**Automata theory** is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science and discrete mathematics a subject of study in both mathematics and computer science. The word *automata* the plural of *automaton* comes from the Greek word αὐτόματα, which means "self-making".

The figure at right illustrates a finite-state machine, which belongs to a well-known type of automaton. This automaton consists of states represented in the figure by circles and transitions represented by arrows. As the automaton sees a symbol of input, it makes a transition or jump to another state, according to its transition function, which takes the current state and the recent symbol as its inputs.

Automata theory is closely related to formal language theory. An automaton is a finite representation of a formal language that may be an infinite set. Automata are often classified by the class of formal languages they can recognize, typically illustrated by the Chomsky hierarchy, which describes the relations between various languages and kinds of formalized logics.

Automata play a major role in theory of computation, compiler construction, artificial intelligence, parsing and formal verification.

Following is an introductory definition of one type of automaton, which attempts to help one grasp the essential concepts involved in automata theory/theories.

An automaton is a construct made of *states* designed to determine if the input should be accepted or rejected. It looks a lot like a basic board game where each space on the board represents a state. Each state has information about what to do when an input is received by the machine again, rather like what to do when you land on the *Jail* spot in a popular board game. As the machine receives a new input, it looks at the state and picks a new spot based on the information on what to do when it receives that input at that state. When there are no more inputs, the automaton stops and the space it is on when it completes determines whether the automaton accepts or rejects that particular set of inputs.

An automaton *runs* when it is given some sequence of *inputs* in discrete individual *time steps* or steps. An automaton processes one input picked from a set of *symbols* or *letters*, which is called an *alphabet*. The symbols received by the automaton as input at any step are a finite sequence of symbols called *words*. An automaton has a finite set of *states*. At each moment during a run of the automaton, the automaton is *in* one of its states. When the automaton receives new input it moves to another state or transitions based on a function that takes the current state and symbol as parameters. This function is called the *transition function*. The automaton reads the symbols of the input word one after another and transitions from state to state according to the transition function until the word is read completely. Once the input word has been read, the automaton is said to have stopped. The state at which the automaton stops is called the final state. Depending on the final state, it's said that the automaton either *accepts* or *rejects* an input word. There is a subset of states of the automaton, which is defined as the set of *accepting states*. If the final state is an accepting state, then the automaton *accepts* the word. Otherwise, the word is *rejected*. The set of all the words accepted by an automaton is called the *language recognized by the automaton*.

In short, an automaton is a ], automata theory plays a crucial role in computational theory.

Automata are defined to study useful machines under mathematical formalism. So, the definition of an automaton is open to variations according to the "real world machine", which we want to model using the automaton. People have studied many variations of automata. The most standard variant, which is described above, is called a deterministic finite automaton. The following are some popular variations in the definition of different components of automata.

Different combinations of the above variations produce many classes of automaton.

Automata theory is a subject matter that studies properties of various types of automata. For example, the following questions are studied about a given type of automata.

Automata theory also studies the existence or nonexistence of any effective algorithms to solve problems similar to the following list:

The following is an incomplete list of types of automata.

Normally automata theory describes the states of abstract machines but there are continuous automata or hybrid discrete-continuous automata, which use analog data, continuous time, or both.

The following is an incomplete hierarchy in terms of powers of different types of virtual machines. The hierarchy reflects the nested categories of languages the machines are able to accept.

same power same power Nondeterministic Finite Automaton NFA above is weaker below is stronger Deterministic Push Down Automaton DPDA-I with 1 push-down store Nondeterministic Push Down Automaton NPDA-I with 1 push-down store Linear Bounded Automaton LBA Deterministic Push Down Automaton DPDA-II with 2 push-down stores Nondeterministic Push Down Automaton NPDA-II with 2 push-down stores Deterministic Turing Machine DTM Nondeterministic Turing Machine NTM Probabilistic Turing Machine PTM Multitape Turing Machine MTM Multidimensional Turing Machine

Each model in automata theory plays important roles in several applied areas. Finite automata are used in text processing, compilers, and hardware design. Context-free grammar CFGs are used in programming languages and artificial intelligence. Originally, CFGs were used in the study of the human languages. Cellular automata are used in the field of biology, the most common example being John Conway's Game of Life. Some other examples which could be explained using automata theory in biology include mollusk and pine cones growth and pigmentation patterns. Going further, a theory suggesting that the whole universe is computed by some sort of a discrete automaton, is advocated by some scientists. The idea originated in the work of Konrad Zuse, and was popularized in America by Edward Fredkin. Automata also appear in the theory of finite fields: the set of irreducible polynomials which can be written as composition of degree two polynomials is in fact a regular language.

Automata simulators are pedagogical tools used to teach, learn and research automata theory. An automata simulator takes as input the description of an automaton and then simulates its working for an arbitrary input string. The description of the automaton can be entered in several ways. An automaton can be defined in a symbolic language or its specification may be entered in a predesigned form or its transition diagram may be drawn by clicking and dragging the mouse. Well known automata simulators include Turing's World, JFLAP, VAS, TAGS and SimStudio.

One can define several distinct sequential machines or *sequential automata*, and Turing machines with automata homomorphisms defining the arrows between automata is a Cartesian closed category, it has both categorical limits and colimits. An automata homomorphism maps a quintuple of an automaton *A*_{i} onto the quintuple of another automaton
* A*_{j}. Automata homomorphisms can also be considered as *automata transformations* or as semigroup homomorphisms, when the state space, * S*, of the automaton is defined as a semigroup

One could also define a *variable automaton*, in the sense of Norbert Wiener in his book on *The Human Use of Human Beings* *via* the endomorphisms . Then, one can show that such variable automata homomorphisms form a mathematical group. In the case of non-deterministic, or other complex kinds of automata, the latter set of endomorphisms may become, however, a *variable automaton categories of groupoids or groupoid categories. Moreover, the category of reversible automata is then a
2-category, and also a subcategory of the 2-category of groupoids, or the groupoid category.*

The automata theory was developed in the mid-20th century in connection with finite automata.