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Balance Machines

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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
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