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# dfa - LANGUAGES AND AUTOMATA Tuesday February 9 2010 DATA...

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LANGUAGES AND AUTOMATA Tuesday, February 9, 2010

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DATA MODEL Σ : a finite alphabet, e.g. { 0 , 1 } , { a, b, c } . string or word : a finite sequence of concatenated symbols of Σ . empty string: length: | 0110 | = 4, | abbccbcac | = 9, | | = 0. 0110 abbccbcac Convention: a, b, c, · · · Σ used for symbols u, v, w, x, y, z used for strings Tuesday, February 9, 2010
STRING OPERATIONS concatenation: x = abbc , y = bccba xy = abbcbccba yx = bccbaabbc If X and Y are two sets of strings, then XY = { xy : x X, y Y } Example. If X = { ab, abca } and Y = { , ab, ca } , then XY = { ab, abab, abca, abcaab, abcaca } Σ n = { w : | w | = n } : set of all strings of length n . We can also define it inductively: Σ n = { } if n = 0 Σ n 1 Σ if n > 0 Example. for Σ = { 0 , 1 } : Σ 0 = { } , Σ 1 = { 0 , 1 } , Σ 2 = { 00 , 01 , 10 , 11 } , . . . Tuesday, February 9, 2010

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KLEENE STAR Σ = n N Σ n NOTE: Σ is infinite and contains arbitrarily long strings but it contains no infinite strings . Example. If Σ = { 0 , 1 } : Σ = { , 0 , 1 , 00 , 01 , 10 , 11 , 000 , 001 , 010 , . . . } If w Σ , then for i N we define w i = if i = 0 ww i 1 if i > 0 We also define: Σ + = n 1 Σ n = ΣΣ Tuesday, February 9, 2010
LANGUAGES Any L Σ is called a language .

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dfa - LANGUAGES AND AUTOMATA Tuesday February 9 2010 DATA...

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