Lecture 20Languages & Automata

U - Up

D - Down

L - Left

R - Right

Introduction to Finite State Machines

Formally, a finite Automaton (FA) is defined as a 5-tuple (Q, S, d, q0, F) where,

(1) Q is a finite set of states.

(2) S is a finite set of symbols or the alphabet.

(3) d: Q x S -> Q is the transition function

(4) q0 is an element of Q called the start state, and

(5) F is a subset of Q called the set of accept states.

Definition of a Finite State Machine

Example: Maze FSM

Example DFA

The Execution of an NFA

The NFA will have zero or more transitions from every state for each input symbol.

All active states are maintained until the input string is consumed or there are no active states.

Equivalent ways Recognize Regular Languages

Finite Automata as Output Devices

We can add output functions to FSA in two different ways. We can have the FSA generate an output symbol for each transition from one state to another, or we can generate an output each time control enters a

state.

input: X Y Z X X Z Y X Z Y Y Z

output: B A A B A A A A A A A A

Moore Machine

Example: Moore Machine

input: X Y Z X X Z Y X Z Y Y Z

output: B A A B A A A A A A A A

Mealy Machine

Let's look at two such machines to find the number of occurences of the substring 'bab' in any strings over the

alphabet S = {a,b}.

Another Example

Mealy Machine Moore Machine

Integer String Recognizer

Build a finite state machine that can be used to recognize character string

representations of integer values.

Valid Integers Not Integers

123 123.456

123456 12+345

-543 Hello There

9 9 3 5

Integer String Recognizer Design

01010000 reject

01001110 accept

0111 accept

110000 reject

FSM that accepts strings containing at least three 1's

FSM that accepts strings containing at least three consecutive 1's

0000110011001100 reject

0001110000000000 accept

1010101010000000 reject

11111111 accept

Substring Detectors

Building a Bit String Recognizerdetect bit string "1101"

Moore Machine: Bit String Recognizer ("1101")Sometimes we need to decide whether overlapping substrings are accepted

Mealy Machine: Bit String Recognizer ("1101")

1101101101101101101

FSM to Recognize if a Binary Encoded Value is Divisible by Four

Parity Testercounting 0's and counting 1's

Recognizing Binary Strings with the Same Number of 1's and 0's.

A Partial FSM to Recognize Palindromesbinary strings

Detecting Binary Encoded Values Divisible by Three

00000

00001

00010

00011

00100

00101

00110

00111

01000

01001

01010

01011

01100

01101

01110

01111

Consider the substrings that leave you in the same state

regardless of which state you are in when the substring is

encountered

Such substrings can be removed from the candidate string

without affecting the final state. The substrings '11', '00' and

'1001' are three such "reducible" substrings.

Reduce the following string by removing these substrings.

11011010000000001111000100010010011010

11011010000000001111000100010010011010

0010001010

101010 yes