11 research outputs found
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
Simplicity via Provability for Universal Prefix-free Turing Machines
Universality is one of the most important ideas in computability theory.
There are various criteria of simplicity for universal Turing machines.
Probably the most popular one is to count the number of states/symbols. This
criterion is more complex than it may appear at a first glance. In this note we
review recent results in Algorithmic Information Theory and propose three new
criteria of simplicity for universal prefix-free Turing machines. These
criteria refer to the possibility of proving various natural properties of such
a machine (its universality, for example) in a formal theory, PA or ZFC. In all
cases some, but not all, machines are simple
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
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
Small Turing universal signal machines
This article aims at providing signal machines as small as possible able to
perform any computation (in the classical understanding). After presenting
signal machines, it is shown how to get universal ones from Turing machines,
cellular-automata and cyclic tag systems. Finally a halting universal signal
machine with 13 meta-signals and 21 collision rules is presented
Small weakly universal Turing machines
We give small universal Turing machines with state-symbol pairs of (6,2), (3,3) and (2,4). These machines are weakly universal, which means that they have an infinitely repeated word to the left of their input and another to the right. They simulate Rule 110 and are currently the smallest known weakly universal Turing machines. Despite their small size these machines are efficient polynomial time simulators of Turing machines