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0.0.4 Z2adic.txt in place of Zadic.txt with improved numeral form
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orcmid committed Feb 5, 2023
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Zadic.txt 0.0.3 UTF-8 2023-02-05
Z2adic.txt 0.0.4 UTF-8 2023-02-05
----|----1----|----2----|----3----|----4----|----5----|----6----|----7----|--*

The Miser Project Code
Expand All @@ -7,15 +7,17 @@ Zadic.txt 0.0.3 UTF-8 2023-02-05
<https://github.com/orcmid/miser/blob/master/oMiser/Zadic.txt>


Z INTEGER ARITHMETIC IN oMISER
------------------------------
Z 2'S COMPLEMENT INTEGER ARITHMETIC IMPLEMENTION
------------------------------------------------

The interpretation of Integer Arithmetic is based on the ‹ba›
simulation of Boolean Algebra using ob.bvx for arbitrary length
Boolean vectors. See boole.txt for details.

For Zadic, ob.bvx structures are interpreted as 2's complement binary
numerals. The terminal element is taken as the sign. It is also
For Z2adic, ob.bvx structures are interpreted as 2's complement binary
numerals. The terminal element, followed by a ":" is taken as the
sign. It is called a bumper and it stands for an indefinite sequence
of the same digit. The numerals are taken to have
taken as being an extension of that bit indefinitely. The notation
is little-endian. The first element is the lowest-order bit of the
numeral.
Expand All @@ -28,39 +30,45 @@ Zadic.txt 0.0.3 UTF-8 2023-02-05
1. SYNOPSIS: THE FUNDAMENTAL REPRESENTATION

For the arithmetic interpretation, write 0 and 1 instead of the
corresponding bot, ⊥, and top, ⊤. [1, 0:] represents 1, [0, 1, 0:]
represents 2, [1, 1, 0:] represents 3, and [0, 0, 1, 0:] represents 4.

The list [0:] is the representation of the numeral 0. It also
signifies as many 0 digits as are needed. It is shorthand for, e.g.,
[0, 0, 0:]. The canonical form is by removing every 0 to the left
up until any 1. So [0:] for [0, 0, ..., 0:] with any number of 0-s.

The list [1:] is the representation of the numeral -1. It also
signifies as many 1 digits as neeed. It is shorthand for, e.g.,
[1, 1, 1:]. The canonical form is by removing every 1 to the left
up until any 0. So [1:] for [1, 1, ..., 1:] with any number of 1-s.

The two's complement representation of negation follows the usual
rules of adding 1 (an incoming carry) to the 1-s complement. This
can be done in a single scan, as follows, starting at the little end.
corresponding bot, ⊥, and top, ⊤. 1:0 represents 1, 01:0
represents 2, 11:0 represents 3, and 001:0 represents 4. Notice
that the left-to-right reading of the bits is from low-order bit to
successively higher ones.

The form :0 is a "bumper." It signifies an inexhaustible sequence of
0 digits (bits). It is a shorthand for as many 0 digits as are
useful. The unique canonical form of a :0-terminated sequence is
achieved by removing every 0 the left of the :0 bumper up until any
1 bit. So :0 is the canonical form for representing integer 0.

The form :1 is the bumper for an inexhaustible sequence of 1 bits.
The canonical form of of a :1-terminated sequence is achieved by
removing ever 0 to the left of the :1 bumper up until any 0. So
:1 for 111...:1 with any number of 1's forming 111... .

The binary numerals are taken as 2's-complement binary numerals
in which :1 represents integer -1, 0:1 represents -2, 10:1 for -3,
and so on.

In this scheme the negation of a number is expressed by adding 1 (an
incoming carry) to the 1's-complement of the Z2adic expression of the
given number. Ths can be done in a single left-to-right scan of the
original numeral, replacing digits as follows.

1. a. If there is a carry in, a 0 remains 0 and there's a carry out.
b. If there is no carry in, a 0 becomes 1 with no carry out.

2. a. If there is a carry in, a 1 remains 1 with no carry out.
b. If there is no carry in, a 1 becomes a 0 with no carry out.

3. a. If there is a carry in, 0: remains 0: and we're done.
b. If there is no carry in, 0: becomes 1: and we're done.
3. a. If there is a carry in, :0 remains :0 and we're done.
b. If there is no carry in, :0 becomes :1 and we're done.

4 a. If there is a carry in, 1: becomes 1, 0: and we're done.
b. If there is no carry in, 1: becomes 0: and we're done.
4 a. If there is a carry in, :1 becomes the pair 1:0 and we're done.
b. If there is no carry in, :1 becomes :0 and we're done.

In this scheme, the 2's complement of [0:] is [0:] as expected. And
the 2's complement of [1:] is [1, 0:] as expected. Accordingly, the
2's complement of [1,0:] is [1,1:] which simplifies to [1:]. Note
that the 2's complement of [1,1:] is [1,0:], also as required.
In this scheme, the 2's complement of :0 is :0 as expected. And
the 2's complement of :1 is 1:0 as expected.

[EDITOR'S NOTE: THE RULE 3a DEPENDS ON A KIND OF JUST-SHORT-OF-
INFINITARY ARGUMENT. THAT NEEDS TO BE DEALT WITH SOMEWHERE BELOW.]
Expand Down Expand Up @@ -209,7 +217,7 @@ BOILERPLATE BELOW HERE - REWRITING FOR mathZ PENDING
After [Forster2003], an interpretation where "="s correspond is termed
an implementation. We will make that the case.

2. MathZ IMPLEMENTATION OF ‹Z›
2. Z2adic IMPLEMENTATION OF ‹Z›

3.4 ‹Z› Implementation in ‹ob›: Treatment of ob.bvx

Expand Down Expand Up @@ -469,7 +477,7 @@ BOILERPLATE BELOW HERE - REWRITING FOR mathZ PENDING
ATTRIBUTION

Hamilton, Dennis E. Zadic Integer Arithmetic in oMiser. Miser Theory
Conception text file mathZ.txt version 0.0.3 dated 2023-02-05, available
Conception text file mathZ.txt version 0.0.4 dated 2023-02-05, available
on the Internet as a version of
<https://github.com/orcmid/miser/blob/master/oMiser/mathZ.txt>

Expand Down Expand Up @@ -522,11 +530,14 @@ BOILERPLATE BELOW HERE - REWRITING FOR mathZ PENDING

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

0.0.4 2023-02-05T05:48Z Adjust to name Z2adic.txt and introduce a short-hand
numerical-bumpered notation that is more compact than the list forms
employed in the oMiser computational interpretation.
0.0.3 2023-02-05T02:41Z Small change commited before changing the name again.
0.0.2 2023-02-03T16:48Z Rename mathZ to Zadic and touch up accordingly.
0.0.1 2023-02-03T16:04Z Ponderings on 2-adic and what that means, with new
references and some more scrapping of the boilerplate.
0.0.0 2023-01-30T23:22Z Draft synopsis over boilerplate of boole.txt version
0.5.3

*** end of Zadic.txt ***
*** end of Z2adic.txt ***

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