Regular Languages and Regular Expressions

1. Regular Languages and Regular Expressions • Regular languages • Inductive definitions • Regular expressions • syntax • semantics

2. Regular Languages(Regular Expressions)

3. Regular Languages • New language class • Elements are languages • We will show that this language class is identical to LFSA • Language class to be defined by Finite State Automata (FSA) • Once we have shown this, we will use the term “regular languages” to refer to this language class

4. Inductive Definition of Integers * • Base case definition • 0 is an integer • Inductive case definition • If x is an integer, then • x+1 is an integer • x-1 is an integer • Completeness • Only numbers generated using the above rules are integers

5. Inductive Definition of Regular Languages • Base case definition • Let S denote the alphabet • {} is a regular language • {l} is a regular language • {a} is a regular language for any character a in S • Inductive case definition • If L1 and L2 are regular languages, then • L1 union L2 is a regular language • L1 concatenate L2 is a regular language • L1* is a regular language • Completeness • Only languages generated using above rules are regular languages

6. Proving a language is regular * • Prove that {aa, bb} is a regular language • {a} and {b} are regular languages • base case of definition • {aa} = {a}{a} is a regular language • concatenation rule • {bb} = {b}{b} is a regular language • concatenation rule • {aa, bb} = {aa} union {bb} is a regular language • union rule • Typically, we will not go through this process to prove a language is regular

7. Regular Expressions • How do we describe a regular language? • Use set notation • {aa, bb, ab, ba}* • {a}{a,b}*{b} • Use regular expressions R • Inductive def of regular languages and regular expressions on page 72 • (aa+bb+ab+ba)* • a(a+b)*b

8. R and L(R) * • How we interpret a regular expression • What does a regular expression R mean to us? • aaba represents the regular language {aaba} • f represents the regular language {} • aa+bb represents the regular language {aa, bb} • We use L(R) to denote the regular language represented by regular expression R.

9. Precedence rules • What is L(ab+c*)? • Possible answers: • {a}({b} union {c}*} • ({a}{b,c})* • ({ab} union {c})* • {ab} union {c}* • Must know precedence rules • * first, then concatenation, then +

10. Precedence rules continued • Precedence rules similar to those for arithmetic expressions • ab+c2 • (a times b) + (c times c) • exponentiation first, then multiplication, then addition • Think of Kleene closure as exponentiation, concatenation as multiplication, and union as addition and the precedence rules are identical

11. Regular expressions are strings * • Let L be a regular language over the alphabet S • A regular expression R for L is just a string over the alphabet S union {(, ), +, *, f, l}. • The set of legal regular expressions is itself a language over the alphabet S union {(, ), +, *} • f, a*aba are strings in the language of legal reg. exp. • )(, *a* are strings NOT in the language of legal reg. exp.

12. Semantics * • We give a regular expression R meaning when we interpret it to represent L(R). • aaba is just a string • we interpret it to represent the language {aaba}. • We do similar things with arithmetic expressions • 10+72 is just a string • We interpret this string to represent the number 59

13. Key fact * • A language L is a regular language iff there exists a reg. exp. R such that L(R) = L • When I ask for a proof that a language L is regular, rather than going through the inductive proof we saw earlier, I expect you to give me a regular expression R s.t. L(R) = L

14. Summary • Regular expressions are strings • syntax for legal regular expressions • semantics for interpreting regular expressions • Regular languages are a new language class • A language L is regular iff there exists a regular expression R s.t. L(R) = L • We will show that the regular languages are identical to LFSA