Regular Languages and Regular Expressions Regular languages Inductive

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

Regular Languages (Regular Expressions) 2

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 3

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 4

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 L 1 and L 2 are regular languages, then • L 1 union L 2 is a regular language • L 1 concatenate L 2 is a regular language • L 1* is a regular language • Completeness – Only languages generated using above rules are regular languages 5

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 6

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 7

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. 8

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 + 9

Precedence rules continued • Precedence rules similar to those for arithmetic expressions – ab+c 2 • (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 10

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. 11

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 12

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 13

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 14