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| In [[computer science]] and [[mathematical logic]], an '''alphabet''' is a non-empty set of ''[[symbol (programming)|symbols]]'' or ''letters'', e.g. characters or digits.<ref>{{Citation |last1=Ebbinghaus |first1=H.-D. |last2=Flum |first2=J. |last3=Thomas |first3=W. |doi= |title=Mathematical Logic |url=http://www.springer.com/mathematics/book/978-0-387-94258-2 |publisher=[[Springer Science+Business Media|Springer]] |location=[[New York City|New York]] |edition=2nd |isbn=0-387-94258-0 |year=1994|page=11|quote=By an ''alphabet'' <math>\mathcal{A}</math> we mean a nonempty set of ''symbols''.}} </ref> For example a common alphabet is {0,1}, the '''binary alphabet'''. A finite [[String (computer science)|string]] is a finite sequence of letters from an alphabet; for instance a binary string is a string drawn from the alphabet {0,1}. An infinite [[Infinite sequence#Infinite sequences in theoretical computer science|sequence]] of letters may be constructed from elements of an alphabet as well.
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| Given an alphabet <math>\Sigma</math>, we write <math>\Sigma^*</math> to denote the set of all finite strings over the alphabet <math>\Sigma</math>. Here, the <math>{}^*</math> denotes the [[Kleene star]] operator, so <math>\Sigma^*</math> is also called the Kleene closure of <math>\Sigma</math>. We write <math>\Sigma^\infty</math> (or occasionally, <math>\Sigma^\N</math> or <math>\Sigma^\omega</math>) to denote the set of all infinite sequences over the alphabet <math>\Sigma</math>.
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| For example, if we use the binary alphabet {0,1}, the strings (ε, 0, 1, 00, 01, 10, 11, 000, etc.) would all be in the Kleene closure of the alphabet (where ε represents the [[empty string]]).
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| Alphabets are important in the use of [[formal languages]], [[Automata theory|automata]] and [[semiautomaton|semiautomata]]. In most cases, for defining instances of automata, such as [[deterministic finite automaton|deterministic finite automata]] (DFAs), it is required to specify an alphabet from which the input strings for the automaton are built.
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| ==See also==
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| *[[Combinatorics on words]]
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| ==References==
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| {{Reflist}}
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| ==Literature==
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| * John E. Hopcroft and Jeffrey D. Ullman, ''[[Introduction to Automata Theory, Languages, and Computation]]'', Addison-Wesley Publishing, Reading Massachusetts, 1979. ISBN 0-201-02988-X.
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| {{DEFAULTSORT:Alphabet (Computer Science)}}
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| [[Category:Formal languages]]
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| [[Category:Combinatorics on words]]
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| {{Comp-sci-theory-stub}}
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