Slides for Rosen, 5th edition

1
12.2 Finite State Machines with Output
2

• Remember the general picture of a computer as
  being a transition function TSI?SO?
• If the state set S is finite (not infinite), we
  call this system a finite state machine.
• If the domain SI 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.

3
Size of FSMs

• The information capacity of an FSM is C IS
  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 2C
  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 CPUs internal
  state), is then roughly 512223 60233
  519,691,042,816 b.
• How many states would be needed to write out the
  machines entire transition function graph?

2519,691,042,816 A number having gt1.7 trillion
decimal digits!
4
One Problem with 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.

5
Other Problems with FSM Model

• It ignores many important physical realities
• How is the transition functions structure to be
  encoded in physical hardware?
• How much hardware complexity is required to do
  this?
• How close in physical space is one bits worth of
  the machines 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 machines performance?
• Lets consider a basic example.

6
Vending Machine Example

• Suppose a certain vending machine accepts
  nickels, dimes, and quarters.
• If gt30 is deposited, change isimmediately
  returned.
• If the coke button is pressed,the machine
  drops a coke.
• Can then accept a new payment.

Ignore any otherbuttons, bills,out of
change,etc.
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Modeling the Machine

• Input symbol set I nickel, dime, quarter,
  button
• We could add nothing or ? as an additional
  input symbol if we want.
• Representing no input at a given time.
• Output symbol set O ?, 5, 10, 15, 20,
  25, coke.
• State set S 0, 5, 10, 15, 20, 25, 30.
• Representing how much money has been taken.

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Transition Function Table
Old state Input New state Output
0 n 5 ?
0 d 10 ?
0 q 25 ?
0 b 0 ?
5 n 10 ?
5 d 15 ?
5 q 30 ?
5 b 5 ?
Old state Input New state Output
10 n 15 ?
10 d 20 ?
10 q 30 5
10 b 10 ?
15 n 20 ?
15 d 25 ?
15 q 30 10
15 b 5 ?
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Transition Function Table cont.
Old state Input New state Output
20 n 25 ?
20 d 30 ?
20 q 30 15
20 b 20 ?
25 n 30 ?
25 d 30 5
25 q 30 20
25 b 25 ?
Old state Input New state Output
30 n 30 5
30 d 30 10
30 q 30 25
30 b 0 coke
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Another Format State Table
Each entryshowsnew state,output symbol
Old state Input Symbol Input Symbol Input Symbol Input Symbol
Old state n d q b
0 5,? 10,? 25,? 0,?
5 10,? 15,? 30,? 5,?
10 15,? 20,? 30,5 10,?
15 20,? 25,? 30,10 15,?
20 25,? 30,? 30,15 20,?
25 30,? 30,5 30,20 25,?
30 30,5 30,10 30,25 0,coke
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Directed-Graph State Diagram

• As you can see, these can get kind of busy.

q,5
d,5
q
q
q,20
d
d
d
n
n
n
n
n
n
0
5
10
15
20
25
30
n,5
b
b
b
b
b
b
d,10
q,25
q,15
b,coke
q,10
12
Formalizing FSMs

• Just like the general transition-function
  definition from earlier, but with the output
  function separated from the transition function,
  and with the various sets added in, along with an
  initial state.
• A finite-state machine M(S, I, O, f, g, s0)
• S is the state set.
• I is the alphabet (vocabulary) of input
  symbols
• O is the alphabet (vocabulary) of output symbols
• f is the state transition function
• g is the output function
• s0 is the initial state.
• Our transition function from before is T (f,g).

13
Construct a state table for the finite-state
machine in Fig. 3.
Find the output string for the input 101011
Answer 001000