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 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-acting".

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 logic.

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 supposed to run on some given sequence of inputs in discrete time steps. An automaton gets one input every time step that is picked up from a set of symbols or letters, which is called an alphabet. At any time, the symbols so far fed to the automaton as input, form a finite sequence of symbols, which finite sequences are called words. An automaton contains a finite set of states. At each instance in time of some run, the automaton is in one of its states. At each time step when the automaton reads a symbol, it jumps or transitions to another state that is decided by 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 and the state at which automaton has stopped 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 of that automaton". Any subset of the language of an automaton is a language recognized by that automaton.

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