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An unconventional computational model A self-regulating balance The Balance-Machine infinite source filler + Z spiller x y X Y X+Y Addition x+y=Z INPUT pan OUTPUT pan (fixed weight) (variable weight) Schematic representation + Z x y Addition x+y=Z represents a balance; weights on both sides must balance + represents combination of two weights that add up. (The weights needn’t balance each other.) small letters, numerals represent fixed weights (inputs) capital letters represent variable weights (outputs) The balance can compute! Addition Increment + Z x+y=Z + Z=x+1 Z x y x 1 Subtraction x+Y=z Decrement + z Y=z-x X+1=z + x Y z X=z-1 X 1 The balance can compute! Weights (or pans) themselves can take the form of a balance-machine. Example 1: Multiplication by 2 1) a + B = A 2) a = B A Therefore, A = 2a. a B output input Example 2: Division by 2 1) A + B = a 2) A = B Therefore, A = a/2. a A B Note: The weight of a balance-machine is the sum of the individual weights on its pans. The balance can compute! Multiplication by 4 A = 4d A output B C d input Division by 4 D = a/4 a input B C D output Sharing pans between balances Example: Solving simultaneous equations X+Y=8 X–Y=2 1 2 + 8 + X2 X1 Y1 Y2 2 outputs 3 4 X1 X2 Y1 Y2 Computation universality of balances NOT(x) x + Y = 15 + 15 5 + 10 = 15 10 + 5 = 15 x Y NOTE: Input true = 10; false = 5; Output Interpreted as 1, if > 5 and as 0, otherwise. Computation universality of balances AND(x,y) x + y = Z + 10 + + 5+5 = 0 + 10 5 + 10 = 5 + 10 Z x y 10 10 + 5 = 5 + 10 10 + 10 = 10 + 10 OR(x,y) x+y=Z+5 + + 5+5 =5+5 Z 5 + 10 = 10 + 5 x y 5 10 + 5 = 10 + 5 10 + 10 = 15 + 5 NOTE: Input true = 10; false = 5; Output Interpreted as 1, if > 5 and as 0, otherwise. Computation universality of balances (1) (2) (3) Balance as a transmission line Balance (2) acts as transmission line, feeding output from (1) into the input of (3). Solving SAT with balances Consider the satisfiability of (a + b) (~a + b) Assumptions + + + + • true = 10; false = 5 • Fluid let out in “drops” (of 5 units) A B 10 5 Extra1 A’ B 10 5 Extra2 • Max. weight held by pan = 10 units (1) (2) a b (~a+b)(a+b) 0 0 0 + 15 1 0 0 A A’ 0 1 1 (3) 1 1 1 Machines 1-3 work together, sharing the variables A, B, and A’. The only possible configuration in which they can “stop” is one of the satisfiable configurations, if any. If the machine keeps “staggering” after a fixed time, then one might conclude that the expression is not satisfiable. Balance Machine – features The balance machine is a closed system unlike TMs. It is a closed system with a negative feed-back. The balance machine’s way of “computing” is very human. Does not require quantification in order to solve problems. Future research Balance-machine as a language recognizer Balance-machine as an artificial neuron

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posted: | 6/19/2013 |

language: | English |

pages: | 13 |

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