# Automata theory

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Classes of automata
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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".

An abstract machine, also called an abstract computer, is a theoretical model of a computer hardware or software system used in automata theory. Abstraction of computing processes is used in both the computer science and computer engineering disciplines and usually assumes a discrete time paradigm.

An automaton is a self-operating machine, or a machine or control mechanism designed to automatically follow a predetermined sequence of operations, or respond to predetermined instructions. Some automata, such as bellstrikers in mechanical clocks, are designed to give the illusion to the casual observer that they are operating under their own power.

In theoretical computer science, a computational problem is a mathematical object representing a collection of questions that computers might be able to solve. For example, the problem of factoring

## Contents

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.

A finite-state machine (FSM) or finite-state automaton, finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine 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 external inputs and/or a condition is satisfied; the change from one state to another is called a transition. An FSM is defined by a list of its states, its initial state, and the conditions for each transition. Finite state machines are of two types – deterministic finite state machines and non-deterministic finite state machines. A deterministic finite-state machine can be constructed equivalent to any non-deterministic one.

In information technology and computer science, a system is described as stateful if it is designed to remember preceding events or user interactions; the remembered information is called the state of the system.

In theoretical computer science, a transition system is a concept used in the study of computation. It is used to describe the potential behavior of discrete systems. It consists of states and transitions between states, which may be labeled with labels chosen from a set; the same label may appear on more than one transition. If the label set is a singleton, the system is essentially unlabeled, and a simpler definition that omits the labels is possible.

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.

In mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.

In the formal languages of computer science and linguistics, the Chomsky hierarchy is a containment hierarchy of classes of formal grammars.

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

In theoretical computer science and mathematics, the theory of computation is the branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm. The field is divided into three major branches: automata theory and languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?".

In computer science, artificial intelligence (AI), sometimes called machine intelligence, is intelligence demonstrated by machines, in contrast to the natural intelligence displayed by humans. Leading AI textbooks define the field as the study of "intelligent agents": any device that perceives its environment and takes actions that maximize its chance of successfully achieving its goals. Colloquially, the term "artificial intelligence" is often used to describe machines that mimic "cognitive" functions that humans associate with the human mind, such as "learning" and "problem solving".

Parsing, syntax analysis, or syntactic analysis is the process of analysing a string of symbols, either in natural language, computer languages or data structures, conforming to the rules of a formal grammar. The term parsing comes from Latin pars (orationis), meaning part.

## Automata

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

### Very informal description

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.

### Informal description

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.

A logical symbol is a fundamental concept in logic, tokens of which may be marks or a configuration of marks which form a particular pattern. Although the term "symbol" in common use refers at some times to the idea being symbolized, and at other times to the marks on a piece of paper or chalkboard which are being used to express that idea; in the formal languages studied in mathematics and logic, the term "symbol" refers to the idea, and the marks are considered to be a token instance of the symbol. In logic, symbols build literal utility to illustrate ideas.

In short, an automaton is a mathematical object that takes a word as input and decides whether to accept it or reject it. Since all computational problems are reducible into the accept/reject question on inputs, (all problem instances can be represented in a finite length of symbols)[ citation needed ], automata theory plays a crucial role in computational theory.

A mathematical object is an abstract object arising in mathematics. The concept is studied in philosophy of mathematics.

### Formal definition

Automaton

#### definition of finite state automata

A deterministic finite automaton is represented formally by a 5-tuple <Q, Σ, δ,q0,F>, where:
• Q is a finite set of states.
• Σ is a finite set of symbols , called the alphabet of the automaton.
• δ is the transition function, that is, δ: Q × Σ  Q.
• q0 is the start state, that is, the state of the automaton before any input has been processed, where q0∈ Q.
• F is a set of states of Q (i.e. F⊆Q) called accept states.
Input word
An automaton reads a finite string of symbols a1,a2,...., an , where ai  Σ, which is called an input word. The set of all words is denoted by Σ*.
Run
A sequence of states q0,q1,q2,...., qn, where qi  Q such that q0 is the start state and qi = δ(qi-1,ai) for 0 < i  n, is a run of the automaton on an input word w = a1,a2,...., an  Σ*. In other words, at first the automaton is at the start state q0, and then the automaton reads symbols of the input word in sequence. When the automaton reads symbol ai it jumps to state qi = δ(qi-1,ai). qn is said to be the final state of the run.
Accepting word
A word w  Σ* is accepted by the automaton if qn  F.
Recognized language
An automaton can recognize a formal language. The language L  Σ* recognized by an automaton is the set of all the words that are accepted by the automaton.
Recognizable languages
The recognizable languages are the set of languages that are recognized by some automaton. For the above definition of automata the recognizable languages are regular languages. For different definitions of automata, the recognizable languages are different.

## Variant definitions of automata

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.

Input
• Finite input: An automaton that accepts only finite sequence of symbols. The above introductory definition only encompasses finite words.
• Infinite input: An automaton that accepts infinite words (ω-words). Such automata are called ω-automata .
• Tree word input: The input may be a tree of symbols instead of sequence of symbols. In this case after reading each symbol, the automaton reads all the successor symbols in the input tree. It is said that the automaton makes one copy of itself for each successor and each such copy starts running on one of the successor symbols from the state according to the transition relation of the automaton. Such an automaton is called a tree automaton.
• Infinite tree input : The two extensions above can be combined, so the automaton reads a tree structure with (in)finite branches. Such an automaton is called an infinite tree automaton
States
• Finite states: An automaton that contains only a finite number of states. The above introductory definition describes automata with finite numbers of states.
• Infinite states: An automaton that may not have a finite number of states, or even a countable number of states. For example, the quantum finite automaton or topological automaton has uncountable infinity of states.
• Stack memory: An automaton may also contain some extra memory in the form of a stack in which symbols can be pushed and popped. This kind of automaton is called a pushdown automaton
Transition function
• Deterministic: For a given current state and an input symbol, if an automaton can only jump to one and only one state then it is a deterministic automaton .
• Nondeterministic: An automaton that, after reading an input symbol, may jump into any of a number of states, as licensed by its transition relation. Notice that the term transition function is replaced by transition relation: The automaton non-deterministically decides to jump into one of the allowed choices. Such automata are called nondeterministic automata.
• Alternation: This idea is quite similar to tree automaton, but orthogonal. The automaton may run its multiple copies on the same next read symbol. Such automata are called alternating automata . Acceptance condition must satisfy all runs of such copies to accept the input.
Acceptance condition
• Acceptance of finite words: Same as described in the informal definition above.
• Acceptance of infinite words: an omega automaton cannot have final states, as infinite words never terminate. Rather, acceptance of the word is decided by looking at the infinite sequence of visited states during the run.
• Probabilistic acceptance: An automaton need not strictly accept or reject an input. It may accept the input with some probability between zero and one. For example, quantum finite automaton, geometric automaton and metric automaton have probabilistic acceptance.

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.

• Which class of formal languages is recognizable by some type of automata? (Recognizable languages)
• Are certain automata closed under union, intersection, or complementation of formal languages? (Closure properties)
• How expressive is a type of automata in terms of recognizing a class of formal languages? And, their relative expressive power? (Language hierarchy)

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

• Does an automaton accept any input word? (Emptiness checking)
• Is it possible to transform a given non-deterministic automaton into deterministic automaton without changing the recognizable language? (Determinization)
• For a given formal language, what is the smallest automaton that recognizes it? (Minimization)

## Classes of automata

The following is an incomplete list of types of automata.

AutomatonRecognizable language
Nondeterministic/Deterministic Finite state machine (FSM) regular languages
Deterministic pushdown automaton (DPDA) deterministic context-free languages
Pushdown automaton (PDA) context-free languages
Linear bounded automaton (LBA) context-sensitive languages
Turing machine recursively enumerable languages
Deterministic Büchi automaton ω-limit languages
Nondeterministic Büchi automaton ω-regular languages
Rabin automaton, Streett automaton, Parity automaton, Muller automaton ω-regular languages

### Discrete, continuous, and hybrid automata

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

## Hierarchy in terms of powers

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. [1]

Automaton
Deterministic Finite Automaton (DFA) -- Lowest Power

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

## Applications

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. [2]

## Automata simulators

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. [3]

## Connection to category theory

One can define several distinct categories of automata [4] following the automata classification into different types described in the previous section. The mathematical category of deterministic automata, sequential machines or sequential automata, and Turing machines with automata homomorphisms defining the arrows between automata is a Cartesian closed category, [5] [6] it has both categorical limits and colimits. An automata homomorphism maps a quintuple of an automaton Ai onto the quintuple of another automaton Aj. [7] 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 Sg. Monoids are also considered as a suitable setting for automata in monoidal categories. [8] [9] [10]

Categories of variable automata

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 ${\displaystyle A_{i}\to A_{i}}$. 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 groupoid . Therefore, in the most general case, categories of variable automata of any kind are 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.

## History

The automata theory was developed in the mid-20th century in connection with finite automata. [11]

## Related Research Articles

In the theory of computation, a branch of theoretical computer science, a pushdown automaton (PDA) is a type of automaton that employs a stack.

In theoretical computer science and formal language theory, a regular language is a formal language that can be expressed using a regular expression, in the strict sense of the latter notion used in theoretical computer science.

In computer science and automata theory, a Büchi automaton is a type of ω-automaton, which extends a finite automaton to infinite inputs. It accepts an infinite input sequence if there exists a run of the automaton that visits one of the final states infinitely often. Büchi automata recognize the omega-regular languages, the infinite word version of regular languages. It is named after the Swiss mathematician Julius Richard Büchi who invented this kind of automaton in 1962.

In the theory of computation, a branch of theoretical computer science, a deterministic finite automaton (DFA)—also known as deterministic finite acceptor (DFA), deterministic finite state machine (DFSM), or deterministic finite state automaton (DFSA)—is a finite-state machine that accepts or rejects strings of symbols and only produces a unique computation of the automaton for each input string. Deterministic refers to the uniqueness of the computation. In search of the simplest models to capture finite-state machines, Warren McCulloch and Walter Pitts were among the first researchers to introduce a concept similar to finite automata in 1943.

In automata theory, a finite state machine is called a deterministic finite automaton (DFA), if

In automata theory, an alternating finite automaton (AFA) is a nondeterministic finite automaton whose transitions are divided into existential and universal transitions. For example, let A be an alternating automaton.

A finite-state transducer (FST) is a finite-state machine with two memory tapes, following the terminology for Turing machines: an input tape and an output tape. This contrasts with an ordinary finite-state automaton, which has a single tape. An FST is a type of finite-state automaton that maps between two sets of symbols. An FST is more general than a finite-state automaton (FSA). An FSA defines a formal language by defining a set of accepted strings, while an FST defines relations between sets of strings.

In the theory of computation and automata theory, the powerset construction or subset construction is a standard method for converting a nondeterministic finite automaton (NFA) into a deterministic finite automaton (DFA) which recognizes the same formal language. It is important in theory because it establishes that NFAs, despite their additional flexibility, are unable to recognize any language that cannot be recognized by some DFA. It is also important in practice for converting easier-to-construct NFAs into more efficiently executable DFAs. However, if the NFA has n states, the resulting DFA may have up to 2n states, an exponentially larger number, which sometimes makes the construction impractical for large NFAs.

In automata theory, a permutation automaton, or pure-group automaton, is a deterministic finite automaton such that each input symbol permutes the set of states.

In automata theory, a deterministic pushdown automaton is a variation of the pushdown automaton. The class of deterministic pushdown automata accepts the deterministic context-free languages, a proper subset of context-free languages.

In computer science, in particular in automata theory, a two-way finite automaton is a finite automaton that is allowed to re-read its input.

In quantum computing, quantum finite automata (QFA) or quantum state machines are a quantum analog of probabilistic automata or a Markov decision process. They are related to quantum computers in a similar fashion as finite automata are related to Turing machines. Several types of automata may be defined, including measure-once and measure-many automata. Quantum finite automata can also be understood as the quantization of subshifts of finite type, or as a quantization of Markov chains. QFAs are, in turn, special cases of geometric finite automata or topological finite automata.

In automata theory, a Muller automaton is a type of an ω-automaton. The acceptance condition separates a Muller automaton from other ω-automata. The Muller automaton is defined using Muller acceptance condition, i.e. the set of all states visited infinitely often must be an element of the acceptance set. Both deterministic and non-deterministic Muller automata recognize the ω-regular languages. They are named after David E. Muller, an American mathematician and computer scientist, who invented them in 1963.

A read-only Turing machine or Two-way deterministic finite-state automaton (2DFA) is class of models of computability that behave like a standard Turing machine and can move in both directions across input, except cannot write to its input tape. The machine in its bare form is equivalent to a Deterministic finite automaton in computational power, and therefore can only parse a regular language.

In automata theory, DFA minimization is the task of transforming a given deterministic finite automaton (DFA) into an equivalent DFA that has a minimum number of states. Here, two DFAs are called equivalent if they recognize the same regular language. Several different algorithms accomplishing this task are known and described in standard textbooks on automata theory.

In computer science, more specifically in automata and formal language theory, nested words are a concept proposed by Alur and Madhusudan as a joint generalization of words, as traditionally used for modelling linearly ordered structures, and of ordered unranked trees, as traditionally used for modelling hierarchical structures. Finite-state acceptors for nested words, so-called nested word automata, then give a more expressive generalization of finite automata on words. The linear encodings of languages accepted by finite nested word automata gives the class of visibly pushdown languages. The latter language class lies properly between the regular languages and the deterministic context-free languages. Since their introduction in 2004, these concepts have triggered much research in that area.

In computer science and mathematical logic, an infinite-tree automaton is a state machine that deals with infinite tree structures. It can be seen as an extension of top-down finite-tree automata to infinite trees or as an extension of infinite-word automata to infinite trees.

In automata theory, a branch of theoretical computer science, an ω-automaton is a variation of finite automatons that runs on infinite, rather than finite, strings as input. Since ω-automata do not stop, they have a variety of acceptance conditions rather than simply a set of accepting states.

In automata theory, a timed automaton is a finite automaton extended with a finite set of real-valued clocks. During a run of a timed automaton, clock values increase all with the same speed. Along the transitions of the automaton, clock values can be compared to integers. These comparisons form guards that may enable or disable transitions and by doing so constrain the possible behaviors of the automaton. Further, clocks can be reset. Timed automata are a sub-class of a type hybrid automata.

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2. Ferraguti, A.; Micheli, G.; Schnyder, R. (2018), Irreducible compositions of degree two polynomials over finite fields have regular structure, The Quarterly Journal of Mathematics, 69, Oxford University Press, pp. 1089–1099, arXiv:, doi:10.1093/qmath/hay015
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