1,718 research outputs found
The Busy Beaver Competition: a historical survey
Tibor Rado defined the Busy Beaver Competition in 1962. He used Turing
machines to give explicit definitions for some functions that are not
computable and grow faster than any computable function. He put forward the
problem of computing the values of these functions on numbers 1, 2, 3, ... More
and more powerful computers have made possible the computation of lower bounds
for these values. In 1988, Brady extended the definitions to functions on two
variables. We give a historical survey of these works. The successive record
holders in the Busy Beaver Competition are displayed, with their discoverers,
the date they were found, and, for some of them, an analysis of their behavior.Comment: 70 page
Problems in number theory from busy beaver competition
By introducing the busy beaver competition of Turing machines, in 1962, Rado
defined noncomputable functions on positive integers. The study of these
functions and variants leads to many mathematical challenges. This article
takes up the following one: How can a small Turing machine manage to produce
very big numbers? It provides the following answer: mostly by simulating
Collatz-like functions, that are generalizations of the famous 3x+1 function.
These functions, like the 3x+1 function, lead to new unsolved problems in
number theory.Comment: 35 page
The Busy Beaver Competition: a historical survey
70 pagesTibor Rado defined the Busy Beaver Competition in 1962. He used Turing machines to give explicit definitions for some functions that are not computable and grow faster than any computable function. He put forward the problem of computing the values of these functions on numbers 1, 2, 3, ... More and more powerful computers have made possible the computation of lower bounds for these values. In 1988, Brady extended the definitions to functions on two variables. We give a historical survey of these works. The successive record holders in the Busy Beaver Competition are displayed, with their discoverers, the date they were found, and, for some of them, an analysis of their behavior
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
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