CSC 4170 Theory of Computation Finite Automata Section 1.1

How a finite automaton works 1 q0 q2 1 1 q1 0 1 1 0 0

The language of a machine 1.1.b 1 q0 q2 1 1 q1 L(M), “the language of M”, or “the language recognized by M” --- the set all strings that the machine M accepts What is the language recognized by our automaton A? L(A) =

Formal definition of a finite automaton 1.1.c A finite automaton is a 5-tuple (Q, , , s, F), where: Q is a finite set called the states,  is a finite set called the alphabet,  is a function of the type Q  Q called the transition function, s is an element of Q called the start state, F is a subset of Q called the set of accept states.

Our automaton formalized 1 q0 q2 Q: : : s: F: 1 1 q1 A = (Q, , , s, F)

Formal definition of accepting M = (Q, , , s, F) 1 q0 q2 1 1 q1 M accepts the string u1 u2 … un iff there is a sequence r1, r2, …, rn, rn+1 of states such that: r1=s ri+1 = (ri,ui), for each i with 1 i  n rn+1  F u1 u2 … un 0 1 1 0 0 r1, r2, …, rn, rn+1

Designing finite automata Task: Design an automaton that accepts a bit string iff it contains an even number of “1”s.

Designing finite automata Task: Design an automaton that accepts a bit string iff the number of “1”s that it contains is divisible by 3.

Designing finite automata 1.1.h Task: Let L2={w | w is a string of 0s whose length is divisible by 2} and L3={w | w is a string of 0s whose length is divisible by 3} Design an automaton that recognizes L2L3

Designing finite automata Task: Let L2={w | w is a string of 0s whose length is divisible by 2} and L3={w | w is a string of 0s whose length is divisible by 3} Design an automaton that recognizes L2L3

Designing finite automata 1.1.j Task: Design an automaton that recognizes the language X={w | w is a string of 0s whose length is divisible neither by 2 nor by 3} Definition: Let L be a language over an alphabet . The complement of L is the language {w | w is a string over  such that wL}. X is the complement of what language?