Regular Expressions

1. Regular Expressions Lecture 12 Section 1.3 Mon, Sep 17, 2007

2. Regular Expressions • Regular expressions are like algebraic expressions except that they describe regular languages.

3. Regular Expressions • The basicregular expressions are • The symbols of the alphabet, • , • .

4. Regular Expressions • For a regular expression R, let L(R) be the language of R. • L(a) = {a}. • L() = {}. • L() =.

5. Regular Operators • The regular operators are union, concatenation, and star. • If R1 and R2 are regular expressions, then so are • R1R2, • R1R2, • R1*

6. Regular Operators • For the regular operators, • L(R1R2) = L(R1) L(R2). • L(R1R2) = L(R1) L(R2). • L(R1*) = (L(R1))*.

7. The Empty String • The regular expression  is not essential because L(*) =  . • But we will use it because is so convenient.

8. Examples • If  = {a, b}, then L(ab) = . • Furthermore, (a b)* represents all strings. • We will call this * for short.

9. Examples • Write regular expressions for the following regular sets. • All strings ending with a. • All strings containing aaa. • All strings containing aaa or bbb. • All strings not containing aaa. • See more examples on p. 65.

10. Examples • Let D = {0, 1, 2, …, 9} • Let L = {A, B, …, Z, a, b, …, z} • Write regular expressions for • C++ identifiers • C++ ints • C++ floats

11. Regular Languages and Regular Expressions • Theorem: A language is regular if and only if it is the language of some regular expression. • Proof (): • The basic languages L(a), L(), and L() are regular. • Union, concatenation, and star of regular languages is regular.

12. Regular Languages and Regular Expressions • Proof (): • Let L be a regular language. • We need to construct a regular expression R that describes it. • We will use a “GNFA,” a Generalized NFA

13. Generalized Nondeterministic Finite Automata • A GNFA is like an NFA except that the labels are regular expressions. • To make a transition to another state, we must read a string described by the regular expression that is the label.

14. Converting a DFA to a Regular Expression • Now begin with a transition diagram for L. • Replace each symbol label with the equivalent regular expression label. • Use the language L = {x | x has an even no. of a’s} to illustrate the proof.

15. Converting a DFA to a Regular Expression • Add a new start state that has no transitions into it, but it has one -move to the original start state. • Add a new accept state that has no transitions out of it, but has -moves into it from all of the original accept states. • Remove all states from which the accept state is inaccessible.

16. Converting a DFA to a Regular Expression • Let k be the number of states in the NFA. • Note that k 3. • We will reduce the number of states down to 2, at which point we will have the regular expression.

17. Converting a DFA to a Regular Expression • Choose a non-initial, non-final state q to be removed. • For every state p with a transition into q and for every state r with a transition from q, • Create a transition from p to r.

18. Converting a DFA to a Regular Expression • Let • R1 be the label on the transition p q. • R2 be the label on a loop q  q, if there is one. • R3 be the label on the transition from q  r. • Apply the label R1(R2)*R3 to the new transition.

19. Converting a DFA to a Regular Expression • After doing this for every combination of transitions into q and out of q, remove state q and all of its transitions. • Repeat this process until only the initial and final states remain. • The label on that transition is the regular expression.

20. Example • Find a regular expression for the language of all strings containing a string with an even number of a’s followed by a string with an even number of b’s.