INFO 2950

Prof. Carla Gomes [email protected]

Module Modeling Computation: Finite State Machines with Output Rosen, Chapter 12.2

1

Remember the general picture of a computer as being a transition function

T

:

S

×

I→S

×

O

?

– If the state set

S

is finite (not infinite), we call this system a

finite state machine

.

If the domain

S

×

I

is reasonably small, then we can specify

T

explicitly by writing out its complete graph.

– However, this is practical only for machines that have a very small information capacity.

Size of FSMs

The information capacity of an FSM is

C

=

I

[

S

] = log |

S

| .

– Thus, if we represent a machine having an information capacity of

C

bits as an FSM, then its state transition graph will have |

S

| = 2

C

nodes.

E

.

g

. suppose your desktop computer has a 512MB memory, and 60GB hard drive.

– Its information capacity, including the hard drive and memory (and ignoring the CPU’s internal state), is then roughly ~512×2 23 + 60×2 33 = 519,691,042,816 b .

– How many states would be needed to write out the machine’s entire transition function graph?

2 519,691,042,816 = A number having >1.7 trillion decimal digits!

FSMs as Models

The FSM diagram of a reasonably-sized computer is more than astronomically huge.

– Yet, we are able to design and build these computers using only a modest amount of industrial resources.

• Why is this possible?

Answer:

A real computer has

regularities

in its transition function that are

not captured

if we just write out its FSM transition function explicitly.

–

I.e.

, a transition function can have a small, simple, regular description, even if its domain is enormous.

Limitations with FSM Model

It ignores many important physical realities: – How is the transition function’s structure to be encoded in physical hardware?

• How much hardware complexity is required to do this?

– How close in physical space is one bit’s worth of the machine’s information capacity to another?

• How long does it take to communicate information from one part of the machine to another?

– How much energy gets dissipated to heat when the machine updates its state?

• How fast can the heat be removed, and how much does this impact the machine’s performance?

Let’s consider a basic example.

Applications

Finite-State Machines are used in a variety of applications.

– Spell checking programs – Grammar checking – Indexing and searching large text files – Speech/Language recognition – Network Protocols

Types of Finite-State Machines

Finite-State Machines with Output – –

Mealy

: Output determined by state and input

Moore

: Output determined by state alone Finite-State Machines with No Output – Also known

as finite-state automata

– There are two types of finite-state automata •

Deterministic

: Each state-input pair dictates a unique transition into another state •

Non-deterministic

: Each state-input pair can lead to several possible states

Finite State Machines with Output

We will focus on only

Mealy machines

Since we will always refer to

finite-state machines with no output

as

finite-state automata

, we will use the term

finite-state machine

to mean

finite state machine with output

.

The best way to understand finite-state machines is probably with an example.

Perhaps the best example deals with a device most of use are very familiar with—vending machines.

Example: Candy Machine

Consider a vending machine that – Accepts nickels (5 cents), dimes (10 cents), and quarters (25 cents), crediting the amount.

– If the total credit is more than 25, it returns the difference so only 25 cents credit remains. – Dispenses a candy bar if the candy button is pushed and there is 20 cents credit.

– Dispenses a candy bar and returns 5 cents if the candy button is pushed and there is 25 cents credit.

– Dispenses a soda if the soda button is pushed and there is 25 cents credit.

Candy Machine States

The vending machine can be in different states based on the amount of money that has been credited to the user.

Change is returned after 25 cents, and all coins are multiples of 5.

Thus, the machine can be in the following states: – 0 cents credit (state

S 0

) – 5 cents credit (state

S 1

) – 10 cents credit (state

S 2

) – 15 cents credit (state

S 3

) – 20 cents credit (state

S 4

) – 25 cents credit (state

S 5

)

Candy Machine Input/Output

The machine can accept the following inputs – A dime – A nickel (10 cents) inserted (5 cents) inserted – A quarter (25 cents) inserted – Candy button pushed (CB) – Soda button pushed (SB) The machine has the following possible outputs – A dime – A nickel (10 cents) returned (5 cents) returned – A quarter (25 cents) returned – A candy bar (C) dispensed – A soda – Nothing (S) dispensed (n) is returned or dispensed

Candy Machine FSM

We now have enough information to construct our finite-state machine.

For each possible input and each possible state, we need to know what to output (if anything) and what state the machine should go to.

For instance: – If the machine is in state

S 3

(15 cents credit) and – a quarter (25 cents) is input – the machine should transition into state

S 5

(25 cents credit) and – 15 cents (a dime and nickel) should be output. We can construct a state diagram and/or a state table by considering every possible input on every possible state.

Candy Machine State Diagram

Candy Machine State Table

State S 0 S 1 S 2 S 3 S 4 S 5 5 S 1 S 2 S 3 S 4 S 5 S 5 Next State Input 10 25 CB SB S 2 S 3 S 4 S 5 S 5 S 5 S S S S S 5 S 5 5 5 5 5 S S S S S 0 S 0 0 1 2 3 S 0 S 1 S 2 S 3 S 4 S 0 5 n n n n n 5 10 n n n n 5 10 Output Input 25 n 5 10 CB n n n 15 20 25 n Candy Candy,5 SB n n n n n Soda

FSM Definition

Definition:

A

finite-state machine

is a 6-tuple

M

=(

S

,

I

,

O

,

f

,

g

,

S 0

) where –

S

is a finite set of

states

– – – –

I

is a finite

input alphabet O

is a finite

output alphabet f

:

S



I



S

is a

transition function

from each state-input pair to a state

g

:

S



I



O

is a

output function

output from each state-input pair to an –

S 0

is the

initial state

FSM Representations

As we have already seen, there are two common ways of representing finite-state machines – A

state table

is used to represent a finite-state machine by giving the values of the functions

f

and

g

.

– A

state diagram

is a directed graph representation of a finite-state machine.

State Tables

State S 0 S 1 S 2 S 3 S 4 S 5 5 S 1 S 2 S 3 S 4 S 5 S 5 A

state table

is organized as follows The rows are indexed by the states .

The columns are split into two groups: •The first half are indexed by the

inputs

•The entries in the table give the value of the function

f

– that is the

new states

10 S 2 S 3 S 4 S 5 S 5 S 5 Next State Input 25 S 5 S 5 S 5 S 5 S 5 S 5 CB S 0 S 1 S 2 S 3 S 0 S 0 SB S 0 S 1 S 2 S 3 S 4 S 0 5 n n n n n 5 10 n n n n 5 10 10 15 20 25 Output Input 25 n 5 CB n n n n Candy Candy,5 SB n n n n n Soda •The second half are also indexed by the

inputs

•The entries in the table give the value of the function

g

– that is, the

outputs

.

State Diagram

A

state diagram

is organized as follows – The

nodes

in the graph represent the states .

The

edges

in the graph represent the

transitions

.

An from

edge

(

S i ,S j

)

S i

to

S j

Each

edge

occurs if some

input

causes a transition is labeled with a pair (

x

,

y

), where •

x

is the

input

which (along with the state) causes the transition •

y

is the

output

state-input pair.

triggered by the

Example: Unit Delay

In some electronic devices, it is necessary to use a

unit-delay machine

.

That is, whatever is time.

input from the machine, but into the machine should be output delayed by a specific amount of For instance, given a string of binary numbers

x 1

,

x 2

, …,

x n

, the machine should produce the string 0,

x 1

,

x 2

, …,

x n-1

.

We want to use a finite state machine to model the behavior of a

unit-delay machine

.

What should a state in this machine represent?

One possibility is that a state represents the last input bit.

Thus we need a state for “1” and a state for “0” We also need start state.

Unit Delay States

We will use the following states (that memorize last bit; except S 0 ) – State

S 0

– State

S 1

– State

S 2

is the start state occurs if the previous input was 1 occurs if the previous input was 0 We can easily construct the state table for the unit delay machine by realizing that – When the input is 0, we always transition to state

S 2

– When the input is 1, we always transition to state

S 1

– When we are in state

S 1

previous input was 1) we always output 1 (since the – When we are in state

S 2

previous input was 0) we always output 0 (since the – When we are in state

S 0

always output 0 first) we always output 0 (since we

Unit Delay State Table/Diagram

Here is the state table and state diagram based on our previous observations.

State S 0 S 1 S 2 Next State Input 0 S 2 S 2 S 2 1 S 1 S 1 S 1 0 0 1 0 Output Input 1 0 1 0 What’s output of 101011?

Example: Binary Adder

We want to construct a finite state machine that will add two numbers.

The input is two binary numbers, (

x n

…

x 1 x 0

) 2 and (

y n

…

y 1 y 0

) 2 At each step, we can compute (

x i +y i

) starting with (

x 0 +y 0

).

– If (

x i +y i

)=0, we output 0. – If (

x i +y i

)=1, we output 1. – If (

x i +y i

)=2, we have a problem. The problem is we need a carry bit.

In fact, our computation needs to know the carry bit at each step (so we compute

x i +y i

+c

i

give it to the next step.

at each step), and be able to We can take care of this by using states to represent the carry bit.

Binary Adder States

We will use the following states – State S 0 – State

S 1

occurs if the carry bit is 0 occurs if the carry bit is 1 Since when we begin the computation, there is no carry, we can use

S 0

as the start state, So, how does which state we are in affect the output?

If we are in state

S 0

(we have a carry of 0) – If (

x i +y i

)=0 , we output 0, and stay in state

S 0

– If (

x i +y i

)=1, we output 1, and stay in state

S 0

– If (

x i +y i

)=2, we output 0, and go to state

S 1

If we are in state

S 1

(we have a carry of 1) – If (

x i +y i

+1)=1, we output 1, and go to state

S 0

– If (

x i +y i

+1)=2, we output 0, and stay in state

S 1

– If (

x i +y i

+1)=3, we output 1, and stay in state

S 1

Binary Adder State Table/Diagram

From the previous observations, we can construct the state table and state diagrams for the binary adder Next State Input State 00 01 10 S 0 S 1 S 0 S 0 S 0 S 1 S 0 S 1 Output Input 11 00 01 10 11 S 1 S 1 0 1 1 0 1 0 0 1

Construct a state table for the finite-state machine in Fig. 3.

Input, output

Find the output string for the input 101011 Answer: 001000

Output of 101011?

001000

Language Recognizer

We want to construct an FSM that outputs 1 iff the string read so far has 111.

S0 – this state remembers that the previous input value (if it exists) was not a 1 S1 – this state remembers that the previous input value was a 1, but the input before (if it exists) was not a 1 S2 – this state remembers that the previous two input values were 1 26

0,0 S0 0,0 S1 1,0 1,0 0,0 S2

Language Recognizer

1,1 S0 – this state remembers that the previous input value (if it exists) was not a 1 S1 – this state remembers that the previous input value was a 1, but the input before (if it exists) was not a 1 S2 – this state remembers that the previous two input values were 1 This finite-state machine recognizes the set of bit strings that end in 111.

27

FSM as a Language Recognizer

Definition:

Let

M

=(

S

,

I

,

O

,

f

,

g

,

S 0

) be a FSM and L  I*.

We say that M recognizes (or accepts) L, iff for any input string x that belongs to L, M produces a 1 as an output

Next Finite state machines with no output Finite Automata 29