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

Regular Language When we say: We are given a Regular Language We mean: Language is in a standard representation

Standard Representations Regular Languages DFAs Regular Grammars NFAs Regular Expressions

Membership Question Question: Given regular language and string how can we check if ? Answer: Take the DFA that accepts and check if is accepted

Membership Question DFA DFA

Elementary Question Question: Given regular language how can we check if is empty: ? Take the DFA that accepts Check if there is a path from the initial state to a final state Answer:

Elementary Question DFA

Elementary Question Question: Given regular language how can we check if is finite? Answer: Take the DFA that accepts Check if there is a walk with cycle from the initial state to a final state

Elementary Question DFA is infinite DFA is finite

Elementary Question Question: Given regular languages and how can we check if ? Answer: Find if

Elementary Question and

Elementary Question or

Non-regular languages

Prove language is Not Regular How can we prove that a language is not regular? Prove that there is no DFA that accepts Problem: this is not easy to prove Solution: Pumping Lemma !!!

The Pigeonhole Principle pigeons pigeonholes

The Pigeonhole Principle A pigeonhole must contain at least two pigeons

The Pigeonhole Principle pigeons ........... pigeonholes ...........

The Pigeonhole Principle pigeons pigeonholes There is a pigeonhole with at least 2 pigeons ...........

Pigeonhole and DFA DFA with states

Pigeonhole and DFA In walks of strings: no state is repeated

Pigeonhole and DFA a state is repeated In walks of strings:

Pigeonhole and DFA If the walk of string has length then a state is repeated

Pigeonhole principle for any DFA If in a walk of a string transitions states of DFA then a state is repeated