Finite Automata - Lecture 4 Sections 3.6 - 3 -...

Finite AutomataLecture 4

Sections 3.6 - 3.7

Robb T. Koether

Hampden-Sydney College

Wed, Jan 21, 2015

Robb T. Koether (Hampden-Sydney College) Finite Automata Wed, Jan 21, 2015 1 / 23

1 Nondeterministic Finite Automata

2 Deterministic Finite Automata

3 Converting an NFA to a DFA

4 Assignment


Outline




4 Assignment


Nondeterministic Finite Automata

Definition (Nondeterministic Finite Automaton)A nondeterministic finite automaton, or NFA, consists of

A finite set S of states.An input alphabet Σ.A transition function δ that maps to each state-symbol pair a set ofnext states. The “symbol” may be ε.A set F of final states (or accepting states), where F ⊆ S.


The Transition Function

Example (The Transition Function)

a | b

a b b1 432

The transition function can be represented as a transition diagram.


The Transition Function

Example (The Transition Function)

State Next State1 {1,2}2 {3}3 {4}4 ∅

The transition function can also be represented as a transitiontable.


Acceptance of Strings

A string w is accepted by an NFA if there exists a path through theNFA that w can follow that leads to acceptance.The string is rejected if all possible paths lead to rejection.

In the previous example, is there a path leading to acceptance forthe string bbabb?In the previous example, is there a path leading to rejection for thestring bbabb?How about the string ababab?



A string w is accepted by an NFA if there exists a path through theNFA that w can follow that leads to acceptance.The string is rejected if all possible paths lead to rejection.In the previous example, is there a path leading to acceptance forthe string bbabb?

In the previous example, is there a path leading to rejection for thestring bbabb?How about the string ababab?



A string w is accepted by an NFA if there exists a path through theNFA that w can follow that leads to acceptance.The string is rejected if all possible paths lead to rejection.In the previous example, is there a path leading to acceptance forthe string bbabb?In the previous example, is there a path leading to rejection for thestring bbabb?

How about the string ababab?



A string w is accepted by an NFA if there exists a path through theNFA that w can follow that leads to acceptance.The string is rejected if all possible paths lead to rejection.In the previous example, is there a path leading to acceptance forthe string bbabb?In the previous example, is there a path leading to rejection for thestring bbabb?How about the string ababab?


Outline




4 Assignment


Deterministic Finite Automata

Definition (Deterministic Finite Automaton)A determimistic finite automaton, or DFA, is the same as an NFAexcept

The empty string ε is not included in the set of symbols. (Noε-moves.)For each state and for each symbol, there is exactly one nextstate.


Deterministic Finite Automata

Example (Deterministic Finite Automata)Design a transition diagram for a DFA that accepts

L((a | b)∗abb).

Write the transition table for this DFA.


Outline




4 Assignment


NFAs vs. DFAs

In general, it is easier to design an NFA than it is to design a DFA.

Therefore, a common strategy is to design an NFA for a language.But how can a computer simulate an NFA, given that NFAs areinherently nondeterministic?There is an algorithm that will convert an NFA into an equivalentDFA.


NFAs vs. DFAs

In general, it is easier to design an NFA than it is to design a DFA.Therefore, a common strategy is to design an NFA for a language.

But how can a computer simulate an NFA, given that NFAs areinherently nondeterministic?There is an algorithm that will convert an NFA into an equivalentDFA.


NFAs vs. DFAs

In general, it is easier to design an NFA than it is to design a DFA.Therefore, a common strategy is to design an NFA for a language.But how can a computer simulate an NFA, given that NFAs areinherently nondeterministic?

There is an algorithm that will convert an NFA into an equivalentDFA.


NFAs vs. DFAs

In general, it is easier to design an NFA than it is to design a DFA.Therefore, a common strategy is to design an NFA for a language.But how can a computer simulate an NFA, given that NFAs areinherently nondeterministic?There is an algorithm that will convert an NFA into an equivalentDFA.


Converting an NFA to a DFA

Definition (ε-Closure)Let S be the states of a NFA and let s ∈ S be a state. The ε-closure ofs is the set of all states that are reachable from s (including s itself)through any sequences of ε-moves.


Converting an NFA to a DFA

Given an NFA, define the states of a DFA to be P(S), i.e., sets ofstates in the NFA.For every state s ∈ P(S) and every symbol a ∈ Σ, the transitionfunction δ(s,a) is the ε-closure of all states in the NFA that arereached from states in s by reading a.That is,

First find all states reached from s by following a-moves.Then find the ε-closure of that set of states.


Example

Example (Converting an NFA to a DFA)Draw a transition diagram for an NFA that accepts the language ofall strings over {a,b} that either contain an even number of a’s orcontain the substring bab.It is easy to draw if we use ε-moves.


Example

Example (Converting an NFA to a DFA)

a

b b

2

1

4

3a

5 6 7ab bε

ε

a | b a | b


Example

Example (Converting an NFA to a DFA)The diagram has 7 states. That is, |S| = 7.Therefore, |P(S)| = 27 = 128.Fortunately, we will use very few of these states.


Example


State ε-closure1 {1,2,4}2 {2}3 {3}4 {4}5 {5}6 {6}7 {7}


Example

Example (Converting an NFA to a DFA)The start state is ε-closure(1) = {1,2,4}.The transition function, for example, includes

δ({1,2,4},a) = ε-closure({3,4}) = {3,4}δ({1,2,4},b) = ε-closure({2,4,5}) = {2,4,5}

Work out the remaining transitions.


Example


a

a124 34

a

24

345

246245

346 3457 2467 2457

347 247

3467

b

a

b

a

b

a

bbb

a

b

a b

a

b

a

ba b

ab

b

a


Example


a124 34

24

345

246245

346

b

a

b

a a

b

b

a

ab

b

a b

a | b

A reduced DFA


Example


a1 2

5

3

64

7

b

a

b

a a

b

b

a

ab

b

a b

a | b8

Relabel the states


Outline




4 Assignment


Assignment

AssignmentRead Sections 3.6 - 3.7.p. 151: 2(i), 3, 4, 5(a).p. 166: 1.


Finite Automata - Lecture 4 Sections 3.6 - 3 -...

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