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Finite Automata

CSC 4170 Theory of Computation. Finite Automata. Section 1.1. 1.1.a. How a finite automaton works. 1. q 0. 0. q 2. 1. 1. 0. q 1. 0. 0 1 1 0 0. 1.1.b. The language of a machine. 1. q 0. 0. q 2. 1. 1. 0. q 1. 0.

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Finite Automata

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  1. CSC 4170 Theory of Computation Finite Automata Section 1.1

  2. 1.1.a How a finite automaton works 1 q0 0 q2 1 1 0 q1 0 0 1 1 0 0

  3. 1.1.b The language of a machine 1 q0 0 q2 1 1 0 q1 0 L(M), “the language ofM”, 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) =

  4. 1.1.c Formal definition of a finite automaton • 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.

  5. 1.1.d Our automaton formalized 1 q0 0 q2 Q: : : s: F: 1 1 0 q1 0 A = (Q, , , s, F)

  6. 1.1.e Formal definition of accepting M = (Q, , , s, F) 1 q0 0 q2 1 1 0 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 0 u1 u2 … un 0 1 1 0 0 r1, r2,…, rn, rn+1

  7. 1.1.f Task: Design an automaton that accepts a bit string iff it contains an even number of “1”s. Designing finite automata

  8. 1.1.g 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

  9. 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

  10. 1.1.i 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

  11. 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} Designing finite automata 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?

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