Computer-Simulation Model Theory (P= NP is not provable)

TL;DR

The paper introduces Computer-Simulation Model Theory (CSMT), a new logical reasoning framework inspired by the simulation hypothesis, to analyze the P vs NP problem by constructing models where P=NP does not hold.

Contribution

It proposes CSMT as an independent reasoning method based on computer simulations, extending mathematical model theory to include computational and simulation aspects.

Findings

Constructed a simulation model where P≠NP holds.

Demonstrated CSMT's applicability to the P vs NP problem.

Provides a new perspective on logical reasoning independent of the simulation hypothesis.

Abstract

The simulation hypothesis says that all the materials and events in the reality (including the universe, our body, our thinking, walking and etc) are computations, and the reality is a computer simulation program like a video game. All works we do (talking, reasoning, seeing and etc) are computations performed by the universe-computer which runs the simulation program. Inspired by the view of the simulation hypothesis (but independent of this hypothesis), we propose a new method of logical reasoning named "Computer-Simulation Model Theory", CSMT. Computer-Simulation Model Theory is an extension of Mathematical Model Theory where instead of mathematical-structures, computer-simulations are replaced, and the activity of reasoning and computing of the reasoner is also simulated in the model. (CSMT) argues that: For a formula $\phi$, construct a computer simulation model $S$, such that 1- $\phi$ does not hold in $S$, and 2- the reasoner $I$ $($human being, the one who lives inside the reality$)$ cannot distinguish $S$ from the reality $(R)$, then $I$ cannot prove $\phi$ in reality. Although $\mathrm{CSMT}$ is inspired by the simulation hypothesis, but this reasoning method is independent of the acceptance of this hypothesis. As we argue in this part, one may do not accept the simulation hypothesis, but knows $\mathrm{CSMT}$ a valid reasoning method. As an application of Computer-Simulation Model Theory, we study the famous problem P vs NP. We let $\phi \equiv\mathrm{ [P= NP]} $ and construct a computer simulation model $E$ such that $\mathrm{P= NP}$ does not hold in $E$.

Equations

$$U=(TBOX,SBOX,INST,CONF)$$

$$U_e=(TBO{X}_e,SBO{X}_e,INS{T}_e,CON{F}_e)$$

$$\{C_{i,w}\in S(w)\mid \exists x\in E(w)\cup D(w)(C_{i,w}=(h,x\underline{△}))\}.$$

Full text

Inspired by the view of the simulation hypothesis (but independent of this hypothesis), we propose a new method of logical reasoning named ”Computer-Simulation Model Theory”, CSMT.

Computer-Simulation Model Theory is an extension of Mathematical Model Theory where instead of mathematical-structures, computer-simulations are replaced, and the activity of reasoning and computing of the reasoner is also simulated in the model. (CSMT) argues that:

for a formula $\phi$, construct a computer simulation model $S$ such that

• 1-

$\phi$* does not hold in $S$, and*

• 2-

*the reasoner $I$ $($human being, the one who lives inside the reality$)$ cannot distinguish $S$ from the reality $(R)$, *

then

$I$* cannot prove $\phi$ in reality.*

Although $\mathrm{CSMT}$ is inspired by the simulation hypothesis, but this reasoning method is independent of the acceptance of this hypothesis. As we argue in this part, one may do not accept the simulation hypothesis, but knows $\mathrm{CSMT}$ a valid reasoning method.

As an application of Computer-Simulation Model Theory, we study the famous problem P vs NP. We let $\phi\equiv\mathrm{[P=NP]}$ and construct a computer simulation model $E$ such that $\mathrm{P=NP}$ does not hold in $E$.

1 Simulation Hypothesis

Human’s worldview changes with the times. At the time of Omar Khayyam, the industry was limited to the Pottery, and Omar Khayyam (inspired by the pottery) assumed the universe as a water vessel and asked ”hey you! where is the potter?”

Our worldview changes as our industry changes and our current industry are artificial intelligence, virtual reality, robotics, and video games. Using artificial intelligence, we are going to make new creatures even more intelligent than us. Sophia Robot 222$https://en.wikipedia.org/wiki/Sophia_{(}robot)$ is one of these creatures.

So Nick Bostrom (inspired by the computer industry) asks ”Are you living in a computer simulation?” (see [1]); we might be all creatures in a video game developed by a programmer. We almost are going to believe this new worldview as Elon Musk says that he feels ”We’re Probably Living in a Simulation”.

Briefly, the simulation hypothesis is the following sentence

we are living in a simulation program (a video game) which we, inhabitants, call it Reality. This simulation program is run on a computer device (similar to your PC or your laptop) which we call it universe-computer (UC).

Regarding reality as a computer simulation means that all we (human beings) do (thinking, walking, reasoning, talking, jumping, and etc) are computations and procedures performed by the ”Universe-Computer” (UC). When I think, it is UC that computes, when I prove a theorem, it is UC that computes, when I breathe, it is again UC, and etc. Physics, Biology, Logic and etc are all computational process which UC performs.

So, we may name human beings (the one who lives in) to be the computist as his/her thinking, breathing, walking and etc are all computations done by UC.

In the next section, we discuss our proposed reasoning method $\mathrm{CSMT}$. For justification of our reasoning method, we do not need the simulation hypothesis to be ”True”, we just need it to be ”Possible”. So one may do not accept the simulation hypothesis, but accepts the $\mathrm{CSMT}$ as a valid reasoning method.

2 Computer-Simulation Model Theory

Computer-Simulation Model Theory is an extension of the Model Theory [2], where the reasoner also lives inside the model and is an inhabitant of the model.

Model theory is a branch of mathematical logic that studies that mathematical-structures satisfy which formulas. Given a set of formulas $\Gamma$, and a formula $\phi$, a way to reason that $\phi$ is not logically derivable from $\Gamma$ is simply by constructing a mathematical-structure $M$, such that all formulas in $\Gamma$ holds in $M$ ($M\models\Gamma$) but $\phi$ does not hold in the world $M$ ($M\not\models\phi$). This reasoning is true, because of the soundness of deduction system 333For example, suppose that we want to show that a formula $\phi$ is not True for all graphs. We only need to construct a graph structure which does not satisfy $\phi$..

In Computer-Simulation Model Theory, instead of mathematical-structure, we deal with computer-simulations. In model theory, the reasoner (the one who thinks and reasons) is out of the mathematical-structure, while In CSMT, the reasoner (the one who thinks, computes, and …) lives and simulated inside the computer-simulation.

In a computer-simulation model, a formula $\psi$ could be provable for the creator of the computer-simulation but not provable for the computist. But we assume soundness, that is if the computist proves a formula $\phi$ then the formula is true in the computer-simulation.

The Computer-Simulation Model Theory argues that:

to show that the reasoner (the computist) cannot prove a sentence $\phi$ in the reality ($R$), construct a computer simulation $S$ such that

• (1)

the reasoner (computist) cannot distinguish $S$ from the reality $(R)$, and

• (2)

$\phi$ does not hold in $S$.

If our deduction system is sound then all we prove must hold in reality. So, if $S$ is a computer simulation that we cannot say whether we live in $S$ or in the reality, then if we can prove $\phi$ in the reality, the statement $\phi$ must also be true at $S$.

Note that our activity of proving and reasoning are computations performed by the UC of reality. This property differs CSMT from model theory.

The activity of the reasoning and computing of the computist is also performed by the simulation, and this activity can affect the model.

We are in the age of artificial intelligence, and constructing a computer simulation exactly similar to the reality is completely plausible. Using artificial intelligence, we can simulate Physics, Biology, our thinking, our reasoning and etc.

The mathematical Model theory considers the reasoner outside of the model. Model theory is not a complete and real reasoning method since in reality, we (the reasoner) are inhabitant and inside the model. In CSMT, an extension of the mathematical model theory, the process of reasoning is also considered in the model, and this process may affect other parts of the model.

2.1 Computer-Simulation Model

A computer-simulation model $S$ consists of the three following parts

• i.

The UC of $S$.

• ii.

The Computist.

• iii.

A set of instruction $INST$ that the computist (the inhabitants in $S$) interacts with UC through them. [when we walk, reason, breathe, and etc, we are asking UC to perform the computation of walking, reasoning, breathing, and etc].

All we do (walking, reasoning, and etc) are procedures performed by UC of the simulation. So, to formally describe a computer-simulation model, we just need to define

• •

what a procedure is?

• •

how the computist using the procedures interacts with the UC of the computer-simulation model?

In the next definitions 2.1, 2.2, and 2.3, we clarify these notions.

Definition 2.1

The UC of a computer-simulation model $S$ is a tuple

[TABLE]

where

• 1.

$INST$* is a nonempty set (the set of all instructions), and $INST_{0}\subseteq INST$ is a nonempty subset called the set of starting instructions.*

• 2.

$CONF$* is a nonempty set called the set of configurations such that to each $x\in\{0,1\}^{*}$,*

• –

a unique configuration $C_{0,x}\in CONF$ is associated as the start configuration, and

• –

to each $C\in CONF$, a unique string $y_{C}\in\{0,1\}^{*}$ is associated.

• 3.

The transition box*, $TBOX$, is a total function from $CONF\times INST$ to $CONF\cup\{\bot\}$444undefined symbol.. The function $TBOX$ is executable by a computer device 555We assume that the computer device is equipped and extended with memory cards as much as needed and we never face the shortage of memory, and thus Turing machines are also executable by computer devices..*

• 4.

The successful box*, $SBOX$, is a total function from $CONF$ to $\{YES,NO\}$. The function $SBOX$ is executable by a computer device.*

Note that when we say a computer-simulation model, then the simulation program of the model must be executable by a computer device and because of this, two functions $SBOX$ and $TBOX$ must be executable by a computer device.

Definition 2.2

• i.

A procedure (an algorithm) is a finite set $M\subseteq INST$ (a finite set of instructions), satisfying the following condition

• The determination condition*: for every $C\in CONF$ either for all $\iota\in M$, $TBOX(C,\iota)=\bot$, or at most there exists one instruction $\tau\in M$ such that $TBOX(C,\iota)\in CONF$.*

We refer to the set of all procedures by the symbol $\Xi$.

• ii.

We let $\upsilon:\Xi\times CONF\rightarrow INST\cup\{\bot\}$ be a total function such that for each procedure $M$ and $C\in CONF$, if $\upsilon(M,C)\in INST$ then

• –

$\upsilon(M,C)\in M$, and

• –

$TBOX(C,\upsilon(M,C))\in CONF$.

The function $\upsilon$ controls that which instruction of a procedure $M$ must apply on a given configuration $C$.

Definition 2.3

For every computer-simulation model $S$, we consider the followings to be true:

• $c1.$

Free will of Inhabitants:* The computist is free to do the following things in any order that he wants:*

• $1-$

he can freely choose an arbitrary instruction $\iota\in INST$ and an arbitrary configuration $C\in CONF$ to apply the $TBOX$ on $(C,\iota)$, and

• $2-$

he can freely choose an arbitrary configuration $C\in CONF$ to apply the $SBOX$ on.

• $c2.$

Computable Languages:* A string $x\in\Sigma^{*}$, $\Sigma=\{0,1\}$, is in the language of a procedure $M$, denoted by $L(M)$, whenever the computist can construct a sequence $C_{0}C_{1},...,C_{n}$ of configurations in $CONF$ such that*

• –

$C_{0}=C_{0,x}$,

• –

each $C_{i}$, $i\geq 1$, is obtained by applying $TBOX$ on $(C_{i-1},\upsilon(M,C_{i-1}))$,

• –

the $SBOX$ outputs $YES$ for $C_{n}$,

• –

and either $\upsilon(M,C_{n})=\bot$ or $TBOX(C_{n},\upsilon(M,C_{n}))=\bot$.

The computist calls $C_{0}C_{1},...,C_{n}$ the successful computation path of $M$ on $x$. The length of a computation path is the number of configurations appeared in.

• $c3.$

Computable Functions:* A partial function $f:\Sigma^{*}\rightarrow\Sigma^{*}$, $\Sigma=\{0,1\}$, is computed by a procedure $M\in\Xi$, whenever for $x\in\Sigma^{*}$, the computist can construct a sequence $C_{0}C_{1},...,C_{n}$ of configurations in $CONF$ such that*

• –

$C_{0}=C_{0,x}$,

• –

each $C_{i}$, $i\geq 1$, is obtained by applying $TBOX$ on $(C_{i-1},\upsilon(M,C_{i-1}))$,

• –

the $SBOX$ outputs $YES$ for $C_{n}$,

• –

and either $\upsilon(M,C_{n})=\bot$ or $TBOX(C_{n},\upsilon(M,C_{n}))=\bot$,

• –

$y_{C_{n}}=f(x)$.

• $c4.$

Parallel use of UC:* The computist may start to apply the UC on a procedure $M$ and a string $x$, however, he does not have to keep on the computation until the successful box outputs $Yes$. The computist can leave the computation $M$ on $x$ at any stage of his activity and choose freely any other procedure $M^{\prime}$ and any other string $x^{\prime}$ to apply UC on them.*

• $c5.$

Time Complexity:* The time complexity of computing a procedure $M$ on an input string $x$, denoted by $time_{M}(x)$, is $n$, for some $n\in\mathbb{N}$, whenever the computist constructs a successful computation path of the syntax-procedure $M$ on $x$ with length $n$.*

• $c6.$

Time Complexity:* Let $f:\mathbb{N}\rightarrow\mathbb{N}$ and $L\subseteq\Sigma^{*}$. The computist says that the time complexity of the computation of the language $L$ is less than $f$ whenever there exists a procedure $M\in\Xi$ such that the language defined by the computist via $M$, i.e., $L(M)$, is equal to $L$, and for all $x\in L$, $time_{M}(x)<f(|x|)$.*

• $c7.$

Complexity Classes:* The computist defines the time complexity class $\mathrm{P}\subseteq 2^{\Sigma^{*}}$ to be the set of all languages that he/she can computes in polynomial time. He/She also defines the complexity class $\mathrm{NP}\subseteq 2^{\Sigma^{*}}$ as follows:*

• $L\in\mathrm{NP}$* iff there exists $J\in\mathrm{P}$ and a polynomial function $q$ such that for all $x\in\Sigma^{*}$,*

$x\in L\Leftrightarrow\exists y\in\Sigma^{*}(|y|\leq q(|x|)\wedge(x,y)\in J)$.

• $c8.$

Turing Computability:* The UC of the simulation is sufficiently powerful such that the computist can compute all partial recursive (Turing computable) function using the UC. That is, for every Turning machine $T$, there exists a procedure $M\in\Xi$ that is $L(M)=L(T)$.*

• $c9.$

Black Box:* The computist lives inside the simulation $S$ and does not access to the structure of $TBOX$ and $SBOX$. For the computist, $TBOX$ and $SBOX$ are black boxes.*

• $c10.$

Deduction System:* There exist a procedure $G\in\Xi$ such that for every proof $z$, and every formula $\psi$, $G(\langle z,\psi^{\prime}\rangle)=1$ means $z$ is the proof of $\psi$. The computist using the procedure $G$ does his logical reasoning. Note that the procedure $G$ is also computed by the universe-computer UC.*

• $c11.$

Soundness:* If the computist can prove a sentence $\phi$, then $\phi$ is True in the simulation.*

• $c12.$

The universe-computer of the simulation, $UC$, works in linear time. Also, the computist knows that the UC works in linear time. There exists a universal clock in the simulation $S$, shown by $Clock_{S}$, such that the computist uses it and measures passing time. That is when the computist insert a configuration $C$ to $SBOX$ ($TBOX$) the number of clocks of $Clock_{S}$ that the computist waits to receive the output is linear with respect to the length of $C$.

In c10, we insist on proof procedure $G$, we could similarly talk about ”walking procedure”, ”breathing procedure” and etc, but we disregard them as we aim to construct a counter-model for $\mathrm{[P=NP]}$. As an example of a computer-simulation model, one may see the example 7.1.

2.2 Indistinguishability

In the following definitions, we formally describe ”indistinguishability” between computer-simulation models.

Definition 2.4

Suppose $S$ is a simulation and $M$ is a procedure in $S$. The experience of the computist on the procedure $M$, denoted by $EXP_{S}(M)$, is defined to be the set of all pairs $(x,y)$ where the computist ran procedure $M$ on input $x$ and received output $y$. Note that always, $EXP_{S}(M)$ is a finite set (although new pairs are always added to it) since, at each stage of time, the computist could only run the procedure on only finite inputs.

Definition 2.5

Suppose $S$ and $S^{\prime}$ are two computer-simulation model where UC is the universe-computer of $S$ and $UC^{\prime}$ is the universe-computer of $S^{\prime}$. We say that the computist $A$ who is an inhabitant in $S$ cannot say if he lives inside $S$ or $S^{\prime}$, $S\sim_{A}S^{\prime}$, whenever

• 1-

$INST_{S}$* = $INST_{S^{\prime}}$,*

• 2-

At each stage of time, the computist $A$ based on his experiences of the procedures cannot say that the universe-computer of $S$ is UC or $UC^{\prime}$.

We formally defined what we mean by a computer-simulation model and indistinguishability of models. In computer-simulation model theory, CSMT, the reasoner (the computist) is also involved in the model. To show the usefulness and importance of CSMT, we study the famous problem P vs NP. Using CSMT, in the following of this paper, we show that there exists no proof for $\mathrm{P=NP}$.

• •

We introduce two notions: non-predetermined functions and persistently evolutionary Turing machines.

• •

We construct a computer simulation world, named $E$, which $\mathrm{P\neq NP}$ in this world. The UC of $E$ is a persistently evolutionary Turing machine.

• •

We discuss that the reality is not distinguishable from a persistently evolutionary model.

3 Non-predetermined functions

The most important and fundamental notion of mathematics is function. A function is a process associating each element $x$ of a set $X$, to a single element $f(x)$ of another set $Y$. Classically, we assumed that all functions in mathematics are pre-determined.

In this section, we discuss functions that are not pre-determined and they are eventually determined through the way we start to associate $f(x)$ for every element $x\in X$.

We introduce Persistently Evolutionary Turing machines that compute non-predetermined functions.

We also discuss that if the UC of our reality persistently evolves then our reality will have alternate (alternate realities are worlds that could exist next to, in parallel of, or in place of our own, if we interacted with the world differently).

Let $f$ be a process that associates elements of a set $X$ to the elements of another set $Y$. If the process $f$ works well-defined then we know $f$ as a mathematical function. But being well-defined does not force the process $f$ to be predetermined.

Suppose that $x_{1}$ and $x_{2}$ are two different elements of $X$. I want to use the process $f$ to determine the value of $f$ for $x_{1}$ and $x_{2}$. It is up to me to first perform the process $f$ on $x_{1}$ or $x_{2}$.

If $f$ is predetermined the it does not matter to perform the process on ordering $x_{1}x_{2}$ or ordering $x_{2}x_{1}$. But if $f$ is non-predetermined then different order of inputs causes different alternate functions which one of them is the function that we are constructing.

Alternate functions are functions that could exist in place of our function (if we interacted with different ordering of inputs, those alternate could happen).

For example, consider the following process $g$:

• •

$W$* is a set which is initially empty.*

• •

for a given natural number $n$, if there exists a pair $(n,z)\in W$ then output $g(n)=z$, else update $W=W\cup\{(n,|W|+1)\}$ and output $g(n)=|W|+1$.

The function $g$ is a non-predetermined function over natural numbers. I input $7,9,1,11$ and the process will associates the following: $g(7)=1$, $g(9)=2$, $g(1)=3$, and $g(11)=4$. The value of other numbers are yet non-predetermined and as soon as I perform process $g$ on each number the value is determined.

The function $g$ is not predetermined. It is determined eventually, but it is always undetermined for some numbers.

• -

The function $g$ is well-defined, and associates to each input a single output.

• -

For every natural number, the function $g$ is definable.

• -

If I inputted $9,1,7,11$, I would have an alternate $g$ which would associate: $g(9)=1$, $g(1)=2$, $g(7)=3$, and $g(11)=4$.

4 Persistently Evolutionary Turing machines

Persistently Evolutionary Turing machines are an extension of the notion of Turing machines in which the structure of the machine can evolve through each computation.

A Turing machine consists of a set of states $Q$, and a table of transitions $\delta$ which both are fixed and remain unchanged forever. In Persistently Evolutionary Turing machines, we allow the set of states and the table of transitions changes through each computation.

As a Persistently Evolutionary Turing Machine $PT$ computes on an input string $x$, the machine $PT$ can add or remove some of its states and transitions, and thus after the computation on the input $x$ is completed, the sets $Q$ and $\delta$ changed.

However these changes are persistent, that is, if we already input a string $x$ and the machine outputs $y$, then whenever we again input $x$ the machine outputs the same $y$, and the changes of states and transitions does not violate well-definedness.

One may consider that we have a BOX and we set a Turing machine in the box with some rules of adding and removing of states and transitions. Then, We input strings to the BOX and for each string, the BOX outputs a single string. The machine in the BOX changes itself but the behavior of the BOX is well-defined.

Persistently Evolutionary Turing Machines computes non-predetermined functions.

In the following example, we introduce a persistently evolutionary nondeterministic finite automate [2].

Example 4.1

*(In the sequel of the paper, we will refer to the persistently evolutionary machine introduced in this example by $PT_{1}$). ***

Define $\mathrm{Evolve}:\mathrm{NFA}_{1}\times\Sigma^{*}\rightarrow\mathrm{NFA}_{1}$ as follows666$\mathrm{NFA}_{1}$ is the class of all nondeterministic finite automata $M=\langle Q,\Sigma=\{0,1\},\delta,q_{0},F\rangle$, where for each state $q\in Q$, and $a\in\Sigma$, there exists at most one transition from $q$ with label $a$.:

Let $M\in\mathrm{NFA}_{1}$, $M=\langle Q,q_{0},\Sigma=\{0,1\},\delta:Q\times\Sigma\rightarrow Q,F\subseteq Q\rangle$777F is the set of accepting states, and $x\in\Sigma^{*}$. Suppose $x=a_{0}a_{1}\cdots a_{k}$ where $a_{i}\in\Sigma$. Applying the automata $M$ on $x$, one of the three following cases may happen:

• case1.

The automata $M$ reads all $a_{0},a_{1}\cdots,a_{k}$ successfully and stops in an accepting state. In this case, the structure of the automata does not change and let $\mathrm{Evolve}(M,x)=M$.

• case2.

The automata $M$ reads all $a_{0},a_{1}\cdots,a_{k}$ successfully and stops in a state $p$ which is not an accepting state.

• –

*If the automata $M$ can transit from the state $p$ to an accepting state by reading *one ** alphabet, then let $\mathrm{Evolve}(M,x)=M$.

• –

If it cannot transit (from $p$ to an accepting state) then let $\mathrm{Evolve}(M,x)$ to be a new automata $M^{\prime}=\langle Q,q^{\prime}_{0},\Sigma=\{0,1\},\delta^{\prime}:Q^{\prime}\times\Sigma\rightarrow Q^{\prime},F^{\prime}\subseteq Q^{\prime}\rangle$, where $Q^{\prime}=Q$, $\delta^{\prime}=\delta$, $F^{\prime}=F\cup\{p\}$.

• case3.

The automata $M$ cannot read all $a_{0},a_{1}\cdots,a_{k}$ successfully,and after reading a part of $x$, say $a_{0}a_{1}\cdots a_{i}$, $0\leq i\leq k$, it crashes in a state $q$ that $\delta(q,a_{i+1})$ is not defined. In this case, we let $\mathrm{Evolve}(M,x)$ be a new automata $M^{\prime}=\langle Q,q^{\prime}_{0},\Sigma=\{0,1\},\delta^{\prime}:Q^{\prime}\times\Sigma\rightarrow Q^{\prime},F^{\prime}\subseteq Q^{\prime}\rangle$, where $Q^{\prime}=Q\cup\{s_{i+1},s_{i+2},\cdots,s_{k}\}$ (all $s_{i+1},s_{i+2},\cdots,s_{k}$ are new states that does not belong to $Q$), $\delta^{\prime}=\delta\cup\{(q,a_{i+1},s_{i+1}),(s_{i+1},a_{i+2},s_{i+2}),\cdots,(s_{k-1},a_{k},s_{k})\}$, and $F^{\prime}=F\cup\{s_{k}\}$.

The machine $PT_{1}$ persistently evolve, that is, if it (rejected) accepted a string $x$ already, then it would (reject) accept the string $x$ for any future trials as well. The language $L(M)$ is not predetermined and it eventually is determined.

For example, assume that initially $M$ is $Q=\{q_{0}\}$, $F=\emptyset$, $\delta=\emptyset$. Now I input the string $101$ and according to case 3, the machine $M$ evolves and new states $q_{1},q_{2},q_{3}$ and transitions $(q_{0},1,q_{1}),(q_{1},0,q_{2}),(q_{2},1,q_{3})$ are added and also $F=F\cup\{q_{3}\}$. Now if I input the string $10$ then according to case 2, $M$ rejects it. However, If at first I inputted $10$ to the machine then it would accept it.

4.1 Time complexity of Evolutionary Turing machines

The time-complexity [1] of Persistent Evolutionary Turing Machines is defined similar to the time-complexity of Turing machines except that for each (adding) removing of states or transitions, we count one extra clock.

Proposition 4.2

The time complexity of the machine $PT_{1}$ in example 4.1 is linear.

Proof. It is straightforward. $\dashv$

4.2 Executable By Computer Devices

Persistently Evolutionary Turing machines (similar to Turing machines) are executable by computer devices if as soon as the device needed extra memory resources, we are ready to add memory cards to the motherboard of the device.

Two simulations $V$ and $E$ introduced in example 7.1, and in definition 7.4 are computer simulations. The $SBOX$ and $TBOX$ of the $V$ are Turing computable and the $SBOX$ and $TBOX$ od $E$ are Persistently Evolutionary Turing computable.

5 Alternate Reality

Alternate realities are worlds that could exist next to, in parallel of, or in place of our own, if we interacted with the world differently.

If the UC of our reality persistently evolves then we have alternate realities. If this morning, I first had breakfast and then washed my hands, I would be in another alternate reality, but since I first washed my hands and then had breakfast, I am in the alternate world which we call it ”reality”.

Although there are lots of alternate realities, only one of them is actualized. UC evolves as human beings (the one who lives in this simulation) interact with it. The future is not predetermined and via our interactions, we always are moving to new alternate realities.