1. Formal Language & Automata Theory
Shyamanta M Hazarika
Computer Sc. & Engineering
Tezpur University

2. 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

3. 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!

4. 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

5. 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.

6. 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 +

7. 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

8. 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.

9. 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

10. 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

11. 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