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{{Redirect|State machine|infinite state machines|State transition system|fault-tolerance methodology|State machine replication}}
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A '''finite-state machine''' ('''FSM''') or '''finite-state automaton''' (plural: ''automata''), or simply a '''state machine''', is a mathematical [[model of computation]] used to design both [[computer programs]] and [[sequential logic]] circuits.  It is conceived as an [[abstract machine]] that can be in one of a finite number of ''[[state (computer science)|states]]''. The machine is in only one state at a time; the state it is in at any given time is called the ''current state''.   It can change from one state to another when initiated by a triggering event or condition; this is called a ''transition''. A particular FSM is defined by a list of its states, and the triggering condition for each transition.
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The behavior of state machines can be observed in many devices in modern society which perform a predetermined sequence of actions depending on a sequence of events with which they are presented. Simple examples are [[vending machine]]s which dispense products when the proper combination of coins is deposited, [[elevator]]s which drop riders off at upper floors before going down, [[traffic light]]s which change sequence when cars are waiting, and [[combination lock]]s which require the input of combination numbers in the proper order.
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Finite-state machines can model a large number of problems, among which are [[electronic design automation]], [[communication protocol]] design, language [[parsing]] and other engineering applications.  In [[biology]] and [[artificial intelligence]] research, state machines or hierarchies of state machines have been used to describe [[neurology|neurological systems]] and in [[linguistics]]—to describe the [[grammar]]s of natural [[languages]].
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


Considered as an abstract model of computation, the finite state machine is weak; it has less computational power than some other models of computation such as the [[Turing machine]].<ref name="Belzer">{{cite book 
<!--'''PNG'''  (currently default in production)
  | last = Belzer
:<math forcemathmode="png">E=mc^2</math>
  | first = Jack 
  | authorlink =  
  |first2=Albert George |last2=Holzman |first3=Allen |last3=Kent
  | title = Encyclopedia of Computer Science and Technology, Vol. 25
  | publisher = CRC Press
  | year = 1975
  | location = USA
  | pages = 73
  | url = http://books.google.com/?id=W2YLBIdeLIEC&printsec=frontcover&dq=%22finite+state+machine%22#v=onepage&q=%22finite%20state%20machine%22&f=false
  | doi =
  | id =
  | isbn = 0824722752}}</ref>   That is, there are tasks which no FSM can do, but some Turing machines can.  This is because the FSM has limited [[Computer memory|memory]].  The memory is limited by the number of states.


FSMs are studied in the more general field of [[automata theory]].
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== Example: a turnstile ==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
[[Image:Turnstile state machine colored.svg|thumb|upright=1.5|State diagram for a turnstile]]
[[Image:Torniqueterevolution.jpg|thumb|upright=0.5|A turnstile]]
An example of a very simple mechanism that can be modeled by a state machine is a [[turnstile]].<ref name="Koshy">{{cite book 
  | last = Koshy
  | first = Thomas
  | authorlink =
  | title = Discrete Mathematics With Applications
  | publisher = Academic Press
  | year = 2004
  | location =
  | pages = 762
  | url = http://books.google.com/?id=90KApidK5NwC&pg=PA762&dq=state+machine+turnstile#v=onepage&q=state%20machine%20turnstile&f=false
  | doi =
  | id =
  | isbn = 0124211801}}</ref><ref name="">{{cite web
  | last = Wright
  | first = David R.
  | authorlink =
  | title =  Finite State Machines
  | work = CSC215 Class Notes
  | publisher = Prof. David R. Wright website, N. Carolina State Univ.
  | year = 2005
  | url = http://www4.ncsu.edu/~drwrigh3/docs/courses/csc216/fsm-notes.pdf
  | format =
  | doi =
  | accessdate = July 14, 2012}}</ref>  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, barring the entry, preventing customers from passing through.  Depositing a coin or [[Token coin|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 states: '''''Locked''''' and '''''Unlocked'''''.<ref name="Koshy" />  There are two 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'''''.
==Demos==


The turnstile state machine can be represented by a [[state transition table]], showing for each state the new state and the output (action) resulting from each input
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
{| style="background:#f5f5f5;"  border="1" cellpadding="5" cellspacing="0"
|-
! scope="col" style="background:#cfcfcf;"  | Current State
! scope="col" style="background:#cfcfcf;"  | Input
! scope="col" style="background:#cfcfcf;"  | Next State
! scope="col" style="background:#cfcfcf;"  | Output
|-
! rowspan="2" scope="row" style="background:#dfdfdf;"  | Locked
| coin || Unlocked || Unlock turnstile so customer can push through
|-
| push || Locked || None
|-
! rowspan="2" scope="row"  style="background:#dfdfdf;" | Unlocked
| coin || Unlocked || None
|-
| push || Locked || When customer has pushed through lock turnstile
|}
It can also be represented by a [[directed graph]] called a [[state diagram]] ''(above)''. Each of the states is represented by a [[node (graph theory)|node]] (''circle'').  Edges (''arrows'') show the transitions from one state to another.  Each arrow is labeled with the input that triggers that transition.  Inputs that don't cause a change of state (such as a '''''coin''''' input in the '''''Unlocked''''' state) are 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:
* accessibility:
* Entry action: performed ''when entering'' the state,
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
* Exit action: performed ''when exiting'' the state.
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


==Representations==
==Test pages ==
[[File:UML state machine Fig5.png|thumb|Fig. 1 UML state chart example (a toaster oven)]]
[[Image:SdlStateMachine.png|thumb|Fig. 2 SDL state machine example]]
[[File:Finite state machine example with comments.svg|thumb|Fig. 3 Example of a simple finite state machine]]


''For an introduction, see [[State diagram]].''
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


===State/Event table===
*[[Inputtypes|Inputtypes (private Wikis only)]]
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. A FSM definition including the full actions information is possible using [[Virtual finite state machine#State Table|state tables]] (see also [[virtual finite-state machine]]).
*[[Url2Image|Url2Image (private Wikis only)]]
 
==Bug reporting==
{| class="wikitable" style="text-align:center; margin-left:auto; margin-right:auto"
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
|+ State transition table
|-
! Current state&nbsp;→ <br /> Input&nbsp;↓ || State A || State B || State C
|-
! Input X
| ... || ... || ...
|-
! Input Y
| ... || State C || ...
|-
! Input Z
| ... || ... || ...
|}
 
===UML state machines===
The [[Unified Modeling Language]] has a notation for describing state machines. [[UML state machine]]s overcome the limitations of traditional finite state machines while retaining their main benefits. UML state machines introduce the new concepts of [[UML state machine#Hierarchically nested states|hierarchically nested states]] and [[UML state machine#Orthogonal regions|orthogonal regions]], while extending the notion of [[UML state machine#Actions and transitions|actions]]. UML state machines have the characteristics of both [[Mealy machine]]s and [[Moore machine]]s. They support [[UML state machine#Actions and transitions|actions]] that depend on both the state of the system and the triggering [[UML state machine#Events|event]], as in Mealy machines, as well as [[UML state machine#Entry and exit actions|entry and exit actions]], which are associated with states rather than transitions, as in Moore machines.
 
===SDL state machines===
The [[Specification and Description Language]] is a standard from [[ITU]] which 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 system]]s presented here, finite state automata are significant in many different areas, including [[electrical engineering]], [[linguistics]], [[computer science]], [[philosophy]], [[biology]], [[mathematic]]s, and [[logic]]. Finite state machines are a class of automata studied in [[automata theory]] and the [[theory of computation]].
In computer science, finite state machines are widely used in modeling of application behavior, design of [[Digital electronics|hardware digital systems]], [[software engineering]], [[compiler]]s, [[network protocol]]s, and the study of computation and languages.
 
==Classification==
The state machines can be subdivided into Transducers, Acceptors, Classifiers and Sequencers.<ref>{{cite book |last=Keller |first= Robert M. |title=Computer Science: Abstraction to Implementation |url=http://www.cs.hmc.edu/~keller/cs60book/%20%20%20All.pdf |year=2001 |publisher=Harvey Mudd College |page= 480| chapter=Classifiers, Acceptors, Transducers, and Sequencers| chapterurl =http://www.cs.hmc.edu/~keller/cs60book/12%20Finite-State%20Machines.pdf}}</ref>
 
===Acceptors and recognizers===
[[File:Fsm parsing word nice.svg|thumb|Fig. 4 Acceptor FSM: parsing the string "nice"]]
 
'''Acceptors''' (also '''recognizers''' and '''sequence detectors''') produce a binary output, saying either ''yes'' or ''no'' to answer whether the input is accepted by the machine or not. All states of the FSM are said to be either accepting or not accepting. At the time when all input is processed, if the current state is an accepting state, the input is accepted; otherwise it is rejected. As a rule the input are symbols (characters); actions are not used. The example in figure 4 shows a finite state machine which accepts the string "nice". In this FSM the only accepting state is number 7.
 
The machine can also be described as defining a language, which would contain every string accepted by the machine but none of the rejected ones; we say then that the language is ''accepted'' by the machine. By definition, the languages accepted by FSMs are the [[regular language]]s—that is, a language is regular if there is some FSM that accepts it.
 
The problem of determining the language accepted by a given FSA 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]].<ref name="PoulyKohlas2012">{{cite book|first1=Marc |last1=Pouly |first2=Jürg |last2=Kohlas |title=Generic Inference: A Unifying Theory for Automated Reasoning|year=2011|publisher=John Wiley & Sons|isbn=978-1-118-01086-0|at=Chapter 6. Valuation Algebras for Path Problems, p. 223 in particular}}</ref><ref name="Storer2001">{{cite book|first=J. A. |last=Storer |title=An Introduction to Data Structures and Algorithms|url=http://books.google.com/books?id=S-tXjl1hsUYC&pg=PA337|year=2001|publisher=Springer Science & Business Media|isbn=978-0-8176-4253-2|page=337}}</ref><ref>http://www.iam.unibe.ch/~run/talks/2008-06-05-Bern-Jonczy.pdf, p. 34</ref>
 
====Start state====
The start state is usually shown drawn with an arrow "pointing at it from any where" (Sipser (2006) p.&nbsp;34).
 
====Accept (or final) states{{anchor |Accept state or final state}}====
[[File:DFAexample.svg|thumb|Fig. 5: Representation of a finite-state machine; this example shows one that determines whether a binary number has an even number of 0s, where <math>S_1</math> is an '''accepting state'''.]]
 
'''Accept states''' (also referred to as '''accepting''' or '''final''' states) are those at which the machine reports that the input string, as processed so far, is a member of the language it accepts. It is usually represented by a double circle.
 
The start state can also be a final state, in which case the automaton accepts the empty string. If the start state is not an accepting state and there are no connecting edges to any of the accepting states, then the automaton is accepting nothing.
 
An example of an accepting state appears in Fig.5: a [[deterministic finite automaton]] (DFA) that detects whether the [[Binary numeral system|binary]] input string contains an even number of 0s.
 
''S''<sub>1</sub> (which is also the start state) indicates the state at which an even number of 0s has been input. S<sub>1</sub> is therefore an accepting state. This machine 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 DFA are [[ε]] (the [[empty string]]), 1, 11, 11..., 00, 010, 1010, 10110, etc...
 
'''Classifier''' is a generalization that, similar to acceptor, produces single output when terminates but has more than two terminal states.
 
===Transducers===
 
{{Main |Finite state transducer }}
 
[[Finite state transducer|Transducers]] generate 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]]: The FSM uses only entry actions, i.e., output depends only on the 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".
 
[[File:Fsm mealy model door control.jpg|thumb|Fig. 7 Transducer FSM: Mealy model example]]
;[[Mealy machine]]: The FSM uses only 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 finite state machine|virtual FSM]] but not for [[event driven finite state machine|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.
 
More details about the differences and usage of Moore and Mealy models, including an executable example, can be found in the external technical note [http://www.stateworks.com/technology/TN10-Moore-Or-Mealy-Model/ "Moore or Mealy model?"]
 
===Generators===
The '''sequencers''' or '''generators''' are a subclass of aforementioned types that have a [[Arity#Nullary|single-letter input alphabet]]. They produce only one sequence, which can be interpreted as output sequence of transducer or classifier outputs.
 
===Determinism===
A further distinction is between '''deterministic''' ([[Deterministic finite state machine|DFA]]) and '''non-deterministic''' ([[Nondeterministic finite automaton|NFA]], [[GNFA]]) automata. In deterministic automata, every state has exactly one transition for each possible input. In non-deterministic automata, an input can lead to one, more than one or no transition for a given state. This distinction is relevant in practice, but not in theory, as there exists an algorithm (the [[powerset construction]]) which can transform any NFA into a more complex DFA with identical functionality.
 
The FSM with only one state is called a combinatorial FSM and uses only input actions. This concept is useful in cases where a number of FSM are required to work together, and where it is convenient to consider a purely combinatorial part as a form of FSM to suit the design tools.<ref>Brutscheck, M., Berger, S., Franke, M., Schwarzbacher, A., Becker, S.: Structural Division Procedure for Efficient IC Analysis. IET Irish
Signals and Systems Conference, (ISSC 2008), pp.18-23. Galway, Ireland, 18–19 June 2008.</ref>
 
==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.<ref>[http://www.csl.sri.com/users/tiwari/papers/stateflow.pdf Tiwari, A. (2002). Formal Semantics and Analysis Methods for Simulink Stateflow Models.]</ref> They combine [[UML state machine#Hierarchically nested states|hierarchical state machines]], flow graphs, and [[truth table]]s into one language, resulting in a different formalism and set of semantics.<ref>{{cite conference | id = {{citeseerx|10.1.1.89.8817}} | last = Hamon | first = G. | year = 2005 | title =  A Denotational Semantics for Stateflow | conference = International Conference on Embedded Software | pages = 164–172 | location = Jersey City, NJ | publisher = ACM }}</ref>  Figure 8 illustrates this mix of state machines and flow graphs with a set of states to represent the state of a stopwatch and a flow graph to control the ticks of the watch.  These charts, like Harel's original state machines,<ref>[http://www.fceia.unr.edu.ar/asist/harel01.pdf Harel, D. (1987). A Visual Formalism for Complex Systems. Science of Computer Programming , 231–274.]</ref> support hierarchically nested states, [[UML state machine#Orthogonal regions|orthogonal regions]], state actions, and transition actions.<ref>[http://drona.csa.iisc.ernet.in/~kanade/publications/symbolic_analysis_for_improving_simulation_coverage_of_simulink_stateflow_models.pdf Alur, R., Kanade, A., Ramesh, S., & Shashidhar, K. C. (2008). Symbolic analysis for improving simulation coverage of Simulink/Stateflow models. Internation Conference on Embedded Software (pp. 89–98). Atlanta, GA: ACM.]</ref>
 
==FSM logic==
[[File:Finite State Machine Logic.svg|thumb|Fig. 8 FSM Logic (Mealy)]]
 
The next state and output of an FSM is a function of the input and of the current state. The FSM logic is shown in Figure 8.
 
==Mathematical model==
In accordance with the general classification, the following formal definitions are found:
* A ''deterministic finite state machine'' or ''acceptor deterministic finite state machine'' is a [[Tuple|quintuple]] <math>(\Sigma, S, s_0, \delta, F)</math>, where:
**<math>\Sigma</math> is the input [[alphabet (computer science)|alphabet]] (a finite, non-empty set of symbols).
**<math>S</math> is a finite, non-empty set of states.
**<math>s_0</math> is an initial state, an element of <math>S</math>.
**<math>\delta</math> is the state-transition function: <math>\delta: S \times \Sigma \rightarrow S</math> (in a [[nondeterministic finite automaton]] it would be <math>\delta: S \times \Sigma \rightarrow \mathcal{P}(S)</math>, i.e., <math>\delta</math> would return a set of states).
**<math>F</math> is the set of final states, a (possibly empty) subset of <math>S</math>.
 
For both deterministic and non-deterministic FSMs, it is conventional to allow <math>\delta</math> to be a [[partial function]], i.e. <math>\delta(q,x)</math> does not have to be defined for every combination of <math>q \isin S</math> and <math>x \isin \Sigma</math>. If an FSM <math>M</math> is in a state <math>q</math>, the next symbol is <math>x</math> and <math>\delta(q,x)</math> is not defined, then <math>M</math> 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 is a restricted [[Turing machine]] where the head can only perform "read" operations, and always moves from left to right.<ref>{{cite journal
|last=Black
|first=Paul E
|date=12 May 2008
|title=Finite State Machine
|journal=Dictionary of Algorithms and Data Structures
|publisher=U.S. [[National Institute of Standards and Technology]]
|url=http://www.nist.gov/dads/HTML/finiteStateMachine.html}}</ref>
 
* A ''[[finite state transducer]]'' is a [[sextuple]] <math>(\Sigma, \Gamma, S, s_0, \delta, \omega)</math>, where:
** <math>\Sigma</math> is the input [[alphabet (computer science)|alphabet]] (a finite non-empty set of symbols).
** <math>\Gamma</math> is the output alphabet (a finite, non-empty set of symbols).
** <math>S</math> is a finite, non-empty set of states.
** <math>s_0</math> is the initial state, an element of <math>S</math>. In a [[nondeterministic finite automaton]], <math>s_0</math> is a set of initial states.
** <math>\delta</math> is the state-transition function: <math>\delta: S \times \Sigma \rightarrow S</math>.
** <math>\omega</math> is the output function.
 
If the output function is a function of a state and input alphabet (<math>\omega: S \times \Sigma \rightarrow \Gamma</math>) that definition corresponds to the '''Mealy model''', and can be modelled as a [[Mealy machine]]. If the output function depends only on a state (<math>\omega: S \rightarrow \Gamma</math>) that definition corresponds to the '''Moore model''', and can be modelled as a [[Moore machine]]. A finite-state machine with no output function at all is known as a [[semiautomaton]] or [[transition system]].
 
If we disregard the first output symbol of a Moore machine, <math>\omega (s_0)</math>, 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.<ref name="AndersonHead2006">{{cite book |first1=James Andrew |last1=Anderson |first2=Thomas J. |last2=Head |title=Automata theory with modern applications |url=http://books.google.com/books?id=ikS8BLdLDxIC&pg=PA105 |year=2006 |publisher=Cambridge University Press |isbn=978-0-521-84887-9 |pages=105–108}}</ref>
 
==Optimization==
{{Main|DFA minimization}}
Optimizing an FSM means finding the machine with the minimum number of states that performs the same function. The fastest known algorithm doing this is the [[DFA minimization#Hopcroft.27s algorithm|Hopcroft minimization algorithm]].<ref>{{cite report |last=Hopcroft |first=John E. |year=1971 |title= An ''n'' log ''n'' algorithm for minimizing states in a finite automaton |volume=CS-TR-71-190 |type=Technical Report |url=ftp://reports.stanford.edu/pub/cstr/reports/cs/tr/71/190/CS-TR-71-190.pdf |publisher=Stanford Univ.}}</ref><ref>{{cite report|last1=Almeida|first1= Marco|last2= Moreira|first2= Nelma|last3= Reis|first3= Rogerio |year=2007|title= On the performance of automata minimization algorithms |url= http://www.dcc.fc.up.pt/dcc/Pubs/TReports/TR07/dcc-2007-03.pdf |type=Technical Report |volume=DCC-2007-03 |publisher=Porto Univ.}}</ref> Other techniques include using an [[implication table]], or the [[Moore reduction procedure]]. Additionally, acyclic FSAs can be minimized in linear time.<ref>{{cite journal|last=Revuz |first=D. |title=Minimization of Acyclic automata in Linear Time| journal= Theoretical Computer Science |volume=92 |date=1992| pages= 181–189 |publisher= Elsevier}}</ref>
 
==Implementation==
 
===Hardware applications===
[[File:4 bit counter.svg|thumbnail|Fig. 9 The [[circuit diagram]] for a 4-bit [[Transistor-transistor logic|TTL]] counter, a type of state machine]]
In a [[digital circuit]], an FSM may be built using a [[programmable logic device]], a [[programmable logic controller]], [[logic gate]]s and [[Flip-flop (electronics)|flip flops]] or [[relay]]s. More specifically, a hardware implementation requires a [[processor register|register]] to store state variables, a block of [[combinational logic]] which 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]].
 
A particular case of Moore FSM, when output is directly connected to the state flip-flops, that is when output function is simple identity, is known as Medvedev FSM.<ref>{{cite book |last= Kaeslin|first= Hubert |title=Digital Integrated Circuit Design: From VLSI Architectures to CMOS Fabrication |url=http://books.google.com/?id=gdRStcYgf2oC&pg=PA787&dq=medvedev+fsm#v=onepage&q=medvedev%20fsm&f=false |year=2008  |publisher=Cambridge University Press |page=787 | chapter=Mealy, Moore, Medvedev-type and combinatorial output bits |isbn= 9780521882675}}</ref> It is advised in chip design that no logic is placed between primary I/O and registers to minimize interchip delays, which are usually long and limit the FSM frequencies.
 
===Software applications===
The following concepts are commonly used to build software applications with finite state machines:
*[[Automata-based programming]]
*[[Event-driven finite-state machine|Event-driven FSM]]
*[[Virtual finite state machine|Virtual FSM (VFSM)]]
 
===Finite automata and compilers===
Finite automata are often used in the [[Compilers#Front end|frontend]] of programming language compilers. Such a frontend may comprise several finite state machines that implement a [[lexical analysis|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.<ref>{{Cite book |authorlink1=Alfred V. Aho |last1=Aho |first1=Alfred V. |authorlink2 = Ravi Sethi |last2=Sethi |first2=Ravi |authorlink3=Jeffrey D. Ullman |last3=Ullman |first3=Jeffrey D. |title=[[Compilers: Principles, Techniques, and Tools]] |isbn=9780201100884 |publisher=[[Addison-Wesley]] |year=1986 |edition=1st}}</ref>
 
==See also==
*[[Abstract state machines]] (ASM)
*[[Artificial intelligence]] (AI)
*[[Abstract State Machine Language]] (AsmL)
*[[Communicating finite-state machine]]
*[[Control system]]
*[[Control table]]
*[[Decision table]]s
*[[DEVS]]: Discrete Event System Specification
*[[Extended finite-state machine]] (EFSM)
*[[FSMD|Finite state machine with datapath]]
*[[Hidden Markov model]]
*[[Petri net]]
*[[Pushdown automaton]]
*[[Quantum finite automata]] (QFA)
*[[Recognizable language]]
*[[Sequential logic]]
*[[Specification and Description Language]]
*[[State diagram]]
*[[SCXML]]
*[[Transition system]]
*[[Tree automaton]]
*[[Turing machine]]
*[[UML state machine]]
*[[YAKINDU Statechart Tools]]
 
==References==
{{Reflist}}
 
==Further reading==
 
===General===
* {{Cite book | last=Sakarovitch | first=Jacques | title=Elements of automata theory | others=Translated from the French by Reuben Thomas | publisher=[[Cambridge University Press]] | year=2009 | isbn=978-0-521-84425-3 | zbl=1188.68177 | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}} }}
*Wagner, F., "Modeling Software with Finite State Machines: A Practical Approach", Auerbach Publications, 2006, ISBN 0-8493-8086-3.
*ITU-T, [http://www.itu.int/rec/T-REC-Z.100-200711-I/en ''Recommendation Z.100 Specification and Description Language (SDL)'']
*Samek, M., [http://www.state-machine.com/psicc/index.php ''Practical Statecharts in C/C++''], CMP Books, 2002, ISBN 1-57820-110-1.
*Samek, M., [http://www.state-machine.com/psicc2/index.php ''Practical UML Statecharts in C/C++, 2nd Edition''], Newnes, 2008, ISBN 0-7506-8706-1.
*Gardner, T., [http://www.troyworks.com/cogs/ ''Advanced State Management''], 2007
*Cassandras, C., Lafortune, S., "Introduction to Discrete Event Systems". Kluwer, 1999, ISBN 0-7923-8609-4.
*Timothy Kam, ''Synthesis of Finite State Machines: Functional Optimization''. Kluwer Academic Publishers, Boston 1997, ISBN 0-7923-9842-4
*Tiziano Villa, ''Synthesis of Finite State Machines: Logic Optimization''. Kluwer Academic Publishers, Boston 1997, ISBN 0-7923-9892-0
*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===
*{{cite book | last = Arbib | first = Michael A. | title = Theories of Abstract Automata  | edition = 1st | publisher = Prentice-Hall, Inc. | location = Englewood Cliffs, N.J. | year = 1969 | isbn = 0-13-913368-2}}
*{{cite book | last =Bobrow | first = Leonard S. |first2=Michael A. |last2=Arbib| title = Discrete Mathematics: Applied Algebra for Computer and Information Science  | edition = 1st | publisher = W. B. Saunders Company, Inc. | location = Philadelphia  | year = 1974 | isbn =0-7216-1768-9}}
*{{cite book | last = Booth| first = Taylor L. | title = Sequential Machines and Automata Theory | edition = 1st | publisher = John Wiley and Sons, Inc. | location = New York | year = 1967| id = Library of Congress Card Catalog Number 67-25924}}
*{{cite book | last = Boolos | first = George |first2=Richard |last2=Jeffrey | title = Computability and Logic | edition = 3rd | publisher = Cambridge University Press | location = Cambridge, England | origyear = 1989| year = 1999| isbn = 0-521-20402-X}}
*{{cite book | last = Brookshear | first = J. Glenn | title = Theory of Computation: Formal Languages, Automata, and Complexity | year = 1989 | publisher = Benjamin/Cummings Publish Company, Inc. | location = Redwood City, California | isbn = 0-8053-0143-7}}
*{{cite book | last = Davis| first = Martin |first2=Ron |last2=Sigal |first3=Elaine J. |last3=Weyuker | title = Computability, Complexity, and Languages and Logic: Fundamentals of Theoretical Computer Science | edition = 2nd | publisher = Academic Press, Harcourt, Brace & Company| location = San Diego | year = 1994 | isbn = 0-12-206382-1}}
*{{cite book | last = Hopcroft | first = John |first2=Jeffrey |last2=Ullman| title = [[Introduction to Automata Theory, Languages, and Computation]] | edition = 1st | publisher = Addison-Wesley | location =  Reading Mass | year = 1979 | isbn = 0-201-02988-X }}
*{{cite book | last = Hopcroft | first = John E.|first2=Rajeev |last2=Motwani |first3=Jeffrey D. |last3=Ullman | title = Introduction to Automata Theory, Languages, and Computation| edition = 2nd | publisher = Addison-Wesley | location =  Reading Mass | year = 2001 | isbn = 0-201-44124-1}}
*{{cite book | last = Hopkin | first = David |first2=Barbara |last2=Moss| title = Automata | edition = | publisher = Elsevier North-Holland | location = New York | year = 1976| isbn = 0-444-00249-9 }}
*{{cite book | last = Kozen | first = Dexter C. | title = Automata and Computability  | edition = 1st | publisher = Springer-Verlag | location = New York | year = 1997|isbn = 0-387-94907-0}}
*{{cite book | last = Lewis| first = Harry R.| authorlink = Harry R. Lewis |first2=Christos H. |last2=Papadimitriou |author2-link=Christos H. Papadimitriou| title = Elements of the Theory of Computation | edition = 2nd | publisher = Prentice-Hall | location = Upper Saddle River, New Jersey | year = 1998| isbn = 0-13-262478-8 }}
*{{cite book | last = Linz| first = Peter| title = Formal Languages and Automata | edition = 4th | publisher = Jones and Bartlett | location = Sudbury, MA | year = 2006| isbn = 978-0-7637-3798-6 }}
*{{cite book | last = Minsky| first = Marvin| title = Computation: Finite and Infinite Machines| edition = 1st | publisher = Prentice-Hall | location = New Jersey | year = 1967| id = }}
* {{cite book |first=Christos |last=Papadimitriou |authorlink=Christos Papadimitriou | year = 1993 | title = Computational Complexity | publisher = Addison Wesley | edition = 1st | isbn = 0-201-53082-1}}
*{{cite book | last = Pippenger| first = Nicholas| title = Theories of Computability| edition = 1st | publisher = Cambridge University Press| location = Cambridge, England | year = 1997| isbn = 0-521-55380-6 }}
*{{cite book | last = Rodger| first = Susan |first2=Thomas |last2=Finley| title = JFLAP: An Interactive Formal Languages and Automata Package | edition = 1st | publisher = Jones and Bartlett | location = Sudbury, MA | year = 2006| isbn = 0-7637-3834-4 }}
*{{cite book | last = Sipser| first = Michael | title = Introduction to the Theory of Computation | edition = 2nd | publisher = Thomson Course Technology | location = Boston Mass | year = 2006| isbn = 0-534-95097-3 }}
*{{cite book | last = Wood| first = Derick | title = Theory of Computation | edition = 1st | publisher = Harper & Row, Publishers, Inc.| location = New York | year = 1987| isbn = 0-06-047208-1 }}
 
===Abstract state machines in theoretical computer science===
*{{cite journal|first=Yuri |last=Gurevich |title=Sequential Abstract State Machines Capture Sequential Algorithms |work=ACM Transactions on Computational Logic| volume= 1| issue=1 |date=July 2000 |pages= 77–111 |url=http://research.microsoft.com/~gurevich/Opera/141.pdf }}
 
===Machine learning using finite-state algorithms===
*{{cite book | last = Mitchell| first = Tom M. | title = Machine Learning | edition = 1st | publisher = WCB/McGraw-Hill Corporation | location = New York | year = 1997| isbn = 0-07-042807-7}}
 
===Hardware engineering: state minimization and synthesis of sequential circuits===
*{{cite book | last = Booth| first = Taylor L. | title = Sequential Machines and Automata Theory | edition = 1st | publisher = John Wiley and Sons, Inc. | location = New York | year = 1967| id = Library of Congress Card Catalog Number 67-25924}}
*{{cite book | last = Booth| first = Taylor L. | title = Digital Networks and Computer Systems| edition = 1st | publisher = John Wiley and Sons, Inc. | location = New York | year = 1971| isbn = 0-471-08840-4}}
*{{cite book | last = McCluskey| first = E. J. | title = Introduction to the Theory of Switching Circuits | edition = 1st | publisher = McGraw-Hill Book Company, Inc. | location = New York | year = 1965| id = Library of Congress Card Catalog Number 65-17394}}
*{{cite book | last = Hill| first = Fredrick J. |first2=Gerald R. |last2=Peterson | title = Introduction to the Theory of Switching Circuits | edition = 1st | publisher = McGraw-Hill Book Company | location = New York | year = 1965| id = Library of Congress Card Catalog Number 65-17394}}
 
===Finite Markov chain processes===
::"We may think of a [[Markov chain]] as a process that moves successively through a set of states ''s<sub>1</sub>'', ''s<sub>2</sub>'', ..., ''s<sub>r</sub>''. ... if it is in state ''s<sub>i</sub>'' it moves on to the next stop to state ''s<sub>j</sub>'' with probability ''p<sub>ij</sub>''. 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]].
 
*{{cite book | last = Booth| first = Taylor L. | title = Sequential Machines and Automata Theory | edition = 1st | publisher = John Wiley and Sons, Inc. | location = New York | year = 1967| id = Library of Congress Card Catalog Number 67-25924}}
*{{cite book | last = Kemeny| first = John G. |first2=Hazleton |last2=Mirkil |first3=J. Laurie |last3=Snell |first4=Gerald L. |last4=Thompson  | title = Finite Mathematical Structures| edition = 1st | publisher = Prentice-Hall, Inc. | location = Englewood Cliffs, N.J. | year = 1959| id = Library of Congress Card Catalog Number 59-12841}} Chapter 6 "Finite Markov Chains".
 
==External links==
{{Commons category|Finite state machine}}
* {{dmoz|Computers/Computer_Science/Theoretical/Automata_Theory/Finite_State_Automata/|Finite State Automata}}
* [http://web.archive.org/web/20130203032845/http://blog.manuvra.com/modeling-a-simple-ai-behavior-using-a-finite-state-machine/ ''Modeling a Simple AI behavior using a Finite State Machine''] Example of usage in Video Games
* [http://foldoc.doc.ic.ac.uk/foldoc/foldoc.cgi?query=finite+state+machine Free On-Line Dictionary of Computing] description of Finite State Machines
* [http://www.nist.gov/dads/HTML/finiteStateMachine.html NIST Dictionary of Algorithms and Data Structures]  description of Finite State Machines
* [http://teahlab.com/Moore_Finite_State_Machine_Control_Circuit/ Interactive FSM: Control Circuit], demonstrates the logic flow of the Finite State Machines.
* [http://ivanzuzak.info/noam/webapps/fsm_simulator/#tab2 FSM simulator], simulates DFAs, NFAs and ε-NFAs, including generated by regular expression.
{{Formal languages and grammars}}
<!-- [[Finite state machine]] -->
 
{{use dmy dates|date=January 2012}}
 
{{DEFAULTSORT:Finite-State Machine}}
[[Category:Automata theory]]
[[Category:Models of computation]]
[[Category:Digital electronics]]
[[Category:Formal languages]]

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