5 research outputs found
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
The complexity of small universal Turing machines: a survey
We survey some work concerned with small universal Turing machines, cellular
automata, tag systems, and other simple models of computation. For example it
has been an open question for some time as to whether the smallest known
universal Turing machines of Minsky, Rogozhin, Baiocchi and Kudlek are
efficient (polynomial time) simulators of Turing machines. These are some of
the most intuitively simple computational devices and previously the best known
simulations were exponentially slow. We discuss recent work that shows that
these machines are indeed efficient simulators. In addition, another related
result shows that Rule 110, a well-known elementary cellular automaton, is
efficiently universal. We also discuss some old and new universal program size
results, including the smallest known universal Turing machines. We finish the
survey with results on generalised and restricted Turing machine models
including machines with a periodic background on the tape (instead of a blank
symbol), multiple tapes, multiple dimensions, and machines that never write to
their tape. We then discuss some ideas for future work
Complexity of Small Universal Turing Machines: A Survey
We survey some work concerned with small universal Turing machines, cellular automata, tag systems, and other simple models of computation. For example it has been an open question for some time as to whether the smallest known universal Turing machines of Minsky, Rogozhin, Baiocchi and Kudlek are efficient (polynomial time) simulators of Turing machines. These are some of the most intuitively simple computational devices and previously the best known simulations were exponentially slow. We discuss recent work that shows that these machines are indeed efficient simulators. In addition, another related result shows that Rule 110, a well-known elementary cellular automaton, is efficiently universal. We also discuss some old and new universal program size results, including the smallest known universal Turing machines. We finish the survey with results on generalised and restricted Turing machine models including machines with a periodic background on the tape (instead of a blank symbol), multiple tapes, multiple dimensions, and machines that never write to their tape. We then discuss some ideas for future work