# lect19 - Summary A language L is a set of strings over some...

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1 Summary . A language L is a set of strings over some alphabet Σ . To describe a regular language we can use: set operations regular expressions DFA that recognizes the language. Any of this specification must be unambiguous (but may be not unique) A language can be regular if it satisfies some restrictions. Set notations can be used to specify what strings belong to language.

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2 Example. Find the DFA to recognize the language L , which consists of all strings over the alphabet Σ = { a , b }, L { a , b } * that include a substring aba . Set notation: L = { a , b } * { aba } { a , b } * Regular expression: ( a + b ) * aba ( a + b ) * q 0 b q 1 a q 2 b b q 3 a a , b a DFA
3 Transition function δ : Q × Σ Q δ ( q 0 , a )= q 1 δ ( q 0 , b )= q 0 δ ( q 1 , a )= q 1 δ ( q 1 , b )= q 2 δ ( q 2 , a )= q 3 δ ( q 2 , b )= q 0 δ ( q 3 , a )= q 3 δ ( q 3 , b )= q 3 Configurations for the input string w = aaaabab ( q 0 , aaaabab ) ( q 1 , aaabab ) ( q 1 , aabab ) ( q 1 , abab ) ( q 1 , bab ) ( q 2 , ab ) ( q 3 , b ) ( q 3 , λ ) Q = { q 0 , q 1 , q 2 , q 3 } Σ ={ a , b } q 0 b q 1 a q 2 b b q 3 a a , b a accepting state start state aaaabab L

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4 What language L { a , b } * is recognized by the following DFA? Give regular expression describing the language. 0 a a , b a b b b 1 2 3 a L = λ + aa * b + aa * baa * b + aa * baa * baa * b +… = λ + aa * b + ( aa * b ) 2 + ( aa * b ) 3 +… = ( aa * b ) * L is the set of strings over { a , b } for which each occurrence of b is preceded by at least one a and the only string with no b ’s is λ .
5 Inductive proofs on strings. Usually induction is done on the length of a string | w | =n , or the number of repetition of some pattern. Prove that the regular expression

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lect19 - Summary A language L is a set of strings over some...

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