karel as a turing machine

Karel as a Turing Karel as a Turing MachineMachine

CSE-111 12/03/04CSE-111 12/03/04

FactsFacts

Any programming language which Any programming language which satisfies Boehm & Jacopini's conditions can satisfies Boehm & Jacopini's conditions can be expressed by means of a Turing be expressed by means of a Turing machine.machine.

The reverse is also true i.e. any The reverse is also true i.e. any programming language can be used to programming language can be used to program a Turing Machineprogram a Turing Machine

Turing Machine-RecapitulationTuring Machine-Recapitulation

A tape of (infinite) A tape of (infinite) squares with a tape squares with a tape head.head.

Squares contain 0 , Squares contain 0 , 11

Tape head


Performs the Performs the following actionsfollowing actions

Tape head


1.1. Move-left-one-Move-left-one-squaresquare


Tape head



2.2. Move-right-one-Move-right-one-squaresquare


Tape head




3.3. Print-0-at-current-Print-0-at-current-squaresquare


Tape head







Tape head






5.5. EraseErase


Tape head

Karel as a TMKarel as a TM

For Karel to work as a TMFor Karel to work as a TM

find an infinite tape.find an infinite tape.

A way to represent 0 and 1.A way to represent 0 and 1.

Program Karel to imitate the 5 Program Karel to imitate the 5 functions or verbsfunctions or verbs

Infinite Tape?Infinite Tape?

Karel has Karel has infinite number infinite number

of streetsof streets infinite number infinite number

of avenuesof avenues

Infinite TapeInfinite Tape

The 1The 1stst street and street and

11stst avenue can be avenue can be

used as the tape.used as the tape. Each corner can be Each corner can be

used as a square on used as a square on the tapethe tape

Representation of 0 and 1Representation of 0 and 1

2 beeper on 2 beeper on corner can corner can represent a represent a square with a 0square with a 0

1 beeper on 1 beeper on corner can corner can represent a represent a square with a 1square with a 1

A corner with no A corner with no beepers is a blank beepers is a blank squaresquare

Programming Karel to become a TMProgramming Karel to become a TM

Define 5 new instructions in Karel for each Define 5 new instructions in Karel for each of the 5 functions/verbs of Turing Machine.of the 5 functions/verbs of Turing Machine.

Define all other instructions that may be Define all other instructions that may be required (by the 5 verb-instructions or the required (by the 5 verb-instructions or the main program) .main program) .

Write the main program which should Write the main program which should perform the task of the Turing machineperform the task of the Turing machine

Move-left-one-squareMove-left-one-square

face-westface-west If on first street, If on first street,

(except on 1(except on 1stst ave) ave)

front should be clear front should be clear so move.so move.



(except on 1(except on 1stst ave) ave)

front should be clear front should be clear so move.so move.

If on first avenue,If on first avenue,

face-north and move.face-north and move.



(except on 1(except on 1stst ave) ave)front should be clear front should be clear so move.so move.

If on first avenue,If on first avenue,face-north and move.face-north and move.

If on a corner with no If on a corner with no beeper, print -0 or put beeper, print -0 or put 2 beepers2 beepers

define-new-instruction move-left as begin define-new-instruction move-left as begin face-west; face-west; if front-is-clear then if front-is-clear then

move move else else begin begin

face-north; face-north; move move

end; end; if not-next-to-a-beeper then if not-next-to-a-beeper then

print-0;print-0; end; end;


define-new-instruction move-right as begin define-new-instruction move-right as begin face-south; face-south; if front-is-clear then if front-is-clear then

move move else else begin begin

face-east; face-east; move move

end; end; if not-next-to-a-beeper then if not-next-to-a-beeper then

print-0;print-0; end; end;

Move-right-one-squareMove-right-one-square

Print-0Print-0

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

define-new-instruction define-new-instruction print-0 as print-0 as

begin begin

pickallbeeperpickallbeeper; ; putbeeper; putbeeper;

putbeeper putbeeper

end; end;

Print-0Print-0


Put 2 beepersPut 2 beepers..


begin begin

pickallbeeper; pickallbeeper; putbeeper; putbeeper;

putbeeper;putbeeper;

end; end;

Print-1Print-1



begin begin

pickallbeeperpickallbeeper; ; putbeeper; putbeeper;

end; end;

Print-0Print-0


Put 1 beeperPut 1 beeper..


begin begin

pickallbeeper; pickallbeeper; putbeeper; putbeeper;

end; end;

EraseEraseCheck whether at left endCheck whether at left end

• Move left but don’t print 0(or put beeper)Move left but don’t print 0(or put beeper)• if not on a beeper (i.e indeed on left end)if not on a beeper (i.e indeed on left end)

Move right but don’t print 0 ( go back to previous Move right but don’t print 0 ( go back to previous corner) corner)

Pick all the beepers (erase)Pick all the beepers (erase)Move right but don’t print 0(move to square on right)Move right but don’t print 0(move to square on right)

• if on a beeperif on a beeperMove right but don’t print 0Move right but don’t print 0

Check whether at right endCheck whether at right end

EraseEraseCheck whether at right endCheck whether at right end

• Move right but don’t print 0(or put beeper)Move right but don’t print 0(or put beeper)• if not on a beeper (i.e indeed on right end)if not on a beeper (i.e indeed on right end)

Move left but don’t print 0 ( go back to previous Move left but don’t print 0 ( go back to previous corner) corner)

Pick all the beepers (erase)Pick all the beepers (erase)Move left but don’t print 0(move to square on left)Move left but don’t print 0(move to square on left)

• if on a beeperif on a beeperMove left but don’t print 0Move left but don’t print 0

EraseErasedefine-new-instruction erase asdefine-new-instruction erase as begin begin

move-left-without-move-left-without-putting-beeper; putting-beeper;

if not-next-to-a-beeper if not-next-to-a-beeper then then

begin begin moveright-without-moveright-without-putting-beeper; putting-beeper;

pickallbeeper;pickallbeeper;moveright-without-moveright-without-putting-beeper putting-beeper end end

else else begin begin

moveright-without-moveright-without-putting-beeper;putting-beeper;

moveright-without-moveright-without-putting-beeper;putting-beeper;

if not-next-to-a-beeper if not-next-to-a-beeper then then

begin begin moveleft-moveleft-without-puttingwithout-putting-beeper; -beeper; pickallbeeper; pickallbeeper; moveleft-moveleft-without-puttingwithout-putting-beeper -beeper

end end elseelse

moveleft-moveleft-without-without- putting-putting-beeper beeper endendend; end;

How to write a Turing Machine How to write a Turing Machine flowchart in Karelflowchart in Karel

Example: Example: Negation of Negation of

a binary digit a binary digit

I/P: 1 squareI/P: 1 square

O/P: 2 squareO/P: 2 square


Example: Example: Negation of Negation of

a binary digit a binary digit

I/P: 1 squareI/P: 1 square

O/P: 2 squareO/P: 2 square

Start

Is 0?

Move right Move right

Print 1Stop

FalseTrue


For Karel:For Karel:

See whether he is See whether he is on the corner with on the corner with

one or two beepers,one or two beepers,

move right , and move right , and place beepers place beepers accordinglyaccordingly

Start

Is 0?


Print 1Stop

FalseTrue


beginning-of-executionbeginning-of-execution

if next-to-a-beeper then if next-to-a-beeper then begin begin

pickbeeper;pickbeeper; if next-to-a-beeper then if next-to-a-beeper then begin begin

putbeeper; putbeeper; move-right; move-right; print-1 print-1

end end else else begin begin

putbeeper;putbeeper;move-right move-right

endendend; end; turnoff; turnoff; end-of-execution end-of-execution

Start

Is 0?


Print 1Stop

FalseTrue

karel as a turing machine

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