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 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.Comment: 18 pages, 1 figure, 1 table, Conference: Turing in context II - History and Philosophy of Computing, 201

On the boundaries of solvability and unsolvability in tag systems. Theoretical and Experimental Results

Several older and more recent results on the boundaries of solvability and unsolvability in tag systems are surveyed. Emphasis will be put on the significance of computer experiments in research on very small tag systems

On the time complexity of 2-tag systems and small universal Turing machines

We show that 2-tag systems efficiently simulate Turing machines. As a corollary we find that the small universal Turing machines of Rogozhin, Minsky and others simulate Turing machines in polynomial time. This is an exponential improvement on the previously known simulation time overhead and improves a forty year old result in the area of small universal Turing machines.Comment: Slightly expanded and updated from conference versio