Deterministic finite automaton: Difference between revisions

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{{Redirect-synonym|DFSA|[[drug-facilitated sexual assault]]}}

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[[File:DFA example multiplies of 3.svg|thumb|upright=1.1|An example of a deterministic finite automaton that accepts only binary numbers that are multiples of 3. The state ''S''₀ is both the start state and an accept state. For example, the string "1001" leads to the state sequence ''S''₀, ''S''₁, ''S''₂, ''S''₁, ''S''₀, and is hence accepted.]]

In the [[theory of computation]], a branch of [[theoretical computer science]], a '''deterministic finite automaton''' ('''DFA''')—also known as '''[[Finite-state machine#Acceptors (recognizers)|deterministic finite acceptor]]''' ('''DFA'''), '''deterministic finite-state machine''' ('''DFSM'''), or '''deterministic finite-state automaton''' ('''DFSA''')—is a [[finite-state machine]] that accepts or rejects a given [[String (computer science)|string]] of symbols, by running through a state sequence uniquely determined by the string.<ref name="{{sfn|Hopcroft 2001">[[#HMU|Hopcroft 2001]]:</ref>Motwani|Ullman|2006}} ''Deterministic'' refers to the uniqueness of the computation run. In search of the simplest models to capture finite-state machines, [[Warren McCulloch]] and [[Walter Pitts]] were among the first researchers to introduce a concept similar to finite automata in 1943.<ref>[[#MP43{{sfn|McCulloch and| Pitts (|1943)]]:</ref><ref>[[#RS59}}{{sfn|Rabin and| Scott (|1959)]]:</ref>}}

The figure illustrates a deterministic finite automaton using a [[state diagram]]. In this example automaton, there are three states: S₀, S₁, and S₂ (denoted graphically by circles). The automaton takes a finite [[sequence]] of 0s and 1s as input. For each state, there is a transition arrow leading out to a next state for both 0 and 1. Upon reading a symbol, a DFA jumps ''deterministically'' from one state to another by following the transition arrow. For example, if the automaton is currently in state S₀ and the current input symbol is 1, then it deterministically jumps to state S₁. A DFA has a ''start state'' (denoted graphically by an arrow coming in from nowhere) where computations begin, and a [[set (mathematics)|set]] of ''accept states'' (denoted graphically by a double circle) which help define when a computation is successful.

A DFA is defined as an abstract mathematical concept, but is often implemented in hardware and software for solving various specific problems such as [[lexical analysis]] and [[pattern matching]]. For example, a DFA can model software that decides whether or not online user input such as email addresses are syntactically valid.<ref>{{Citationcitation

| last1 = Bai | lastfirst1 = GoudaGina R.

| last2 = Clee | first2 = Brian

| first = Prabhakar

| last3 = Shrestha | first3 = Nischal

| title = Application of Finite automata

| last4 = Chapman | first4 = Carl

}}</ref>

| last5 = Wright | first5 = Cimone

| last6 = Stolee | first6 = Kathryn T.

| editor1-last = Guéhéneuc | editor1-first = Yann-Gaël

| editor2-last = Khomh | editor2-first = Foutse

| editor3-last = Sarro | editor3-first = Federica

| contribution = Exploring tools and strategies used during regular expression composition tasks

| contribution-url = https://par.nsf.gov/servlets/purl/10100320

| doi = 10.1109/ICPC.2019.00039

| pages = 197–208

| publisher = IEEE / ACM

| title = Proceedings of the 27th International Conference on Program Comprehension, ICPC 2019, Montreal, QC, Canada, May 25-31, 2019

| year = 2019| isbn = 978-1-7281-1519-1

}}</ref>

DFAs have been generalized to ''[[nondeterministic finite automata]] (NFA)'' which may have several arrows of the same label starting from a state. Using the [[powerset construction]] method, every NFA can be translated to a DFA that recognizes the same language. DFAs, and NFAs as well, recognize exactly the set of [[regular language]]s.<ref name="{{sfn|Hopcroft 2001"/>|Motwani|Ullman|2006}}

==Formal definition==

A deterministic finite automaton {{mvar|M}} is a 5-[[n-tuple|tuple]], {{math|(''Q'', Σ, ''δ'', ''q''₀, ''F'')}}, consisting of

* a finite [[Set (mathematics)|set]] of [[State (computer science)|states]] {{mvar|Q}}

* a finite set of input symbols called the [[Alphabet (computer science)|alphabet]] {{math|Σ}}

* a transition [[function (mathematics)|function]] {{math|''δ'' : ''Q'' × Σ → ''Q''}}

* an initial (or [[Finite-state machine#Start state|start) state]] <math>q_0 \in Q</math>

* a set of [[Finite-stateaccepting machine#Accept .28or(or final.29 states|accept) states]] <math>F \subseteq Q</math>

Let {{math|1=''w'' = ''a''₁''a''₂…...''a_n''}} be a string over the alphabet {{math|Σ}}. The automaton {{mvar|M}} accepts the string {{mvar|w}} if a sequence of states, {{math|''r''₀, ''r''₁, …..., ''r_n''}}, exists in {{mvar|Q}} with the following conditions:

# {{math|1=''r''₀ = ''q''₀}}

# {{math|1=''r''_''i''+1 = ''δ''(''r_i'', ''a''_''i''+1)}}, for {{math|1=''i'' = 0, …..., ''n'' − 1}}

# <math>r_n \in F</math>.

In words, the first condition says that the machine starts in the start state {{math|''q''₀}}. The second condition says that given each character of string {{mvar|w}}, the machine will transition from state to state according to the transition function {{mvar|δ}}. The last condition says that the machine accepts {{mvar|w}} if the last input of {{mvar|w}} causes the machine to halt in one of the accepting states. Otherwise, it is said that the automaton ''rejects'' the string. The set of strings that {{mvar|M}} accepts is the [[Formal language|language]] ''recognized'' by {{mvar|M}} and this language is denoted by {{math|''L''(''M'')}}.

A deterministic finite automaton without accept states and without a starting state is known as a [[transition system]] or [[semiautomaton]].

tlttt..t4;lt5t.t4,;45p66o76bl yl6plypyplyh65y.y[yp56ly.

For more comprehensive introduction of the formal definition see [[automata theory]].

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*{{mvar|δ}} is defined by the following [[state transition table]]:

:{| border="1" cellpadding="1" cellspacing="0"

| || <{{center>|'''0'''</center>}} || <{{center>|'''1'''</center>}}

|-

|'''''S''₁''' || ''S''₂ || ''S''₁

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===Local automata===

A '''local automaton''' is a DFA, not necessarily complete, for which all edges with the same label lead to a single vertex. Local automata accept the class of [[Local language (formal language)|local languages]], those for which membership of a word in the language is determined by a "sliding window" of length two on the word.<ref name=Law129>{{sfn|Lawson (|2004) |p.129</ref><ref name=Sak228>129}}{{sfn|Sakarovitch (|2009) |p.=228</ref>}}

A '''Myhill graph''' over an alphabet ''A'' is a [[directed graph]] with [[Vertex (graph theory)|vertex set]] ''A'' and subsets of vertices labelled "start" and "finish". The language accepted by a Myhill graph is the set of directed paths from a start vertex to a finish vertex: the graph thus acts as an automaton.<ref name{{sfn|Lawson|2004|p=Law129/>129}} The class of languages accepted by Myhill graphs is the class of local languages.<ref name=Law128>{{sfn|Lawson (|2004)| p.=128</ref>}}

===Randomness===

When the start state and accept states are ignored, a DFA of {{mvar|n}} states and an alphabet of size {{mvar|k}} can be seen as a [[Directed graph|digraph]] of {{mvar|n}} vertices in which all vertices have {{mvar|k}} out-arcs labeled {{math|1, …..., ''k''}} (a {{mvar|k}}-out digraph). It is known that when {{math|''k'' ≥ 2}} is a fixed integer, with high probability, the largest [[strongly connected component]] (SCC) in such a {{mvar|k}}-out digraph chosen uniformly at random is of linear size and it can be reached by all vertices.<ref name=Grusho>{{cite journal|last1=Grusho|first1=A. A.|title=Limit distributions of certain characteristics of random automaton graphs|journal=Mathematical Notes of the Academy of Sciences of the USSR|date=1973|volume=4|pages=633–637|doi=10.1007/BF01095785|s2cid=121723743|ref=Grusho1973}}</ref> It has also been proven that if {{mvar|k}} is allowed to increase as {{mvar|n}} increases, then the whole digraph has a phase transition for strong connectivity similar to [[Erdős–Rényi model]] for connectivity.<ref name=Cai>{{cite journal |last1=Cai |first1=Xing Shi |last2=Devroye |first2=Luc |title=The graph structure of a deterministic automaton chosen at random |journal=Random Structures & Algorithms |date=October 2017 |volume=51 |issue=3 |pages=428–458 |doi=10.1002/rsa.20707|arxiv=1504.06238 |s2cid=13013344 }}</ref>

In a random DFA, the maximum number of vertices reachable from one vertex is very close to the number of vertices in the largest [[strongly connected component|SCC]] with high probability.<ref name=Grusho /><ref>{{cite conference |last1=Carayol |first1=Arnaud |last2=Nicaud |first2=Cyril |date=February 2012 |title=Distribution of the number of accessible states in a random deterministic automaton |volume=14 |pages=194–205 |conference=STACS'12 (29th Symposium on Theoretical Aspects of Computer Science) |___location=Paris, France |url=https://hal.archives-ouvertes.fr/hal-00678213}}</ref> This is also true for the largest [[Glossary of graph theory#Subgraphs|induced sub-digraph]] of minimum in-degree one, which can be seen as a directed version of [[Degeneracy (graph theory)#k-Cores|{{math|1}}-core]].<ref name=Cai />

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{{columns-list|colwidth=30em|

*Union

*Intersection{{sfn|Hopcroft|Ullman|1979|pp=59–60}} (see picture)

*Intersection<ref>{{cite book | isbn=0-201-02988-X | author=John E. Hopcroft and Jeffrey D. Ullman | title=Introduction to Automata Theory, Languages, and Computation | url=https://archive.org/details/introductiontoau00hopc | url-access=registration | ___location=Reading/MA | publisher=Addison-Wesley | year=1979 }}</ref>{{rp|59–60}} (see picture)

*Concatenation

*[[Complementation of automata#With deterministic finite automata|Complement]]

*Negation

*[[Kleene closure]]

*Reversal<ref name=rose/>

*Quotient<ref name=rose>{{cite journal

*Init

| last = Rose | first = Gene F.

*Quotient{{citation needed|date=January 2015}}

| doi = 10.1016/S0022-0000(68)80029-7

*Substitution{{citation needed|date=January 2015}}

| issue = 2

*Homomorphism{{citation needed|date=January 2015}}

| journal = [[Journal of Computer and System Sciences]]

| pages = 148–168

| title = Closures which Preserve Finiteness in Families of Languages

| volume = 2

| year = 1968}}</ref>

*Substitution<ref name=spanier>{{cite journal

| last = Spanier | first = E.

| doi = 10.1080/00029890.1969.12000214

| journal = American Mathematical Monthly

| jstor = 2316423

| mr = 241205

| pages = 335–342

| title = Grammars and languages

| volume = 76

| year = 1969| issue = 4

}}</ref>

*Homomorphism<ref name=rose/><ref name=spanier/>

}}

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* the DFA with a minimum number of states for a particular regular language (Minimization Problem)

DFAs are equivalent in computing power to [[nondeterministic finite automata]] (NFAs). This is because, firstly any DFA is also an NFA, so an NFA can do what a DFA can do. Also, given an NFA, using the [[powerset construction]] one can build a DFA that recognizes the same language as the NFA, although the DFA could have exponentially larger number of states than the NFA.<ref name=Sak105>{{sfn|Sakarovitch (|2009) |p.=105</ref><ref name=Law63>}}{{sfn|Lawson (|2004) |p.=63</ref>}} However, even though NFAs are computationally equivalent to DFAs, the above -mentioned problems are not necessarily solved efficiently also for NFAs. The non-universality problem for NFAs is [[PSPACE complete]] since there are small NFAs with shortest rejecting word in exponential size. A DFA is universal if and only if all states are final states, but this does not hold for NFAs. The Equality, Inclusion and Minimization Problems are also PSPACE complete since they require forming the complement of an NFA which results in an exponential blow up of size.<ref>{{Cite web |last1=Esparza Estaun |first1=Francisco Javier |last2=Sickert |first2=Salomon |last3=Blondin |first3=Michael |date=16 November 2016 |orig-date= |title=Operations and tests on sets: Implementation on DFAs |url=https://www7.in.tum.de/um/courses/auto/ws1718/slides1718/04-Implementations_sets.pdf {{Bare|url-status=dead URL|archive-url=https://web.archive.org/web/20180808171506/https://www7.in.tum.de/um/courses/auto/ws1718/slides1718/04-Implementations_sets.pdf PDF|archive-date=March8 August 2018 |website=Automata and Formal Languages 2017/18 2022}}</ref>

On the other hand, finite-state automata are of strictly limited power in the languages they can recognize; many simple languages, including any problem that requires more than constant space to solve, cannot be recognized by a DFA. The classic example of a simply described language that no DFA can recognize is bracket or [[Dyck language]], i.e., the language that consists of properly paired brackets such as word "(()())". Intuitively, no DFA can recognize the Dyck language because DFAs are not capable of counting: a DFA-like automaton needs to have a state to represent any possible number of "currently open" parentheses, meaning it would need an unbounded number of states. Another simpler example is the language consisting of strings of the form ''aⁿbⁿ'' for some finite but arbitrary number of ''a''&apos;{{'}}s, followed by an equal number of ''b''&apos;{{'}}s.<ref name=Law46>{{sfn|Lawson (|2004) |p.=46</ref>}}

==DFA identification from labeled words==

{{main|Induction of regular languages}}

Given a set of ''positive'' words <math>S^+ \subset \Sigma^*</math> and a set of ''negative'' words <math>S^- \subset \Sigma^*</math> one can construct a DFA that accepts all words from <math>S^+</math> and rejects all words from <math>S^-</math>: this problem is called ''DFA identification'' (synthesis, learning).

While ''some'' DFA can be constructed in linear time, the problem of identifying a DFA with the minimal number of states is NP-complete.<ref name="Complexity of Automaton Identificat">{{cite journal|last1=Gold|first1=E. M.|author1-link=E. Mark Gold|title=Complexity of Automaton Identification from Given Data|journal=Information and Control|volume=37|issue=3|pages=302–320|year=1978|doi=10.1016/S0019-9958(78)90562-4|ref=Gold78|doi-access=free}}</ref>

The first algorithm for minimal DFA identification has been proposed by Trakhtenbrot and Barzdin in<ref>{{Cite book | url=https://books.google.com/books?id=h5XOBQAAQBAJ&pg=PP1 |title = Finite Automata: Behavior and Synthesis|isbn = 9781483297293|last1 = De Vries|first1 = A.|date = 28 June 2014| publisher=Elsevier }}</ref> and is called the ''TB-algorithm''.

However, the TB-algorithm assumes that all words from <math>\Sigma</math> up to a given length are contained in either <math>S^+ \cup S^-</math>.

Later, K. Lang proposed an extension of the TB-algorithm that does not use any assumptions about <math>S^+</math> and <math>S^-</math>, the ''Traxbar'' algorithm.<ref>{{Cite book |doi = 10.1145/130385.130390|chapter = Random DFA's can be approximately learned from sparse uniform examples|title = Proceedings of the fifth annual workshop on Computational learning theory - COLT '92|pages = 45–52|year = 1992|last1 = Lang|first1 = Kevin J.|isbn = 089791497X|s2cid = 7480497}}</ref>

However, Traxbar does not guarantee the minimality of the constructed DFA.

In his work<ref name="Complexity of Automaton Identificat"/> E.M. Gold also proposed a heuristic algorithm for minimal DFA identification.

Gold's algorithm assumes that <math>S^+</math> and <math>S^-</math> contain a ''[[Characteristic Samples|characteristic set]]'' of the regular language; otherwise, the constructed DFA will be inconsistent either with <math>S^+</math> or <math>S^-</math>.

Other notable DFA identification algorithms include the RPNI algorithm,<ref>{{Cite book | doi=10.1142/9789812797902_0004| chapter=Inferring Regular Languages in Polynomial Updated Time| title=Pattern Recognition and Image Analysis| volume=1| pages=49–61| series=Series in Machine Perception and Artificial Intelligence| year=1992| last1=Oncina| first1=J.| last2=García| first2=P.| isbn=978-981-02-0881-3}}</ref> the Blue-Fringe evidence-driven state-merging algorithm,<ref>{{Cite book |doi = 10.1007/BFb0054059|chapter = Results of the Abbadingo one DFA learning competition and a new evidence-driven state merging algorithm|title = Grammatical Inference|volume = 1433|pages = 1–12|series = Lecture Notes in Computer Science|year = 1998|last1 = Lang|first1 = Kevin J.|last2 = Pearlmutter|first2 = Barak A.|last3 = Price|first3 = Rodney A.|isbn = 978-3-540-64776-8|url = http://eprints.maynoothuniversity.ie/10250/1/BP-Results-1998.pdf}}</ref>

and Windowed-EDSM.<ref>{{Cite book | url=https://dl.acm.org/doi/abs/10.5555/645519.655966 | title=Beyond EDSM &#124;{{pipe}} Proceedings of the 6th International Colloquium on Grammatical Inference: Algorithms and Applications| date=23 September 2002| pages=37–48| isbn=9783540442394| last1=Adriaans| first1=Pieter| last2=Fernau| first2=Henning| last3=Zaanen| first3=Menno van| publisher=Springer}}</ref>

Another research direction is the application of [[evolutionary algorithmsalgorithm]]s: the smart state labeling evolutionary algorithm<ref>{{Cite journal |doi = 10.1109/TPAMI.2005.143|pmid = 16013754|title = Learning deterministic finite automata with a smart state labeling evolutionary algorithm|journal = IEEE Transactions on Pattern Analysis and Machine Intelligence|volume = 27|issue = 7|pages = 1063–1074|year = 2005|last1 = Lucas|first1 = S.M.|last2 = Reynolds|first2 = T.J.|s2cid = 14062047}}</ref> allowed to solve a modified DFA identification problem in which the training data (sets <math>S^+</math> and <math>S^-</math>) is ''noisy'' in the sense that some words are attributed to wrong classes.

Yet another step forward is due to application of [[Boolean satisfiability problem|SAT]] solvers by [[Marijn Heule|Marjin J. H. Heule]] and S. Verwer: the minimal DFA identification problem is reduced to deciding the satisfiability of a Boolean formula.<ref name=HW1>{{cite conference|last1=Heule|first1=M. J. H.|title=Grammatical Inference: Theoretical Results and Applications |author-link=Marijn Heule|chapter=Exact DFA Identification Using SAT Solvers|series=Lecture Notes in Computer Science |conference=Grammatical Inference: Theoretical Results and Applications. ICGI 2010. Lecture Notes in Computer Science|date=2010|volume=6339|pages=66–79|doi=10.1007/978-3-642-15488-1_7|isbn=978-3-642-15487-4 |ref=Heule2010}}</ref> The main idea is to build aan augmented prefix-tree acceptor (a [[trie]] containing all input words with corresponding labels) based on the input sets and reduce the problem of finding a DFA with <math>C</math> states to ''coloring'' the tree vertices with <math>C</math> states in such a way that when vertices with one color are merged to one state, the generated automaton is deterministic and complies with <math>S^+</math> and <math>S^-</math>.

Though this approach allows finding the minimal DFA, it suffers from exponential blow-up of execution time when the size of input data increases.

Therefore, Heule and Verwer's initial algorithm has later been augmented with making several steps of the EDSM algorithm prior to SAT solver execution: the DFASAT algorithm.<ref>{{Cite journal |doi = 10.1007/s10664-012-9222-z|title = Software model synthesis using satisfiability solvers|journal = Empirical Software Engineering|volume = 18|issue = 4|pages = 825–856|year = 2013|last1 = Heule|first1 = Marijn J. H.|author-link=Marijn Heule|last2 = Verwer|first2 = Sicco|hdl = 2066/103766|s2cid = 17865020|url = https://lirias.kuleuven.be/handle/123456789/370182|hdl-access = free}}</ref>

This allows reducing the search space of the problem, but leads to loss of the minimality guarantee.

Another way of reducing the search space has been proposed inby Ulyantsev et al.<ref>{{Cite book |doi = 10.1007/978-3-319-15579-1_48|chapter = BFS-Based Symmetry Breaking Predicates for DFA Identification|title = Language and Automata Theory and Applications|volume = 8977|pages = 611–622|series = Lecture Notes in Computer Science|year = 2015|last1 = Ulyantsev|first1 = Vladimir|last2 = Zakirzyanov|first2 = Ilya|last3 = Shalyto|first3 = Anatoly|isbn = 978-3-319-15578-4}}</ref> by means of new symmetry breaking predicates based on the [[breadth-first search]] algorithm:

the sought DFA's states are constrained to be numbered according to the BFS algorithm launched from the initial state. This approach reduces the search space by <math>C!</math> by eliminating isomorphic automata.

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'''Read-only right-moving Turing machines''' are a particular type of [[Turing machine]] that only moves right; these

are almost exactly equivalent to DFAs.<ref name=RORMTM>{{cite book | last = Davis| first = Martin |author2=Ron Sigal |author3=Elaine J. Weyuker | title = Second Edition: Computability, Complexity, and Languages and Logic: Fundamentals of Theoretical Computer Science | edition = 2nd | publisher = Academic Press, Harcourt, Brace & Company| ___location = San Diego | year = 1994| ISBNisbn =0-12-206382-1}}</ref>

The definition based on a singly infinite tape is ] a 7-[[tuple]]

: <math>M = \langle Q, \Gamma, b, \Sigma, \delta, q_0, F \rangle,</math>

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==References==

* {{Hopcroft and Ullman 1979|author-link=no|title-link=no}}{{sfn whitelist|CITEREFHopcroftUllman1979}}

* {{cite book

* {{Hopcroft, Motwani, and Ullman 2006}}{{sfn whitelist|CITEREFHopcroftMotwaniUllman2006}}

|last1= Hopcroft |first1= John E. | author-link1=John Hopcroft

|last2=Motwani |first2=Rajeev | author-link2=Rajeev Motwani

|last3=Ullman |first3=Jeffrey D. | author-link3=Jeffrey Ullman

|title= Introduction to Automata Theory, Languages, and Computation

|edition=2

|url=http://www.pearsonhighered.com/educator/product/Introduction-to-Automata-Theory-Languages-and-Computation/9780201441246.page

|access-date=19 November 2012

|publisher= [[Addison Wesley]]

|year= 2001

|isbn= 0-201-44124-1

|ref=HMU}}

* {{cite book | last=Lawson | first=Mark V. | title=Finite automata | publisher=Chapman and Hall/CRC | year=2004 | isbn=1-58488-255-7 | zbl=1086.68074 }}

* {{cite journal

Line 257 ⟶ 277:

|issue= 4

|pages= 115–133

|ref=MP43

|doi= 10.1007/BF02478259

|pmid= 2185863

Line 272 ⟶ 291:

|year= 1959

|pages= 114–125

|ref=RS59

|url=https://www.researchgate.net/publication/230876408|doi= 10.1147/rd.32.0114

}}

* {{cite book | last=Sakarovitch | first=Jacques | title=Elements of automata theory | others=Translated from the French by Reuben Thomas | ___location=Cambridge | publisher=[[Cambridge University Press]] | year=2009 | isbn=978-0-521-84425-3 | zbl=1188.68177 }}

===Further reading===

* {{cite book | first=Michael | last=Sipser | title=Introduction to the Theory of Computation | publisher=PWS | ___location=Boston | year=1997 | isbn=0-534-94728-X | url-access=registration | url=https://archive.org/details/introductiontoth00sips }}. Section 1.1: Finite Automata, pp.&nbsp;31&ndash;47. Subsection "Decidable Problems Concerning Regular Languages" of section 4.1: Decidable Languages, pp.&nbsp;152&ndash;155.4.4 DFA can accept only regular language

* {{Sipser 1997}} — '''1.1''': "Finite Automata" pp.&nbsp;31&ndash;47. '''4.1''': "Decidable Languages - Decidable Problems Concerning Regular Languages" pp.&nbsp;152&ndash;155. '''4.4''': DFA can accept only regular language

{{Formal languages and grammars|state=collapsed}}

{{DEFAULTSORT:Deterministic Finite-State Machine}}

[[Category:Finite-state automatamachines]]