Algebras for Deterministic Computation Are Inherently Incomplete

TL;DR

This paper proves that the deterministic fragment of Kleene Algebra with Tests cannot be finitely generated by a small set of control flow operations, highlighting inherent limitations in algebraic representations of deterministic computation.

Contribution

It demonstrates that no finite set of regular control flow operations can generate the deterministic fragment of KAT, extending previous results on control flow expressivity.

Findings

Deterministic fragment of KAT is not finitely generated.

Traditional control flow operations are insufficient for deterministic computation.

Generalizes earlier results on control flow expressivity.

Abstract

Kleene Algebra with Tests (KAT) provides an elegant algebraic framework for describing non-deterministic finite-state computations. Using a small finite set of non-deterministic programming constructs (sequencing, non-deterministic choice, and iteration) it is able to express all non-deterministic finite state control flow over a finite set of primitives. It is natural to ask whether there exists a similar finite set of constructs that can capture all deterministic computation. We show that this is not the case. More precisely, the deterministic fragment of KAT is not generated by any finite set of regular control flow operations. This generalizes earlier results about the expressivity of the traditional control flow operations, i.e., sequential composition, if-then-else and while.