As You Know, The World Of Artificial Intelligence And Its Subfields Such As Machine Learning, Deep Learning, Data Mining, Natural Language Processing, Image Processing, Etc. Are All Based On Mathematical Principles?
Artificial Intelligence, you may attend a training course and learn about machine learning and artificial intelligence. Still, when you enter the real world, you will find that most problems are solved based on mathematical and statistical principles, and algorithms are based on principles.
In this regard, there are important theories used directly and indirectly in the world of artificial intelligence. In this article, we will get acquainted with some important cases of these theories.
Chaos Theory
The common use of the term chaos means a state of disorder. In chaos theory, however, the term has a more precise definition. Although there is no exact mathematical definition for turbulence, the most commonly used definition is given by Robert L. Diwani formulated.
“For a dynamic system to be classified as chaotic, it must have one of three properties that must be sensitive to the initial conditions, must be topologically transient, or must have dense alternating circuits,” the definition says.
Based on this definition, we must say that Chaos Theory is a branch of mathematics that focuses on the study of dynamic, chaotic systems, systems that seem to be random and unordered, but in practice, are governed by hidden definite patterns and rules that are Severely sensitive to initial conditions.
Chaos theory is an interdisciplinary branch that states:
Despite their apparent randomness, there are patterns, interconnections, feedback loops, repetition, self-similarity, fractals, and self-organization in complex systems.
The propeller effect is the underlying principle of chaos theory. It describes the phenomenon of how small changes in a definite and nonlinear system can lead to large differences in one of the following states (i.e., sensitive dependence on the initial conditions). That flies in Texas is capable of creating a storm in China.
Small changes in the initial conditions, such as changes due to measurement errors due to rounding of numbers in calculations, can cause large divergences of the outputs of such systems, making a long-term prediction of their behavior in general impossible.
This can happen even though the systems are down.
Definitive means that their future behavior follows a unique evolutionary path and is completely determined by the initial conditions, and no random elements are involved in determining it. In other words, the definite nature of these systems does not make them predictable. This type of behavior is called definite chaos or just chaos.
The theory was summed up by Edward Lorentz as follows: “Chaos: When the present determines the future, but the approximate present cannot approximate the future.”
Fractal
Fractals are geometrically defined as having three properties: self-similar, complexity at the micro-scale, and not an integer (for example, 2). Fractal is a geometric structure obtained by enlarging each part of this structure in a certain proportion, the same original structure. In other words, a fractal is a structure in which every part of it is the same as a hat.
The fractal is seen from the same distance and close. This feature is called self-similarity. Fractals are one of the most important tools in computer graphics and can be used in many ways.
Automata theory
An automated machine runs on several inputs from a sequence or string in discrete time steps. At each time stage, the machine receives an input taken from a set of symbols or characters, called the Alphabet. A machine contains a finite set of modes. It can be in one or more of its modes at any time, depending on the machine.
At each step, when the machine reads a symbol, it jumps to or fro according to the current state and the symbol being read. This function is called the current state and the input symbol of the transmitter function.
The machine reads the input symbol by symbol in sequence until a complete input is read and switches from state to state based on the transition function. When the final input is read, the machine stops, which is called the final state. Based on the final state, it is said that the machine has accepted or rejected an input.
There is a subset of machine modes that is defined as an acceptable set of modes.
If the final state is acceptable, the input machine is accepted. Otherwise, the input is rejected. The set of inputs accepted by the machine is called machine-recognizable language. Automata theory, or machine theory, is the mathematical study of abstract computing machines and their ability to solve problems.
These abstract machines are called automata. This theory is very close to formal language theory as officially recognizable languages often categorize them.
Automation plays a key role in compiler design and parsing. The languages examined by these machines are formal. If the final state is acceptable, the input machine is accepted.
Otherwise, the input is rejected.
The set of inputs accepted by the machine is called machine-recognizable language. Automata theory, or machine theory, is the mathematical study of abstract computing machines and their ability to solve problems.
These abstract machines are called automata. This theory is very close to the theory of formal language.
As they are often categorized by officially recognizable languages. Automation plays a key role in compiler design and parsing. The languages examined by these machines are formal. If the final state is acceptable, the input machine is accepted.
Otherwise, the input is rejected. The set of inputs accepted by the machine is called machine-recognizable language. Automata theory or machine theory is the mathematical study of abstract computing machines and their ability to solve problems.
These abstract machines are called automata.
This theory is very close to formal language theory as recognizable official language categories often categorize them. Automation plays a key role in compiler design and parsing.
The languages examined by these machines are formal. Automata theory or machine theory is the mathematical study of abstract computing machines and their ability to solve problems. These abstract machines are called automata. This theory is very close to the theory of formal language.
As they are often categorized by recognizable official language categories. Automation plays a key role in compiler design and parsing. The languages examined by these machines are formal.
Automata theory, or machine theory, is the mathematical study of abstract computing machines and their ability to solve problems.
These abstract machines are called automata.
This theory is very close to formal language theory as officially recognizable languages often categorize them. Automation plays a key role in compiler design and parsing. The languages examined by these machines are formal.
Information theory in Artificial Intelligence
Information theory deals with the quantification, storage, and transmission of information. This theory provides a mathematical model of the conditions and factors affecting the processing and transmission of information (data).
This theory focuses on the fundamental limitations of information transmission and processing and less on how information transfer and processing methods work and are implemented.
The theory was developed in 1948 by Claude Shannon. The entropy of a random variable is obtained by measuring the probabilities of that random variable.
The concept of entropy has changed over the decades and has led to important applications in other disciplines, including data compression, neuroscience, telecommunications engineering, and channel coding.
Information theory is used specifically by telecommunications engineers, although some of its concepts are also used in other disciplines such as psychology, linguistics, librarianship and information science, and cognitive sciences.
Computational number theory
The study of algorithms for implementing computations related to number theory in mathematics and computer science is called computational number theory or algorithmic number theory.
Social choice theory in Artificial Intelligence
Social choice theory is about how the individual and society relate. In particular, the realm of social choice can be considered as the study of the possibility of aggregating interests, preferences, judgments, and individual welfare status to achieve social interests, preferences, and well-being.
Social choice theory is a theoretical framework for measuring tastes, values, and well-being as a collective decision. The theory of social choice began with the formulation of the Condorcet voting paradox.
Game theory
Game theory uses mathematical models to analyze the methods of cooperation or competition of rational and intelligent beings. At the same time, Game theory is a branch of applied mathematics used in the social sciences, especially in economics, biology, engineering, political science, international relations, computer science, marketing, philosophy, and gambling.
This theory attempts to use mathematics to estimate behavior in strategic situations or in a game in which one’s success depends on the choices of others.
It attempts to model the mathematical behavior of a strategic situation (conflict of interest). This situation arises when a person’s success depends on the strategies that others choose.
The ultimate goal of this knowledge is to find the optimal strategy for the players. Initially, game theory was equivalent to a zero-sum game. One participant’s profit (or loss) is equal to the other participants’ losses (or profits), and players gain what another player has lost.
Today, game theory is a mother word for sciences that analyze the logical interaction of humans, animals, and computers.
Approximation theory
In mathematics, approximation theory follows how a function is best approximated to a similar function.
String Theory
String theory provides a theoretical framework in which particle physics particle points are replaced by one-dimensional objects called strings. This theory describes how strings are propagated in space and interact with each other.
At scales larger than the dimensions of the strings, the strings are like point particles whose mass, the vibrational state of each string determines charge and other properties.
In string theory, it is one of several vibrational states corresponding to graviton; A particle in quantum mechanics that carries the force of gravity. Therefore, string theory is also a kind of quantum gravitational theory.
Theory of operators in Artificial Intelligence
In mathematics, the study and operation of operators theory is a central part of functional analysis. Linear operators make up a large part of the theory of operators. The theory of linear operators is divided into two main parts.
The first part is based solely on the algebraic properties of linear operators, and the second part is expressed based on their operating properties.
Coding theory
Coding theory (not to be confused with cryptography or cryptography) examines information coding methods and is one of the most important topics in various fields of science (such as information theory, electrical engineering, mathematics, and computer science); In this way, it is possible to design reliable methods for data transfer so that unnecessary duplication, elimination, and errors are reduced.
Control theory
Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of input dynamic systems. The input applied to a system is called a command or reference.
When one or more system outputs follow a specific reference over time, a controller (compensator added to the primary system) manipulates the system input to make appropriate changes to the system output and the system behavior. Optimal user behavior gets closer and closer.
Usually, control theory aims to find appropriate answers to implement the optimal compensation of the system behavior by the controller so that the stability of the system and the quietness of its output or outputs around a working point and no fluctuation of outputs around this point.
Most of the time, a set of differential equations defines the relationship between the inputs and outputs.
If these equations are linear differential equations with constant coefficients, a conversion function that describes the relationship between system inputs and outputs can obtain by calculating their Laplace transform.
If the sets of differential equations are nonlinear but have a definite answer, the system conversion function can obtain by linearizing them around a working point and recalculating the Laplace transform.
The conversion function, also called the system function or network function is a mathematical description of the relationship between the input and output of a linear solution that does not change the time of the set of differential equations representing a system.
One way to express and understand a control system is to display it using a block diagram in which the relationship between inputs and outputs and conversion functions is expressed visually.
If the sets of differential equations are nonlinear but have a definite answer, the system conversion function can obtain by linearizing them around a working point and recalculating the Laplace transform.
The conversion function also called the system function or network function, is a mathematical description of the relationship between the input and output of a linear solution that does not change with the time of the set of differential equations representing a system.
One way to express and understand a control system is to display it using a block diagram in which the relationship between inputs and outputs and conversion functions is expressed visually.
If the sets of differential equations are nonlinear but have a definite answer, the system conversion function can obtain by linearizing them around a working point and recalculating the Laplace transform.
The conversion function also called the system function or network function, is a mathematical description of the relationship between the input and output of a linear solution that does not change with the time of the set of differential equations representing a system.
One way to express and understand a control system is to display it using a block diagram in which the relationship between inputs and outputs and conversion functions is expressed visually.
Linear control theory in Artificial Intelligence
Control theory is divided into the following two branches:
Linear control theory: This branch deals with systems made of devices that follow the principle of superposition. That is, the output is approximately proportional to the input.
They are a function of linear differential equations. A major subclass of systems with parameters that do not change over time is called linear time (LTI) systems.
These systems are acceptable mathematical techniques with high-frequency amplitude with high popularity, such as Laplace transform, Fourier transform, Z conversion, Bode diagram, root source, and Nyquist stability criterion.
Nonlinear control theory: This includes a broader class of systems that do not follow the principle of superposition and apply to more realistic systems because all real control systems are nonlinear. Nonlinear differential equations often govern these systems.
Graph Theory in Artificial Intelligence
Graph theory is a branch of mathematics that discusses graphs. This topic is, in fact, a branch of topology that is strongly related to algebra and matrix theory.
Unlike other branches of mathematics, graph theory has a definite starting point: publishing a paper by the Swiss mathematician Leonard Euler to solve the Koenigsberg Stairs problem in 1736.
Recent advances in mathematics, especially in its applications, have led to the dramatic expansion of graph theory, so that graph theory is now a handy tool for research in various fields such as coding theory, operations research, statistics, electrical networks, computer science, chemistry, biology, Social sciences, and other fields.
Engineering graph theory
A geometric graph is a graph in which vertices and edges are associated with geometric objects. The simplest example is a random geometric graph.
Node theory in Artificial Intelligence
The study of mathematical nodes in topology is called node theory. While this concept is inspired by the knots made in the shoelace or rope in everyday life, the mathematical knots are different and do not untie.
In mathematical terms, a node is a sleeping circle in Euclidean Heravi’s three-dimensional space that has always been deformed.