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A finite-state machine (FSM) or finite-state automaton (FSA, plural: ''automata''), finite automaton, or simply a state machine, is a mathematical
model of computation In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes how ...
. It is an
abstract machine An abstract machine is a computer science theoretical model that allows for a detailed and precise analysis of how a computer system functions. It is analogous to a mathematical function in that it receives inputs and produces outputs based on pr ...
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''. An FSM is defined by a list of its states, its initial state, and the inputs that trigger 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. The behavior of state machines can be observed in many devices in modern society that perform a predetermined sequence of actions depending on a sequence of events with which they are presented. Simple examples are
vending machine A vending machine is an automated machine that provides items such as snacks, beverages, cigarettes, and lottery tickets to consumers after cash, a credit card, or other forms of payment are inserted into the machine or otherwise made. The ...
s, which dispense products when the proper combination of coins is deposited,
elevator An elevator or lift is a cable-assisted, hydraulic cylinder-assisted, or roller-track assisted machine that vertically transports people or freight between floors, levels, or decks of a building, vessel, or other structure. They a ...
s, whose sequence of stops is determined by the floors requested by riders,
traffic light Traffic lights, traffic signals, or stoplights – known also as robots in South Africa are signalling devices positioned at road intersections, pedestrian crossings, and other locations in order to control flows of traffic. Traffic light ...
s, which change sequence when cars are waiting, and combination locks, which require the input of a sequence of numbers in the proper order. The finite-state machine has less computational power than some other models of computation such as the
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine 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 algori ...
. The computational power distinction means there are computational tasks that a Turing machine can do but an FSM cannot. This is because an FSM's
memory Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembered ...
is limited by the number of states it has. A finite-state machine has the same computational power as a Turing machine that is restricted such that its head may only perform "read" operations, and always has to move from left to right. FSMs are studied in the more general field of
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. The word ''automata'' comes from the Greek word αὐτόματο ...
.


Example: coin-operated turnstile

An example of a simple mechanism that can be modeled by a state machine is a
turnstile A turnstile (also called a turnpike, gateline, baffle gate, automated gate, turn gate in some regions) is a form of gate which allows one person to pass at a time. A turnstile can be configured to enforce one-way human traffic. In addition, a ...
. A turnstile, used to control access to subways and amusement park rides, is a gate with three rotating arms at waist height, one across the entryway. Initially the arms are locked, blocking the entry, preventing patrons from passing through. Depositing a coin or token in a slot on the turnstile unlocks the arms, allowing a single customer to push through. After the customer passes through, the arms are locked again until another coin is inserted. Considered as a state machine, the turnstile has two possible states: ''Locked'' and ''Unlocked''. There are two possible inputs that affect its state: putting a coin in the slot (''coin'') and pushing the arm (''push''). In the locked state, pushing on the arm has no effect; no matter how many times the input ''push'' is given, it stays in the locked state. Putting a coin in – that is, giving the machine a ''coin'' input – shifts the state from ''Locked'' to ''Unlocked''. In the unlocked state, putting additional coins in has no effect; that is, giving additional ''coin'' inputs does not change the state. However, a customer pushing through the arms, giving a ''push'' input, shifts the state back to ''Locked''. The turnstile state machine can be represented by a state-transition table, showing for each possible state, the transitions between them (based upon the inputs given to the machine) and the outputs resulting from each input: :: The turnstile state machine can also be represented by a
directed graph In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs. Definition In formal terms, a directed graph is an ordered pa ...
called a
state diagram A state diagram is a type of diagram used in computer science and related fields to describe the behavior of systems. State diagrams require that the system described is composed of a finite number of states; sometimes, this is indeed the case, ...
''(above)''. Each state is represented by a
node In general, a node is a localized swelling (a "knot") or a point of intersection (a vertex). Node may refer to: In mathematics * Vertex (graph theory), a vertex in a mathematical graph *Vertex (geometry), a point where two or more curves, lines ...
(''circle''). Edges (''arrows'') show the transitions from one state to another. Each arrow is labeled with the input that triggers that transition. An input that doesn't cause a change of state (such as a ''coin'' input in the ''Unlocked'' state) is represented by a circular arrow returning to the original state. The arrow into the ''Locked'' node from the black dot indicates it is the initial state.


Concepts and terminology

A ''state'' is a description of the status of a system that is waiting to execute a ''transition''. A transition is a set of actions to be executed when a condition is fulfilled or when an event is received. For example, when using an audio system to listen to the radio (the system is in the "radio" state), receiving a "next" stimulus results in moving to the next station. When the system is in the "CD" state, the "next" stimulus results in moving to the next track. Identical stimuli trigger different actions depending on the current state. In some finite-state machine representations, it is also possible to associate actions with a state: * an entry action: performed ''when entering'' the state, and * an exit action: performed ''when exiting'' the state.


Representations


State/Event table

Several state-transition table types are used. The most common representation is shown below: the combination of current state (e.g. B) and input (e.g. Y) shows the next state (e.g. C). The complete action's information is not directly described in the table and can only be added using footnotes. An FSM definition including the full action's information is possible using state tables (see also
virtual finite-state machine A virtual finite-state machine (VFSM) is a finite-state machine (FSM) defined in a Virtual Environment. The VFSM concept provides a software specification method to describe the behaviour of a control system using assigned names of input control ...
).


UML state machines

The
Unified Modeling Language The Unified Modeling Language (UML) is a general-purpose, developmental modeling language in the field of software engineering that is intended to provide a standard way to visualize the design of a system. The creation of UML was originally m ...
has a notation for describing state machines.
UML state machine UML state machine, also known as UML statechart, is an extension of the mathematical concept of a finite automaton in computer science applications as expressed in the Unified Modeling Language (UML) notation. The concepts behind it are abou ...
s overcome the limitations of traditional finite-state machines while retaining their main benefits. UML state machines introduce the new concepts of hierarchically nested states and orthogonal regions, while extending the notion of actions. UML state machines have the characteristics of both
Mealy machine In the theory of computation, a Mealy machine is a finite-state machine whose output values are determined both by its current state and the current inputs. This is in contrast to a Moore machine, whose output values are determined solely by its cu ...
s and
Moore machine In the theory of computation, a Moore machine is a finite-state machine whose current output values are determined only by its current state. This is in contrast to a Mealy machine, whose output values are determined both by its current state and ...
s. They support actions that depend on both the state of the system and the triggering
event Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of e ...
, as in Mealy machines, as well as entry and exit actions, which are associated with states rather than transitions, as in Moore machines.


SDL state machines

The
Specification and Description Language Specification and Description Language (SDL) is a specification language targeted at the unambiguous specification and description of the behaviour of reactive and distributed systems. Overview The ITU-T has defined SDL in Recommendations Z.100 t ...
is a standard from ITU that includes graphical symbols to describe actions in the transition: * send an event * receive an event * start a timer * cancel a timer * start another concurrent state machine * decision SDL embeds basic data types called "Abstract Data Types", an action language, and an execution semantic in order to make the finite-state machine executable.


Other state diagrams

There are a large number of variants to represent an FSM such as the one in figure 3.


Usage

In addition to their use in modeling reactive systems presented here, finite-state machines are significant in many different areas, including electrical engineering,
linguistics Linguistics is the science, scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure ...
,
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to practical disciplines (includi ...
, philosophy,
biology Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
, mathematics, video game programming, and
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premise ...
. Finite-state machines are a class of automata studied in
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. The word ''automata'' comes from the Greek word αὐτόματο ...
and the
theory of computation In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how algorithmic efficiency, efficiently they can be solved or t ...
. In computer science, finite-state machines are widely used in modeling of application behavior (
control theory Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
), design of hardware digital systems,
software engineering Software engineering is a systematic engineering approach to software development. A software engineer is a person who applies the principles of software engineering to design, develop, maintain, test, and evaluate computer software. The term '' ...
,
compiler In computing, a compiler is a computer program that translates computer code written in one programming language (the ''source'' language) into another language (the ''target'' language). The name "compiler" is primarily used for programs tha ...
s,
network protocol A communication protocol is a system of rules that allows two or more entities of a communications system to transmit information via any kind of variation of a physical quantity. The protocol defines the rules, syntax, semantics and synchroniza ...
s, and computational linguistics.


Classification

Finite-state machines can be subdivided into acceptors, classifiers, transducers and sequencers.


Acceptors

Acceptors (also called detectors or recognizers) produce binary output, indicating whether or not the received input is accepted. Each state of an acceptor is either ''accepting'' or ''non accepting''. Once all input has been received, if the current state is an accepting state, the input is accepted; otherwise it is rejected. As a rule, input is a
sequence of symbols In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable. The latter may allow its elements to be mutated and the length changed, or it may be fixed (after creation). ...
(characters); actions are not used. The start state can also be an accepting state, in which case the acceptor accepts the empty string. The example in figure 4 shows an acceptor that accepts the string "nice". In this acceptor, the only accepting state is state 7. A (possibly infinite) set of symbol sequences, called a
formal language In logic, 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. The alphabet of a formal language consists of sy ...
, is a
regular language In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to ...
if there is some acceptor that accepts ''exactly'' that set. For example, the set of binary strings with an even number of zeroes is a regular language (cf. Fig. 5), while the set of all strings whose length is a prime number is not. An acceptor could also be described as defining a language that would contain every string accepted by the acceptor but none of the rejected ones; that language is ''accepted'' by the acceptor. By definition, the languages accepted by acceptors are the
regular language In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to ...
s. The problem of determining the language accepted by a given acceptor is an instance of the algebraic path problem—itself a generalization of the shortest path problem to graphs with edges weighted by the elements of an (arbitrary)
semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...
. An example of an accepting state appears in Fig. 5: a
deterministic finite automaton 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 autom ...
(DFA) that detects whether the
binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two digits (0 and 1) * Binary function, a function that takes two arguments * Binary operation, a mathematical operation that ta ...
input string contains an even number of 0s. ''S''1 (which is also the start state) indicates the state at which an even number of 0s has been input. S1 is therefore an accepting state. This acceptor will finish in an accept state, if the binary string contains an even number of 0s (including any binary string containing no 0s). Examples of strings accepted by this acceptor are ε (the
empty string In formal language theory, the empty string, or empty word, is the unique string of length zero. Formal theory Formally, a string is a finite, ordered sequence of characters such as letters, digits or spaces. The empty string is the special cas ...
), 1, 11, 11..., 00, 010, 1010, 10110, etc.


Classifiers

Classifiers are a generalization of acceptors that produce ''n''-ary output where ''n'' is strictly greater than two.


Transducers

Transducers produce output based on a given input and/or a state using actions. They are used for control applications and in the field of computational linguistics. In control applications, two types are distinguished: ;
Moore machine In the theory of computation, a Moore machine is a finite-state machine whose current output values are determined only by its current state. This is in contrast to a Mealy machine, whose output values are determined both by its current state and ...
: The FSM uses only entry actions, i.e., output depends only on state. The advantage of the Moore model is a simplification of the behaviour. Consider an elevator door. The state machine recognizes two commands: "command_open" and "command_close", which trigger state changes. The entry action (E:) in state "Opening" starts a motor opening the door, the entry action in state "Closing" starts a motor in the other direction closing the door. States "Opened" and "Closed" stop the motor when fully opened or closed. They signal to the outside world (e.g., to other state machines) the situation: "door is open" or "door is closed". ;
Mealy machine In the theory of computation, a Mealy machine is a finite-state machine whose output values are determined both by its current state and the current inputs. This is in contrast to a Moore machine, whose output values are determined solely by its cu ...
: The FSM also uses input actions, i.e., output depends on input and state. The use of a Mealy FSM leads often to a reduction of the number of states. The example in figure 7 shows a Mealy FSM implementing the same behaviour as in the Moore example (the behaviour depends on the implemented FSM execution model and will work, e.g., for virtual FSM but not for event-driven FSM). There are two input actions (I:): "start motor to close the door if command_close arrives" and "start motor in the other direction to open the door if command_open arrives". The "opening" and "closing" intermediate states are not shown.


Sequencers

Sequencers (also called generators) are a subclass of acceptors and transducers that have a single-letter input alphabet. They produce only one sequence, which can be seen as an output sequence of acceptor or transducer outputs.


Determinism

A further distinction is between deterministic ( DFA) and non-deterministic ( NFA, GNFA) automata. In a deterministic automaton, every state has exactly one transition for each possible input. In a non-deterministic automaton, an input can lead to one, more than one, or no transition for a given state. The
powerset construction 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 sa ...
algorithm can transform any nondeterministic automaton into a (usually more complex) deterministic automaton with identical functionality. A finite-state machine with only one state is called a "combinatorial FSM". It only allows actions upon transition ''into'' a state. This concept is useful in cases where a number of finite-state machines are required to work together, and when it is convenient to consider a purely combinatorial part as a form of FSM to suit the design tools.


Alternative semantics

There are other sets of semantics available to represent state machines. For example, there are tools for modeling and designing logic for embedded controllers. They combine hierarchical state machines (which usually have more than one current state), flow graphs, and
truth table A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
s into one language, resulting in a different formalism and set of semantics. These charts, like Harel's original state machines, support hierarchically nested states, orthogonal regions, state actions, and transition actions.


Mathematical model

In accordance with the general classification, the following formal definitions are found. A ''deterministic finite-state machine'' or ''deterministic finite-state acceptor'' is a quintuple (\Sigma, S, s_0, \delta, F), where: * \Sigma is the input
alphabet An alphabet is a standardized set of basic written graphemes (called letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a syllab ...
(a finite non-empty set of symbols); * S is a finite non-empty set of states; * s_0 is an initial state, an element of S; * \delta is the state-transition function: \delta: S \times \Sigma \rightarrow S (in a
nondeterministic finite automaton In automata theory, a finite-state machine is called a deterministic finite automaton (DFA), if * each of its transitions is ''uniquely'' determined by its source state and input symbol, and * reading an input symbol is required for each state ...
it would be \delta: S \times \Sigma \rightarrow \mathcal(S), i.e. \delta would return a set of states); * F is the set of final states, a (possibly empty) subset of S. For both deterministic and non-deterministic FSMs, it is conventional to allow \delta to be a
partial function In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
, i.e. \delta(s, x) does not have to be defined for every combination of s \isin S and x \isin \Sigma. If an FSM M is in a state s, the next symbol is x and \delta(s, x) is not defined, then M can announce an error (i.e. reject the input). This is useful in definitions of general state machines, but less useful when transforming the machine. Some algorithms in their default form may require total functions. A finite-state machine has the same computational power as a
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine 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 algori ...
that is restricted such that its head may only perform "read" operations, and always has to move from left to right. That is, each formal language accepted by a finite-state machine is accepted by such a kind of restricted Turing machine, and vice versa. A ''
finite-state transducer 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. ...
'' is a
sextuple In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
(\Sigma, \Gamma, S, s_0, \delta, \omega), where: * \Sigma is the input
alphabet An alphabet is a standardized set of basic written graphemes (called letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a syllab ...
(a finite non-empty set of symbols); * \Gamma is the output alphabet (a finite non-empty set of symbols); * S is a finite non-empty set of states; * s_0 is the initial state, an element of S; * \delta is the state-transition function: \delta: S \times \Sigma \rightarrow S; * \omega is the output function. If the output function depends on the state and input symbol (\omega: S \times \Sigma \rightarrow \Gamma) that definition corresponds to the ''Mealy model'', and can be modelled as a
Mealy machine In the theory of computation, a Mealy machine is a finite-state machine whose output values are determined both by its current state and the current inputs. This is in contrast to a Moore machine, whose output values are determined solely by its cu ...
. If the output function depends only on the state (\omega: S \rightarrow \Gamma) that definition corresponds to the ''Moore model'', and can be modelled as a
Moore machine In the theory of computation, a Moore machine is a finite-state machine whose current output values are determined only by its current state. This is in contrast to a Mealy machine, whose output values are determined both by its current state and ...
. A finite-state machine with no output function at all is known as a
semiautomaton In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set ''Q'' of states, a set Σ called the input alphabet, and a function ''T'': ''Q'' × Σ → ''Q'' ...
or transition system. If we disregard the first output symbol of a Moore machine, \omega(s_0), then it can be readily converted to an output-equivalent Mealy machine by setting the output function of every Mealy transition (i.e. labeling every edge) with the output symbol given of the destination Moore state. The converse transformation is less straightforward because a Mealy machine state may have different output labels on its incoming transitions (edges). Every such state needs to be split in multiple Moore machine states, one for every incident output symbol.


Optimization

Optimizing an FSM means finding a machine with the minimum number of states that performs the same function. The fastest known algorithm doing this is the Hopcroft minimization algorithm. Other techniques include using an implication table, or the Moore reduction procedure. Additionally, acyclic FSAs can be minimized in linear time.


Implementation


Hardware applications

In a digital circuit, an FSM may be built using a programmable logic device, a programmable logic controller, logic gates and flip flops or
relay A relay Electromechanical relay schematic showing a control coil, four pairs of normally open and one pair of normally closed contacts An automotive-style miniature relay with the dust cover taken off A relay is an electrically operated switch ...
s. More specifically, a hardware implementation requires a
register Register or registration may refer to: Arts entertainment, and media Music * Register (music), the relative "height" or range of a note, melody, part, instrument, etc. * ''Register'', a 2017 album by Travis Miller * Registration (organ), th ...
to store state variables, a block of
combinational logic In automata theory, combinational logic (also referred to as time-independent logic or combinatorial logic) is a type of digital logic which is implemented by Boolean circuits, where the output is a pure function of the present input only. This ...
that determines the state transition, and a second block of combinational logic that determines the output of an FSM. One of the classic hardware implementations is the Richards controller. In a ''Medvedev machine'', the output is directly connected to the state flip-flops minimizing the time delay between flip-flops and output. Through
state encoding for low power State encoding assigns a unique pattern of ones and zeros to each defined state of a finite-state machine (FSM). Traditionally, design criteria for FSM synthesis were speed, area or both. Following Moore's law, with technology advancement, density a ...
state machines may be optimized to minimize power consumption.


Software applications

The following concepts are commonly used to build software applications with finite-state machines: *
Automata-based programming Automata-based programming is a programming paradigm in which the program or part of it is thought of as a model of a finite-state machine (FSM) or any other (often more complicated) formal automaton (see automata theory). Sometimes a potentiall ...
* Event-driven finite-state machine *
Virtual finite-state machine A virtual finite-state machine (VFSM) is a finite-state machine (FSM) defined in a Virtual Environment. The VFSM concept provides a software specification method to describe the behaviour of a control system using assigned names of input control ...
* State design pattern


Finite-state machines and compilers

Finite automata are often used in the frontend of programming language compilers. Such a frontend may comprise several finite-state machines that implement a lexical analyzer and a parser. Starting from a sequence of characters, the lexical analyzer builds a sequence of language tokens (such as reserved words, literals, and identifiers) from which the parser builds a syntax tree. The lexical analyzer and the parser handle the regular and context-free parts of the programming language's grammar.


See also

* Abstract state machines * Alternating finite automaton * Communicating finite-state machine * Control system *
Control table Control tables are tables that control the control flow or play a major part in program control. There are no rigid rules about the structure or content of a control table—its qualifying attribute is its ability to direct control flow in some w ...
* Decision tables *
DEVS ''Devs'' is an American science fiction thriller television miniseries created, written, and directed by Alex Garland. It premiered on March 5, 2020, on FX on Hulu. Lily Chan (Sonoya Mizuno) is a software engineer for Amaya, a quantum computing ...
*
Hidden Markov model A hidden Markov model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process — call it X — with unobservable ("''hidden''") states. As part of the definition, HMM requires that there be an o ...
*
Petri net A Petri net, also known as a place/transition (PT) net, is one of several mathematical modeling languages for the description of distributed systems. It is a class of discrete event dynamic system. A Petri net is a directed bipartite graph that ...
*
Pushdown automaton In the theory of computation, a branch of theoretical computer science, a pushdown automaton (PDA) is a type of automaton that employs a stack. Pushdown automata are used in theories about what can be computed by machines. They are more capab ...
*
Quantum finite automaton In quantum computing, quantum finite automata (QFA) or quantum state machines are a quantum analog of probabilistic automata or a Markov decision process. They provide a mathematical abstraction of real-world quantum computers. Several types of a ...
* SCXML *
Semiautomaton In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set ''Q'' of states, a set Σ called the input alphabet, and a function ''T'': ''Q'' × Σ → ''Q'' ...
*
Semigroup action In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup (using t ...
* Sequential logic *
State diagram A state diagram is a type of diagram used in computer science and related fields to describe the behavior of systems. State diagrams require that the system described is composed of a finite number of states; sometimes, this is indeed the case, ...
* Synchronizing word *
Transformation semigroup In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations ( functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transfo ...
* Transition system * Tree automaton *
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine 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 algori ...
*
UML state machine UML state machine, also known as UML statechart, is an extension of the mathematical concept of a finite automaton in computer science applications as expressed in the Unified Modeling Language (UML) notation. The concepts behind it are abou ...


References


Further reading


General

* * Wagner, F., "Modeling Software with Finite State Machines: A Practical Approach", Auerbach Publications, 2006, . * ITU-T
''Recommendation Z.100 Specification and Description Language (SDL)''
* Samek, M.
''Practical Statecharts in C/C++''
CMP Books, 2002, . * Samek, M.
''Practical UML Statecharts in C/C++, 2nd Edition''
Newnes, 2008, . * Gardner, T.
''Advanced State Management''
, 2007 * Cassandras, C., Lafortune, S., "Introduction to Discrete Event Systems". Kluwer, 1999, . * Timothy Kam, ''Synthesis of Finite State Machines: Functional Optimization''. Kluwer Academic Publishers, Boston 1997, * Tiziano Villa, ''Synthesis of Finite State Machines: Logic Optimization''. Kluwer Academic Publishers, Boston 1997, * Carroll, J., Long, D.,
Theory of Finite Automata with an Introduction to Formal Languages
'. Prentice Hall, Englewood Cliffs, 1989. * Kohavi, Z., ''Switching and Finite Automata Theory''. McGraw-Hill, 1978. * Gill, A., ''Introduction to the Theory of Finite-state Machines''. McGraw-Hill, 1962. * Ginsburg, S., ''An Introduction to Mathematical Machine Theory''. Addison-Wesley, 1962.


Finite-state machines (automata theory) in theoretical computer science

* * * * * * * * * * * * * * * * * *


Abstract state machines in theoretical computer science

*


Machine learning using finite-state algorithms

*


Hardware engineering: state minimization and synthesis of sequential circuits

* * * *


Finite Markov chain processes

::"We may think of a Markov chain as a process that moves successively through a set of states ''s1'', ''s2'', …, ''sr''. … if it is in state ''si'' it moves on to the next stop to state ''sj'' with probability ''pij''. These probabilities can be exhibited in the form of a transition matrix" (Kemeny (1959), p. 384) Finite Markov-chain processes are also known as subshifts of finite type. * * Chapter 6 "Finite Markov Chains".


External links

*
''Modeling a Simple AI behavior using a Finite State Machine''
Example of usage in Video Games
Free On-Line Dictionary of Computing
description of Finite-State Machines

description of Finite-State Machines
A brief overview of state machine types
comparing theoretical aspects of Mealy, Moore, Harel & UML state machines. {{digital systems