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From: cowan (John Cowan)
Subject: MEX paper, draft 1.0, part #4 of 5
To: lojban-list
Date: Tue, 21 May 91 13:46:16 EDT
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5.  Reverse Polish Notation

So far, the Lojban notational conventions have mapped fairly familiar kinds
of mathematical discourse.  The use of forethought operators may have seemed
odd when applied to "+", but when applied to "f(" they appear as familiar
functional notation.  Now comes a sharp break.  Reverse Polish notation
represents something completely different; even mathematicians don't use
it much.  (The two main uses are in certain kinds of mathematical logic and
-- perhaps more familiar -- in certain brands of hand-held calculators).

In reverse Polish notation, the operator follows the operands, and there
are always exactly two operands.  No parentheses are required or permitted.
Reverse Polish notation is explicitly marked, requiring a "fu'a" at the start;
there is no terminator.  Here is a simple example:

	li fu'a reboi ci su'i du li mu
	the-number (reverse-Polish!) two three plus equals the-number five.

The operands are "re" and "ci".

Here is a complex example:

	li fu'a reboi ci pi'i voboi mu pi'i su'i du li rexa
	the-number (reverse-Polish!) two three times four five times plus
		equals the-number two-six

Here the operands of the first "pi'i" are "re" and "ci"; the operands of
the second "pi'i" are "vo" and "mu" (with "boi" inserted where needed),
and the operands of the "su'i" are "reboi ci pi'i", or 6, and "voboi mu
pi'i", or 20.  As you can see, it is easy to get lost in the world of
reverse Polish notation; on the other hand, it is especially easy for
a mechanical listener (who has a deep mental stack and doesn't get lost)
to comprehend.

The operands of a reverse Polish operator can be any legal operand,
including parenthesized mekso that can contain any valid syntax, whether
more reverse Polish or something more conventional.


6.  Other mekso bridi

So far we have only seen bare numbers as sentences (which are legal)
and equation bridi involving "du".  What about inequalities such as "x < 5"?
The answer is to use a bridi with an appropriate selbri, thus:

	li xy. mleca li mu
	the-number x is-less-than the-number 5

Other useful selbri in mathematical bridi include:

	zmadu	x1 is greater than x2
	trami'u	x1 is similar to x2 [shape-same]
	surmi'u	x1 is isomorphic to x2 [structure-same]
	cmima	x1 is a member of set x2
	gripau	x1 is a subset of set x2 [set-part]
	jbimi'u	x1 is approximately equal to [near-same]
	terci'e	x1 is a component with function x2 of system x3
	dunli	x1 is equal/congruent to x2 in/on property/quality/
			dimension/quantity x3

Note the difference between "dunli" and "du"; "dunli" has a third place
that specifies the kind of equality that is meant.  "du" is more like
"mintu", which has the same two-place structure "x1 is the same as x2".


7.  Logical And Non-Logical Connectives Within mekso

As befits a logical language, Lojban has extensive provision for logical
connectives within both operators and operands.  Operands are connected
in afterthought with selma'o A and in forethought with selma'o GA, just
like sumti.  Operators are connected in afterthought with selma'o JA
and in forethought with selma'o GUhA, just like selbri.  This parallelism
is no accident.

In addition, A+BO and A+KE constructs are allowed for operand grouping,
and KE...KEhE is allowed for operator grouping, although only one operator
is allowed between KE and KEhE: there are no analogues of tanru among the
operators.

Only one example of each kind of logical connection will be given.
Despite the number of rules required to support this feature, it is
relatively minor in the mekso scheme of things.

	vei ci .a vo ve'o prenu cu klama le zarci
	vei ge ci gi vo ve'o prenu cu klama le zarci
	(3 or 4) people go to the market.

	mi ka'e morji vei ze su'i je vu'u re ve'o sidbo
	mi ka'e morji vei ze gu'e su'i gi vu'u re ve'o sidbo
	I can remember (7 plus or minus 2) concepts.

Note that the mekso here are being used as quantifiers.  Lojban requires
that any mekso other than a simple number be enclosed in parentheses
when used as a quantifier.  This prevents ambiguities that do not exist
when using "li".  (By the way, "li" has an elidable terminator, "lo'o",
which is needed only when a "li" sumti is followed by a logical connective
that could seem to be within the mekso.)

Non-logical connection with JOI and BIhI is permitted between operands and
between operators.  The only known use for this is to connect operands
with BIhI to create intervals:

	li no ke'i bi'i ga'o pa
	the-number zero (inclusive) from-to (exclusive) one