CS 3240: Languages and Computation

Introducing Regular Languages:

Deterministic Finite Automata

and Regular Expressions

Automata Finite Automata model many design and

analysis tasks, e.g. Lexical analyzer in a compiler Digital cicuit design Keywork searching in texts or on the web. Software for verifying finite state systems, such as communication protocols. Your ATM, vendig machine, Etc.

1950-59 McCulloch&Pitts, Rabin&Scott, Moore, Huffman

Finite State Machine and Finite Automata

Deterministic Finite Automata

A simplest model for computingDeterministic: Machine is in a state. Upon

receipt of a symbol will go to a unique state.Finite: Have a finite number of statesAutomata: (pl. of automaton) Self-operating

machine DFA: finite-state machine without ambiguity

DFA and Strings

DFA can recognize stringsString is input If DFA ends at accept state, string is recognized

A language is called a regular language if some finite automaton recognizes it

Let us look at a few example before giving formal mathematical definition

DFA Examples

1 2

0 11

0

Accept all stringsthat end in 1

Note: The alphabet for this example is {0, 1}. Each statehas a transition for every symbolin the alphabet

accept state

transition

start state

DFA Examples

Accept strings of 'a's and 'b's that begin and end with same symbol

5

3

4

2

1a

a

a

a

a

b

bb

b

b

DFA Examples

2

Keep running count of total of symbols readin mod 3. Accept on 0.

1

00 1

2

0

1

2

0

1

2

Start

DFA Examples

Strings with an oddnumber of ones.

Even Odd

0 0

1

1

DFA Examples

Strings containingthe substring 001

'001'

0 01

1

'0' '00'

0

10,1

Formal Definition of DFA A DFA consists of:

Alphabet A set of states Q A transition function δ : Q Q One start state q0

One or more accepting states F Q

Language accepted by a DFA is the set of strings such that DFA ends at an accepting state Each string is c1c2…cn with ci States are qi = δ(qi-1,ci ) for i=1…n

qn is an accepting state

Can DFA's be designed to accept any string?

No!

Examples

Design a DFA to recognize strings that start out with k zeros followed by k ones. Impossible

Design a DFA to recognize strings with an equal number of ones and zeros. Impossible

Design a DFA to recognize strings with an equal number of strings "01" and "10".Possible!

Actually the thirdone is regular!

DFA to recognize strings with an equal number of strings "01" and "10"

0

0

0

0

0

1

1

1

1

1

1

0

1

0

DFA More examples

DFA More examples

A,B are the input into which themarble is dropped. The x-levers cause fall either to left or right, but lever reversesupon a marble passingAccept if marble exits through D

Non-Deterministic Finite Automata

NFA and -NFA

Nondeterministic Finite AutomataSame input may produce multiple pathsAllows transition with an empty string or

transition from one state to different states given a character

q1 q2

empty string transition

q1

q21

q31

nondeterministic transition