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Fall 2004COMP 3351 Regular Expressions

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Fall 2004COMP 3352 Regular Expressions Regular expressions describe regular languages Example: describes the language

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Fall 2004COMP 3353 Recursive Definition Are regular expressions Primitive regular expressions: Given regular expressions and

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Fall 2004COMP 3354 Examples A regular expression: Not a regular expression:

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Fall 2004COMP 3355 Languages of Regular Expressions : language of regular expression Example:

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Fall 2004COMP 3356 Definition For primitive regular expressions :

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Fall 2004COMP 3357 Definition (continued) For regular expressions and

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Fall 2004COMP 3358 Example Regular expression:

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Fall 2004COMP 3359 Example Regular expression

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Fall 2004COMP 33510 Example Regular expression

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Fall 2004COMP 33511 Example Regular expression = {all strings with at least two consecutive 0}

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Fall 2004COMP 33512 Example Regular expression = { all strings without two consecutive 0 }

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Fall 2004COMP 33513 Equivalent Regular Expressions Definition: Regular expressions and are equivalent if

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Fall 2004COMP 33514 Example = { all strings without two consecutive 0 } and are equivalent Reg. expressions

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Fall 2004COMP 33515 Regular Expressions and Regular Languages

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Fall 2004COMP 33516 Theorem Languages Generated by Regular Expressions Regular Languages

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Fall 2004COMP 33517 Theorem - Part 1 1. For any regular expression the language is regular Languages Generated by Regular Expressions Regular Languages

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Fall 2004COMP 33518 Theorem - Part 2 Languages Generated by Regular Expressions Regular Languages 2. For any regular language, there is a regular expression with

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Fall 2004COMP 33519 Proof - Part 1 1. For any regular expression the language is regular Proof by induction on the size of

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Fall 2004COMP 33520 Induction Basis Primitive Regular Expressions: NFAs regular languages

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Fall 2004COMP 33521 Inductive Hypothesis Assume for regular expressions and that and are regular languages

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Fall 2004COMP 33522 Inductive Step We will prove: are regular Languages.

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Fall 2004COMP 33523 By definition of regular expressions:

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Fall 2004COMP 33524 By inductive hypothesis we know: and are regular languages Regular languages are closed under: Union Concatenation Star We also know:

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Fall 2004COMP 33525 Therefore: Are regular languages

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Fall 2004COMP 33526 And trivially: is a regular language

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Fall 2004COMP 33527 Proof – Part 2 2. For any regular language there is a regular expression with Proof by construction of regular expression

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Fall 2004COMP 33528 Since is regular, take an NFA that accepts it Single final state

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Fall 2004COMP 33529 From, construct an equivalent Generalized Transition Graph in which transition labels are regular expressions Example:

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Fall 2004COMP 33530 Another Example:

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Fall 2004COMP 33531 Reducing the states:

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Fall 2004COMP 33532 Resulting Regular Expression:

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Fall 2004COMP 33533 In General Removing states:

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Fall 2004COMP 33534 The final transition graph: The resulting regular expression:

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