DISCRETE MATHEMATICS Lecture 22

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DISCRETE MATHEMATICSLecture 22

• Dr. Kemal Akkaya
• Department of Computer Science

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Modeling Computation

• We learned earlier the concept of an algorithm.
• A description of a computational procedure.
• Now, how can we model the computer itself, and
  what it is doing when it carries out an
  algorithm?
• For this, we want to model the abstract process
  of computation itself.

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Early Models of Computation

• Recursive Function Theory
• Kleene, Church, Turing, Post, 1930s
• Turing Machines Turing, 1940s
• RAM Machines von Neumann, 1940s
• Cellular Automata von Neumann, 1950s
• Finite-state machines, pushdown automata
• various people, 1950s
• VLSI models 1970s
• Parallel RAMs, etc. 1980s

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Turing Finite State Machines

• Any algorithm can be implemented in a Turing
  machine
• Alan Turing
• Topic of Theory of Computation Programming
  Languages
• We will give definition of Turing machines
• Finite State machines with external memory
• And briefly look at Finite State Machines

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Turing Machines (TM)

• Alan M. Turing (19121954)

• In 1936, Turing introduced hisabstract model
  for computation inhis article On Computable
  Numbers, with an
• application to the Entscheidungsproblem.

• At the same time, Alonzo Church published
  similar ideas and results.

• However, the Turing model has become
  thestandard model in theoretical computer
  science.

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Informal Description TM
At every step, the head of the TM M reads a
letter xi from the one-way infinite tape.
Depending on its state and the letter xi, the TM
- writes down a letter, - moves its read/write
head left or right, and - jumps to a new state.
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Motivation to use Finite State Machines

• Many applications of FSMs
• Control systems of all sorts
• GUI interaction model
• Elevator, vending machine, car-building robot
• Compilers, interpreters
• Simple parsers for languages
• Let M be a FSM. We say that M accepts a language
  L (of finite strings from Ms input alphabet S)
  if and only if
• For all t in L, FSM on t ends in a final state
• For all t not in L, FSM on t does not end in a
  final state

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12.2 - What is a Finite State Machine?

• Lets define the idea of a machine
• organism (real or synthetic) that responds to a
  countable (finite) set of stimuli (events) by
  generating predictable responses (outputs) based
  on a history of prior events (current state).
• A finite state machine (fsm) is a computational
  model of a machine with primitive memory
• Similar to Combinational Circuits
• But this time their output also depend on state
• Input is provided
• Base on the state an output is given
• Any circuit with memory is a FSM
• Serial adder example

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Serial Adder

• Gets inputs as bits
• The state here is
• Having a carry bit
• If there is a carry, add it to the sum
• Otherwise do nothing
• So INPUT STATE determine together the OUPUT

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Serial Adder FSM
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Elements of a FSM

• States represent the particular configurations
  that our machine can assume.
• Events define the various inputs that a machine
  will recognize
• Transitions represent a change of state from a
  current state to another (possibly the same)
  state that is dependent upon a specific event.
• The Start State is the state of the machine
  before it has received any events

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Size of FSMs

• The information capacity of an FSM is C 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.
• Computers can be viewed as huge FSMs
• 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!
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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.

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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
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Transition Function Table cont.
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Another Format State Table
Each entryshowsnew state,output symbol
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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
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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 (vocab.) 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).