Wang's B machines are efficiently universal, as is Hasenjaeger's small universal electromechanical toy

TL;DR

This paper demonstrates that Wang's B machines can simulate Turing machines efficiently with polynomial slowdown, and consequently, Hasenjaeger's small electromechanical machine is both small and fast, improving previous exponential slowdown results.

Contribution

The paper introduces a new polynomial-time simulation algorithm for Wang's B machines and shows that Hasenjaeger's machine also simulates Turing machines efficiently.

Findings

Wang's B machines simulate Turing machines with polynomial slowdown.

Hasenjaeger's machine efficiently simulates Turing machines in polynomial time.

Hooper's small universal Turing machine also achieves polynomial simulation, improving previous results.

Abstract

In the 1960's Gisbert Hasenjaeger built Turing Machines from electromechanical relays and uniselectors. Recently, Glaschick reverse engineered the program of one of these machines and found that it is a universal Turing machine. In fact, its program uses only four states and two symbols, making it a very small universal Turing machine. (The machine has three tapes and a number of other features that are important to keep in mind when comparing it to other small universal machines.) Hasenjaeger's machine simulates Hao Wang's B machines, which were proved universal by Wang. Unfortunately, Wang's original simulation algorithm suffers from an exponential slowdown when simulating Turing machines. Hence, via this simulation, Hasenjaeger's machine also has an exponential slowdown when simulating Turing machines. In this work, we give a new efficient simulation algorithm for Wang's B machines by showing that they simulate Turing machines with only a polynomial slowdown. As a second result, we find that Hasenjaeger's machine also efficiently simulates Turing machines in polynomial time. Thus, Hasenjaeger's machine is both small and fast. In another application of our result, we show that Hooper's small universal Turing machine simulates Turing machines in polynomial time, an exponential improvement.