Wang’s B machines are efficiently universal,as is Hasenjaeger’s small universal electromechanical toy |
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Authors: | Turlough Neary Damien Woods Niall Murphy Rainer Glaschick |
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Institution: | 1. Institute for Neuroinformatics, University of Zürich & ETH Zürich, Switzerland;2. California Institute of Technology, Pasadena, CA 91125, USA;3. Facultad de Informática, Universidad Politécnica de Madrid & CEI-Moncloa UPM-UCM, Spain;4. Microsoft Research, Cambridge, CB1 2FB, UK1;5. Paderborn, Germany |
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Abstract: | In the 1960s 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. |
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Keywords: | Polynomial time Computational complexity Small universal Turing machines Wang&rsquo s B machine Non-erasing Turing machines Models of computation |
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