Araara ċb — Apar
This is the twelfth in a series of expository blog posts on Araara. In Alungo, we extended the numbers we can write. Now let’s give a name to a value we have not yet chosen.
We will call this Apar, after the Globasa word parametre.
A name that cancels itself
We can make variable names from Akurto expressions. Take an expression and append its inverse, with the inverse letters in reverse order. This works for representations of positive integers, negative integers, and zero.
For example, three can be written as ba. The inverses of b and a are p and e. Reversing their order gives ep, so we write:
baep
Under Akurto evaluation, this is 2 + 1 − 1 − 2 = 0. The middle ae cancels first, leaving bp, which also cancels. As Apar, however, we keep the whole expression as a variable name.
The first half need not be in normal form. Different representations give different names:
| Value | Akurto representation | Apar |
|---|---|---|
| 0 | o | oo |
| 1 | a | ae |
| 2 | b | bp |
| 3 | ba | baep |
| 3 | ab | abpe |
| 3 | aaa | aaaeee |
baep, abpe, and aaaeee therefore name different variables, although their first halves all represent three. The number three helped us construct these names; it does not determine the variables’ values.
A place for a value
A variable also needs a place in which its value can live. We will call this its ambient type. Depending on what we are studying, we might want variables for Akurto, Asoho, Apeundeku, or another type we introduce later.
Apar leaves this choice open. We specify the ambient type in the discussion, rather than fixing it once for every variable.
The spelling can suggest an interpretation without prescribing one. For example, if we call the variable named baep x₃ in the English metalanguage, we might choose the following readings:
| Apar expression | Meaning |
|---|---|
| baep | x3 |
| BAEP | 1/x3 |
| peab | −x3 |
The spellings suggest these readings: capitals recall Asoho, while dual letters recall negation. We are free to interpret the variables this way, but Apar does not require it. Their meanings, including any relationships between them, belong to the semantic interpretation we choose.
Polynomials with integer coefficients
We can choose a polynomial algebra as the ambient type. Then our variable names can stand for indeterminates, and expressions can describe how they combine.
Recall the uppercase-followed-by-lowercase notation for multiplication in Apeundeku. We can give the same pattern a polynomial interpretation: the lowercase part supplies a coefficient, while concatenating the terms adds them.
For example, interpret PB, NM, and EA as variables P, N, and E. Our expression for a neutral carbon-12 atom becomes:
PBca NMca EAca = 6P + 6N + 6E
This is a linear polynomial. This keeps the three contributions separate until we choose values for the variables. The variables can carry the proton, neutron, and electron interpretations chosen in our referencing convention.
Something familiar
The same idea fits Unicode normal form. Interpret ui as a variable u, and keep the following lowercase letters as an added integer. Since cca represents eleven, we can read:
uicca = u + 11
This is another linear polynomial: a variable plus a constant. Under ordinary Akurto evaluation, ui still contributes zero and the expression evaluates to eleven. Under our Unicode interpretation, we retain ui as the marker that tells us what the eleven refers to.
Think of a place card at a table. The card gives the place a name. The table sets the context. Who sits there can be decided later.
In the next post, we will introduce the Fraktur alphabet.
