2-dimensional Turing machines:
The TM with a 2-d ‘tape’, or a multi-dimensional grid of the cells, is no more powerful than the usual TM with a 1-dimensional tape. The unbounded tape consists of the capacity to store all info on a grid and to encode the geometry of the grid in the form of space filling curve. Apparently, 2-d layouts of the data can speed-up computation as compared to the 1-d layout, and the programs are simpler to write. We exemplify this with illustrations of arithmetic. E, N, W, S represent motions of the read-write head to neighboring square East, North, West, South. We permit multiple motions in a single transition, like NE for North-East.
Illustration: Define a 2-d TM which adds 2 binary integers x, y to generate the sum z. Select a convenient layout.
Illustration: Multiplication by doubling and halving: x * y = 2x * (y div 2) [+ x when y is odd], y > 0
Doubling and halving in the binary notation is attained by shifting. In this condition, doubling adds a new least significant 0, halving drops least significant bit. When the bit dropped is 1, that is, when y is odd, we have to add up a correction term according to the formula x * y = 2x * (y div 2) + x.
Outline of solution 1): The illustration below exhibits the multiplication 9 * 26 in decimal and in binary. In binary we ask for y to be written in the reversed order: 26 = 11010, 26rev = 01011. The reason is that the modification in both x and y takes place at least significant bit: as x gains a new bit, y loses a bit and the only modifications take place near the # which separates the least significant bits of x and y.
9 * 26 1001 # 01011 18 * 13 10010 # 1011 36 * 6 100100 # 011 72 * 3 1001000 # 11144 * 1 10010000 # 1--- --------234 11101010
Solution 2 (courtesy of Michael Breitenstein): x * y = (x div 2) * 2y [+ y if x is odd], y > 0
Idea: The multiplication ensues on two lines: the top line includes the present product x * y, the bottom line the present partial sum. Illustration 27*7 = 189, intermediate stages displayed from left to right.
27 * 7 13 * 14 6 * 28 3 * 56 1 * 1120 7 21 21 77 189
Solutions to such problems, and many other illustrations, can be found in our software system Turing Kara.
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