Chapter 2: Finite Automata

Chapter 2: Finite Automata...  Mastering the following concepts: – Deterministic Finite Accepter DFA – Nondeterministic Finite Accepter NFA – DFA and NFA Equivalence... Example 2.23 Let

Chapter 2: Finite Automata

 Mastering the following concepts:

– Deterministic Finite Accepter (DFA)

– Nondeterministic Finite Accepter (NFA)

– DFA and NFA Equivalence

Deterministic Finite Accepter (DFA)

M = (Q, ∑, δ, q0, F)

Q: finite set of internal states

∑: finite set of symbols - input alphabet

δ: Q × ∑ → Q transition function

q0∈Q: initial state

F⊆Q: set of final states

Example 2.2

3

Letter Digit

Letter or Digit

Letter or Digit 2

Extended Transition Function

δ*(q, λ) = q

δ*(q, wa) = δ(δ*(q, w), a)

Example 2.2: δ(q0, a) = q1 & δ(q1, b) = q2 ⇒ δ*(q0, ab) = q2

Languages and DFAs

M = (Q, ∑ , δ , q0, F) L(M) = {w ∈∑* | δ*(q0, w) ∈ F} L(M) = {w ∈∑* | δ*(q0, w) ∉ F}

b

Example 2.4:

trap state

a, b a

b

Transition Table

 A table used to represent an automaton

 Rows headers: states

 Columns headers: input symbols

 Entries: next states

b

Regular Languages

Nondeterministic Finite Accepter (NFA)

M = (Q, ∑, δ, q0, F)

Q: finite set of internal states

∑: finite set of symbols - input alphabet

δ: Q × (∑ ∪ {λ}) → 2Q transition function

q0∈Q: initial state

F⊆Q: set of final states

Extended Transition Function

δ*(qi, w) = Qj

δ*(qi, w) contains qj iff there is a walk

labelled w from qi to qj

How to calculate δ*

Evaluate all walks of length at most Λ + (1 + Λ )|w|; Λ is the

number of λ -edges

) ( )

, (

| { )

q closure T

*(

) 1

|

|, )};

, ( { (

) ,

(

*

b q p

c c b a

c p closure

a

q

δ

δ λ

δ

∈

= +

=

−

=

0,1

NFA and Languages

M = (Q, ∑ , δ , q0, F) L(M) = {w ∈∑* | δ*(q0, w) ∩ F ≠ ∅ }

Equivalence of DFA and NFA

 Two automata are equivalent if they accept the same language

 For a given language, we can usually find unlimited accepters (both deterministic and nondeterministic)

NFA and DFA

NFA: δ*(q, w) = {qi, qj, , qk}  label of one state in DFA

Theorem 2.2

Given MN = (QN, ∑ , δN, q0N, FN)

there exists MD = (QD, ∑ , δD, q0D, FD) such that L(MD) = L(MN)

3 Every state of GD containing qf ∈ FN is a final vertex.

4 If λ ∈ L(MN) then {q0N} is also a final vertex.

• Exercises: 1, 5, 6, 11, 14, 17, 15, 17 of Section 2.1 - Linz’s book

• Exercises: 3, 4, 6, 7, 9, 10 of Section 2.2 - Linz’s book

• Exercises: 3, 8, 11, 12 of Section 2.3 - Linz’s

book