Non-Deterministic Finite Automata The Conversion Theorem Closure Properties Summary Inf2A: Converting from NFAs to DFAs and Closure Properties Stuart Anderson School of Informatics University of Edinburgh October 13, 2009 university-logo 1/43
Non-Deterministic Finite Automata The Conversion Theorem Closure Properties Summary Starter Questions Can you devise a way of testing for any FSM M whether 1 L ( M ) = ∅ ? Can you devise a way of testing for any pair of FSMs M 2 and M ′ whether L ( M ) = L ( M ′ ) ? university-logo 2/43
Non-Deterministic Finite Automata The Conversion Theorem Closure Properties Summary Outline Non-Deterministic Finite Automata 1 NFA - the definitions NFA: Example The Conversion Theorem 2 Converting from NFAs to DFAs Example: Converting from an NFA to a DFA Closure Properties 3 Demonstrating Closure under various operators Closure: Example Summary 4 university-logo 3/43
Non-Deterministic Finite Automata The Conversion Theorem NFA - the definitions Closure Properties NFA: Example Summary A non-deterministic automaton a,b a a b 1 0 2 3 Formally: � { 0 , 1 , 2 , 3 } , { a , b } , 0 , { 3 } , ∆ � N = , where the transition relation ∆ is given by the following table: ∆ a b 0 { 0 , 1 } { 0 } 1 ∅ { 2 } { 3 } ∅ 2 3 ∅ ∅ university-logo 4/43
Non-Deterministic Finite Automata The Conversion Theorem NFA - the definitions Closure Properties NFA: Example Summary Non-deterministic Finite Automata Definition: A nondeterministic finite automaton (or NFA ) is a tuple N = ( Q , Σ , q 0 , F , ∆) consisting of: a finite set Q of states , 1 a finite alphabet Σ , 2 a distinguished starting state q 0 ∈ Q , 3 a set F ⊆ Q of final states (the ones that indicate 4 acceptance), a description ∆ of all the possible transitions . 5 ∆ must be given by a table that answers the following question: “Given a state q and an input symbol a , what are the possible next states?” university-logo ∆ is called the transition function or transition relation . 6/43
Non-Deterministic Finite Automata The Conversion Theorem NFA - the definitions Closure Properties NFA: Example Summary The Language of an NFA As opposed to a DFA, an NFA may have several possible computations for the same input string — or it may get stuck and have no (complete) computation at all. An NFA N is said to accept a string x if there is at least one computation of N on input x that leads to an accepting final state. The language recognized by an NFA N = ( Q , Σ , q 0 , F , ∆) is the language x ∈ Σ ∗ | N accepts x � � L ( N )= . university-logo 7/43
Non-Deterministic Finite Automata The Conversion Theorem NFA - the definitions Closure Properties NFA: Example Summary Example a,b a a b 1 2 3 0 Two possible computations of this NFA on input aaaba : a a a b a 0 − → 0 − → 0 − → 0 − → 0 − → 0 a a a b a 0 − → 0 − → 0 − → 1 − → 2 − → 3 The NFA accepts the language � xaba | x ∈ { a , b } ∗ � . university-logo 9/43
Non-Deterministic Finite Automata The Conversion Theorem NFA - the definitions Closure Properties NFA: Example Summary Example 1 0 2 0−9 0 1 1 0 2 1 1 1 0 university-logo 10/43
Non-Deterministic Finite Automata The Conversion Theorem NFA - the definitions Closure Properties NFA: Example Summary NFAs with ε -Transitions ε -transitions are transitions not triggered by any input symbol. If in a state q an ε -transition to a state q ′ is possible, then whenever the automaton reaches state q in a computation it can immediately proceed to state q ′ without needing any further input. To descibe this formally the function ∆ is defined for Σ ∪ { ε } . Example: ε ε ε ε a b a c 3 4 0 1 2 An NFA with ε -transitions recognizing the language { ε, c , ac , bac , abac } . university-logo 11/43
Non-Deterministic Finite Automata The Conversion Theorem Converting from NFAs to DFAs Closure Properties Example: Converting from an NFA to a DFA Summary The languages of NFAs Obvious fact 1: Any language recognized by some DFA can also be recognized by some NFA. Obvious fact 2: Any language recognized by some NFA can also be recognized by some NFA with ε -transitions. university-logo 12/43
Non-Deterministic Finite Automata The Conversion Theorem Converting from NFAs to DFAs Closure Properties Example: Converting from an NFA to a DFA Summary The Conversion Theorem Theorem: Any language accepted by some NFA with ε -transitions is also accepted by some DFA. Therefore, every language accepted by an NFA, with or without ε -transitions, is regular. university-logo 14/43
Non-Deterministic Finite Automata The Conversion Theorem Converting from NFAs to DFAs Closure Properties Example: Converting from an NFA to a DFA Summary Proof: The Powerset Construction Let N = ( Q , Σ , q 0 , F , ∆) be an NFA. Idea: Simulate all possible computations of N “in parallel”. Construction: Let M = ( S , Σ , s 0 , F ′ , δ ) , where S = P ( Q ) = { s | s ⊆ Q } (the powerset of Q ) s 0 = { q 0 } ∪ { q | q can be reached from q 0 using only ε -transitions } δ is defined by δ ( s , a ) = set of all states reachable from a state in s by one a -transition and ε -transitions. F ′ = { s ∈ S | s ∩ F � = ∅} . university-logo 15/43
Non-Deterministic Finite Automata The Conversion Theorem Converting from NFAs to DFAs Closure Properties Example: Converting from an NFA to a DFA Summary The conversion algorithm Construct the start state s 0 consisting of q 0 and all states 1 of N that can be reached from q 0 by one or several ε -transitions. Mark s 0 as “unfinished”. While there still are “unfinished” states: 2 Take an “unfinished” state s . 1 For each a ∈ Σ , let 2 δ ( s , a ) = set of all states reachable from a state in s by one a -transition and ε -transitions � = q ∈ s set of all states reachable from q by one a -transition and ε -transitions If δ ( s , a ) is neither “finished” nor “unfinished” yet, then mark δ ( s , a ) as “unfinished”. Mark s as “finished”. 3 Mark all states that contain a final state from N as final 3 university-logo states of M . 16/43
Non-Deterministic Finite Automata The Conversion Theorem Converting from NFAs to DFAs Closure Properties Example: Converting from an NFA to a DFA Summary An example conversion 1 0 1 3 ε ε ε 0 7 6 ε 0 ε 1 0 2 4 5 ε The start state: s 0 = { 0 , 1 , 2 } . university-logo 18/43
Non-Deterministic Finite Automata The Conversion Theorem Converting from NFAs to DFAs Closure Properties Example: Converting from an NFA to a DFA Summary Computing the transition table δ δ 0 1 0 1 { 0 , 1 , 2 } { 0 , 1 , 2 } { 3 , 4 , 6 } ∅ { 3 , 4 , 6 } ∅ δ δ 0 1 0 1 { 0 , 1 , 2 } { 3 , 4 , 6 } ∅ { 0 , 1 , 2 } { 3 , 4 , 6 } ∅ { 3 , 4 , 6 } { 6 , 7 } { 2 , 3 , 5 , 6 } { 3 , 4 , 6 } { 6 , 7 } { 2 , 3 , 5 , 6 } ∅ ∅ ∅ ∅ { 6 , 7 } { 6 , 7 } { 2 , 3 , 5 , 6 } { 2 , 3 , 5 , 6 } university-logo 19/43
Non-Deterministic Finite Automata The Conversion Theorem Converting from NFAs to DFAs Closure Properties Example: Converting from an NFA to a DFA Summary δ 0 1 { 0 , 1 , 2 } { 3 , 4 , 6 } ∅ { 3 , 4 , 6 } { 6 , 7 } { 2 , 3 , 5 , 6 } ∅ ∅ ∅ { 6 , 7 } { 6 , 7 } ∅ { 2 , 3 , 5 , 6 } δ 0 1 { 0 , 1 , 2 } { 3 , 4 , 6 } ∅ { 3 , 4 , 6 } { 6 , 7 } { 2 , 3 , 5 , 6 } ∅ ∅ ∅ { 6 , 7 } { 6 , 7 } ∅ { 2 , 3 , 5 , 6 } { 4 , 6 , 7 } { 3 , 6 } { 4 , 6 , 7 } { 3 , 6 } university-logo 20/43
Non-Deterministic Finite Automata The Conversion Theorem Converting from NFAs to DFAs Closure Properties Example: Converting from an NFA to a DFA Summary δ 0 1 { 0 , 1 , 2 } { 3 , 4 , 6 } ∅ { 3 , 4 , 6 } { 6 , 7 } { 2 , 3 , 5 , 6 } ∅ ∅ ∅ { 6 , 7 } { 6 , 7 } ∅ { 2 , 3 , 5 , 6 } { 4 , 6 , 7 } { 3 , 6 } { 4 , 6 , 7 } { 6 , 7 } { 2 , 5 , 6 } { 3 , 6 } { 2 , 5 , 6 } university-logo 21/43
Non-Deterministic Finite Automata The Conversion Theorem Converting from NFAs to DFAs Closure Properties Example: Converting from an NFA to a DFA Summary δ 0 1 { 0 , 1 , 2 } { 3 , 4 , 6 } ∅ { 3 , 4 , 6 } { 6 , 7 } { 2 , 3 , 5 , 6 } ∅ ∅ ∅ { 6 , 7 } { 6 , 7 } ∅ { 2 , 3 , 5 , 6 } { 4 , 6 , 7 } { 3 , 6 } { 4 , 6 , 7 } { 6 , 7 } { 2 , 5 , 6 } { 3 , 6 } { 6 , 7 } { 3 , 6 } { 2 , 5 , 6 } university-logo 22/43
Non-Deterministic Finite Automata The Conversion Theorem Converting from NFAs to DFAs Closure Properties Example: Converting from an NFA to a DFA Summary δ 0 1 { 0 , 1 , 2 } { 3 , 4 , 6 } ∅ { 3 , 4 , 6 } { 6 , 7 } { 2 , 3 , 5 , 6 } ∅ ∅ ∅ { 6 , 7 } { 6 , 7 } ∅ { 2 , 3 , 5 , 6 } { 4 , 6 , 7 } { 3 , 6 } { 4 , 6 , 7 } { 6 , 7 } { 2 , 5 , 6 } { 3 , 6 } { 6 , 7 } { 3 , 6 } { 2 , 5 , 6 } { 4 , 6 , 7 } ∅ The final states: F ′ = { s ∈ S | s ∩ { 7 } � = ∅} � � = { 6 , 7 } , { 4 , 6 , 7 } . university-logo 23/43
Non-Deterministic Finite Automata The Conversion Theorem Converting from NFAs to DFAs Closure Properties Example: Converting from an NFA to a DFA Summary The resulting DFA 0,1 1 {0,1,2} φ 0 1 1 0 0 {3,4,6} {2,5,6} {6,7} 0 1 1 0 0 1 1 {2,3,5,6} {3,6} {4,6.7} university-logo 0 24/43
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