# 3-re - CS 476 3 Regular Expressions 1 Regular expressions...

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CS 476 – 3. Regular Expressions 1 Regular expressions An alternative way to describe regular languages Common in scripting languages (perl, python, etc.) & UNIX utilities (awk, sed, grep, etc.) Defined by the “regular operations”: Union ( or + ) Concatenation ( · or nothing) Star ( * ). * Note: + is similar to star; except that it cannot include empty set: R + = R * - {∅} . Example 0 * 10 * = { w | w contains exactly one 1 } Σ * * = { w | w contains at least one 1 } Σ * 001Σ * = { w | w contains 001 as a substring } (ΣΣ) * = { w | w contains an even number of characters } ΣΣ * = { w | w 6 = } 01 + 11 = { 01 , 11 } = (0 + 1)1 • ∅ * = { } Precedence of operators 1. Star 2. Concatenation 3. Union Formal Definition: R is a regular expression if R is one of the following: a , a Σ • ∅ R 1 + R 2 R 1 · R 2 R * 1

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1.1 Equivalence of REs and FAs Theorem 1.1 A language is regular iff it can be generated by some Regular Expression. This will be proven in two parts: Lemma 1.2 Every language generated by a RE can be generated by some DFA/NFA. Lemma 1.3 Every language generated by a DFA/NFA can be generated by some RE. Proof of Lemma 1.2. We will give an NFA for each of the 6 cases that define a RE. R = a for some a Σ q 0 start q 1 a R = q 0 start R = q 0 start The NFA for the remainder can be constructed just in the same way as the NFA constructions we gave for the regular language operators: R 1 + R 2 R 1 · R 2 R * 1 Example 0 : q 00 start q 01 0 1 : q 10 start q 11 1 01 : q 00 start q 01 q 10 q 11 0 1 2
01 + 0 : q S start q 00 q 01 q 10 q 11 q 0 00 q 0 01 0 1 0 To prove Lemma 1.3, we first define “Generalized NFA” (GNFA) which takes regular expressions for its transitions: e.g: p q (01 + 1) * A GNFA also has the following properties: The start state does not have any incoming transitions. The accepting state (there is only one accepting state) does not have any outgoing transitions. Except for these, there is exactly one transition from every state to every other state. Proof of Lemma 1.3. We will: 1. Convert the DFA into a GNFA (generalized NFA). 2. Reduce the number of states in the GNFS one by one until just 2 states remain. q s start q a R which is equivalent to regular expression R . Definition : A GNFA is a 5-tuple ( Q, Σ , δ, q s , q a ) : Q is a finite set of states.

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• Spring '16
• Can Alkan
• Formal language, Regular expression, Regular language, Nondeterministic finite state machine, qs

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