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Automata
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‘0101010’
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01111?
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1100?
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11010?
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000?
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00110?
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AcceptNot Accept 010101001111 1100000 1101000110
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AcceptNot Accept 0101010010101001111 1100000 1101000110
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Accept 되는 가장 짧은 String?
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‘’
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ε
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(Q, Σ, δ, q 0, A)
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a finite set of states.
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(Q, Σ, δ, q 0, A) a finite set of states.
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(Q, Σ, δ, q 0, A) a finite set of states. {s 1, s 2 }
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(Q, Σ, δ, q 0, A)
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a finite set of symbols
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(Q, Σ, δ, q 0, A) a finite set of symbols : alphabet
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(Q, Σ, δ, q 0, A) a finite set of symbols : alphabet
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(Q, Σ, δ, q 0, A) a finite set of symbols : alphabet {0, 1}
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(Q, Σ, δ, q 0, A)
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the transition function
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(Q, Σ, δ, q 0, A) the transition function : δ: Q × Σ → Q
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(Q, Σ, δ, q 0, A) the transition function : δ: Q × Σ → Q
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(Q, Σ, δ, q 0, A) the transition function : δ: Q × Σ → Q { ((s 1, 0), s 2 ), ((s 1, 1), s 1 ), ((s 2, 0), s 1 ), ((s 2, 1), s 2 ) }
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(Q, Σ, δ, q 0, A) the transition function : δ: Q × Σ → Q 01 s1s1 s2s2 s1s1 s2s2 s1s1 s2s2
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(Q, Σ, δ, q 0, A)
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the start state
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(Q, Σ, δ, q 0, A) the start state
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(Q, Σ, δ, q 0, A) the start state s1s1
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(Q, Σ, δ, q 0, A)
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accept states
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(Q, Σ, δ, q 0, A) accept states
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(Q, Σ, δ, q 0, A) accept states {s 1 }
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(Q, Σ, δ, q 0, A) : {s 1, s 2 } : {0, 1} : {((s 1, 0), s 2 ), ((s 1, 1), s 1 ), ((s 2, 0), s 1 ), ((s 2, 1), s 2 ) } : s 1 : {s 1 } QΣδq0AQΣδq0A
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(Q, Σ, δ, q 0, A) QΣδq0AQΣδq0A
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: {1, 2, 3, 4, 0} : {a, b, c} : { ((1, a), 2), ((1, b), 0), ((1, c), 0), ((2, a), 0), ((2, b), 3), ((2, c), 0), ((3, a), 0), ((3, b), 0), ((3, c), 4), ((4, a), 2), ((4, b), 0), ((4, c), 4) } : 1 : {4} QΣδq0AQΣδq0A ?
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(Q, Σ, δ, q 0, A) : {1, 2, 3, 4, 0} : {a, b, c} : { ((1, a), 2), ((1, b), 0), ((1, c), 0), ((2, a), 0), ((2, b), 3), ((2, c), 0), ((3, a), 0), ((3, b), 0), ((3, c), 4), ((4, a), 2), ((4, b), 0), ((4, c), 4), ((0, a), 0), ((0, b), 0), ((0, c), 0) } : 1 : {4} QΣδq0AQΣδq0A
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(Q, Σ, δ, q 0, A) : {1, 2, 3, 4, 0} : {a, b, c} : : 1 : {4} QΣδq0AQΣδq0A abc 1200 2030 3004 4204 0000
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(Q, Σ, δ, q 0, A) : {1, 2, 3, 4, 0} : {a, b, c} : : 1 : {4} QΣδq0AQΣδq0A abc 12 23 34 424
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Dead state
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(Q, Σ, δ, q 0, A) : {q 0, q 1, q 2, q 3, q 4 } : {0, 1} : { ((q 0, 0), q 0 ), ((q 0, 1), q 0 ), ((q 0, 0), q 3 ), ((q 0, 1), q 1 ), ((q 3, 0), q 4 ), ((q 1, 1), q 2 ), ((q 4, 0), q 4 ), ((q 4, 1), q 4 ), ((q 2, 0), q 2 ), ((q 2, 1), q 2 ) } : q 0 : {q 2, q 4 } QΣδq0AQΣδq0A
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(Q, Σ, δ, q 0, A) : {q 0, q 1, q 2, q 3, q 4 } : {0, 1} : : q 0 : {q 2, q 4 } QΣδq0AQΣδq0A 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q4q4 q4q4 q4q4 q4q4
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(Q, Σ, δ, q 0, A) the transition function : δ: Q × Σ → Q
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Q × Σ : { (q 0, 0), (q 0, 1), (q 1, 0), (q 1, 1), (q 2, 0), (q 2, 1), (q 3, 0), (q 3, 1), (q 4, 0), (q 4, 1) } δ : { ((q 0, 0), q 0 ), ((q 0, 1), q 0 ), ((q 0, 0), q 3 ), ((q 0, 1), q 1 ), ((q 3, 0), q 4 ), ((q 1, 1), q 2 ), ((q 4, 0), q 4 ), ((q 4, 1), q 4 ), ((q 2, 0), q 2 ), ((q 2, 1), q 2 ) }
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q0 q3 0 0 둘 다 가능 ! 두 개의 경로로 다 해본다. 하나라도 accept 되면 accept.
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‘011’ : q0 → q3 Not Accept
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‘011’ : q0 → q0 → q1 → q2
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‘001’ : q0 → q0 → q0 → q3 Not Accept
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‘001’ : q0 → q3 → q4 → q4
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‘001’ : q0 → q3 → q4 → q4 ‘011’ : q0 → q0 → q1 → q2
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‘001’ : q0 → q3 → q4 → q4 ‘011’ : q0 → q0 → q1 → q2
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‘001’ : q0 → q3 → q4 → q4 ‘011’ : q0 → q0 → q1 → q2 Nondeterministic
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Nondeterministic Finite Automata
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No Input
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NFA
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DFA
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Deterministic Finite Automata
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NFA ⊃ DFA
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NFA ⇒ DFA
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Nondeterministic Finite Automata
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⇒ Deterministic Finite Automata
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01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q4q4 q4q4 q4q4 q4q4
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01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q4q4 q4q4 q4q4 q4q4
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0 1 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2
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0 1 01 q0q0 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2
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01 q0q0 q 0, q 3 q0, q1q0, q1 01 q0q0 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2 0 1
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01 q0q0 q 0, q 3 q0, q1q0, q1 q0, q1q0, q1 01 q0q0 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2 0 1
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01 q0q0 q 0, q 3 q0, q1q0, q1 q 0, q 2, q 3 q0, q1q0, q1 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2 0 1
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01 q0q0 q 0, q 3 q0, q1q0, q1 q 0, q 2, q 3 q0, q1q0, q1 q0, q1q0, q1 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2 0 1
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01 q0q0 q 0, q 3 q0, q1q0, q1 q 0, q 2, q 3 q0, q1q0, q1 q0, q1q0, q1 q0, q3q0, q3 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2 0 1
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01 q0q0 q 0, q 3 q0, q1q0, q1 q 0, q 2, q 3 q0, q1q0, q1 q0, q1q0, q1 q0, q3q0, q3 q 0, q 1, q 2 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2 0 1
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01 q0q0 q 0, q 3 q0, q1q0, q1 q 0, q 2, q 3 q0, q1q0, q1 q0, q1q0, q1 q0, q3q0, q3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2 0 1
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01 q0q0 q 0, q 3 q0, q1q0, q1 q 0, q 2, q 3 q0, q1q0, q1 q0, q1q0, q1 q0, q3q0, q3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2 0 1
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01 q0q0 q 0, q 3 q0, q1q0, q1 q 0, q 2, q 3 q0, q1q0, q1 q0, q1q0, q1 q0, q3q0, q3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2 0 1
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01 q0q0 q 0, q 3 q0, q1q0, q1 q 0, q 2, q 3 q0, q1q0, q1 q0, q1q0, q1 q0, q3q0, q3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2 0 1
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01 q0q0 q 0, q 3 q0, q1q0, q1 q 0, q 2, q 3 q0, q1q0, q1 q0, q1q0, q1 q0, q3q0, q3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2 0 1
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0 1 01 q0q0 q 0, q 3 q0, q1q0, q1 q 0, q 2, q 3 q0, q1q0, q1 q0, q1q0, q1 q0, q3q0, q3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2
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0 1 01 q0q0 q 0, q 3 q0, q1q0, q1 q 0, q 2, q 3 q0, q1q0, q1 q0, q1q0, q1 q0, q3q0, q3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2
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0 1 01 q0q0 q 0, q 3 q0, q1q0, q1 q 0, q 2, q 3 q0, q1q0, q1 q0, q1q0, q1 q0, q3q0, q3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 01 q0q0 q 0, q 3 q 0, q 1 q1q1 q2q2 q2q2 q2q2 q2q2 q3q3 q2q2
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01 q0q0 q 0, q 3 q0, q1q0, q1 q 0, q 2, q 3 q0, q1q0, q1 q0, q1q0, q1 q0, q3q0, q3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2 q 0, q 2, q 3 q 0, q 1, q 2
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01 {q 0 }{q 0, q 3 }{q 0, q 1 } {q 0, q 3 }{q 0, q 2, q 3 }{q 0, q 1 } {q 0, q 3 }{q 0, q 1, q 2 } {q 0, q 2, q 3 } {q 0, q 1, q 2 } {q 0, q 2, q 3 }{q 0, q 1, q 2 }
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1 0 {q 0, q 3 } {q 0 } {q 0, q 1 } {q 0, q 2, q 3 } {q 0, q 1, q 2 } 1 0 0 1 01 {q 0 }{q 0, q 3 }{q 0, q 1 } {q 0, q 3 }{q 0, q 2, q 3 }{q 0, q 1 } {q 0, q 3 }{q 0, q 1, q 2 } {q 0, q 2, q 3 } {q 0, q 1, q 2 } {q 0, q 2, q 3 }{q 0, q 1, q 2 }
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1 0 {q 0, q 3 } {q 0 } {q 0, q 1 } {q 0, q 2, q 3 } {q 0, q 1, q 2 } 1 0 0 1 01 {q 0 }{q 0, q 3 }{q 0, q 1 } {q 0, q 3 }{q 0, q 2, q 3 }{q 0, q 1 } {q 0, q 3 }{q 0, q 1, q 2 } {q 0, q 2, q 3 } {q 0, q 1, q 2 } {q 0, q 2, q 3 }{q 0, q 1, q 2 }
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01 {q 0 }{q 0, q 3 }{q 0, q 1 } {q 0, q 3 }{q 0, q 2, q 3 }{q 0, q 1 } {q 0, q 3 }{q 0, q 2, q 3 } 1 0 {q 0, q 3 } {q 0 } {q 0, q 1 } {q 0, q 2, q 3 } {q 0, q 1, q 2 } 1 0 0 1 0 1 {q 0, q 2, q 3 }
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0 1 1 0 {q 0, q 3 } {q 0 } {q 0, q 1 } {q 0, q 2, q 3 } {q 0, q 1, q 2 } 1 0 0 1 0 1 {q 0, q 2, q 3 } NFA DFA
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abc
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abccccccccccc
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abcccccccccccabc
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abcc...cabcc...cabccc...cc
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(abc + ) +
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Regular Expression
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|, (), *, +, ε
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OR
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|, (), *, +, ε OR gray|grey -> {‘gray’, ‘grey’}
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|, (), *, +, ε
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scope
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|, (), *, +, ε scope gray|grey = gr(a|e)y
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|, (), *, +, ε
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zero or more of the preceding element
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|, (), *, +, ε zero or more of the preceding element ab*c →
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|, (), *, +, ε zero or more of the preceding element ab*c → {"ac", "abc", "abbc", "abbbc", …}
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|, (), *, +, ε
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one or more of the preceding element
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|, (), *, +, ε one or more of the preceding element ab + c →
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|, (), *, +, ε one or more of the preceding element ab + c → {"abc", "abbc", "abbbc", …}, but not "ac”
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|, (), *, +, ε one or more of the preceding element ab + c = abb*c
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|, (), *, +, ε one or more of the preceding element ab + c = abb*c = ab*bc
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|, (), *, +, ε
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empty string
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|, (), *, +, ε empty string : “”
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|, (), *, +, ε empty string : “” a(b|ε)c →
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|, (), *, +, ε empty string : “” a(b|ε)c → {"abc", "ac"}
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abc +
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(abc + ) +
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((0|1)*00(0|1)*) | ((0|1)*11(0|1)*)
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(0|1)* (00|11) (0|1)*
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( 1 | 01*0 )*
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{01, 0011, 000111, 00001111,…}
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0n1n0n1n
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Is it Regular?? 0n1n0n1n
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0 1 {01}
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0 1 0 1 1 {01, 0011}
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0 1 0 1 1 0 1 1 {01, 0011, 000111}
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0 1 0 1 1 0 1 1 0 1 1 {01, 0011, 000111, 00001111}
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{01, 0011, 000111, 00001111, …} 0 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1
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0 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 cannot be FINITE automata 0n1n0n1n
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NOT regular 0 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0n1n0n1n
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Pumping Lemma
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concept
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Finite Automata Infinite Sentence &
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0 1 0 1 1 ={01, 0011} L(01|0011)
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1 ={01, 011, 0111, 01111, …} 0 L(01*)
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1 ={01, 011, 0111, 01111, …} 0 L(01*)
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1 0 ={0, 10, 00, 110, 0100, …} L( (0|1)*0 ) 1 0
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L( a(bc)*d ) ={ad, abcd, abcbcd, …}
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Pumping Lemma
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∀ : for all ∃ : exist
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p a(bc)*d If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p a(bc)*d If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 a(bc)*d If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 a(bc)*d If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w = {abcd, abcbcd, …} a(bc)*d If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w = {abcd, abcbcd, …} a(bc)*d If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = d If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = d If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = d | y | = 2 If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = d If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = d xy = abc | xy |=3 If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = d If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = d xy i z = a(bc)*d If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w = {abcd, abcbcd, …} a(bc)*d If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = bcd If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = bcd If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = bcd | y | = 2 If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = bcd If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = bcd xy = abc | xy |=3 If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = bcd If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = bcd xy i z = a(bc) + d If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p p=3 w ∈ {abcd, abcbcd, …} a(bc)*d x = a y = bc z = d If L is regular
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Pumping Lemma Why?
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Pumping Lemma to prove not regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p If L is regular
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p L is not regular if
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∃ p ≥ 1 s.t. ∀ w ∈ L, | w | > p L is not regular if
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if
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w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if
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∀ x, y, z w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if
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∀ x, y, z w = xyz | y | ≥ 1 | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if
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∀ x, y, z w = xyz | y | = 0 or | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if
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∀ x, y, z w = xyz | y | = 0 or | xy | ≤ p ∀ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if
211
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∀ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if
212
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∀ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if
213
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if
214
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if
215
Pumping Lemma to prove not regular
216
0n1n0n1n
217
0n1n0n1n ∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if
218
0n1n0n1n ∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if
219
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n
220
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n
221
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p
222
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p
223
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p ⅰ)ⅱ)ⅰ)ⅱ)
224
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p ⅰ ) x = 0 p y = ε z = 0 p-m-n 1 p ⅱ )
225
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p ⅰ ) x = 0 p y = ε z = 0 p-m-n 1 p ⅱ )
226
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p ⅰ ) x = 0 m y = 0 n (n>0) z = 0 p-m-n 1 p ⅱ )
227
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p ⅰ ) x = 0 m y = 0 n (n>0) z = 0 p-m-n 1 p ⅱ )
228
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p ⅰ ) x = 0 m y = 0 n (n>0) z = 0 p-m-n 1 p ⅱ )
229
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p ⅰ ) x = 0 m y = 0 n (n>0) z = 0 p-m-n 1 p ⅱ ) i =0, xy i z =
230
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p ⅰ ) x = 0 m y = 0 n (n>0) z = 0 p-m-n 1 p ⅱ ) i =0, xy i z = 0 p-n 1 p
231
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p ⅰ ) x = 0 m y = 0 n (n>0) z = 0 p-m-n 1 p ⅱ ) i =0, xy i z = 0 p-n 1 p ∈ L (0 n 1 n )
232
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p ⅰ ) ⅱ ) x = 0 p y = 1 n (n>0) z = 1 p-n
233
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p ⅰ ) ⅱ ) ?
234
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p ⅰ ) x = 0 p y = 0 n (n>0) z = 0 p-m-n 1 n
235
∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p 0n1n0n1n let w =0 p 1 p ⅰ ) x = 0 p y = 0 n (n>0) z = 0 p-m-n 1 n ⅱ ) other wise | y | = 0 or | xy | > p
236
(01) n 1 n
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∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p (01) n 1 n let w =(01) p 1 p ⅰ ) x = (01) m y = (01) n (n>0) z = (01) p-m-n 1 p ⅱ ) i =0, xy i z = (01) p-n 1 p
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∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p (01) n 1 n let w =(01) p 1 p ⅰ ) ⅱ ) x = (01) m y = (01) n 0 (n>0) z = 1(01) p-m-n-1 1 p i =0, xy i z = (01) p-n 1 p
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∀ x, y, z w = xyz | y | = 0 or | xy | > p or ∃ i ≥ 0, xy i z ∈ L ∀ p ≥ 1 ∃ w ∈ L, | w | > p s.t. L is not regular if assume p (01) n 1 n let w =(01) p 1 p ⅰ)ⅱ)ⅲ)ⅳ)ⅰ)ⅱ)ⅲ)ⅳ) i =0, xy i z = (01) p-n 1 p
240
? regular
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Context Free regular
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Context Free Grammar
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S → 0S1 S → ε
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S → 0S1 S → ε L(S) = 0 n 1 n
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S → 0S1 | ε L(S) = 0 n 1 n
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S → S + S S → S - S S → S * S S → S / S S → ( S )
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S → x | y | z S → S + S S → S - S S → S * S S → S / S S → ( S )
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S → A S → S + S S → S - S S → S * S S → S / S S → ( S ) A → 0|1|2|3|4|5|6|7|8|9
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S → A S → S + S S → S - S S → S * S S → S / S S → ( S ) A → 0|1|2|3|4|5|6|7|8|9|AA
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S → B S → S + S S → S - S S → S * S S → S / S S → ( S ) A → 0|1|2|3|4|5|6|7|8|9|AA|ε B → 0 | (1|2|3|4|5|6|7|8|9)A
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S → NP VP NP → A N | N VP → V NP N → � dogs | cats V � → like A � → nice
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(V, Σ, R, S)
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a finite set of non-terminal characters
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(V, Σ, R, S) a finite set of variables
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(V, Σ, R, S) S → NP VP NP → A N | N VP → V NP N → � dogs | cats V � → like A � → nice a finite set of variables
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(V, Σ, R, S) a finite set of variables {S, NP, VP, N, V, A} S → NP VP NP → A N | N VP → V NP N → � dogs | cats V � → like A � → nice
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(V, Σ, R, S)
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a finite set of terminals
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(V, Σ, R, S) a finite set of terminals S → NP VP NP → A N | N VP → V NP N → � dogs | cats V � → like A � → nice
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(V, Σ, R, S) a finite set of terminals {dogs, cats, like, nice} S → NP VP NP → A N | N VP → V NP N → � dogs | cats V � → like A � → nice
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(V, Σ, R, S)
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a finite relation from V to (V ∪ Σ)*
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(V, Σ, R, S) a finite rewrite rules(productions)
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(V, Σ, R, S) a finite rewrite rules(productions) S → NP VP NP → A N | N VP → V NP N → � dogs | cats V � → like A � → nice
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(V, Σ, R, S) a finite rewrite rules(productions) S → NP VP NP → A N | N … A � → nice S → NP VP NP → A N | N VP → V NP N → � dogs | cats V � → like A � → nice
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(V, Σ, R, S)
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a start variable
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(V, Σ, R, S) a start variable S → NP VP NP → A N | N VP → V NP N → � dogs | cats V � → like A � → nice
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(V, Σ, R, S) a start variable S S → NP VP NP → A N | N VP → V NP N → � dogs | cats V � → like A � → nice
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PDA
273
Push Down Automata
274
Pushdown Automata
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State
279
Input Character
280
State
281
Input Character State Change in Stack
282
Input Character State Change in Stack
336
(Q, Σ, Γ, δ, q 0, Z, F)
338
a finite set of states
339
(Q, Σ, Γ, δ, q 0, Z, F) a finite set of states
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(Q, Σ, Γ, δ, q 0, Z, F) a finite set of states {p, q, r}
341
(Q, Σ, Γ, δ, q 0, Z, F)
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a finite set of input character
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(Q, Σ, Γ, δ, q 0, Z, F) input alphabet
344
(Q, Σ, Γ, δ, q 0, Z, F) input alphabet
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(Q, Σ, Γ, δ, q 0, Z, F) {0, 1} input alphabet
346
(Q, Σ, Γ, δ, q 0, Z, F)
347
a finite set of stack character
348
(Q, Σ, Γ, δ, q 0, Z, F) stack alphabet
349
(Q, Σ, Γ, δ, q 0, Z, F) stack alphabet
350
(Q, Σ, Γ, δ, q 0, Z, F) stack alphabet {A, Z}
351
(Q, Σ, Γ, δ, q 0, Z, F)
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Transition relation: Q x (Σ ∪ {ε}) x Г →Г*
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(Q, Σ, Γ, δ, q 0, Z, F) Transition relation: Q x (Σ ∪ {ε}) x Г →Г*
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(Q, Σ, Γ, δ, q 0, Z, F) Transition relation: Q x (Σ ∪ {ε}) x Г →Г* ((p,0,Z), (p,AZ)) ((p,0,A), (p,AA))
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(Q, Σ, Γ, δ, q 0, Z, F) Transition relation: Q x (Σ ∪ {ε}) x Г →Г* ((p,0,Z), (p,AZ)) ((p,0,A), (p,AA)) ((p,ε,A), (q,A)) ((p,ε,Z), (q,Z))
356
(Q, Σ, Γ, δ, q 0, Z, F) Transition relation: Q x (Σ ∪ {ε}) x Г →Г* ((p,0,Z), (p,AZ)) ((p,0,A), (p,AA)) ((p,ε,A), (q,A)) ((p,ε,Z), (q,Z)) ((q,1,A), (q,ε))
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(Q, Σ, Γ, δ, q 0, Z, F) Transition relation: Q x (Σ ∪ {ε}) x Г →Г* ((p,0,Z), (p,AZ)) ((p,0,A), (p,AA)) ((p,ε,A), (q,A)) ((p,ε,Z), (q,Z)) ((q,1,A), (q,ε)) ((q,ε,Z), (r,Z))
358
(Q, Σ, Γ, δ, q 0, Z, F)
359
start state
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(Q, Σ, Γ, δ, q 0, Z, F) start state
361
(Q, Σ, Γ, δ, q 0, Z, F) start state p
362
(Q, Σ, Γ, δ, q 0, Z, F)
363
initial stack symbol
364
(Q, Σ, Γ, δ, q 0, Z, F) initial stack symbol
365
(Q, Σ, Γ, δ, q 0, Z, F) initial stack symbol Z
366
(Q, Σ, Γ, δ, q 0, Z, F)
367
accepting states
368
(Q, Σ, Γ, δ, q 0, Z, F) accepting states
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(Q, Σ, Γ, δ, q 0, Z, F) accepting states {r}
370
0n1n0n1n
371
000….0111….1
372
Z
373
Z
374
A:AAZA:AAZ Z
375
A:AAZA:AAZ
376
A:AAZA:AAZ Z
377
0n1n2n0n1n2n
378
000…0111…1222…2
379
Z
380
Z
381
A:AAZA:AAZ Z
382
A:AAZA:AAZ
383
Z A:AAZA:AAZ
384
Z
385
Z
386
Z
387
Z Z
388
Z Z
389
A:AAZA:AAZ Z Z Z
390
A:AAZA:AAZ Z
391
Z A:AAZA:AAZ Z
392
Z B:BBZB:BBZ A:AAZA:AAZ Z
393
Z B:BBZB:BBZ
394
Z Z Z B:BBZB:BBZ
395
Z Z
396
0n1n2n0n1n2n
397
Turing Machine
398
000111222
399
bbbbbbbbb000111222bbbbbbbbb
400
bbbbbbbbq 0 000111222bbbbbbbb
401
q 0 000111222
402
0q 1 00111222
403
00q 1 0111222
404
000q 1 111222
405
0001q 1 11222
406
00011q 1 1222
407
000111q 1 222
408
000111q 2 222
409
00011q 3 3222
410
0001q 3 13222
411
000q 3 113222
412
00q 3 0113222
413
0q 3 00113222
414
q 3 000113222
415
q 4 000113222
416
bq 1 00113222
417
b0q 1 0113222
418
b00q 1 113222
419
b001q 1 13222
420
b0011q 1 3222
421
b0011q 2 3222
422
b001q 3 33222
423
b00q 3 133222
424
b0q 3 0133222
425
bq 3 00133222
426
bq 4 00133222
427
bbq 1 0133222
428
bb0q 1 133222
429
bb01q 1 33222
430
bb01q 2 33222
431
bb0q 3 333222
432
bbq 3 0333222
433
bbq 4 0333222
434
bbbq 1 333222
435
bbbq 2 333222
436
bbbq 5 333222
437
bbb333222q 5
438
bbb333222q 6
439
bbb33322q 7 b
440
bbbq 8 33322b
441
bbbbq 7 3322b
442
bbbb3322q 7 b
443
bbbbq 8 332bb
444
bbbbb32q 7 bb
445
bbbbbq 8 3bb
446
bbbbbbq 7 bb
447
bbbbbbq 8 bb
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