Languages Given an alphabet , we can make a word or string by concatenating the letters of . Concatenation of “x” and “y” is “xy” Typical example: ={0,1}, the possible words over  are the finite bit strings. A language is a set of words.

More about Languages The empty string  is the unique string with zero length. Concatenation of two langauges: A•B = { xy | xA and yB } Typical examples: L = { x | x is a bit string with two zeros } L = { anbn | n  N } L = {1n | n is prime}

A Word of Warning Do not confuse the concatenation of languages with the Cartesian product of sets. For example, let A = {0,00} then A•A = { 00, 000, 0000 } with |A•A|=3, AA = { (0,0), (0,00), (00,0), (00,00) } with |AA|=4

Recognizing Languages Let L be a language  S a machine M recognizes L if “accept” if and only if xL xS M “reject” if and only if xL

Finite Automaton The most simple machine that is not just a finite list of words. “Read once”, “no write” procedure. Typical is its limited memory. Think cell-phone, elevator door, etc.

A Simple Automaton (0) transition rules states starting state q1 q2 q3 1 0,1 starting state accepting state

A Simple Automaton (1) start accept on input “0110”, the machine goes: q1 q2 q3 1 0,1 start accept on input “0110”, the machine goes: q1  q1  q2  q2  q3 = “reject”

A Simple Automaton (2) on input “101”, the machine goes: q1 q2 q3 1 0,1 on input “101”, the machine goes: q1  q2  q3  q2 = “accept”

A Simple Automaton (3) 010: reject 11: accept 010100100100100: accept q1 q2 q3 1 0,1 010: reject 11: accept 010100100100100: accept 010000010010: reject : reject

Finite Automaton (def.) A deterministic finite automaton (DFA) M is defined by a 5-tuple M=(Q,,,q0,F) Q: finite set of states : finite alphabet : transition function :QQ q0Q: start state FQ: set of accepting states

M = (Q,,,q,F) q1 q2 q3 1 0,1 states Q = {q1,q2,q3} 0,1 states Q = {q1,q2,q3} alphabet  = {0,1} start state q1 accept states F={q2} transition function :

Recognizing Languages (def) A finite automaton M = (Q,,,q,F) accepts a string/word w = w1…wn if and only if there is a sequence r0…rn of states in Q such that: 1) r0 = q0 2) (ri,wi+1) = ri+1 for all i = 0,…,n–1 3) rn  F

Regular Languages The language recognized by a finite automaton M is denoted by L(M). A regular language is a language for which there exists a recognizing finite automaton.

THANK YOU THT Prepared By:- Preeti Gupta Deptt. Of Computer Applications Prepared By:-PP