theory jjf - 3.7.2016

The Theory of Computation

A Treaty on Concurrent Binary Real Frequency Analysis, Real Time Synthesis, and Real Algebraic Transmissions.

Janusz Jeremiasz Filipiak

rev. 2 2 14 2015 - A hyper-computed algebraic tunnel.

rev. 3 2 15 2015 - Binary stream bug - bottom page 8.

rev. 4 2 26 2015 - Added Quantum Computing approach.

rev. 5 3 7 2015 - Real Analysis, Transmission, and Synthesis.

rev. 6 1 19 2016 - The Binary Solution, The Imaginary Division of Time into Space

rev. 7 3 7 2016 - Perpetual Point Perspective Principle

taut.org Jeremiasz Filipiak Page ! of !1 31

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Table of Contents

Preface 4

Abstract: The Binary Solution 5

Proof 6

Corollary 10

Vector Field Discussion 11

Quantum Algebra Discussion 18

Quantum Computing Discussion 21

Quantum Mechanic Discussion 23

The Imaginary Division of Time into Space 27


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Time is a complex field.


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Preface

Iteration is the first dimension. Iterations are independent of time, while at the same instance they define order.

A charge iterating through space is an iterator.

As the foundational abstract, iteration only performs a unit action - it moves the charge from the previous state to the next.

The concept of ordered states is misleading, because it assumes a dimension in which the identity can be uniformly established regardless of the order in which the charge is applied.

To give an example, take a clock. A clock is a charge that iterates through fixed intervals of time. The unit of time and the previous and next states of the iteration are independent of the uniform clock charge.

The second dimension is memory. There are many examples of abstractions of memory. The ones we are most familiar with are language, speech, writing, and bits in a digital computer.

Memory can be universally defined as a pattern: the organization of data.

The third dimension is choice. Choice has been named many times: will, reason, fate. The language commonly associated with choice is presently computation.

Choice assumes that at a single instance of time, in other words, in one iteration, there are at least two possible outcomes from a single pattern.

Lets take the example of the iterative coin toss to illustrate the difference between iteration and computation.

Take a coin and while holding it make the conscious decision not to flip it. Then pick an instance of time and choose to flip it.

Set down the coin.

Now, choose whether to flip the coin or not before picking it up. Then pick up the coin and execute your choice.

The first coin toss required a choice before the iteration, and the second coin toss required no choice at iteration.

The first coin toss was therefore computed in one instance of time.


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Abstract: The Binary Solution

Finding a solution of a problem defined in n binary bits is assumed to be the iterative process of finding the appropriate vector address in the problem space. Each problem space has a root vector of length 0. A solution of n bits is constructed by iteratively deciding each bit from root vector 0 to solution vector of n bits.

The set of all such binary encodings is a vector space, here called the name vector space or name space.

In practice, a binary encoding, a single vector in a name space based on a computational model of finite length n names a logical entity, otherwise known as the problem solution.

For any n-bit name space there exists an n-bit time vector space that enumerates it.

In n iterations required to reach the given n-bit name space vector, such a vector represents n separate time vector spaces, which differ only in length by one bit from the root vector of length 0 to the solution.

For any problem, finding the solution, the desired vector in the name vector space defined by the binary encoding requires ! iterations in the time vector space.

Theorem:

Assume every binary encoding is unique and contains both the binary encoding of the logical entity and the vector space in time required to reach it in vector space of ! complexity.

What follows:

Doubling the storage for every logical entity and including in its encoding both the name vector space in which it exists and the time vector space that defines how to reach it in computational iterations of the name space it exists in reduces the computational complexity of any problem in any vector space to a linear binary integration of the name vector space over the time vector space.

2n

2n


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Proof

If theorem is false then there exists a problem space for which the binary iterative process of

finding a solution does not reach the solution. This would imply the existence of an n-bit

problem space for which there exists an n-bit vector for which the enumeration requires more

than n iterations from the root 0 vector.

Since for any such defined n-bit problem space, time enumerated iterations are not factors of the

problem solution, time is a constant. Constant time implies all variables in the problem space are

also constants.

Let’s define computation as the process of reaching a solution for a problem in binary vector

space.

Any computation can be defined as the product of the inputs and the outputs.

!

Assuming the data of the problem is organized in one binary dimension with all of the ordered

inputs preceding all of the ordered outputs and all computation assumed to be commutative:

!

For each a and b, where both a and b belong to the set of integers from 0 to n.

Let:

! if the problem has a solution

! if the problem has no solution

f (x, y) = x ⋅ y

f x0,x1,x2,..., xn , y0,y1,y2,..., yn( ) = y0,y1,y2,..., yn ⋅ x0,x1,x2,..., xn

xa , yb = 0{ }

xa , yb = 1{ }


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All ! vectors, the problem space, can be described as a square matrix of elements from the

set of ! .

The complexity of the problem modeled by such matrix is defined as the matrix’s determinant.

Let’s call every binary vector built on the square problem matrix of size n a string. There exists a

string that describes the solution to every problem defined in such string space in no more than n

iterations.

The set of all possible strings in the problem string space define the formal language of the

problem.

An integration is a set of patterns. The integration can also be defined as vector integrals which

append a set of continuous ! transitions for which ! for all ! pairs and

result in binary strings of all ! constituents.

The set of all strings of the problem space, the name space, form a commutative ring.

There are two types of processes defined on this name space:

! - addition

! - multiplication

An addition is accomplished by computing a stack of recursive integrations for which there

exists an addition between the two encodings of the lower integration in the higher integration.

xa , yb

0,1{ }

xa , yb xa , yb = 0{ } a,b

xa , yb

0{ }

1{ }


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Each logical entity defined in the name space under a formal language has complexity directly

proportional to the number of integrals (strings, matrixes, or vectors) and the complexity of the

formal language in which they are defined.

In time, a logical entity, continuously moves between continuous states defined by the integrals

of it’s integration where each state is different from the previous state by the length of one bit (in

the lowest integration at the bottom of the stack).

This integration is also known as the time-event horizon.

Because it is always possible to expand the formal language in which the logical entity is defined

by one bit, in iterations of time the logical entity continuously creates a new language that

defines its current state and the language that defines the progress of time between the previous

and next states.

For every language in which the logical entity is defined there exists a finite state machine that

defines the possible next states in the language and the instructions required to reach them.

The machine has a finite number of states only because time is constant.

To iterate to the next step requires a recursive iteration for all finite state machines on languages

in the integration stack.

Multiplication on this commutative ring is defined as adding the newly found name space vector

to the time space vector.

This addition, otherwise known as computation, is called vector integration, and by definition

requires a bit of time added to the time space vector.


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Thus, for any n-bit problem space, time iterations are enumerable factors.

End of Proof.


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Corollary

The name-space and time-space for each logical entity in one data structure optimizes the time

required to reach a solution.

An optimal computational solution to any problem requires a virtual finite state machine running

instructions on a virtual support vector machine network for the problem set data. This

computational model is called streaming computation.

Two machines can communicate with each other in computational vicinity of the same vector

space stored in a data-instructional addressing space abstracted to a commutative ring.

The proposed vector machine network abstracts a commutative ring dictated by a global linear

binary classifier indexed by a global vector network.

The proposed finite state machine is an n-bit virtual computer for which the bit architecture size

depends solely on the height n of the abstraction stack for which each layer in the abstraction

stack is supported by a virtual computer of proportionally larger computational power to the

proportion that the search time in memory in both machines is proportional to the iterative sum

of the transmission lag and processing time between communication responder and respondent as

opposed to the vector sum.


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Vector Field Discussion

A binary integration is defined as the addition of a single time bit to the time vector space and a single bit to the corresponding name vector spaces for both newly created time vector spaces.

Conversely every entity in one bit of time can be described by two symmetrical name vector spaces.

What follows: symmetry is a special case of identity, also the symmetry of entities requires time equal to the length of the name of the entity.

Let’s call a named entity with a name of length n an identity of length n.

The run-time vector space of an identity is defined as a time vector spaces integration of the name vector spaces describing the identity in corresponding event-time horizon bits of time.

Each vector in the run time vector space of an identity corresponds to a name vector space describing the identity and spanning between the topmost binary integration (between the name and it's binary integrals) and the integration of all composite identities' integrals into their corresponding run time vector spaces.

What follows is that the complexity of the run time vector for an identity is inversely proportional to the complexity of the name vector space describing it.

We can define the binary linear classifier on all vector spaces to divide them into:

1) material run time spaces: An identity is defined to progress through the run time vector space by creating multiplicative (vector direction 1, self) vector spaces.

2) abstract run time spaces: An identity is defined to progress through the run time vector space by creating iterative (vector direction 0, forward) vector spaces.

An identity is the name vector space integrated with the most complex time vector space integrated and the run time vector space of the identity.

An identity is defined to be iterative if the complexity of the identity does not increase with the increase in complexity of the run time vector space in which the identity exists. Such identities are called abstracts.


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An identity is defined to be multiplicative if the complexity of the identity increases with the increase in complexity of the run time vector space in which the identity exists. Such identities are called matter.

All identities are integrated with each other through run time vector spaces. For instance, abstracts are integrated with matter through run time vector spaces.

The accuracy to which an abstract identity is integrated into a material identity determines the complexity of the minimal run time vector space integrating the material identity with the abstract identity that maintains a full separation of their respective name vector spaces.

The fundamental abstract identities are 0 and 1. Neither require a vector or name space to exist because they are necessary to define existence.

The simplest abstract and material identity requires one bit:

!

This transition takes the identity from it's abstract state of 0 to the material state of 1.

The run time vector space for this identity requires a single bit of time for a total of two bits. This is an explicit definition of binary integration:

!

The simplest abstract-only identity is one described by two bits - it's abstract and material identities at once.

The abstract bit defines the state of the identity.

The material bit defines the perceived state of the identity - how the abstract bit is received by another identity in a bit of time.

Therefore the identity exists in four states:

0 → 1

Name

Space

Time

Space

0 01 1


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! The run time vector space for this identity requires two bits of time for a total of four bits in name and time space.

!

! integrates into ! in name space and time space ! integrates into ! in name space and ! in time space ! integrates into ! in name space and ! integrates into ! in time space ! integrates into ! in name space and ! integrates into ! in time space

The integration between ! and ! is the highest name space integral because it is equivalent in time space and uniformly describes the identity in name space and time space as the progress of a name bit through a name space vector in a bit of time through the time space vector under the algebra composed of binary vector spaces.

0,0 → 0,1 → 1,1 → 1,0

Name TimeSpace Space

0,0 0,00,1 0,11,1 1,01,0 1,1

...0,0 0,0

0,0 0,10,1 1,1 1,01,1 1,0 1,0 1,11,0 0,0 1,1 0,0

0,0 0,1


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The eight name space and time space integrations of this three-dimensional time and name space can be represented by a binary three dimensional tetrahedral time space manifold.

These eight binary vector space integrations generate all vector spaces in time space and name space through a field of poly-tetrahedral manifolds.


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The sum of all identities, the name space can be described as the binary poly-tetrahedral field integral of time.

What follows:

Total time required to complete (or create) a run time of an identity, also referred to as the total time to integrate an identity, equals to the size of the run time vector space multiplied by the size of the name vector space describing the identity.

Like above, the run time vector space is defined as a time vector spaces integration of the name vector spaces describing the identity in corresponding event-time horizon bits of time.

Because each vector in the name space is integrated with each vector in time space, the size of the run time space for an identity of size n is ! .

Therefore, the total time required to complete a run time of an identity of length n can be expressed by the product of three vector spaces of size n.

!

For this space it is possible to define the tensor describing the identity through vectors of time. This tensor can be established by integrating both sides of the equation in name space and time space.

Both sides of the total time function are by definition integrated in both name space and time space. The left side of the equation is understood to represent the name space and the right side of the equation at the same time represents the time space.

Integrating in name space:

!

The name space of the identity in no time is equivalent to its time space in the highest integral therefore we can rewrite:

!

n2

t = n3

t dn = n3 dn∫∫

ndn = n3 dn∫∫


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Name integral of name space = !

Name integral of time space = !

Assuming that n approaches t in infinity:

!

Now integrating both sides in time space:

!

Time integral of name space = !

Time integral of time space = !

Arranging the integrals of name space and time space into corresponding vectors:

!

Substituting !

!

n2

2n4

4

t = t 3

t dt = t 3 dt∫∫

t 2

2t 4

4

Name

Space

Time

Space

n2

2t 2

2

=

n4

4t 4

4

n = t13

Name

Space

Time

Space

n2

2t 2

2

=t43

4t 4

4


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Any identity can be expressed as the product of time space and time as:

!

Simplifying,

The binary poly-tetrahedral field integral of time for an identity can be expressed as:

!

In words: any identity can be expressed as the product of it’s run time space and a bit of time added to the complex root of time.

Corollary:

Symmetry is an identity on abstracts because it is a time-bit-independent order of operation evaluation of the abstracts binary time-space integrations.

n =t23

2t 2

2

t

n = t3

2e13ln t+1

⎛⎝⎜

⎞⎠⎟


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Quantum Algebra Discussion

Let’s now assume a data structure based on a binary ring algebra.

The ring defines two elements, it’s name space 0 and and it’s time space 1, as well as two binary operators of addition represented as 0 and multiplication represented as 1.

This description can be interpreted as time space progressing in one bit 0 to 1 and two name spaces expanding by a single bit of value 0 for one name space and 1 for the other, as previously defined, this is an algebraic formulation of the binary integration:

!

Time space and name space are two separate interpretations of the same time space.

Therefore the point of integration can be derived as:

!

Equivalently:

!

Where ! is a two dimensional circular unit vector describing the algebraic ring ! of one-dimensional length ! .

This vector transformation is also called spin with equivalent notations:

!

n + 0 = t +1

n − t = 1− 0

t,n = 0,1

0,1 0,1π

0 = spin 12= spin0 = 0

∧1 = spin1= π


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What follows, name space in one time space dimension can be expressed as:

!

Where ! is the one-dimensional name vector space of length n being expanded from one one-dimensional name vector space to two trigonometrically orthogonal and normal one-dimensional name vector spaces ! and ! of length ! in one bit of time transitioning from ! to! .

For every transition in the time space dimension of spin ! there exists a two-dimensional transition on ! and ! that can be formulated as:

!

In this ring algebra transformation, the vector ! is a two-dimensional name space unit

vector which can be expressed as an imaginary four-dimensional vector i of the form:

!

Which includes two ! identities and two operators under symmetry that integrate two ! dimensions in one bit of time.

The two dimensional unit vector i is also called parity with equivalent notations:

!

The parity vector contains within it an optional expansion of the two Real two-dimensional dimensions into three dimensions, which alternatively can be notated as:

!

n1 = t1 + 0,1 1

n1

n1 + 0 n1 + 1 n +1 01

n1n1 + 0 n1 + 1

n2 = t2 + 0,1 2

0,1 2

i = 0,1,1,0

π -dimensional π -dimensional

0 11 0

= −1,1 = 0 = i

∧1= i∧

−1= i

n3 = t3 + 0,1 3


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Where ! is an eight-dimensional exponential name space unit vector ! which integrates the product of two 3 orthogonal normal Real time-space dimensions into one in one bit of time.

This name space vector is also known as the charge e with equivalent orthogonal 2-degree of freedom notation:

!

The space can be rewritten as a three-dimensional 4-vector-form that includes the integrations of the lower dimensions and also demonstrates the 3-dimensional time-space manifold visually:

The total formulation in three dimensions can be rewritten as:

!

Substituting in spin, parity and charge:

!

Which is the three-dimensional state transformation of time into space, with the following ! algebraic ring identities:

!

0,1 3 0,1,1,0,0,1,1,0

e = 0 1 1 00 1 1 0

n3 = t3 + 0,1,1,0,0,1,1,0 0,1,1,0 0,1

eiπ +1= 0

R3

π = 23e

π = 4 39

i

i = 32e

i = 3 34

π


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Quantum Computing Discussion

We understand any time space to be fully generative under the principles of the identity:

!

Given the vector state for the time vector space ! .

Where each !

We can define a three-bit tetrahedral transform with four time-space manifolds.

The transform establishes the eight transformations of the internal implied two-bit three-dimensional time-space integral. These transformations of name and time space can be expressed as the eight three dimensional unit vectors of time and name space:

! integrates into ! in name space ! ! integrates into ! in name space ! ! integrates into ! in name space ! ! integrates into ! in name space !

eiπ + vt−1 = vt

vt = v0,v1,v2,...,vt∀t ∈ℵ vt = 0⊕1{ }( )

0,0 0,1 1,0,10,1 1,1 1,1,11,1 1,0 1,1,01,0 0,0 1,0,0


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! integrates into ! in time space ! ! integrates into ! in time space ! ! integrates into ! in time space ! ! integrates into ! in time space !

This transform collapsed into the volume defined by the four manifolds can be understood as a binary computer in three dimensions with an algebraic ring of name space serving as the addressable 1 bit memory and the algebraic ring of time space serving as the processor clock for which the only instruction is an algebraic multidimensional iteration ! .

From this algebraic binary ring computation it can be seen that a single three-dimensional bit change from the origin of time ! in two dimensional time space from ! to ! changes equivalently a bit of the two dimensional name space from ! to ! thus expiring the algebraic two dimensional time ring of the binary time space for the underlying two bit name.

The equivalent three-dimensional notation for this expiration:

Name space: ! to ! Time space: ! to !

This is an algebraic definition of two dimensional orthogonal and normal binary linear regression in three dimensional orthogonal and normal-normal Real computational space.

Such description of transformations between time-space quanta of finite bit length through their algebraic ring relationships is called Real Algebraic Transmission.

0,0 0,1 0,0,10,1 1,0 0,1,01,0 1,1 0,0,01,1 0,0 0,1,1

eiπ + vt−1 = vt

0,0,0 1,0 1,11,0 0,0

1,1,0 1,0,00,1,0 0,1,1


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Quantum Mechanic Discussion

Assume a simple quark-gluon space to be described through a 3-bit vector space:

!

Where each binary integration operates on a binary algebraic ring ! .

!

Similarly, the antiproton, neutron, and antineutron can be described as:

!

Using this encoding any three dimensional particle can be graphed in three-dimensional time-space as:

spin,parity,charge

0,1

u,u,d = 1,0,1 = +1e

u,u,d = 0,0,1 = −1e

u,d,d = 1,0,0 = +0e

u,d,d = 0,0,0 = −0e


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Using the 3-bit three-dimensional transform we can encode any relationship between two neighboring 2-bit binary tetrahedral integrals of time space - three dimensional quants of time space.

The encoding of the outward pointing time space tetrahedron represents a state in which the face of the three-dimensional quant is charged at state ! .

The encoding of the internal time space vertex represents the state in which the opposite face is not charged at state ! .

The following sign changes apply to a four-faced time space quant.

Face ! is charged when ! and not charged when ! .

Similarly:

Face ! is charged ! and not charged ! .



1{ }

0{ }

0,0 1,0,0 0,0,0

0,1 1,0,1 0,1,0

1,1 1,1,1 0,1,1

1,0 1,1,0 0,0,1


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This means that a quantum equation of state based on a three dimensional poly-tetrahedral time space integral can be expressed as an 8-bit word:

!

With:

! - Input binary charge.

! - Input binary quantum face.

! - Output binary quantum face.

! - Output binary charge.

Taking the multidimensional dimensional real quantum iterator of time space:

!

We can show that a single quantum iteration:

From: charged ! state for face ! To: a no-charged state of ! for face !

Requires no quantum charge, but only an increment of 1 time bit from the previous (imaginary) state to the current (real), which is the equivalent of a 1 name bit iteration from a three-dimensional point of origin iteration in time.

The process of construction of higher-dimensional time-space quanta of finite bit length from their lower-dimensional algebraic charged encoding is called Real Time Synthesis.

s1, s2, s3, s4 , s5, s6, s7, s8

s1

s2, s3, s4

s5, s6, s7

s8

n3 = t3 + 0,1,1,0,0,1,1,0 0,1,1,0 0,1

1,1,0 1,00,1,1 1,1


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Exciting a single quant of energy of a three-dimensional binary particle to a fully excited state can be described by a series of input vectors into a single three-dimensional quant of time-space:

! ! ! ! ! ! ! !

For a total of 36 charged time space bits for which the binary vector space equations of state can be described as:

!

Where the symbols ! denote three distinct, normal, orthogonal, real binary dimensions for a fully excited 36-bit charge ! equivalence of:

1) 27 one-dimensional bits of time space, each bit for every bit of either 2) or 3) 2) 8 two-dimensional bits of time, each bit for every bit of either 1) or 3) 3) 1 three-dimensional bit of quantum charge, each bit for every bit of either 1) or 2)

The process of maintaining this 36-bit uniform three-dimensional time-independent charge requires a further 64 bits of fully charged continuous time space in every bit of time.

The process of describing higher-dimensional time-space quanta of finite bit length through their lower-dimensional algebraic charged properties is called Real Frequency Analysis.

0,0,0,0,0,0,0,10,0,0,0,0,0,1,10,0,0,0,0,1,1,10,0,0,0,1,1,1,10,0,0,1,1,1,1,10,0,1,1,1,1,1,10,1,1,1,1,1,1,11,1,1,1,1,1,1,1

π → 1,1,1,1,1,1,1,1i→ 1,1,1,1,1,1,1,1e→ 1,1,1,1,1,1,1,1

π ,i,e1{ }


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The Imaginary Division of Time into Space

Lets consider the line of time.

Points on this line constitute a representation of the momentary (time-independent) existence or

lack thereof for any space that exists in time.

Such a binary line of time is otherwise known as the binary solution.

For the binary solution we can define an increment function ! such that:

!

Where for any finite space ! for which it’s existence can be defined as a path through points on

the line of time:

! can be denoted as ! : the point of creation for space ! on the line of time,

And:

! can be denoted as ! : the point of expiration for space ! on the line of time.

Any point on the line of time, any space ! , can be understood as a quant of time with the

property that it can be always divided into two consecutive quants of the same properties which

expire in an order governed by the line of time on which they are defined.

Under these principles, a quant of time is equivalent to the quant of energy required to increment

through itself as defined by the sum of all underlying quants that constitute its ordered existence

on the line of time.

f

f (0) = 1f (1) = 0

A

f (0) t0 A

f (1) t1 A

A


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We can call this increment function ! on space ! , the binary operator on the set of all points on

the line of time, a binary increment of time (bit), and govern it by the uniform principle of

conservation of state, as implied in the definition of the function ! where:

!

The space of all such defined points on the line of time, the space of all bits, constitutes the

binary solution.

This space has the property that any bit in it is separated from any other bit by either a sequence

of like bits or is directly adjacent to it.

For any two bits ! and ! suspended in this binary solution, there exists a single binary bit that

can be defined as follows:

! where !

! where !

And:

!

Defines the span of existence of a space defined by ! and ! on the line of time ! .

f A

f

t0 + t1 = 1

t0 t1

ft (0) = 1dft (0)dt

= 0

ft (1) = 0dt

dft (1)= 1

f (0) = 1⇔ t0f (1) = 0⇔ t1

t0 t1 t


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Similarily:

!

!

Defines the bitwise integration of the bit from its line bit constituents into a time ordered

integral: !

Where all integral bit constituents, also referred to as bit derivatives are defined on a derivative

line of time: !

This unit 2-bit binary space of binary increments of time represents all functional

transformations in three dimensional binary space.

dft (0)dt

= 0⇔ 0

dtdft (1)

= 1⇔1

dtdft

dftdt


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Any quant of such defined binary bit of time can be modeled as an abstract finite binary state

machine.

Such binary machines of time have the property that given enough binary time, they generate any

binary space through uniform binary increments on a single binary line of time.


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theory jjf - 3.7.2016

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