Formal Language & Automata Theory Shyamanta M Hazarika Computer Sc. & Engineering Tezpur University http://www.tezu.ernet.in/~smh

Inductive Definition of Regular Languages Base case definition Let S denote the alphabet {} is a regular language {} 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

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 {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. Use Pumping Lemma!

Regular Language using regex 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 (aa+bb+ab+ba)* a(a+b)*b

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.

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 +

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

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,} which follows certain syntactic rules That is, 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.

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

Key fact A language L is a regular language iff there exists a Reg. Exp. R such that L(R) = L For a proof that a language L is regular, rather than going through the inductive proof we saw earlier, give a regular expression R s.t. L(R) = L

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