An unconventional computational model

A self-regulating balance INPUT pan (fixed weight) OUTPUT pan (variable weight) infinite source filler spiller The Balance-Machine + x y Z Addition x + y = Z X + Y X Y

Schematic representation + x y Z 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! + x y Z Addition x + y = Z + x 1 Z Increment Z = x + 1 + x Y z Subtraction x + Y = z Y = z - x + X 1 z Decrement X + 1 = z X = z - 1

The balance can compute! Weights (or pans) themselves can take the form of a balance-machine. a B A Example 1: Multiplication by 2 input output 1) a + B = A 2) a = B Therefore, A = 2a. A B a Example 2: Division by 2 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.

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

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

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

Computation universality of balances + x y Z AND(x,y) 10 x + y = Z + 10 5 + 5 = 0 + 10 5 + 10 = 5 + 10 10 + 5 = 5 + 10 10 + 10 = 10 + 10 + x y OR(x,y) Z 5 x + y = Z + 5 5 + 5 = 5 + 5 5 + 10 = 10 + 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) Max. weight held by pan = 10 units a b (~a+b)(a+b) 0 0 0 1 0 0 0 1 1 1 1 1 + A B 10 (1) 5 Extra1 + A’ B (2) 10 5 Extra2 + A 15 A’ (3) 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