MEX paper, draft 1.0, part #3 of 5

[cowan (John Cowan)]

4.  Forethought Operators (Polish Notation, Functions)

The infix form explained so far is reasonable for many purposes, but it is
limited and rigid.  It works smoothly only where all operators have exactly
two operands, and where precedences can either be assumed from context
or are limited to just two levels, with some help from parentheses.  

But there are many operators which do not have two operands, or which have
a variable number of operands.  Traditionally, some of these (like negation)
have been written in the same infix style as the dyadic operators, and
Lojban can emulate that style through the use of a pair of tricks, the
null operand "tu'o" and the null operator "ge'a".  The first is suitable
when there are too few operands, the second when there are too many.
For example:

	li tu'o va'a ny. du no vu'u ny
	the number (null) additive-inverse n equals the number zero minus n
	-n = 0 - n

The "tu'o" fulfills the grammatical requirement for a left operand for "va'a",
even though semantically none is needed or wanted.

Here is an example using "ge'a", plus several other new features:

	li cinonoki'oki'o du li ci gei bi ge'a pano
	the-number three-zero-comma-comma equals
		the-number eight exponential-notation three to-base one-zero
	                   8
	300,000,000 = 3 x 10

The operator "gei" takes three operands:  an order of magnitude, a value,
and a base, in that order.  (This differs from the standard value, base,
order-of-magnitude order that scientific notation uses.)  Since infix
notation will not allow three operands, the third operand has to be hooked
on with "ge'a".  In addition, the left side of the equation illustrates
the elision of zeros after "ki'o", making "ki'oki'o" a way of saying
"million".

However, getting the "tu'o"s and "ge'a"s right is annoying.  The alternative
is the use of "forethought operators", also known as Polish notation.
In this style of writing mathematics, the operator comes first and the
operands afterwards.  For example:

	li si'u paboi reboi ciboi du li xa
	the-number plus one two three equals the-number six

Note the normally elidable number/lerfu terminator, "boi".  This is required
after "pa" and "re" because otherwise the reading would be "pareci" = 123.
It is not required after "ci" but is inserted here for symmetry.  The only
time "boi" is required is, as in this case, when there are two consecutive
numbers or lerfu strings.

Forethought mekso can use any number of operands, in this case three.
How do we know how many operands there are in ambiguous circumstances?  The
usual Lojban solution is employed:  an elidable terminator, in this case "ku'e".
Example:

	li .abu su'i va'a ny. ku'e su'i zy du li xy
	the-number a plus additive-inverse( n ) plus z equals the-number x
	a + -n + z = x

Here the operator "va'a" and the terminator "ku'e" serve as, in effect,
parentheses.  (The regular parentheses "vei" and "ve'o" are NOT used for
this purpose.)  If the "ku'e" were omitted, the "su'i zy" would be
swallowed up by the "va'a" forethought operator, which would then appear
to have two operands, "ny" and "su'i zy.", where the latter is also a
forethought operator.

Forethought mekso is also useful for matching standard functional notation.
How do we represent "y = f(x)"?  The answer is:

	li .ybu du li ma'o fy.boi xy.
	the-number y equals the-number the-operator f x
	y = f(x)

Again, no parentheses are used.  The construct "ma'o fy.boi" is the equivalent
of an operator.  In mathematics, letters sometimes mean functions and
sometimes mean variables, with only the context to tell which.  Lojban
chooses to accept the variable interpretation as default, and uses the
special flag "ma'o" to mark a lerfu as an operator.  "xy" and "ybu" are
variables, but "fy" is an operator (a function) because "ma'o" is used.
The "boi" is required because otherwise the "xy" would look like part of
the operator name.

When using forethought mekso, the optional marker "pe'o" may be placed
in front of the operator.  This can help avoid confusion by providing
clearly marked "pe'o" and "ku'e" pairs to delimit the operand list.