# What is automata theory?

by Stephen M. Walker II, Co-Founder / CEO

## What is automata theory?

Automata theory is a theoretical branch of computer science and mathematics that studies abstract mathematical machines, known as automata. These machines, when given a finite set of inputs, automatically perform tasks by going through a finite sequence of states. Automata theory is closely related to formal language theory, as both fields deal with the description and classification of formal languages.

Automata are often classified by the class of formal languages they can recognize, as in the Chomsky hierarchy, which describes a nesting relationship between major classes of automata. Automata play a significant role in various areas such as the theory of computation, compiler construction, artificial intelligence, parsing, and formal verification.

There are several types of automata, including finite automata, pushdown automata, linear-bounded automata, and Turing machines. Each type of automaton has its own unique set of features and capabilities, and each can be used to model different types of computational processes.

Finite automata are the simplest type of automata and are used for tasks such as designing the lexical analysis of a compiler, recognizing patterns using regular expressions, and implementing text editors and spell checkers. Pushdown automata are used for designing the parsing phase of a compiler (Syntax Analysis), implementing stack applications, and evaluating arithmetic expressions. Linear-bounded automata are used for implementing genetic programming and recognizing context-sensitive languages. Turing machines, the most general and powerful automata, are used for solving recursively enumerable problems and understanding complexity theory.

Automata theory has wide-ranging applications beyond computer science. For instance, it's used in linguistics as the basis for the theory of formal languages. It's also used in modeling and verification of software, distributed systems, real-time systems, or structured data. Furthermore, it's applied in the analysis and manipulation of actual human language as well as the development of human-computer interactions.

## What is an automaton?

An automaton, in the context of automata theory, is an abstract self-propelled computing device that follows a predetermined sequence of operations. It's a theoretical model of machines that perform computations on an input by moving through a series of states. At each state of the computation, a transition function determines the next configuration based on a finite portion of the present state. Once the computation reaches an accepting configuration, it accepts that input.

Automata are used as finite representations of formal languages that may be infinite. They are often classified by the class of formal languages they can recognize, as in the Chomsky hierarchy, which describes a nesting relationship between major classes of automata. Automata play a major role in the theory of computation, compiler construction, artificial intelligence, parsing, and formal verification.

There are several types of automata, including finite state machines, pushdown automata, and Turing machines. Finite state machines are the simplest, while Turing machines are the most complex and powerful.

Automata theory is a branch of theoretical computer science and mathematics. It helps computer scientists understand how machines compute functions and solve problems, and what it means for a function to be defined as computable or for a question to be described as decidable.

## What are the different types of automata?

Automata theory is a branch of computer science that deals with abstract self-operating computing devices, known as automata, which follow a predetermined sequence of operations automatically. There are four major types of automata:

1. Finite-state machine (FSM) — This is the simplest type of automaton. It is a mathematical model of computation that can be in exactly one of a finite number of states at any given time. The FSM can change from one state to another in response to some inputs; the change from one state to another is called a transition. Finite-state machines are widely used in modeling application behavior, design of hardware digital systems, software engineering, compilers, network protocols, and computational linguistics.

2. Pushdown automata (PDA) — This type of automaton employs a stack, allowing it to remember an infinite amount of information. It reads a given input string from left to right, choosing a transition by indexing a table by input symbol, current state, and the symbol at the top of the stack. PDAs are more capable than finite-state machines but less capable than Turing machines. They are often used in theories about what can be computed by machines and in parser design.

3. Linear-bounded automata (LBA) — These are a type of Turing machine where the amount of working memory is linearly bounded by the length of the input. They are more complex than pushdown automata and are used to recognize context-sensitive languages.

4. Turing machine — This is the most general and powerful automaton. It is a mathematical model of computation that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algorithm. Turing machines allow us to make statements about algorithms which will theoretically hold forever regardless of advances in conventional computing machine architecture.

Each type of automaton is used to recognize and generate different classes of languages: finite automata generate regular languages, pushdown automata recognize context-free languages, linear-bounded automata recognize context-sensitive languages, and Turing machines recognize recursively enumerable languages.

## What are the properties of automata?

Automata theory studies properties of various types of automata, which are abstract models of machines that perform computations on an input by moving through a series of states. The properties of automata can be broadly categorized into the following:

1. Inputs and Outputs — Automata process sequences of symbols selected from a finite set of input signals. Every automaton defines a language, i.e., a set of strings it accepts.

2. States and Transitions — Automata have a finite set of states and rules for moving from one state to another, based on the input. This is often referred to as the transition function. The initial state is where any input is processed, and there are also final states that signify the acceptance of an input string.

3. Closure Properties — Automata theory studies whether certain automata are closed under operations such as union, intersection, or complementation of formal languages.

4. Language Hierarchy — The theory also investigates how expressive a type of automata is in terms of recognizing a class of formal languages, and their relative expressive power.

5. Algorithmic Properties — Automata theory also studies the existence or nonexistence of any effective algorithms to solve problems such as emptiness checking (does an automaton accept at least one input word?), determinization (is it possible to transform a given non-deterministic automaton into a deterministic automaton?), and minimization (for a given formal language, what is the smallest automaton that recognizes it?).

6. Deterministic Nature — A quintessential property of a Finite Automata is its deterministic nature. That is, given a certain state and input, it clearly defines the next state.

These properties help computer scientists understand how machines compute functions and solve problems, and what it means for a function to be defined as computable or for a question to be described as decidable.

## More terms

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### What is approximation error?

Approximation error refers to the difference between an approximate value or solution and its exact counterpart. In mathematical and computational contexts, this often arises when we use an estimate or an algorithm to find a numerical solution instead of an analytical one. The accuracy of the approximation depends on factors like the complexity of the problem at hand, the quality of the method used, and the presence of any inherent limitations or constraints in the chosen approach.