Probabilistic methods of bypassing the maze using stones and a random number sensor

TL;DR

This paper investigates the capabilities of finite state machines with random bit generators, stones, and flags to bypass various dimensions of integer spaces, revealing specific thresholds where bypassing is possible or impossible.

Contribution

It establishes new theoretical bounds on the dimensions of integer spaces that robots with different tools can bypass, extending previous work on maze bypassing with finite state machines.

Findings

Robots with a random bit generator can bypass up to dimension 4 but not 5.

Robots with a stone can bypass up to dimension 6 but not 7.

Robots with a stone and flags can bypass up to dimension 8 but not 9.

Abstract

In this paper, some open questions that are posed in Ajans' dissertation continue to be addressed: a robot bypass with a generator of random bits of integer spaces in the presence of a stone and a subspace of flags. This work is devoted to bypassing the maze with a finite state machine with a random bit generator. This task is part of the rapidly evolving theme of bypassing the maze by various finite state machines. or their teams, which is closely related to problems from the theory of computational complexity and probability theory. In this paper, it is shown at what dimensions a robot with a random bit generator and a stone can bypass integer space with a subspace of flags. In this paper, we will study the behavior of a finite state machine with a random bit generator on integer spaces. In particular, it was proved that the robot bypasses $ \ zs ^ 2 $ and cannot bypass $ \ zs ^ 3 $; a robot with a stone bypasses $ \ zs ^ 4 $ and cannot bypass $ \ zs ^ 5 $; a robot with a stone and a flag bypasses $ \ zs ^ 6 $ and cannot bypass $ \ zs ^ 7 $; a robot with a stone and a plane of flags bypasses $ \ zs ^ 8 $ and cannot bypass $ \ zs ^ 9 $.