Theory of Systems

By Luis Felipe Gomes da Silva

Foreword

All my life I have sought the unification of certain scientific and philosophical principles of my thought. My science, or knowledge, is an association between intuition, analogy, and analysis.

For me, science, or the act of conceiving and knowing through concepts of the understanding, is a path without end. It does not finish in this account. It does not go very far, other than small truths, for the major Truth is the inspiration.

Kurt Gödel, the logician, produced an astonishing proof, from whose consequences we cannot escape. There are no complete systems capable of containing all truths, all propositions, derived from its principles (axioms). There are hidden truths within systems that will remain without a formal proof. Consequently, this small work here is just in its principles and does not represent the totality (nor the great Truth) that we can represent as valid, subject to the mathematical analysis from which my training emerged and from which I can never stray.

The main proposition of this account is the connection of three systems...

Part 1: Definition of Elements

Chapter 1: Introduction

1. I begin with the system $\langle \mathcal{F}, :, +, -, = \rangle$

$$ \begin{array}{|c|c|c|} \hline - & 0 & 1 \\ \hline 0 & 1 & 0 \\ \hline \end{array} $$ $$ \begin{array}{|c|c|c|} \hline : & 0 & 1 \\ \hline 0 & 1 & 1 \\ 1 & 0 & 1 \\ \hline \end{array} $$ $$ \begin{array}{|c|c|c|} \hline + & 1 & 0 \\ \hline 0 & 1 & 0 \\ 1 & 1 & 1 \\ \hline \end{array} $$

The system accepts a field and three logical operators. It remains to be shown that (1)(2)(3) form a complete axiomatic system $(1) \iff (2) \iff (3)$.

2. Next, I will demonstrate the same for the system with the pair $\langle (\mathcal{F},\mathcal{F}) , :, +, -, = \rangle$.

4. We will demonstrate that the subsystem

$\langle (\mathbb{Z_2},\mathbb{Z_2}), : \rangle, \langle (\mathbb{Z_2},\mathbb{Z_2}), + \rangle, (\mathbb{Z_2},\mathbb{Z_2}), - \rangle$, is complete according to the axiomatic system (a) $(P \implies Q)$, (b) $(P \vee \neg Q)$, (c) $\neg Q$

Part 2: Abstract Systems

Chapter 2: Sequences

§ Sequences are always present in our daily lives in such a way that we do not bring them to our reason. It is intuitively that we learn to recognize this structure. In these chapters, we will deal with multiples useful in number theory and in combinatorial analysis.

Section 2.1: Number Sequences

Number sequences are best represented by Peano's axioms. These axioms describe a sequence starting from the term 0 and using a function that returns the successors for each subsequent term.

We can define sequences in a general way using the following axioms. The axioms for a numerical sequence are given as:

  1. 0 belongs to the set of natural numbers $\mathbb{N}$ (0 $\in$ $\mathbb{N}$).
  2. For every natural number m, m + 0 = m.
  3. For all natural numbers m and n, m + n is the successor of n.
  4. For all natural numbers m, n, m', and n', if (m, n) = (m', n') and m + n = m' + n', then m = n.

Section 2.2: Cardinality and Structures

§ Structures are algebraic devices such that: [] := 0; [x] := 1; [x,[x]] = [x,1] := 2 ... [x,n]:= n+1

We can demonstrate the following properties of the structure:

2.2.1 The Identity

[x,x] = [x,[]] = [x,0] = 1

[x,y] = [x,[x]] = [x,1] = 2

$\blacksquare$

2.2.2 Symmetry

[x,y] = x + (n + 1) = (n + 1) + x = [y,x]

$\blacksquare$

2.2.3 Transitivity

[[x,y],z] = [[x,z],[y,z]] $\implies$ [x,z].

Section 2.3: Recurrent and Recursive Structures

Recurrence relations will allow the generation of finite and infinite sequences through a pattern (schema) of repeating a rule in the form of an algebraic identity.

The following relation describes a recurrence relation.

Definition 1:

  1. $\lambda_0 = \emptyset$ the empty relation.
  2. $[ ] =_{def} 0$
  3. $[n]^0 =_{def} 1$
  4. $[n] = n-1$

We can derive the following theorem.

Theorem 1:

Prove the identity $[n]^r = n(n-1)!$.

Be careful not to confuse the concept of recursion with recurrence. Whereas in recursion, the function is defined under a simplified "version" of itself. Recursion is useful when the main problem can be divided into sub-problems that contain the same resolution structure as the problem.

Among the abstract structures, there is one that lies at the midpoint between natural systems and artificial systems called recursion. Recursion is a structure formed by an automorphism where a function f is defined by induction on the partitions of a set D.

Let S(X) be a statement defined recursively on partitions $(Y_1,Y_2,...,Y_n)$ of a set X using the same proposition applied to X.

  • T
    • T1
      • left
      • right
    • T2
      • left
      • right

Chapter 3: Axiomatic Systems

In this section, we will see a type of autonomous machine, that is, an abstract axiomatized system. We start with a simple alphabet, such as the symbols of the I Ching, which we denote $\mathbb{A} = {:,|}$. The complete system can be represented by $\mu :=\langle (\mathbb{A}, =), \alpha, \gamma, \phi, S \rangle$. The pair $(\mathbb{A}, =)$ establishes the identity relation between the character strings of the alphabet. The Greek terms are transformation functions given by tables:

The transformation is equivalent to the logical operator $P \iff Q$.
():|
:|:
|:|
The transformation is equivalent to the logical operator $P \vee Q$.
<>:|
::|
|||
The transformation is equivalent to the logical operator $ \neg Q$.
[]:|
 :|
 :

Section 3.1: Randomness

§ Given initial conditions, the number of 1s and 0s in a sequence will approach a quantity close to the frequency $\dfrac{n({1s})}{N} \approx \dfrac{n({0s})}{N}$ according to the binomial distribution.

3.1.1 Binary Sequence Generators

To generate a random binary sequence of numbers, we can resort to shift registers with feedback. The shift registers are given by the expression:

$$ x_{n+1} = \sum_{n = 0}^{k} x_n \oplus y, \pmod 2 $$ 

/*
This is a linear random bit generator. It is optimized for speed, but not for quality.
To use it, include this file in your project,
and define LCG_USE_LINEAR_RANDOM_BIT_GENERATOR.
Motivation: Create a fast bit generator
Objective A: Generate a random bit sequence of #length <|n|>
Objective B: Test the generator for statistical quality.
Test: using binomial distribution. The number of 1s must be close to 0s in a large number of trials.
*/

#include <stdio.h>

int sum(char *a, int k) {
  int n = 0;
  int sum = 0;
  while (n < k) {
    sum += (a[n] - a[n] + a[n + 1]) % 2;
    n++;
  }
  return sum;
}

void createSequence(char* s, int n, int k) {
  n = 0;
  do {
    int x = sum(&s[0], 7);
    int y = x + s[7];
    for (int i = 7; i > 0; i--) {
      s[i] = s[i - 1] % 2;
    }
    s[0] = y % 2;
    n++;
  } while (n < k);

  printf("Generated sequence: ");
  for (int i = 0; i < 8; i++) {
    printf("%d ", s[i]);
  }
  printf("\n");
}

void booleanShift(char *a, int size) {
  char temp = a[size - 1];
  for (int i = size - 1; i > 0; i--) {
    a[i] = a[i - 1] % 2;
  }
  a[0] = temp;
}

int main() {
  int n = 0, k = 10;
  int number_of_sequences = 0;
  char s[8] = {0, 1, 0, 1, 0, 1, 1, 0};
  while (number_of_sequences < k) {
    printf("Original sequence: ");
    for (int i = 0; i < 8; i++) {
      printf("%d ", s[i]);
    }
    printf("\n");

    booleanShift(s, 8);
    printf("Shifted sequence: ");
    for (int i = 0; i < 8; i++) {
      printf("%d ", s[i]);
    }
    printf("\n");

    createSequence(&s[0],n, k);
    number_of_sequences++;
  }
  return 0;
}

Part 3: Natural Systems

§ Here "natural" signifies physics (φύσισ) which deals with natural phenomena concerning thermodynamics towards an abstract quantum system and its simulation in a probability space. Biological systems are brought in the following part.

Section 3.2: Random Functions and Processes

Almost any process in nature is random, that is, they are governed by chance. Random processes occur when we have conditions where the number of 1s and 0s in a binary sequence are close to 1/2, which is the fair frequency of a binomial event.

We can generate a random sequence of binary numbers using a shift register which can be generalized as:

$$ f(x_n,y) = \sum^{k}_{n = 0} x_n x_{n+1} $$

For example, if we want to calculate the series $(x^2)_n$. The recurrence is:

$$ x^2 = \sum^k_{n=0} x_n y = \dfrac{1}{x^2(1-x^3)} $$

Chapter 4: Machines

Initially, machines were invented to facilitate human tasks as an extension of their capabilities. A water wheel taking the flow of a stream to move the mechanisms of a mill. Thus, a machine exceeds a certain human capacity in power and resistance, according to the inventor's design. The machines known by the common people have no value to serve as an extension neither to the work of force, nor to the extension of their intellectual faculties. If for some reason a machine is not useful today, it is because the inventor left it incomplete, or because the system dissipates its efforts due to inefficiency in producing with the necessary and sufficient order for energy economy. The psychological effects of the relationship between the problem-man-machine-goal are not important for making autonomous mechanical systems that receive and radiate the excess energy in the world, or the misallocation, in the case of a non-autonomous system.

Section 4.1: μMachines

The μMachines are derived from μAlgebra. The following table demonstrates a type of machine that will be used.

The table describes a μ-machine.
μαγφxyzxyzφ
in 1RR-10101000
in -1SS-11(h)alt1-100(h)alt
0TT-110101-11

Part 4: Synthetic Systems

§ Systems called synthetic are systems that are at the crossroads between natural biological systems and natural physical systems. By analogy, one can well recall that there are arts like opera and ballet which are the "mixture" of two arts. Opera is the synthesis of theater and music. Similarly, synthetic systems are the bio-chemical, the physio-chemical, etc. [not very precise]

Chapter 5: Problems and Algorithms

Section 5.1: Problems

To start with problems, they are nothing but conditional systems, a system of general form replacing equality with an inequality. So we have the following:

$$ A[x_i] + Q[y_j] \leq f_{x,y} $$

In this case, it is called an optimization problem. Any optimization problem can be transformed into a decision problem by taking the system as questions about the existence of values that solve the inequality.

For example, the question could be: "Does there exist a combination of $x_i$ and $y_j$ such that $A[x_i] + Q[y_j] \leq f_{x,y}$?" This is a yes/no question, and if the answer is yes, then you have a feasible solution for the optimization problem.

Section 5.2: Algorithms

An algorithm can be described as a process composed of atomic statements that are used in solving certain problems.

Algorithms can be of two types: a) Deterministic algorithms and b) non-deterministic algorithms.

a) Given inputs $(x_1,x_2,...x_n)$, the outputs $(y_1,y_2,...y_n)$ must be the same.

b) On the contrary, non-deterministic ones do not have this necessity.

Section 5.3: The Algorithm and the Three Machines

Section 5.4: μ-Machines

μ-Machines are derived from μ-Algebra which is a kind of algebra for the description of systems.

There are constant machines, linear machines, polynomial machines, and recursive machines.

One can build an algorithm using a μ-Machine, or an association of machines forming a system.

Random machines are also possible.

$\mu_{x,y} := \langle (z_2)^2, f_{xy}, S \rangle$

The machine described above is a constant machine if $f_{xy}$ = a constant.

$\mu_{x,y} := \langle \mathbb{Z_2}\times\mathbb{Z_2}, \langle x, y \rangle, S \rangle$

Part 5: Organic Structures

§ An organism is capable of a) self-replication and b) self-organization. Property (a) is present in synthetic systems. One can see computer viruses or genetic algorithms. Property (b) is present in physical systems that work against entropy, for example, a refrigerator. We can say that artificial organisms are, in a certain way, defined by two structures: 1) the recurrent and 2) the recursive.

Appendices

References