generic computation
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2017 ◽  
Vol 64 ◽  
pp. 67-78 ◽  
Author(s):  
Marion Morel ◽  
Catherine Achard ◽  
Richard Kulpa ◽  
Séverine Dubuisson

2015 ◽  
Vol 80 (4) ◽  
pp. 1290-1314 ◽  
Author(s):  
GREGORY IGUSA

AbstractA generic computation of a subsetAof ℕ is a computation which correctly computes most of the bits ofA, but which potentially does not halt on all inputs. The motivation for this concept is derived from complexity theory, where it has been noticed that frequently, it is more important to know how difficult a type of problem is in the general case than how difficult it is in the worst case. When we study this concept from a recursion theoretic point of view, to create a transitive relationship, we are forced to consider oracles that sometimes fail to give answers when asked questions. Unfortunately, this makes working in the generic degrees quite difficult. Indeed, we show that generic reduction is$\Pi _1^1$―complete. To help avoid this difficulty, we work with the generic degrees of density-1 reals. We demonstrate how an understanding of these degrees leads to a greater understanding of the overall structure of the generic degrees, and we also use these density-1 sets to provide a new a characterization of the hyperartithmetical Turing degrees.


2013 ◽  
Vol 78 (2) ◽  
pp. 511-522 ◽  
Author(s):  
Gregory Igusa

AbstractA generic computation of a subset A of ℕ consists of a computation that correctly computes most of the bits of A, and never incorrectly computes any bits of A, but which does not necessarily give an answer for every input. The motivation for this concept comes from group theory and complexity theory, but the purely recursion theoretic analysis proves to be interesting, and often counterintuitive. The primary result of this paper is that there are no minimal pairs for generic computability, answering a question of Jockusch and Schupp.


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