Results 1 
2 of
2
Short lists with short programs in short time
"... Given a machine U, a cshort program for x is a string p such that U(p) = x and the length of p is bounded by c + (the length of a shortest program for x). We show that for any universal machine, it is possible to compute in polynomial time on input x a list of polynomial size guaranteed to contai ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
(Show Context)
Given a machine U, a cshort program for x is a string p such that U(p) = x and the length of p is bounded by c + (the length of a shortest program for x). We show that for any universal machine, it is possible to compute in polynomial time on input x a list of polynomial size guaranteed to contain a O(logx)short program for x. We also show that there exist computable functions that map every x to a list of size O(x  2) containing a O(1)short program for x and this is essentially optimal because we prove that such a list must have size Ω(x  2). Finally we show that for some machines, computable lists containing a shortest program must have length Ω(2 x ).
On approximate decidability of minimal programs
, 2014
"... An index e in a numbering of partialrecursive functions is called minimal if every lesser index computes a different function from e. Since the 1960’s it has been known that, in any reasonable programming language, no effective procedure determines whether or not a given index is minimal. We invest ..."
Abstract
 Add to MetaCart
(Show Context)
An index e in a numbering of partialrecursive functions is called minimal if every lesser index computes a different function from e. Since the 1960’s it has been known that, in any reasonable programming language, no effective procedure determines whether or not a given index is minimal. We investigate whether the task of determining minimal indices can be solved in an approximate sense. Our first question, regarding the set of minimal indices, is whether there exists an algorithm which can correctly label 1 out of k indices as either minimal or nonminimal. Our second question, regarding the function which computes minimal indices, is whether one can compute a short list of candidate indices which includes a minimal index for a given program. We give some negative results and leave the possibility of positive results as open questions.