Karel as a Turing
Machine
CSE-111
12/03/04
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
Turing Machine-Recapitulation


A tape of (infinite)
squares with a tape
head.
Squares contain 0 ,
1
Tape head
Turing Machine-Recapitulation

Performs the
following actions
Tape head
Turing Machine-Recapitulation

1.
Performs the
following actions
Move-left-onesquare
Tape head
Turing Machine-Recapitulation

1.
2.
Performs the
following actions
Move-left-onesquare
Move-right-onesquare
Tape head
Turing Machine-Recapitulation

1.
2.
3.
Performs the
following actions
Move-left-onesquare
Move-right-onesquare
Print-0-at-currentsquare
Tape head
Turing Machine-Recapitulation

1.
2.
3.
4.
Performs the
following actions
Move-left-onesquare
Move-right-onesquare
Print-0-at-currentsquare
Print-1-at-currentsquare
Tape head
Turing Machine-Recapitulation

1.
2.
3.
4.
5.
Performs the
following actions
Move-left-onesquare
Move-right-onesquare
Print-0-at-currentsquare
Print-1-at-currentsquare
Erase
Tape head
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
Infinite Tape?
Karel has
 infinite number
of streets
 infinite number
of avenues
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
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
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
Move-left-one-square


face-west
If on first street,
(except on 1st ave)
front should be clear
so move.
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.
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
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;
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;
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;
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;
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;
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;
Erase
Check whether at left end
• Move left but don’t print 0(or put beeper)
• if not on a beeper (i.e indeed on left end)
Move right but don’t print 0 ( go
back to previous corner)
Pick all the beepers (erase)
Move right but don’t print 0(move to
square on right)
• if on a beeper
Move right but don’t print 0
Check whether at right end
Erase
Check whether at right end
• Move right but don’t print 0(or put beeper)
• if not on a beeper (i.e indeed on right end)
Move left but don’t print 0 ( go back to
previous corner)
Pick all the beepers (erase)
Move left but don’t print 0(move to
square on left)
• if on a beeper
Move left but don’t print 0
Erase
define-new-instruction erase as
begin
move-left-withoutputting-beeper;
if not-next-to-a-beeper
then
begin
moveright-withoutputting-beeper;
pickallbeeper;
moveright-withoutputting-beeper
end
else
begin
moveright-withoutputting-beeper;
moveright-withoutputting-beeper;
if not-next-to-a-beeper
then
begin
moveleftwithout-putting
-beeper;
pickallbeeper;
moveleftwithout-putting
-beeper
end
else
moveleft-withoutputting-beeper
end
end;
How to write a Turing Machine
flowchart in Karel
Example: Negation of
a binary digit
I/P: 1 square
O/P: 2 square
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
Move right
Is 0?
False
Move right
Print 1
Stop
How to write a Turing Machine
flowchart in Karel
For Karel:
See whether he is
on the corner with
one or two beepers,
True
move right , and
place beepers
Move right
accordingly
Start
Is 0?
False
Move right
Print 1
Stop
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
Move right
Is 0?
False
Move right
Print 1
Stop