Karel as a Turing Machine

1
Karel as a Turing Machine

• CSE-111 12/03/04

2
Facts

• Any programming language which satisfies Boehm
  Jacopini's conditions can be expressed by means
  of a Turing machine.
• The reverse is also true i.e. any programming
  language can be used to program a Turing Machine

3
Turing Machine-Recapitulation

• A tape of (infinite) squares with a tape head.
• Squares contain 0 , 1

4
Turing Machine-Recapitulation

• Performs the following actions

5
Turing Machine-Recapitulation

• Performs the following actions
• Move-left-one-square

Tape head
6
Turing Machine-Recapitulation

• Performs the following actions
• Move-left-one-square
• Move-right-one-square

7
Turing Machine-Recapitulation

• Performs the following actions
• Move-left-one-square
• Move-right-one-square
• Print-0-at-current-square

8
Turing Machine-Recapitulation

• Performs the following actions
• Move-left-one-square
• Move-right-one-square
• Print-0-at-current-square
• Print-1-at-current-square

9
Turing Machine-Recapitulation

• Performs the following actions
• Move-left-one-square
• Move-right-one-square
• Print-0-at-current-square
• Print-1-at-current-square
• Erase

10
Karel as a TM

• For Karel to work as a TM
• find an infinite tape.
• A way to represent 0 and 1.
• Program Karel to imitate the 5 functions or verbs

11
Infinite Tape?

• Karel has
• infinite number of streets
• infinite number of avenues

12
Infinite Tape

• The 1st street and
• 1st avenue can be
• used as the tape.
• Each corner can be used as a square on the tape

13
Representation of 0 and 1

• 2 beeper on corner can represent a square with a
  0
• 1 beeper on corner can represent a square with a
  1
• A corner with no beepers is a blank square

14
Programming Karel to become a TM

• Define 5 new instructions in Karel for each of
  the 5 functions/verbs of Turing Machine.
• Define all other instructions that may be
  required (by the 5 verb-instructions or the main
  program) .
• Write the main program which should perform the
  task of the Turing machine

15
Move-left-one-square

• face-west
• If on first street, (except on 1st ave)
• front should be clear so move.

16
Move-left-one-square

• face-west
• If on first street, (except on 1st ave)
• front should be clear so move.
• If on first avenue,
• face-north and move.

17
Move-left-one-square

• face-west
• If on first street, (except on 1st ave)
• front should be clear so move.
• If on first avenue,
• face-north and move.
• If on a corner with no beeper, print -0 or put 2
  beepers

18
Move-left-one-square

• define-new-instruction move-left as begin
• face-west
• if front-is-clear then
• move
• else
• begin
• face-north
• move
• end
• if not-next-to-a-beeper then
• print-0
• end

19
Move-right-one-square

• define-new-instruction move-right as begin
• face-south
• if front-is-clear then
• move
• else
• begin
• face-east
• move
• end
• if not-next-to-a-beeper then
• print-0
• end

20
Print-0

• Pick all the beepers at the current corner (there
  may be 1 or 2 beepers already)

• define-new-instruction print-0 as
• begin
• pickallbeeper putbeeper
• putbeeper
• end

21
Print-0

• Pick all the beepers at the current corner (there
  may be 1 or 2 beepers already)
• Put 2 beepers.

• define-new-instruction print-0 as
• begin
• pickallbeeper putbeeper
• putbeeper
• end

22
Print-1

• Pick all the beepers at the current corner (there
  may be 1 or 2 beepers already)

• define-new-instruction print-1 as
• begin
• pickallbeeper putbeeper
• end

23
Print-0

• Pick all the beepers at the current corner (there
  may be 1 or 2 beepers already)
• Put 1 beeper.

• define-new-instruction print-1 as
• begin
• pickallbeeper putbeeper
• end

24
Erase

• Check whether at left end
• Move left but dont print 0(or put beeper)
• if not on a beeper (i.e indeed on left end)
• Move right but dont print 0 ( go back to
  previous corner)
• Pick all the beepers (erase)
• Move right but dont print 0(move to square on
  right)
• if on a beeper
• Move right but dont print 0
• Check whether at right end

25
Erase

• Check whether at right end
• Move right but dont print 0(or put beeper)
• if not on a beeper (i.e indeed on right end)
• Move left but dont print 0 ( go back to previous
  corner)
• Pick all the beepers (erase)
• Move left but dont print 0(move to square on
  left)
• if on a beeper
• Move left but dont print 0

26
Erase

• if not-next-to-a-beeper then
• begin
• moveleft- without-putting
• -beeper pickallbeeper
• moveleft- without-putting
• -beeper
• end
• else
• moveleft-without- putting-beeper
• end
• end

• define-new-instruction erase as
• begin
• move-left-without- putting-beeper
• if not-next-to-a-beeper then
• begin
• moveright-without- putting-beeper
  pickallbeeper
• moveright-without- putting-beeper
• end
• else
• begin
• moveright-without- putting-beeper
• moveright-without-
• putting-beeper

27
How to write a Turing Machine flowchart in Karel

• Example Negation of a binary digit
• I/P 1 square
• O/P 2 square

28
How to write a Turing Machine flowchart in Karel

• Example Negation of a binary digit
• I/P 1 square
• O/P 2 square

Start
True
Is 0?
False
Move right
Move right
Print 1
Stop
29
How to write a Turing Machine flowchart in Karel

• For Karel
• See whether he is on the corner with
• one or two beepers,
• move right , and place beepers accordingly

Start
True
Is 0?
False
Move right
Move right
Print 1
Stop
30
How to write a Turing Machine flowchart in Karel

• beginning-of-execution
• if next-to-a-beeper then
• begin
• pickbeeper
• if next-to-a-beeper then begin
• putbeeper
• move-right
• print-1
• end
• else
• begin
• putbeeper
• move-right
• end
• end
• turnoff
• end-of-execution

Start
True
Is 0?
False
Move right
Move right
Print 1
Stop