Straight-line instruction sequence completeness for total calculation on cancellation meadows

TL;DR

This paper proves that total functions on cancellation meadows can be computed by straight-line programs with at most five auxiliary variables, integrating program algebra with meadow theory for algebraic computation.

Contribution

It introduces a novel combination of program algebra and meadow theory to establish the completeness of straight-line instruction sequences for total calculation.

Findings

Total functions on cancellation meadows are computable with at most 5 auxiliary variables.

Similar results are achieved for signed meadows.

The approach integrates algebraic structures with program algebra for computation theory.

Abstract

A combination of program algebra with the theory of meadows is designed leading to a theory of computation in algebraic structures which use in addition to a zero test and copying instructions the instruction set $\{x \Leftarrow 0, x \Leftarrow 1, x\Leftarrow -x, x\Leftarrow x^{-1}, x\Leftarrow x+y, x\Leftarrow x\cdot y\}$. It is proven that total functions on cancellation meadows can be computed by straight-line programs using at most 5 auxiliary variables. A similar result is obtained for signed meadows.