4 research outputs found
Busy beavers gone wild
We show some incompleteness results a la Chaitin using the busy beaver
functions. Then, with the help of ordinal logics, we show how to obtain a
theory in which the values of the busy beaver functions can be provably
established and use this to reveal a structure on the provability of the values
of these functions
Godel's Incompleteness Phenomenon - Computationally
We argue that Godel's completeness theorem is equivalent to completability of
consistent theories, and Godel's incompleteness theorem is equivalent to the
fact that this completion is not constructive, in the sense that there are some
consistent and recursively enumerable theories which cannot be extended to any
complete and consistent and recursively enumerable theory. Though any
consistent and decidable theory can be extended to a complete and consistent
and decidable theory. Thus deduction and consistency are not decidable in
logic, and an analogue of Rice's Theorem holds for recursively enumerable
theories: all the non-trivial properties of such theories are undecidable
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
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