Finite automata

Deterministic Finite State Automata (DFA) ,[object Object],[object Object],[object Object],[object Object],[object Object],Finite Control 0 1 1 0 0

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],q 0 q 1 0 0 1 1

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],q 1 q 0 q 2 a b a b c c a/b/c

Inductive Proof (sketch): Base: x a string with |x|=0. state will be q0 => rejected. Inductive hypothesis: |x|=k, rejected -in state q0 (x must have 0 c), rejected – in state q1 (x must have 1 c), accepted – in state q2 (x already with 2 c’s) Inductive step: String xp, for p = a, b and c q0 and, xa or xb : q0->q0 rejected, as should be (no c) q0 and, xc : q0 -> q1 rejected, as should be (1 c) q1 and xa or xb: q1 -> q1 rejected, … q1 and xc: q1-> q2 accepted, as should be ( 2 c’s now) q2 and xa, or xb, or xc: q2 -> q2 accepted, (no change)

Formal Definition of a DFA ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],q 0 q 1 0 0 1 1

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],q 1 q 0 q 2 a b a b c c a/b/c

Extension of δ to Strings ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],q 0 q 1 0 0 1 1

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],q 1 q 0 1 1 0 0 1 0 q 2

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],0 0 q 1 q 0 q 2 1 1 0 1

Definitions for DFAs ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],q 1 q 0 q 2 0/1 0/1 0/1

[object Object],[object Object],[object Object],q 2 q 0 a a/b/c a q 1 b/c b/c

[object Object],[object Object],[object Object],q 2 q 0 a a/b/c b q 1 c b/c a b/c q 3 a

[object Object],[object Object],[object Object],q 0 b q 7 q 5 q 4 q 6 b b b a q 2 q 1 q 3 a a a b a/b b a a a b

[object Object],[object Object],[object Object],0/1 q 0 0/1 q 0 q 1 q 0 0/1 0/1 0/1 q 0 q 1 0/1

Nondeterministic Finite State Automata (NFA) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],q 1 q 0 0 1 0 1 0/1 { q 0 , q 1 } {} {} { q 1 , q 2 } { q 2 } { q 2 } q 2

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],1 { q 0 , q 3 } { q 0 , q 1 } {} { q 2 } { q 2 } { q 2 } { q 4 } {} { q 4 } { q 4 } q 0 0/1 0 0 q 3 q 4 0/1 q 1 q 2 0/1 1

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],0 0 1

[object Object],[object Object],[object Object],[object Object],[object Object],0 1 0

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],q 1 q 0 q 2 a a/b/c b a/b/c

[object Object],[object Object],[object Object],[object Object],[object Object],q 1 q 0 b q 3 a/b a/b q 2 a/b

Extension of δ to Strings and Sets of States ,[object Object],[object Object],[object Object],q 0 0 1 q 1 q 4 q 3 0 1 q 2 0 0 1 0 0

Extension of δ to Strings and Sets of States ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],q 0 0 1 q 1 q 3 0 1 q 2 1 1 0

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Definitions for NFAs ,[object Object],[object Object],[object Object],[object Object],[object Object]

Equivalence of DFAs and NFAs ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],q 1 q 0 q 2 a b a b c c a/b/c

[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],q 1 0 0/1 0 { q 1 } {} {q 0 , q 1 } { q 1 } q 0

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[ ] 1 0 1 [q 1 ] 0 0/1 1 0 [q 0 , q 1 ] [q 0 ]

[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],{ q 0 , q 1 } { } { q 1 } { q 2 } { q 2 } { q 2 }

NFAs with ε Moves ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],q 0 ε 0/1 q 2 1 0 q 1 0 q 3 ε 0 1 { q 0 } { } { q 1 } { q 1 , q 2 } { q 0 , q 3 } { q 2 } { q 2 } { q 2 } { } { } { } { }

Informal Definitions ,[object Object],[object Object],[object Object]

ε -closure ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Extension of δ to Strings and Sets of States ,[object Object],[object Object],[object Object],q 0 ε 0/1 q 2 1 0 q 1 0 q 3 ε 0 1

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Definitions for NFA- ε Machines ,[object Object],[object Object],[object Object],[object Object],[object Object]

Equivalence of NFAs and NFA- εs ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],q 0 ε 0/1 q 2 1 0 q 1 0 q 3 ε 0 1 q 2 q 1 q 3 q 0

[object Object],[object Object],[object Object],q 1 q 3 q 0 ε 0/1 q 2 1 0 q 1 0 q 3 ε 0 1 q 2 q 0

[object Object],[object Object],q 0 ε 0/1 1 0 q 1 0 q 3 ε 0 1 q 1 q 3 0 0 0 q 2 q 2 q 0

[object Object],[object Object],q 0 ε 0/1 1 0 q 1 0 q 3 ε 0 1 q 1 q 3 0/1 0/1 0/1 1 q 2 q 2 q 0

[object Object],[object Object],q 0 ε 0/1 1 0 q 1 0 q 3 ε 0 1 q 1 q 3 0/1 0/1 0/1 1 0 0 q 2 q 2 q 0

[object Object],[object Object],q 0 ε 0/1 1 0 q 1 0 q 3 ε 0 1 q 1 q 3 0/1 0/1 0/1 1 0/1 0/1 1 1 q 2 q 2 q 0

[object Object],[object Object],q 0 ε 0/1 1 0 q 1 0 q 3 ε 0 1 q 1 q 3 0/1 0/1 0/1 1 0/1 0/1 1 1 0 q 2 q 2 q 0

[object Object],[object Object],[object Object],q 0 ε 0/1 1 0 q 1 0 q 3 ε 0 1 q 1 q 3 0/1 0/1 0/1 1 0/1 0/1 1 1 0/1 q 2 q 2 q 0

Finite automata

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Finite automata