A Comment on Budach's Mouse-in-an-Octant Problem

TL;DR

This paper discusses the decidability of Budach's Mouse-in-an-Octant Problem and sketches a proof that an extended version, the super-mouse, is undecidable, contributing to the understanding of computational limits in simple grid-based machines.

Contribution

It provides a proof sketch that the extended super-mouse version of the problem is undecidable, advancing theoretical knowledge in automata and computational geometry.

Findings

Extended super-mouse problem is undecidable

Decidability of the original problem remains open

Contributes to automata theory and computational geometry

Abstract

Budach's Mouse-in-an-Octant Problem (attributed to Lothar Budach in a 1980 article by van Emde Boas and Karpinski) concerns the behaviour of a very simple finite-state machine ("the mouse") moving on the integer two-dimensional grid. Its decidability is apparently still open. This note sketches a proof that an extended version of the problem (a super-mouse) is undecidable.