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

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 ∗This work is the outcome of a visit by the authors to the Isaac Newton Institute for Mathematical Sciences in Cambridge (UK) for the centenary of Alan Turing’s birth, hosted by the 2012 programme “Semantics and Syntax: A Legacy of Alan Turing’. †Turlough Neary was supported by Swiss National Science Foundation grant 200021-141029. ‡Damien Woods was supported by the USA National Science Foundation under grants 0832824 (The Molecular Programming Project), CCF-1219274, and CCF-1162589. §Niall Murphy was supported by a PICATA Young Doctors fellowship from CEI Campus Moncloa, UCM-UPM, Madrid, Spain. ¶Rainer Glaschick was supported by the Heinz Nixdorf MuseumsForum, Germany. 1 ar X iv :1 30 4. 00 53 v2 [ cs .C C ] 5 A ug 2 01 4 Control unit: main program