(Clicking on each layer gets an article on that subject)
The automaton described by thisstate diagram starts in state S1, and changes states following the arrows marked 0 or 1 according to the input symbols as they arrive. The double circle marks S1 as an accepting state. Since all paths from S1 to itself contain an even number of arrows marked 0, this automaton accepts strings containing even numbers of 0s.
Automata theory is the study ofabstract machines andautomata, as well as thecomputational problems that can be solved using them. It is a theory intheoretical computer science with close connections tocognitive science andmathematical logic. The wordautomata comes from theGreek word αὐτόματος, which means "self-acting, self-willed, self-moving". Anautomaton (automata in plural) is an abstract self-propelledcomputing device which follows a predetermined sequence of operations automatically. An automaton with a finite number ofstates is called a finite automaton (FA) orfinite-state machine (FSM). The figure on the right illustrates a finite-state machine, which is a well-known type of automaton. This automaton consists ofstates (represented in the figure by circles) andtransitions (represented by arrows). As the automaton sees a symbol of input, it makes a transition (or jump) to another state, according to itstransition function, which takes the previous state and current input symbol as itsarguments.
The theory of abstract automata was developed in the mid-20th century in connection withfinite automata.[1] Automata theory was initially considered a branch of mathematicalsystems theory, studying the behavior of discrete-parameter systems. Early work in automata theory differed from previous work on systems by usingabstract algebra to describeinformation systems rather thandifferential calculus to describe material systems.[2] The theory of thefinite-state transducer was developed under different names by different research communities.[3] The earlier concept ofTuring machine was also included in the discipline along with new forms of infinite-state automata, such aspushdown automata.
The study oflinear bounded automata led to theMyhill–Nerode theorem,[8] which gives a necessary and sufficient condition for a formal language to be regular, and an exact count of the number of states in a minimal machine for the language. Thepumping lemma for regular languages, also useful in regularity proofs, was proven in this period byMichael O. Rabin andDana Scott, along with the computational equivalence of deterministic and nondeterministic finite automata.[9]
In the 1960s, a body of algebraic results known as "structure theory" or "algebraic decomposition theory" emerged, which dealt with the realization of sequential machines from smaller machines by interconnection.[10] While any finite automaton can be simulated using a universal gate set, this requires that the simulating circuit contain loops of arbitrary complexity. Structure theory deals with the "loop-free" realizability of machines.[5]The theory ofcomputational complexity also took shape in the 1960s.[11][12] By the end of the decade, automata theory came to be seen as "the pure mathematics of computer science".[5]
What follows is a general definition of an automaton, which restricts a broader definition of asystem to one viewed as acting in discrete time-steps, with its state behavior and outputs defined at each step by unchanging functions of only its state and input.[5]
An automatonruns when it is given some sequence ofinputs in discrete (individual)time steps (or juststeps). An automaton processes one input picked from a set ofsymbols orletters, which is called aninputalphabet. The symbols received by the automaton as input at any step are a sequence of symbols calledwords. An automaton has a set ofstates. At each moment during a run of the automaton, the automaton isin one of its states. When the automaton receives new input, it moves to another state (ortransitions) based on atransition function that takes the previous state and current input symbol as parameters. At the same time, another function called theoutput function produces symbols from theoutput alphabet, also according to the previous state and current input symbol. The automaton reads the symbols of the input word and transitions between states until the word is read completely, if it is finite in length, at which point the automatonhalts. A state at which the automaton halts is called thefinal state.
To investigate the possible state/input/output sequences in an automaton usingformal language theory, a machine can be assigned astarting state and a set ofaccepting states. Then, depending on whether a run starting from the starting state ends in an accepting state, the automaton can be said toaccept orreject an input sequence. The set of all the words accepted by an automaton is called thelanguage recognized by the automaton. A familiar example of a machine recognizing a language is anelectronic lock, which accepts or rejects attempts to enter the correct code.
An automaton reads a finitestring of symbols, where, which is called aninput word. The set of all words is denoted by.
Run
A sequence of states, where such that for, is arun of the automaton on an input starting from state. In other words, at first the automaton is at the start state, and receives input. For and every following in the input string, the automaton picks the next state according to the transition function, until the last symbol has been read, leaving the machine in thefinal state of the run,. Similarly, at each step, the automaton emits an output symbol according to the output function.
The transition function is extended inductively into to describe the machine's behavior when fed whole input words. For the empty string, for all states, and for strings where is the last symbol and is the (possibly empty) rest of the string,.[10] The output function may be extended similarly into, which gives the complete output of the machine when run on word from state.
Acceptor
In order to study an automaton with the theory offormal languages, an automaton may be considered as anacceptor, replacing the output alphabet and function and with
, a designatedstart state, and
, a set of states of (i.e.) calledaccept states.
This allows the following to be defined:
Accepting word
A word is anaccepting word for the automaton if, that is, if after consuming the whole string the machine is in an accept state.
Recognized language
The languagerecognized by an automaton is the set of all the words that are accepted by the automaton,.[13]
Recognizable languages
Therecognizable languages are the set of languages that are recognized by some automaton. Forfinite automata the recognizable languages areregular languages. For different types of automata, the recognizable languages are different.
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" that we want to model using the automaton. People have studied many variations of automata. The following are some popular variations in the definition of different components of automata.
Input
Finite input: An automaton that accepts only finite sequences of symbols. The above introductory definition only encompasses finite words.
Infinite input: An automaton that accepts infinite words (ω-words). Such automata are calledω-automata.
Tree input: The input may be atree of symbols instead of sequence of symbols. In this case after reading each symbol, the automatonreads all the successor symbols in the input tree. It is said that the automatonmakes 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 atree 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 aninfinite tree automaton.
States
Single state: An automaton with one state, also called acombinational circuit, performs a transformation which may implementcombinational logic.[10]
Finite states: An automaton that contains only a finite number of states.
Infinite states: An automaton that may not have a finite number of states, or even acountable number of states. Different kinds of abstract memory may be used to give such machines finite descriptions.
Stack memory: An automaton may also contain some extra memory in the form of astack in which symbols can be pushed and popped. This kind of automaton is called apushdown automaton.
Queue memory: An automaton may have memory in the form of aqueue. Such a machine is calledqueue machine and is Turing-complete.
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 adeterministic 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. The term transition function is replaced by transition relation: The automatonnon-deterministically decides to jump into one of the allowed choices. Such automata are callednondeterministic automata.
Alternation: This idea is quite similar to tree automata but orthogonal. The automaton may run itsmultiple copies on thesame next read symbol. Such automata are calledalternating automata. The acceptance condition must be satisfied on all runs of suchcopies to accept the input.
Two-wayness: Automata may read their input from left to right, or they may be allowed to move back-and-forth on the input, in a way similar to aTuring machine. Automata which can move back-and-forth on the input are calledtwo-way finite automata.
Acceptance condition
Acceptance of finite words: Same as described in the informal definition above.
Acceptance of infinite words: anω-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.
Different combinations of the above variations produce many classes of automata.
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 automataclosed 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 anyeffective algorithms to solve problems similar to the following list:
Does an automaton accept at least one input word? (Emptiness checking)
Is it possible to transform a given non-deterministic automaton into a deterministic automaton without changing the language recognized? (Determinization)
For a given formal language, what is the smallest automaton that recognizes it? (Minimization)
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.[14]
Each model in automata theory plays important roles in several applied areas.Finite automata are used intext processing, compilers, andhardware design.Context-free grammar (CFGs) are used inprogramming languages and artificial intelligence. Originally, CFGs were used in the study ofhuman languages.Cellular automata are used in the field ofartificial life, the most famous example beingJohn Conway'sGame of Life. Some other examples which could be explained using automata theory in biology include mollusk and pine cone 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 ofKonrad Zuse, and was popularized in America byEdward Fredkin. Automata also appear in the theory offinite fields: the set ofirreducible polynomials that can be written as composition of degree two polynomials is in fact a regular language.[15]Another problem for which automata can be used is theinduction of regular languages.
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 asymbolic 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.[16]
One can define several distinctcategories of automata[17] following the automata classification into different types described in the previous section. The mathematical category of deterministic automata,sequential machines orsequential automata, and Turing machines withautomata homomorphisms defining the arrows between automata is aCartesian closed category,[18] it has both categoricallimits andcolimits. An automata homomorphism maps a quintuple of an automatonAi onto the quintuple of another automaton Aj. Automata homomorphisms can also be considered asautomata transformations or assemigroup homomorphisms, when the state space,S, of the automaton is defined as a semigroupSg.Monoids are also considered as a suitable setting for automata inmonoidal categories.[19][20][21]
Categories of variable automata
One could also define avariable automaton, in the sense ofNorbert Wiener in his book onThe Human Use of Human Beingsvia 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, avariable automatongroupoid. Therefore, in the most general case, categories of variable automata of any kind arecategories of groupoids orgroupoid categories. Moreover, the category of reversible automata is then a2-category, and also a subcategory of the 2-category of groupoids, or the groupoid category.[citation needed]
^Booth, Taylor (1967).Sequential Machines and Automata Theory. New York: John Wiley & Sons. p. 1-13.ISBN0-471-08848-X.
^Ashby, William Ross (1967-01-15)."The Place of the Brain in the Natural World"(PDF).Currents in Modern Biology.1 (2):95–104.Bibcode:1967BiSys...1...95A.doi:10.1016/0303-2647(67)90021-4.PMID6060865. Archived fromthe original(PDF) on 2023-06-04. Retrieved2021-03-29.: "The theories, now well developed, of the "finite-state machine" (Gill, 1962), of the "noiseless transducer" (Shannon and Weaver, 1949), of the "state-determined system" (Ashby, 1952), and of the "sequential circuit", are essentially homologous."
^Ashby, W. R.; et al. (1956). C.E. Shannon; J. McCarthy (eds.).Automata Studies. Princeton, N.J.: Princeton University Press.
^abcdeArbib, Michael (1969).Theories of Abstract Automata. Englewood Cliffs, N.J.: Prentice-Hall.
^Li, Ming; Paul, Vitanyi (1997).An Introduction to Kolmogorov Complexity and its Applications. New York: Springer-Verlag. p. 84.
^Ferraguti, A.; Micheli, G.; Schnyder, R. (2018),Irreducible compositions of degree two polynomials over finite fields have regular structure, The Quarterly Journal of Mathematics, vol. 69, Oxford University Press, pp. 1089–1099,arXiv:1701.06040,doi:10.1093/qmath/hay015,S2CID3962424
Each category of languages, except those marked by a*, is aproper subset of the category directly above it.Any language in each category is generated by a grammar and by an automaton in the category in the same line.
