Derivation of brain math.

Axioms:
 A1: Brains differ in size.
 A2: Brains are between sensors and motors.
 A3: Brains have more inputs than outputs.
Definitions:
 D1: Sensors write data streams and are outputs.
 D2: Motors read data streams and are inputs.
 D3: Data streams are continuous streams of data.
 D4: HalTree is any function with one output and two or more inputs.
Theorems:
 T1: There is a least brain.
    Proof: Brains can be mapped to N, since A1. So brains can be measured by size. 0 1 2 3 4 5 6... are some possible brain sizes.
 T2: Brains are HalTrees. 
    Proof: A3 and D4.
 T3: Neurons are HalTrees.
    Proof: Replace brain with neuron in the axioms. Anything true of brains is true of neurons.
...
    The least brain (1) reads and writes a single data stream. If O is the output and I is the input then O = F(I). O is a digit in the status number of the animal.
Let O = n(I) ; negate I
    O = not(I); NOT I ; if I = 0 then O = 1 else O = 0
    O = d(I); differential of I
    O = f(I); frequency of I
etc. F(I) = {n, not, d, f, i, z, g, l} are some simple functions from Hal algebra.
Brain 2 is the simplest HalTree. Now two data streams enter and one data stream exits. Two digits become one digit.
Now O = F(I,R).
O = I - R ; CNode ; comparison
O = I + R ; PNode ; aggregation
O = I o R ; ONode ; Choose biggest
O = I a R ; ANode ; Choose smallest
etc. F(I,R) = {-, +, o, a, is, gt, lt} are Hal algebra.
...
    From the above we develop a brain math. Connect all this and you will have Hal algebra. Why do you need it? Consider this: just four eight bit inputs can provide 4,294,967,296 different input values. Can you write that many if-then statements? Let b be the number base of the sensors and d be the number of sensors. The possible input values p are: p = b^d. (b raised to the d power.)


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