## Chapter 18: lojbau mekso: Mathematical Expressions in Lojban

### 16. Reverse Polish notation

The following cmavo is discussed in this section:

`fu'a    FUhA    reverse Polish flag`

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 the usual functional notation. Now comes a sharp break. Reverse Polish (RP) notation represents something completely different; even mathematicians don't use it much. (The only common uses of RP, in fact, are in some kinds of calculators and in the implementation of some programming languages.)

In RP notation, the operator follows the operands. (Polish notation, where the operator precedes its operands, is another name for forethought mekso of the kind explained in c18-§6.) The number of operands per operator is always fixed. No parentheses are required or permitted. In Lojban, RP notation is always explicitly marked by a “fu'a” at the beginning of the expression; there is no terminator. Here is a simple example:

```✥16.1    li fu'a reboi ci su'i
du li mu
the-number (RP!) two, three, plus
equals the-number five.```

The operands are “re” and “ci”; the operator is “su'i”.

Here is a more complex example:

```✥16.2    li fu'a reboi ci pi'i
voboi mu pi'i su'i
du li rexa
the-number (RP!) (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 an RP operator can be any legal mekso operand, including parenthesized mekso that can contain any valid syntax, whether more RP or something more conventional.

In Lojban, RP operators are always parsed with exactly two operands. What about operators which require only one operand, or more than two operands? The null operand “tu'o” and the null operator “ge'a” provide a simple solution. A one-operand operator like “va'a” always appears in a reverse Polish context as “tu'o va'a”. The “tu'o” provides the second operand, which is semantically ignored but grammatically necessary. Likewise, the three-operand version of “gei” appears in reverse Polish as “ge'a gei”, where the “ge'a” effectively merges the 2nd and 3rd operands into a single operand. Here are some examples:

```✥16.3    li fu'a ciboi muboi vu'u
du li fu'a reboi tu'o va'a
The-number (RP!) (three, five, minus)
equals the-number (RP!) two, null, negative-of.
3 - 5 = -2

✥16.4 li cinoki'oki'o du
li fu'a biboi
ciboi panoboi ge'a gei
the-number 30-comma-comma equals
the-number (RP!) 8,
(3, 10, null-op), exponential-notation
30,000,000 = 3 × 108```