54 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
Iteration of functions and contractibility of acyclic 2-complexes
We show that there can be no algorithm to decide whether infinite recursively
described acyclic aspherical 2-complexes are contractible. We construct such a
complex that is contractible if and only if the Collatz conjecture holds.Comment: Third version: more minor misprints remove
Iteration of functions and contractibility of acyclic 2-complexes
We show that there can be no algorithm to decide whether infinite recursively described acyclic aspherical 2-complexes are contractible. We construct such a complex that is contractible if and only if the Collatz conjecture holds
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
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