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An unconventional computational model

Explore the innovative balance-machine model that represents a self-regulating balance; weights must add up on both sides. Learn about its computation universality, solving equations, and SAT problems. Discover its unique features and potential for future research.

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An unconventional computational model

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  1. Anunconventional computational model

  2. The Balance-Machine infinite source filler Z + spiller x y X + Y X Y Addition x + y = Z INPUTpan (fixed weight) OUTPUTpan (variable weight) Aself-regulating balance

  3. 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.) Schematic representation small letters,numerals represent fixed weights (inputs) capital letters represent variable weights (outputs)

  4. Addition Increment x + y = Z Z = x + 1 Z Z x y x 1 + + + + Subtraction x + Y= z Y = z - x Decrement z X + 1 = z X = z - 1 x Y z X 1 The balance can compute!

  5. Example 1: Multiplication by 2 A output a B input Example 2: Division by 2 a A B The balance can compute! Weights (or pans) themselves can take the form of a balance-machine. 1) a + B = A 2) a = B Therefore, A = 2a. 1) A + B = a 2) A = B Therefore, A = a/2. Note:The weight of a balance-machine is the sum of the individual weights on its pans.

  6. Multiplication by 4 A output B C d input Division by 4 a input B C D output The balance can compute! A = 4d D = a/4

  7. Example: Solving simultaneous equations X + Y = 8 X – Y = 2 1 2 8 X2 + + X1 Y1 Y2 2 3 4 X1 X2 Y1 Y2 Sharing pans between balances outputs

  8. NOT(x) x + Y = 15 5 + 10 = 15 10 + 5 = 15 15 x Y + Computation universality of balances NOTE: Input true = 10; false = 5; Output Interpreted as 1, if > 5 and as 0, otherwise.

  9. AND(x,y) x + y = Z + 10 5 + 5 = 0 + 10 5 + 10 = 5 + 10 10 + 5 = 5 + 10 10 + 10 = 10 + 10 10 x y Z + + + + OR(x,y) x + y = Z + 5 5 + 5 = 5 + 5 5 + 10 = 10 + 5 10 + 5 = 10 + 5 10 + 10 = 15 + 5 5 x y Z Computation universality of balances NOTE: Input true = 10; false = 5; Output Interpreted as 1, if > 5 and as 0, otherwise.

  10. (1) (2) (3) Computation universality of balances Balance as a transmission line Balance (2) acts as transmission line, feeding output from (1) into the input of (3).

  11. Assumptions • true = 10; false = 5 • Fluid let out in “drops” (of 5 units) • Max. weight held by pan = 10 units A B 10 5 Extra1 A’ B 10 5 Extra2 (1) (2) + + + + + a b (~a+b)(a+b) 0 0 0 1 0 0 0 1 1 1 1 1 15 A A’ (3) Solving SAT with balances Consider the satisfiability of (a + b) (~a + b) 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.

  12. 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.

  13. Future research • Balance-machine as a language recognizer • Balance-machine as an artificial neuron

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