su'i" and "vu'u",
which are addition and subtraction, might belong to VUhU3 whereas "pi'i" and
"fe'i", which are multiplication and division, might belong to VUhU5.  This
would give times and divide higher precedence.  Changing the table would change
the precedence.  Eventually, the parser could have a semantic routine which
altered the table when a "ti'o" clause was parsed.

The arguments in favor of this change are that it matches standard elementary
mathematics far better, and does not require elaborate hand-waving to explain
easy cases.  The meanings of simple expressions like "ci su'i vo pi'i mu"
come out right automatically.  In unusual circumstances, the precedence could
still be overridden by convention (first method above) or by the explicit
use of declarations (method 2 above).

Bob and Nora and I agreed that the question should be put to the Lojban List.
Please give us your feedback as soon as possible, as Bob is even as we
speak (read, write) preparing JL17, and the mekso paper will hopefully
appear in that issue.

-- 
cowan@snark.thyrsus.com		...!uunet!cbmvax!snark!cowan
		e'osai ko sarji la lojban