[Inquiry] Re: Theme One Program

[Jon Awbrey]

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TOP.  Expository Note 6

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3.1.  Lexical Cacti

The Index section of Theme One embodies an algorithm that "learns"
finite sequences of finite sequences of characters.  In this context,
from now on, I will use the word "sequence" to mean a finite sequence.

A set of sequences of sequences of whatever signs one happens to have
in an alphabet !A! is called a "two-level formal language" over !A!.
So I can say that the program's Index function "learns" a two-level
formal language over a particular alphabet !A!, the characters of
which you can probably guess, more or less.

It is usual to introduce a collection of notations that 
comes along with the language of formal language theory:

An "alphabet" !A! is a finite set.

The "kleene-star" !A!* of an alphabet !A! is the set
of all finite sequences over !A!, that is, the set
of all finite sequences that can be formed from
the elements of !A!.

The "sur-plus" !A!^+ of an alphabet !A! is the set
of all finite sequences over !A! of length greater
than zero.

A "formal language" L over !A! is a subset of !A!*.
Employing the symbol "c" for "subset of", L c !A!*.

A "two-level formal language" L over !A! is a subset L c !A!**
that is specified by giving the first-level language L_1 c !A!*
and the second-level language L = L_2 c L_1* c !A!**.

The elements of L_1 are called "words" or "distinguished strings".
The elements of L_2 are called "phrases" or "distinguished strands".

We refer to L_1 as the "lexical" level and
we refer to L_2 as the "literal" level of
the two-level formal language in question.

For the rest of this discussion, pick an alphabet !A! that contains
all of the usual alphanumerics, plus a finite number of other signs
that may be found convenient, but excluding the blank, the comma,
the left and right parentheses, and the period, as these latter
marks are reserved to special purposes by the Theme One program.

Jon Awbrey

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