Lec 02. Non-deterministic Finite Automata

COMPUTABILITY AND COMPLEXITY, 2020 FALL SEMESTER

Lec 02. Non-deterministic FiniteAutomata

Eunjung Kim

NONDETERMINISM

Figure 1.27, Sipser 2012.

Deterministic FA Nondeterministic FAeach state & symbol one leaving arc multiple arcs or none

labels Σ Σ ∪ {ε}computation history single path multiple paths

COMPUTABILITY & COMPLEXITY 1 / 15

NONDETERMINISM: COMPUTATION

TREE AND ε



EXAMPLES OF NFA




FORMAL DEFINITION OF NFA

NONDETERMINISTIC FA IS A 5-TUPLE (Q,Σ,δ,q0,F )

Q a finite set called the states,

Σ a finite set called the alphabet,

δ a function from Q × Σε to 2Q called the transition function,

q0 ∈ Q the start state,

F ⊆ Q the set of accept states.

Write a formal description of this NFA.


LANGUAGE RECOGNIZED BY NFA

NFA N ACCEPTS w IF

1 w can be written as y1, . . . , ym with yi ∈ Σε,2 there exists a computation path ending in an accept state.

Write an accepting computation pathfor w = baaabaaaa. Is there anotherone?


CLOSURE UNDER REGULAR OPERATION

UNION OPERATION

Let A1 and A2 be two languages recognized by NFA N1 and N2 respectively.Then A1 ∪ A2 is recognized by some NFA.



CONCATENATION OPERATION

Let A1 and A2 be two languages recognized by NFA N1 and N2 respectively.Then A1 ◦ A2 is recognized by some NFA.



KLEENE STAR OPERATION

Let A a languages recognized by NFA N. Then A∗ is recognized by someNFA.


CLOSURE UNDER COMPLEMENTATION

COMPLEMENTATION OPERATION

Let A a languages recognized by NFA N. Then A, that is, Σ∗ − A isrecognized by some NFA.


CLOSURE UNDER INTERSECTION

INTERSECTION OPERATION

Let A1 and A2 be two languages recognized by NFA N1 and N2 respectively.Then A1 ∩ A2 is recognized by some NFA.


EQUIVALENCE OF NFA AND DFA

NFA AND DFA OWN THE SAME COMPUTATIONAL POWER

For every NFA, there exists a deterministic finite automata recognizing thesame language.

Proof outline.Let N = (Q,Σ,δ,q0,F ) be an NFA.

We want to construct a DFA M = (Q′,Σ,δ′,q′0,F ′) such thatL(N) = L(M).

Define Q′ := 2Q , i.e. the family of all subsets of Q.

We shall define δ′, q′0 ∈ 2Q and F ′ ⊆ 2Q ,

...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′ inM, we build an accepting computation history π in N.


PROOF: CONSTRUCTING DFA M

Idea: from a state q ∈ Q of NFA N, any other state q′ that can bereached by reading a string of k ε’s, k ≥ 0,can be aggregated with q asa single state.

Define ext(q) ⊆ N as the set of all states q′ of N such that there is adirected path from q to q′ in (the state diagram of) N each of whose arcscarries the label ε. Extend the definition ext(X ) :=

⋃q∈X ext(q).

transition function δ′ from 2Q × Σ to 2Q :for every R ∈ 2Q (R is a subset of Q) and every symbol a ∈ Σ,

δ′(R,a) := ext(⋃r∈R

δ(r ,a))

Define q′0 := ext(q0) ∈ 2Q . Note that q′0 corresponds to a subset of Q.

Define F ′ ∈ 2Q as the family of all subsets of Q containing at least oneaccept state of N.


PROOF: L(N) ⊆ L(M)

Let π = (q0,w = w0), . . . , (qi ,wi ), . . . , (qs,ws = ε) be an acceptingcomputation history of N for w , that is

qi ∈ δ(qi−1,ai ) for every i ∈ [s] and qs ∈ F ,

where wi = ai ◦ wi−1 with ai ∈ Σ ∪ {ε} for every i ∈ [s].

Let 0 ≤ i0 < · · · < it−1 < s be the indices in π such that the leadingsymbol of wi in (qi ,wi ) is in Σ (not ε). Let it = s.

Let Q0 = q′0. Inductively for each i ∈ [t ], let

Qi = δ′(Qi−1, yi ).

Now we have a computation history for w = y1 · · · yt as follows

π′ = (Q0,w), (Q1, y2 · · · yt ), · · · (Qt , ε).


PROOF: L(N) ⊆ L(M)

It remains to see Qt is an accept state of M, i.e. the subset Qt ⊆ Qcontains at least one accept state of N.

It suffices to prove qij ∈ Qj for each 0 ≤ j ≤ t (this implies the above).

By induction. Base case holds because Q0 = q′0 = ext(q0) and qi0 isreachable from q0 by reading a number of ε’s.

From the partition using i0, . . . , it , qij is reachable from qij−1 by reading asingle symbol aij−1+1 ∈ Σ and additional ε’s. Henceqij ∈ ext(δ(qij−1 ,aij−1+1) ⊆ ext(δ(Qj−1,aij−1+1) = δ′(Qj−1,aij−1+1) = Qj .


PROOF: L(M) ⊆ L(N)

Let π′ = (Q0,w = w0), . . . , (Qi ,wi ), . . . , (Qt ,wt = ε) be an acceptingcomputation history of M for w . By definition of computation historyQi = δ′(Qi−1, xi ), where xi ∈ Σ is the leading symbol of wi−1.

We construct an accepting computation history of N by following thesequence π′ backwardly.

Let qf ∈ Qt be an accept state of N.

Observe: for each q ∈ Qi ⊆ Q, there exists a state q′ ∈ Qi−1 such thatfrom q′ to q there is a computation history consisting of readying a stringconsisting of xi followed by ε’s.

Now starting from qf , we concatenate computation histories witnessedby the previous observation.


Lec 02. Non-deterministic Finite Automata

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