Combinary

From Davidbond.net

Combinary is a mathematical system combining complex numbers and binary. It was created by David Bond to test a theorem combining quantum mechanics and statistical mechanics.

Combinary was developed to constrain probability amplitudes to three possible states. The term "combinary" derives from the terms "complex" and "binary".

In the binary system, a "bit" of information may take one of two values. The "combit" is the combinary equivalent of the bit, with real and imaginary parts and, like the qubit, can have an "unknown" state. Only the "real" part is "observable" in the physical sense. Combits differ from qubits due to constraints on real and imaginary part permutations.

Contents 1 Combits 2 Combers 3 Constraint systems 4 Operations 5 Uses

Combits

Combinary notation extends binary notation, allowing each combit to contain the following values:

• The real part may take the values: 0, 1 and q
• The imaginary part may take the equivalent values: o, i, and <math>\phi</math>.

• 0 : "not 1"
• 1 : "1"
• q : "the superposition of 1 and not 1"
• o : "not the square root of minus 1"
• i : "the square root of minus 1"
• <math>\phi</math> : "the superposition of 'the square root of minus 1' and 'not the square root of minus 1'"

<math>\phi</math> may also be written 'p' so that ASCII may be used when programming computers. The symbols q, <math>\phi</math> and p are chosen as they symbolise "both 0 and 1", containing both a circle and a stroke.

Combers

Just as binary constructs "bytes", "words" etc. from bits, a "combinary number" or "comber" (pronounced like number) contains any number of bits, each more significant than the one to its right by a factor of 2, in both real and imaginary axes.

Examples of simple combers:

• 10 (= 2 decimal)
• q (= unknown)
• <math>\phi</math> (unknown on the imaginary axis)

When writing a comber normal C programming notation for binary, decimal, hexadeciaml etc. are used. Real and imaginary bits may be interleaved, or written next to each other. If not interleaved, capital R and I are used to clearly denote the Real and Imaginary parts, respectively. The following are equivalent:

• 1<math>\phi</math>0i
• 0x2 + 0b<math>\phi</math>i
• 0x2R0bq1I

If the comber contains no imaginary part or Q-values, the comber is a "real" number, and may be implementated in binary as normal. If the comber contains no Q parts AND no <math>\phi</math> parts, it is a standard complex number. In this way, "real" and "complex" numbers are special cases of combers.

Unless otherwise stated, all more-significant, non-written bits are assumed to be 0. Less-significant, non-written bits are assumed to be q (o and <math>\phi</math> respectively on the imaginary axis). If no The following are equivalent:

• 0o0o1i1i
• 1i1i
• 3R3I

The following is NOT equivalent, as precision is not maintained:

• 1i1i.0o

If the combit is not fully specified (i.e. onthe the real or imaginary part is present), 0 or o are assumed. The following are equivalent:

• q
• qo

Precision may vary between real and imaginary parts, as in the following, equivalent examples:

• qiqiqi0o0o.1o1p1p1p
• qqq00.1111R11100.0I

Repeating expansions are denoted as with binary, with a line over the final digits 0 and o, either individually, or together:

Constraint systems

"Constraint systems" limit the combinations of 1, 0, q, i, o and <math>\phi</math> within one combit. It is the constraint systems and their implications that forms the core of Comber studies.

At the heart of all constraint systems is the rule that a combit must be made up of:

• only one of 1, 0 and q
• PLUS
• only one of i, o and <math>\phi</math>

Beyond this, certain combinations are either permitted or forbidden.

Some example constraint systems:

standard combinary

In standard combinary, each combit can be stored using four bits.

• 0) 0o: permitted
• 1) 1o: permitted
• 2) qo: permitted
• 3) 0i: permitted
• 4) 1i: permitted
• 5) qi: permitted
• 6) 0p: permitted
• 7) 1p: permitted
• 8) qp: permitted
• 9-15) Unused

Simple 2-bit combinary

In simple 2-bit combinary, each combit can be stored using two bits. It seems beautiful that complex binary would use two bits to store each combit.

• 0) 0o: permitted
• 1) 1o: permitted
• 2) 0i: permitted
• 3) qp: permitted
• All other combinations: forbidden

Operations

Combers can be operated on as would be expected. The following examples use the standard combinary constraint system:

• 11 + 1qq = 1qqq
• 10 - q = qq
• 1 | q = 1
• 1 & q = q
• (1<math>\phi</math>)² = 0<math>\phi</math>qo

Taking the last case as an example:

• (1<math>\phi</math>)²
• = (1RqI)²
• = (1-q)R+(2q)I
• = qR+(q0)I
• = 0<math>\phi</math>qo

Uses

It is believed that combinary could have relevance in quantum mechanics, at the smallest scales of space and time. Here, Occam's Razor would guide that if you don't need infinitely variable probabilities, dispense with them, constraining yourself to "observed true" (1), "observed false" (0) and "unobserved" (q). It is expected that only the real part is observable, and that the transfer of a bit of information from the imaginary to the real axis during a multiplication function is the process of observation. Processes which, through chain reaction, assemble combits into coherent combers create macro-scale "events" suitable for human observation.