Araara c — Abece

This is the fifth in a series of expository blog posts on Araara. The preceding post introduced Asks: strings formed by concatenating Unicode normal forms in Akurto. The series begins here.

We will return to the uses of Akurto terms for Unicode code points later on. But first: what is an alphabet?

Recall that an Asks is a concatenation of Akurto terms in Unicode normal form (unf): ui followed by the ordinary normal form of each code point. Each ui marks a new code point, so we need no terminators.

Consider the tiny alphabet o a e b p. In that order, its characters have the values 0, 1, −1, 2, −2. These are their values in the alphabet, not their Unicode code points: ‘a’ still has code point 97, while its value here is 1.

An alphabet (Abece in Globasa; Alphabet in English) is just an Asks with a few more conditions imposed on it:

1. An odd number of unf terms, at least three.
2. Each unf term corresponds to an assigned Unicode character.
3. No character occurs more than once.

We require an odd number of unf terms because we want a zero element (the first element), followed by pairs of mutually inverse elements.

From here on, we will write code points using this colouring. Here is code point zero:

uio

Let’s see some examples!

uiggbb uiffda uiffdb uiffdba uiffdbb uiffdc uiffdca uiffdcb uiffdcba uiffdcbb uiffdcc uiffdcca uiffdccb uiffdccba uiffdd uiffdda uiffddb uiffddba uiffddbb uiffddc uiffddca uiffddcb uiffddcba uiffddcbb uiffddcc uiffddcca uig (ø-English)

Decoded: øabcdefghijklmnopqrstuvwxyz

uiffdda uiffda uiffdc uiffdb uiffddb uiffdba uiffdcc uiffdbb uiffddca uiffdca uiffddcba uiffdcb uiffdcca uiffdcba uiffddcc uiffdcbb uiffddcb uiffdccb uiffddbb uiffdccba uiffdd uiffddc uig uiffddcbb uiffddcca (Aglobasa)

Decoded: oaebpcjdtfvgkhxiulrmnszwy

These also have their signatures: the sums of their Unicode code points, expressed in unf. To encode a sum this way, we simply prepend ui to its ordinary normal form. The prefix contributes zero.

uiiihhgfd (ø-English: 3095)

uiiihgfdbb (Aglobasa: 2734)

Later we will see how to find other kinds of signatures that can also encode the order of these alphabets.

Gapped alphabets

A gapped alphabet is an alphabet in which one or more unf terms have been replaced with gaps, indicated by

ui

A gap leaves a position empty; it does not move the later characters forward. Gaps may repeat: the uniqueness condition applies to the characters that remain. Here, □ will stand for a gap in the decoded examples.

Here are some famous gapped alphabets:

uifcb uifcba ui (binary)

Decoded: 0 1 □

uifcb uifcba ui uifcbb ui uifcc ui uifcca ui uifccb ui uifccba ui uifd ui uifda ui uifdb ui uiffda ui uiffdb ui uiffdba ui uiffdbb ui uiffdc ui uiffdca ui (lowercase hexadecimal)

Decoded: 0 1 □ 2 □ 3 □ 4 □ 5 □ 6 □ 7 □ 8 □ 9 □ a □ b □ c □ d □ e □ f □

In binary, 0 occupies the zero position and 1 the first positive position; its inverse position is a gap. In hexadecimal, the characters from 1 to f occupy the successive positive positions, with a gap at each corresponding negative position. This gives us 31 positions in total.

Of course these also have their signatures! A gap is ui, which has value −1094 + 1094 = 0, so gaps do not affect the sum.

uiffda (binary: 97)

uiidd (lowercase hexadecimal: 1122)

Exercises

1. Choose five distinct Unicode characters and arrange them into an alphabet. Which character is zero? Which two pairs are inverses? Write a few terms using your alphabet, then swap two characters. What changes?
2. Now compute its signature. Does swapping the characters change it? The sum depends on which characters you chose, but not their order. Different choices can share a sum too!

In the next post, we will think of shorter ways in which we can write Asks.