06-finiteAutomata2

06-finiteAutomata2 - CMSC 330 Organization of Programming...

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Unformatted text preview: CMSC 330: Organization of Programming Languages Finite Automata 2 2 Last Lecture Finite automata • Alphabet, states… • ( Σ , Q, q , F, δ ) Types • Deterministic (DFA) • Non-deterministic (NFA) Reducing RE to NFA • Concatenation • Union • Closure a b a a ε b ε a a b ε ε ε ε ε ε ε ε a ε 3 This Lecture Reducing NFA to DFA • ε-closure • Subset algorithm Minimizing DFA • Hopcroft reduction Complementing DFA Implementing DFA 4 How NFA Works When NFA processes a string • NFA may be in several possible states h Multiple transitions with same label h ε-transitions Example • After processing “a” h NFA may be in states S1 S2 S3 a ε S1 S2 a S3 5 Reducing NFA to DFA NFA may be reduced to DFA • By explicitly tracking the set of NFA states Intuition • Build DFA where h Each DFA state represents a set of NFA states Example S1 a S1, S2, S3 a ε S1 S2 a NFA DFA S3 a 6 Reducing NFA to DFA (cont.) Reduction applied using the subset algorithm • DFA state is a subset of set of all NFA states Algorithm • Input h NFA ( Σ , Q, q , F n , δ ) • Output h DFA ( Σ , R, r , F d , δ ) • Using h ε-closure(p) h move(p, a) 7 ε-transitions and ε-closure We say p → q • If it is possible to go from state p to state q by taking only ε-transitions • If ∃ p, p 1 , p 2 , … p n , q ∈ Q such that h {p, ε ,p 1 } ∈ δ , {p 1 , ε ,p 2 } ∈ δ , … , {p n , ε ,q} ∈ δ ε-closure(p) • Set of states reachable from p using ε-transitions alone h Set of states q such that p → q h ε-closure(p) = {q | p → q } • Note h ε-closure(p) always includes p h ε-closure( ) may be applied to set of states (take union) ε ε ε 8 ε-closure: Example 1 Following NFA contains • S1 → S2 • S2 → S3 • S1 → S3 ε-closures • ε-closure(S1) = • ε-closure(S2) = • ε-closure(S3) = • ε-closure( { S1, S2 } ) = ε S1 S2 S3 ε ε ε ε { S1, S2, S3 } { S2, S3 } { S3 } a { S1, S2, S3 } ∪ { S2, S3 } 9 ε-closure: Example 2 Following NFA contains • S1 → S3 • S3 → S2 • S1 → S2 ε-closures • ε-closure(S1) = • ε-closure(S2) = • ε-closure(S3) = • ε-closure( { S2,S3 } ) = b S1 S2 S3 a ε ε { S1, S2, S3 } { S2 } { S2, S3 } ε ε { S2 } ∪ { S2, S3 } ε 10 ε-closure: Practice Find ε-closures for following NFA Find ε-closures for the NFA you construct for • The regular expression (0|1*)111(0*|1) 11 Calculating move(p,a) move(p,a) • Set of states reachable from p using exactly one transition on a h Set of states q such that {p, a, q} ∈ δ h move(p,a) = {q | {p, a, q} ∈ δ } • Note move(p,a) may be empty Ø h If no transition from p with label a 12 move(a,p) : Example 1 Following NFA • Σ = { a, b } Move • move(S1, a) = • move(S1, b) = • move(S2, a) = • move(S2, b) = • move(S3, a) = • move(S3, b) = b S1 S2 S3 a { S2, S3 } { S3 } a Ø Ø Ø Ø 13 move(a,p) : Example 2 Following NFA • Σ = { a, b } Move • move(S1, a) = • move(S1, b) = • move(S2, a) = • move(S2, b) =...
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This note was uploaded on 01/13/2012 for the course CMSC 330 taught by Professor Staff during the Fall '08 term at Maryland.

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06-finiteAutomata2 - CMSC 330 Organization of Programming...

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