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# Automata theory

## What is automata theory?

Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them.

It is an exciting, theoretical branch of computer science. It established its roots during the 20th Century, as mathematicians began developing - both theoretically and literally - machines that imitated certain features of man, completing calculations more quickly and reliably. The word automaton itself, closely related to the word "automation", denotes automatic processes carrying out the production of specific processes. Simply stated, automata theory deals with the logic of computation with respect to simple machines, referred to as automata. Through automata, computer scientists are able to understand how machines compute functions and solve problems and more importantly, what it means for a function to be defined as computable or for a question to be described as decidable.

Automatons are abstract models of machines that perform computations on input by moving through a series of states or configurations. At each stage of the computation, a transition function determines the next configuration on the basis of a finite portion of the present configuration. As a result, once the computation reaches an accepting configuration, it accepts that input. The most general and powerful automata is the Turing machine.

## Why is automata theory important?

The major objective of automata theory is to develop methods by which computer scientists can describe and analyze the dynamic behavior of discrete systems, in which signals are sampled periodically. The behavior of these discrete systems is determined by the way that the system is constructed from storage and combinational elements. Characteristics of such machines include:

• Inputs: assumed to be sequences of symbols selected from a finite set I of input signals. Namely, set I is the set {x1, x,2, x3... xk} where k is the number of inputs.
• Outputs: sequences of symbols selected from a finite set Z. Namely, set Z is the set {y1, y2, y3 ... ym} where m is the number of outputs.
• States: finite set Q, whose definition depends on the type of automaton.

There are four major families of automaton:

• Finite-state machine
• Pushdown automata
• Linear-bounded automata
• Turing machine

The families of automata above can be interpreted in a hierarchal form, where the finite-state machine is the simplest automata and the Turing machine is the most complex. The focus of this project is on the finite-state machine and the Turing machine. A Turing machine is a finite-state machine yet the inverse is not true.

## Applications of automata theory

### 1. Language

Automata theory is the basis for the theory of formal languages. A proper treatment of formal language theory begins with some basic definitions:

• A symbol is simply a character, an abstraction that is meaningless by itself.
• An alphabet is a finite set of symbols.
• A word is a finite string of symbols from a given alphabet.
• Finally, a language is a set of words formed from a given alphabet.

The set of words that form a language is usually infinite, although it may be finite or empty as well. Formal languages are treated like mathematical sets, so they can undergo standard set theory operations such as union and intersection. Additionally, operating on languages always produces a language. As sets, they are defined and classified using techniques of automata theory.

Formal languages are normally defined in one of three ways, all of which can be described by automata theory:

• regular expressions
• standard automata
• a formal grammar system

### 2. Biology

To the casual observer, biology is an impossibly complex science. Traditionally, the intricacy and variation found in life science have been attributed to the notion of natural selection. Species become "intentionally" complex because it increases their chance for survival. For example, a camouflage-patterned toad will have a far lower risk of being eaten by a python than a frog colored entirely in orange. This idea makes sense, but automata theory offers a simpler and more logical explanation, one that relies not on random, optimizing mutations but on a simple set of rules.

Basic automata theory shows that simplicity can naturally generate complexity.

### 3. Other Applications

Many other branches of science also involve unbelievable levels of complexity, impossibly large degrees of variation, and apparently random processes, so it makes sense that automata theory can contribute to a better scientific understanding of these areas as well. The modern-day pioneer of cellular automata applications is Stephen Wolfram, who argues that the entire universe might eventually be describable as a machine with finite sets of states and rules and a single initial condition. He relates automata theory to a wide variety of scientific pursuits, including:

• Fluid Flow
• Snowflake and crystal formation
• Chaos theory
• Cosmology
• Financial analysis

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# Automata theory

October 14, 2020

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## What is automata theory?

Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them.

It is an exciting, theoretical branch of computer science. It established its roots during the 20th Century, as mathematicians began developing - both theoretically and literally - machines that imitated certain features of man, completing calculations more quickly and reliably. The word automaton itself, closely related to the word "automation", denotes automatic processes carrying out the production of specific processes. Simply stated, automata theory deals with the logic of computation with respect to simple machines, referred to as automata. Through automata, computer scientists are able to understand how machines compute functions and solve problems and more importantly, what it means for a function to be defined as computable or for a question to be described as decidable.

Automatons are abstract models of machines that perform computations on input by moving through a series of states or configurations. At each stage of the computation, a transition function determines the next configuration on the basis of a finite portion of the present configuration. As a result, once the computation reaches an accepting configuration, it accepts that input. The most general and powerful automata is the Turing machine.

## Why is automata theory important?

The major objective of automata theory is to develop methods by which computer scientists can describe and analyze the dynamic behavior of discrete systems, in which signals are sampled periodically. The behavior of these discrete systems is determined by the way that the system is constructed from storage and combinational elements. Characteristics of such machines include:

• Inputs: assumed to be sequences of symbols selected from a finite set I of input signals. Namely, set I is the set {x1, x,2, x3... xk} where k is the number of inputs.
• Outputs: sequences of symbols selected from a finite set Z. Namely, set Z is the set {y1, y2, y3 ... ym} where m is the number of outputs.
• States: finite set Q, whose definition depends on the type of automaton.

There are four major families of automaton:

• Finite-state machine
• Pushdown automata
• Linear-bounded automata
• Turing machine

The families of automata above can be interpreted in a hierarchal form, where the finite-state machine is the simplest automata and the Turing machine is the most complex. The focus of this project is on the finite-state machine and the Turing machine. A Turing machine is a finite-state machine yet the inverse is not true.

## Applications of automata theory

### 1. Language

Automata theory is the basis for the theory of formal languages. A proper treatment of formal language theory begins with some basic definitions:

• A symbol is simply a character, an abstraction that is meaningless by itself.
• An alphabet is a finite set of symbols.
• A word is a finite string of symbols from a given alphabet.
• Finally, a language is a set of words formed from a given alphabet.

The set of words that form a language is usually infinite, although it may be finite or empty as well. Formal languages are treated like mathematical sets, so they can undergo standard set theory operations such as union and intersection. Additionally, operating on languages always produces a language. As sets, they are defined and classified using techniques of automata theory.

Formal languages are normally defined in one of three ways, all of which can be described by automata theory:

• regular expressions
• standard automata
• a formal grammar system

### 2. Biology

To the casual observer, biology is an impossibly complex science. Traditionally, the intricacy and variation found in life science have been attributed to the notion of natural selection. Species become "intentionally" complex because it increases their chance for survival. For example, a camouflage-patterned toad will have a far lower risk of being eaten by a python than a frog colored entirely in orange. This idea makes sense, but automata theory offers a simpler and more logical explanation, one that relies not on random, optimizing mutations but on a simple set of rules.

Basic automata theory shows that simplicity can naturally generate complexity.

### 3. Other Applications

Many other branches of science also involve unbelievable levels of complexity, impossibly large degrees of variation, and apparently random processes, so it makes sense that automata theory can contribute to a better scientific understanding of these areas as well. The modern-day pioneer of cellular automata applications is Stephen Wolfram, who argues that the entire universe might eventually be describable as a machine with finite sets of states and rules and a single initial condition. He relates automata theory to a wide variety of scientific pursuits, including:

• Fluid Flow
• Snowflake and crystal formation
• Chaos theory
• Cosmology
• Financial analysis