Nondeterministic - Automata and Complexity Theory - Lecture Slides, Slides of Theory of Automata

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Chapter 3

Nondeterministic Finite Automata (NFA)

(include lecture 5 and 6)

• An important notions(or abstraction) in computerscience Nondeterminism

• refer to situations in which the next state of a computation is not uniquely determined by the current state. - Ex: find a program to compute max(x,y): - pr1: case x ≥ y => print x; - y ≥ x => print y - endcase; - Then which branch will be executed when x = y? - ==> don't care nondeterminism - Pr2: do-one-of { - {if x < y fail; print x}, - {if y < x fail, print y} }. - ==>The program is powerful in that it will never choose branches that finally lead to ‘fail’ -- an unrealistic model. - ==> don't know nondeterminism.

• Causes of nodeterminism in real life: – incomplete information about the statenondeterminism (cont'd)

• external forces affecting the course of the computation
• ex: the behavior of a process in a distributed system
• Nondeterministic programs cannot be executed directly but can be simulated by real machine.
• Nondeterminism can be used as a tool for the specification of problem solutions.
• an important tool in the design of efficient algorithms
• There are many problems with efficient nondeterministic algorithm but no known efficient deterministic one.
• the open problem NP = P?
• How to make DFAs become nondeterministic? ==> allow multiple transitions for each state-input-symbol pair ==> modify the transition function δ.

• A NFA is a five-tuple N = (Q,same as in a DFA, except:Formal Definition of NFAsΣ,δ,S,F) where everything is the

• S ⊆ Q is a set of starting states , instead of a single state.
• δ is the transition function δ: Q x Σ -> 2 Q^. For each state p and symbol a, δ(p,a) is the set of all states that N is allowed to move from p in one step under input symbol a.
• diagrammatic notation: p --a--> q Note: δ(p,a) can be the empty set
• The extended transition function ∆ (multi-step version of δ) for NFA can be defined analogously to that of DFAs: ∆: 2Q^ xΣ* -> 2Q^ is defined inductively as follows:

1. Basis: ∆(A, ε) = ____ for every set of states A (6.1)
2. Ind. case: ∆(A, xa) = ____ for every x ∈ Σ* and a ∈ Σ (6.2) Note: Intuitively q ∈ ∆(A,x) means q can be reached from some state ∈ A after scanning input string x.

p a q

Equivalence of FAs

Note: under such definition, every DFA M = (Q,Σ,δ,s,F) is equivalent to an NFA N = (Q,Σ,δ',{s},F) where

• δ'(p,a) = {δ(p,a)} for every state p and input a.
• Problem: Does the converse hold as well?
• i.e. For every NFA N there is a DFA M s.t. L(M) = L(N).
• Ans: ____

Some examples of NFAs

Ex: Find a NFA accepting A = { x ∈ {0,1}* | the fifth symbol from the right is 1 } = {010000, 11111,...}. Sol: 1. (in diagram form)

2: tabular form:

3. tuple form: (Q,Σ,δ,S,F) = (,,,,__).

1 0,1^ 0,1 0,1 0,

0,

Some properties about the

extended transition function ∆

• Lem 6.1: ∆(A,xy) = ∆(∆(A,x),y).
• pf: by induciton on |y|:
  1. |y| = 0 => ∆(A,xε) = ∆(A,x) = ∆(∆(A,x),ε) -- (6.1).

2. y = zc => ∆(A,xzc) = U (^) q ∈ ∆(A,xz) δ(q,c) -- (6.2) = U (^) q ∈ ∆(∆(A,x),z) δ(q,c) -- ind. hyp. = ∆(∆(A,x),zc) -- (6.2)

• Lem 6.2 ∆ commutes with set union:
  • i.e., ∆ (U (^) i ∈ I Ai,x) = U (^) i ∈ I ∆(Ai,x). in particular, ∆(A,x) = U (^) p∈ A ∆({p},x) Docsity.com

The subset construction

• N = (Q (^) N ,Σ,δN ,S (^) N ,F (^) N ) : a NFA.
• M = (Q (^) M,Σ,δM,sM,F (^) M) (denoted 2 N^ ): a DFA where
  • QM = 2 QN
  • δM (A,a) = ∆N (A,a) ( = ⋃q∈ A δN (q,a) ) for every A ⊆ QN.
  • s (^) M = S (^) N and
  • F (^) M = {A ⊆ QN | A∩ F (^) N ≠ {}}.
  • note: States of M are subsets of states of N.
• Lem 6.3: for any A ⊆ Q (^) N. and x in Σ*, ∆M(A,x) = ∆N (A,x). pf: by ind on |x|. if x = ε => ∆ (A,ε) = A = ∆ (A,ε). --Docsity.com

sol: A more human friendly method

1. Copy the transition table
2. add Row(S) /* =def

Sum (^) p∈S Row(p) to table */

3. D={X|X in Row(p).tail } – {S} // S is the initial set of states

1 0,

0,

p q r

0 1

{} {} {} -> {p} {p} {p,q} {q} {r} {r} {r}F {} {} {p} {p} {p,q} {p,q} {p,r} {p,q,r} {p,r}F {p} {p,q} {q,r}F {r} {r} {p,q,r}F {p,r} {p,q,r}

ε-transition

• Another extension of FAs, useful but adds no more power.
• An ε-transition is a transition with label ε, a label standing for the empty string ε.
• The FA can take such a

transition anytime w/o reading an input symbol.

Ex 6.5 : The set accepted by the FA is {b,bb,bbb}.

Ex 6.6 : A NFA-ε accepting the set {x ∈ {a}* | |x| is

p q ε

s

p q

t u

r

b b b

ε ε ε (^) ε

Ex6.

E-closure

• Eclosure(A) is the set of states reachable from states of A without consuming any input symbols,

(i.e., q∈Eclosure(A) iff∃p∈ A s.t. q ∈ ∆(p, εk) for some k ≥ 0 ).

• Eclosure(A) can be computed as follows:
  1. R=F={}; nF=A; //F: frontier; nF: new frontier

2. do { R = R U nF; F = nF; nF={};
3. For each q ∈ F do
4. nF = nF U (δ(q,ε)- R)
5. }while nF ≠ {};

• N = (QThe subset construction for NFA- N ,Σ,δN ,S N ,F N ) : a NFA-ε.where δN : Q x (ΣU {εε}) ->

2 Q^.

• M = (Q (^) M,Σ,δM,sM,F (^) M) (denoted 2 N^ ): a DFA where
  • QM = { EC(A) | A ⊆ QN }
  • δM (A,a) = ⋃q∈ Ec(A) EC(δN (q,a)) for every A ∈ Q (^) M.
  • s (^) M = EC(S (^) N ) and
  • F (^) M = {A ∈ QM | A∩ F (^) N ≠ {}}.
  • note: States of M are subsets of states of N.
• Lem 6.3: for any A ⊆ Q (^) N. and x ∈ Σ*, ∆M(A,x) = ∆N (A,x). pf: by ind on |x|. if x = ε => ∆M(A,ε) = A =EC(A) = ∆N (A,ε). --(def)

M* machine

• M = (Q 1 ,Σ,δ 1 ,S 1 ,F 1 ) : a NFA
• The machine M*, which executes M a nondeterministic number of times, can be defined as follows:
• M* = (^) def (Q, Σ, δ, S, F) where
  • Q = Q U {s,f}, where s and f are two new states ∉Q
  • S = {s}, F = {f},
  • δ = δ 1 U {(s, ε, f)} U {(s,ε,p) | p ∈ S 1 } U {(q,ε,s) | q ∈ F 1 }

Theorem: L(M) = L(M)

ε ε

ε

Μ

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