Proof Trees

• probability vector. $v^{k}=\underbrace{0,0,\ldots,1,0,\ldots,0}_{2^{32}\ \mbox{elements}}$
  In a given node k will be in one state i, $\Rightarrow v_{i}^{k}=1$ and $v_{\not \,i}^{k}=0$

• For each function there are two n³ matrices (one for left and one for its right argument) (e.g. 2 $2^{32}\times 2^{32}\times 2^{32}$)
• Find other argument. Create probability vector o representing it

• Transformation matrix M = o N (N is the function's matrix)
  $\mbox{new state} = \mbox{state}\, M$

  The value at the division node in our example is given by
  

  $u_{y/3} = u_{y} \left(u_{3} N_{/,\mbox{right}}\right)$
  
  where $N_{/,\mbox{right}}$ is the instruction matrix corresponding to division (``right'' indicates ``divide by 3'' rather than ``3 divided by'')