an entire collection of useful table-driven algorithms makes use of a theoretical concept known as a...

An entire collection of useful table-driven algorithms makes use of a theoretical concept known as a finite state machine (FSM).

Example AlgorithmInput a stream of “bits” (‘0’ or ‘1’ characters) from a text stream.Output to the standard output stream, according to the following rules: 1) Write an ‘X’ for every odd-numbered bit (i.e., 1st, 3rd, 5th, etc.) 2) Write a ‘Z’ for every even-numbered bit with value of ‘0’. 3) Write an ‘N’ for every even-numbered bit with value of ‘1’.

Sample input stream

resulting output

0

X

0 0 1 1 1 1 0 0 1

Z

Question

How could we draw a picture of this algorithm? The Object of Data Abstraction

and Structure, David D. Riley© Addison Wesley pub.

A graphical model for the preceding algorithm is shown below.

After Even

After Odd

1 / X

0 / X

0 / Z

1 / N

Notation Each circle is a separate __________.

Each arc represents a potential transition from one state to another. The label a / b denotes an input symbol of a and output of b.

One state has an incoming arc without a state on the opposite end. This state is called the _________________________.

The Object of Data Abstraction and Structure, David D. Riley© Addison Wesley pub.

Sample input stream

resulting output

0 0 0 1 1 1 1 0 0 1

After Even

After Odd

1 / X

0 / X

0 / Z

1 / N


An (Finite State Machine) FSM is a 6-tuple: 1) A set of

states2) A start state3) A set of input symbols4) A set of output symbols5) A next state function

( State Input State)6) An output function

( State Input Output)

NextState(AfterEven, 0) AfterOdd NextState(AfterEven, 1) AfterOddNextState(AfterOdd, 0) AfterEvenNextState(AfterOdd, 1) AfterEven

Output(AfterEven, 0) XOutput(AfterEven, 1) XOutput(AfterOdd, 0) ZOutput(AfterOdd, 1) N

After Even

After Odd

1 / X

0 / X

0 / Z

1 / N

Name the parts below

Actions other than output are also permitted in some FSMs.


How many states?

0 / 0

1 / 1

0 / 2

1 / 3

1 / 30 / 00 /

2

1 / 1

Which is the start state?

What is the input alphabet? What is the output alphabet? Select meaningful state labels.


A 4-Step Strategy 1) Select the

states. 2) Identify the start state. 3) Complete the Next State function. 4) Complete the Output function.


Input consists of individual bits (0 or 1).

Input is partitioned into packets consisting of four consecutive bits. A dash (-) is output for each of the first three bits in a packet. An ‘E’ is output for each packet that has even parity.

A ‘D’ is output for each packet that has odd parity.

Sample

input stream resulting output

0

-

0 0 1 0 0 0 0 0 1

- - D - - - E - -

1

-

1

D

(Note that even parity occurs when the count of 1-valued bits in the packet is even, and odd parity occurs when the count is odd.)




What is a state?A state is a kind of

memory. A state “remembers” the most recent input(s). A state “remembers” a count.

What does the 4-bit parity checker need to remember?

Hints for Selecting States Partition the algorithm into all possible situations (states). Try tracing FSM execution until states appear to repeat.

What are the states of a 4-bit parity checker?


What is the start state of the 4-bit parity checker?




Think of the NextState function as a table.

0 1

input symbol

previous state




Think of the Output function as a table.

0 1

input symbol

previous state

NoBitsEven1

Odd1

Even2

Odd2

Even3

Odd3




Using table-driven code for the NextState and Output functions, makes it possible to create a general-purpose FSM simulator.

currentState = startState();input = readInputSymbol();while (! endOfInput() ) {

performOutput( outputTable[currentState][input] );

currentState = nextStateTable[currentState][input];

input = readInputSymbol();}


an entire collection of useful table-driven algorithms makes use of a theoretical concept known as a...

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