Theory of Computation (Finite Automata), Slides of Computer Science

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Theory of Computation

(Finite Automata)

Pramod Ganapathi

Department of Computer Science State University of New York at Stony Brook

January 24, 2021

Contents

Contents Deterministic Finite Automata (DFA) Regular Languages Regular Expressions Nondeterministic Finite Automata (NFA) Transformations Non-Regular Languages

Electric bulb

Problem Design the logic behind an electric bulb.

Electric bulb

Problem Design the logic behind an electric bulb.

Solution Diagram.

Analysis. States = {nolight , light}, Input = {off , on} Finite Automaton. on

off

Multispeed fan

Problem Design the logic behind a multispeed fan.

Solution Diagram.

2

Off 1 3

Analysis. States = { 0 , 1 , 2 , 3 } Input = { , }

Finite Automaton. 0

Automatic doors

Problem Design the logic behind automatic doors in Walmart.

Basic features of finite automata

A finite automaton is a simple computer with extremely limited memory A finite automaton has a finite set of states Current state of a finite automaton changes when it reads an input symbol A finite automaton acts as a language acceptor i.e., outputs “yes” or “no”

Why should you care?

Deterministic Finite Automata (DFA) are everywhere. ATMs Ticket machines Vending machines Traffic signal systems Calculators Digital watches Automatic doors Elevators Washing machines Dishwashing machines Thermostats Train switches (CS) Compilers (CS) Search engines (CS) Regular expressions

What is a decision problem?

Definition A decision problem is a computational problem with a ‘yes’ or ‘no’ answer. A computer that solves a decision problem is a decider. Input to a decider: A string w Output of a decider: Accept ( w is in the language) or Reject ( w is not in the language) w Decider yes/no

What is a decision problem?

English word Accept

Other word Reject

Language = English language = {milk , food , sleep ,... } B Accept Not in language = {zffgb , cdcapqw ,... } B Reject

How does a DFA work?

Problem Does the DFA accept the string bbab?

start q 0 q 1 q 2

b

a

a, b

a b

How does a DFA work?

Problem Does the DFA accept the string bbab?

start q 0 q 1 q 2

b

a

a, b

a b

Solution The DFA accepts the string bbab. The computation is:

1. Start in state q 0
2. Read b , follow transition from q 0 to q 1.
3. Read b , follow transition from q 1 to q 1.
4. Read a , follow transition from q 1 to q 2.
5. Read b , follow transition from q 2 to q 1.
6. Accept because the DFA is in an accept state q 1 at the end of the input.

How does a DFA work?

Problem Does the DFA accept the string aaba?

start q 0 q 1 q 2

b

a

a, b

a b

Solution The DFA rejects the string aaba. The computation is:

1. Start in state q 0
2. Read a , follow transition from q 0 to q 0.
3. Read a , follow transition from q 0 to q 0.
4. Read b , follow transition from q 0 to q 1.
5. Read a , follow transition from q 1 to q 2.
6. Reject because the DFA is in a reject state q 2 at the end of the input.

How does a DFA work?

q 0 q 1 q 2

b

a

a, b

a b

bbab Accept

q 0 q 1 q 2

b

a

a, b

a b

aaba Reject