A mod B function Turing machine

Question

I'd like to know how can I calculate function A mod B, where A > B and A, B are unary numbers, with a deterministic turing machine with a single tape.

Thanks

Answer 1

Given an input like B111...10111...1BBB..., where the first string of 1s is a unary encoding of a (i.e., 1^a) and the second string of 1s is a unary encoding of b (i.e., 1^b), we can design a single-tape deterministic Turing machine to compute a mod b (by entering halt-accept after leaving the unary representation of a mod b left on the tape).

Note first that a mod b < a, so we can recover the unary representation of a mod b by erasing some of the 1s from a's unary representation, and all of the 1s from b's unary representation. Observe that a mod b = (a - b) mod b, at least when a >= b; when a < b, then a mod b = a. This observation suggests that we can erase b 1s from a's unary representation until we have fewer than b 1s remaining, at which point we erase the 1s from b's representation and halt-accept.

Pseudocode:

move right until you find a blank.
move one step to the left.
you are now looking at the last 1 in b's representation.
mark this as Y and move left until you find 0.
move left until you find a 1 or blank.
you are now looking at the last 1 in a's representation, or blank.
if 1, mark this as X and move right until you find Y.
if blank, a < b; change all Xs to 1s and all 0s, 1s and YBs to blanks. halt-accept.
move one step to the left.
you are now looking at the last 1 in b's representation, or 0.
if 1, continue as above.
if 0, b < a; change all Xs to 0s, all Ys to 1s, and restart from the beginning

Example: 10 mod 3

B11111111110111BBB...
^

B11111111110111BBB...
               ^        move right until you find a blank

B11111111110111BBB...
              ^         move one step to the left. looking at last 1 in b

B1111111111011YBBB...
           ^            mark as Y and move left to 0

B1111111111011YBBB...
          ^             move one step to the left. looking at last 1 in a.

B111111111X011YBBB...
              ^         mark as X and move right to Y

B111111111X011YBBB...
             ^          move one step to the left. looking at last 1 in b.

B111111111X01YYBBB...
           ^            mark as Y and move left to 0

B111111111X01YYBBB...
         ^              move left to 1

B11111111XX01YYBBB...
             ^          mark as X and move right to Y

B11111111XX01YYBBB...
            ^           move one step to the left. looking at last 1 in b

B11111111XX0YYYBBB...
           ^            mark as Y and move left to 0

B11111111XX0YYYBBB...
        ^               move left to 1

B1111111XXX0YYYBBB...   mark as X and move right to Y
            ^

B1111111XXX0YYYBBB...   move one step left. looking at 0; b < a
           ^

B11111110000111BBB...
^                       change Xs to 0s and Ys to 1s; start over.

(above process repeats two more times)

B10000000000111BBB...
^                       erased 3x 1s from a 3x times

B10000000000111BBB...
               ^        move right to blank

B10000000000111BBB...
              ^         move one step left. looking at last 1 in b

B1000000000011YBBB...
           ^            mark as Y and move left to 0.

B1000000000011YBBB...
 ^                      move left to 1

BX000000000011YBBB...
              ^         mark as X and move right to Y

BX000000000011YBBB...
             ^          move one step left. looking at last 1 in b.

BX00000000001YYBBB...
           ^            mark as Y and move left to 0

BX00000000001YYBBB...
^                       move left to blank. a < b.

B1BBBBBBBBBBBBBBBB...
^                       change Xs to 1s and 0s, 1s, Ys to blank. halt-accept