RM-FRS

RM-FRS - Maintain one prime per RM instruction Instruction...

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Register Machines ≤ Factor Replacement Systems Chris Ellis Bruce Meeks, Jr.
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Register Machines Natural numbers stored in finite set of registers (result in n th register) Finite length programs of instructions labeled 1 to m Two types of instructions: INC r [i] : Increment value of register r by 1, jump to instruction 1 DEC r [p,z] : If register r > 0, decrement by 1 and jump to p, otherwise jump to z
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Factor Replacement Systems Single natural number x used for information storage (multiple values encoded as exponents of prime numbers, result stored as exponent of 2) “Instructions” stored as list of rational numbers a / b First fraction a / b for which x is divisible by b is multiplied by x, which replaces the value of x by xa / b
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RM ≤ FRS Show that RMs are no more powerful than FRS Encode state of RM as natural number Simulate behavior of RM by FRS
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Encoding RM State as FRS State Convert value of all registers to a single natural number (prime factorization) <r 0 , r 1 , r 2 , … , r n > p 1 r 0 p 2 r 1 p 3 r 2 …p n r n
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Unformatted text preview: Maintain one prime per RM instruction Instruction 1,2,,m p n+1 ,p n+2 ,,p n+m Keep one last prime to represent halt state p n+m+1 Simulating Behavior of RM For each instruction 1 j m If j is an INC instruction (j. INC k [ l ]) Add FRS rule p n+j x p n+ l p k x If j is a DEC instruction (j. DEC k [p,z]) Add FRS rule p n+j p k x p n+p x Add FRS rule p n+j x p n+z x Clean-up rules 0 i n-1 p n+m+1 p i x p n+m+1 x Finally p n+m+1 x x Example Limited Subtraction Register Machine r2 r1 r2, if r1 r2; 0, otherwise Factor Replacement System Input: 3 x 5 y Output: 2 z where z = x y, if x y; 0, otherwise Example Limited Subtraction 1. DEC2[2,3] 2. INC1[1] 3. DEC3[4,5] 4. DEC1[3,3] 5. 3*7x 11x 7x 13x 11x 2*7x 5*13x 17x 13x 19x 2*17x 13x 17x 13x Associated Primes 7 11 13 17 19...
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RM-FRS - Maintain one prime per RM instruction Instruction...

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