Walk as thread-able expression

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Walk as thread-able expression is a method how we can treat walk in space and thread-able expression in arithmetic as the same thing.

Informal examples

We will give below informal examples, it is based on a variant of Example 2a with [math]\displaystyle{ \mu = 1, \lambda = \ln 2 }[/math] and the assignment is [math]\displaystyle{ A = -\frac{x}{y} }[/math] which labeled as red numbers on below picture.

Walk as expression.png

There are several walks from 1 to 3 marked by different colors

  • the black line encoded the expression [math]\displaystyle{ 1 \times 8 - 5 = 3 }[/math]
  • the purple line encoded the expression [math]\displaystyle{ (1 - \frac{5}{8}) \times 8 = 3 }[/math]
  • the brown line encoded the expression [math]\displaystyle{ (((1 - \frac{1}{8}) \times 2 - \frac{1}{2}) \times 2 - 1) \times 2 = 3 }[/math]
  • the orange line encoded the expression with accumulated infinitesimal by additions and multiplications which is a special integral.

A formal description will involve the completeness of grid systems like above case.