C OMPUTABILITY AND C OMPLEXITY , 2020 F ALL SEMESTER Lec 02. Non-deterministic Finite Automata Eunjung Kim
N ONDETERMINISM Figure 1.27, Sipser 2012. Deterministic FA Nondeterministic FA each state & symbol one leaving arc multiple arcs or none labels Σ Σ ∪ { ǫ } computation history single path multiple paths COMPUTABILITY & COMPLEXITY 1 / 15
N ONDETERMINISM : COMPUTATION TREE AND ǫ Figure 1.27, Sipser 2012. COMPUTABILITY & COMPLEXITY 2 / 15
E XAMPLES OF NFA Figure 1.31, Sipser 2012. Figure 1.32, Sipser 2012. COMPUTABILITY & COMPLEXITY 3 / 15
E XAMPLES OF NFA Figure 1.33, Sipser 2012. Figure 1.36, Sipser 2012. COMPUTABILITY & COMPLEXITY 4 / 15
F ORMAL DEFINITION OF NFA N ONDETERMINISTIC FA IS A 5- TUPLE ( Q , Σ , δ , q 0 , F ) Q a finite set called the states, Σ a finite set called the alphabet, δ a function from Q × Σ ǫ to 2 Q called the transition function, q 0 ∈ Q the start state, F ⊆ Q the set of accept states. Write a formal description of this NFA. COMPUTABILITY & COMPLEXITY 5 / 15
L ANGUAGE RECOGNIZED BY NFA NFA N ACCEPTS w IF 1 w can be written as y 1 , . . . , y m with y i ∈ Σ ǫ , 2 there exists a computation path ending in an accept state. Write an accepting computation path for w = baaabaaaa . Is there another one? COMPUTABILITY & COMPLEXITY 6 / 15
C LOSURE UNDER REGULAR OPERATION U NION OPERATION Let A 1 and A 2 be two languages recognized by NFA N 1 and N 2 respectively. Then A 1 ∪ A 2 is recognized by some NFA. COMPUTABILITY & COMPLEXITY 7 / 15
C LOSURE UNDER REGULAR OPERATION C ONCATENATION OPERATION Let A 1 and A 2 be two languages recognized by NFA N 1 and N 2 respectively. Then A 1 ◦ A 2 is recognized by some NFA. COMPUTABILITY & COMPLEXITY 8 / 15
C LOSURE UNDER REGULAR OPERATION K LEENE STAR OPERATION Let A a languages recognized by NFA N . Then A ∗ is recognized by some NFA. COMPUTABILITY & COMPLEXITY 9 / 15
C LOSURE UNDER COMPLEMENTATION C OMPLEMENTATION OPERATION A , that is, Σ ∗ − A is Then ¯ Let A a languages recognized by NFA N . recognized by some NFA. COMPUTABILITY & COMPLEXITY 10 / 15
C LOSURE UNDER INTERSECTION I NTERSECTION OPERATION Let A 1 and A 2 be two languages recognized by NFA N 1 and N 2 respectively. Then A 1 ∩ A 2 is recognized by some NFA. COMPUTABILITY & COMPLEXITY 11 / 15
E QUIVALENCE OF NFA AND DFA NFA AND DFA OWN THE SAME COMPUTATIONAL POWER For every NFA, there exists a deterministic finite automata recognizing the same language. Proof outline. Let N = ( Q , Σ , δ , q 0 , F ) be an NFA. We want to construct a DFA M = ( Q ′ , Σ , δ ′ , q ′ 0 , F ′ ) such that L ( N ) = L ( M ) . Define Q ′ := 2 Q , i.e. the family of all subsets of Q . 0 ∈ 2 Q and F ′ ⊆ 2 Q , We shall define δ ′ , q ′ ...and for any w ∈ L ( N ) with an accepting computation history π in N , we demonstrate an accepting computation history π ′ in M . ... and conversely, for any w with an accepting computation history π ′ in M , we build an accepting computation history π in N . COMPUTABILITY & COMPLEXITY 12 / 15
P ROOF : CONSTRUCTING DFA M Idea: from a state q ∈ Q of NFA N , any other state q ′ that can be reached by reading a string of k ǫ ’s, k ≥ 0,can be aggregated with q as a single state. Define ext ( q ) ⊆ N as the set of all states q ′ of N such that there is a directed path from q to q ′ in (the state diagram of) N each of whose arcs carries the label ǫ . Extend the definition ext ( X ) := � q ∈ X ext ( q ) . transition function δ ′ from 2 Q × Σ to 2 Q : for every R ∈ 2 Q ( R is a subset of Q ) and every symbol a ∈ Σ , � δ ′ ( R , a ) := ext ( δ ( r , a )) r ∈ R 0 := ext ( q 0 ) ∈ 2 Q . Note that q ′ Define q ′ 0 corresponds to a subset of Q . Define F ′ ∈ 2 Q as the family of all subsets of Q containing at least one accept state of N . COMPUTABILITY & COMPLEXITY 13 / 15
P ROOF : L ( N ) ⊆ L ( M ) Let π = ( q 0 , w = w 0 ) , . . . , ( q i , w i ) , . . . , ( q s , w s = ǫ ) be an accepting computation history of N for w , that is q i ∈ δ ( q i − 1 , a i ) for every i ∈ [ s ] and q s ∈ F , where w i = a i ◦ w i − 1 with a i ∈ Σ ∪ { ǫ } for every i ∈ [ s ] . Let 0 ≤ i 0 < · · · < i t − 1 < s be the indices in π such that the leading symbol of w i in ( q i , w i ) is in Σ (not ǫ ). Let i t = s . Let Q 0 = q ′ 0 . Inductively for each i ∈ [ t ] , let Q i = δ ′ ( Q i − 1 , y i ) . Now we have a computation history for w = y 1 · · · y t as follows π ′ = ( Q 0 , w ) , ( Q 1 , y 2 · · · y t ) , · · · ( Q t , ǫ ) . COMPUTABILITY & COMPLEXITY 14 / 15
P ROOF : L ( N ) ⊆ L ( M ) It remains to see Q t is an accept state of M , i.e. the subset Q t ⊆ Q contains at least one accept state of N . It suffices to prove q i j ∈ Q j for each 0 ≤ j ≤ t (this implies the above). By induction. Base case holds because Q 0 = q ′ 0 = ext ( q 0 ) and q i 0 is reachable from q 0 by reading a number of ǫ ’s. From the partition using i 0 , . . . , i t , q i j is reachable from q i j − 1 by reading a single symbol a i j − 1 + 1 ∈ Σ and additional ǫ ’s. Hence q i j ∈ ext ( δ ( q i j − 1 , a i j − 1 + 1 ) ⊆ ext ( δ ( Q j − 1 , a i j − 1 + 1 ) = δ ′ ( Q j − 1 , a i j − 1 + 1 ) = Q j . COMPUTABILITY & COMPLEXITY 15 / 15
P ROOF : L ( M ) ⊆ L ( N ) Let π ′ = ( Q 0 , w = w 0 ) , . . . , ( Q i , w i ) , . . . , ( Q t , w t = ǫ ) be an accepting computation history of M for w . By definition of computation history Q i = δ ′ ( Q i − 1 , x i ) , where x i ∈ Σ is the leading symbol of w i − 1 . We construct an accepting computation history of N by following the sequence π ′ backwardly. Let q f ∈ Q t be an accept state of N . Observe: for each q ∈ Q i ⊆ Q , there exists a state q ′ ∈ Q i − 1 such that from q ′ to q there is a computation history consisting of readying a string consisting of x i followed by ǫ ’s. Now starting from q f , we concatenate computation histories witnessed by the previous observation. COMPUTABILITY & COMPLEXITY 16 / 15
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