Asymptotic density, immunity and randomness

Abstract

Abstract. In 2012, inspired by developments in group theory and complexity, Jockusch and Schupp introduced generic computability, capturing the idea that an algorithm might work correctly except for a vanishing fraction of cases. However, we observe that their definition of a negligible set is not computably invariant (and thus not well-defined on the 1-degrees), resulting in some failures of intuition and a break with standard expectations in computability theory.To strengthen their approach, we introduce a new notion of intrinsic asymptotic density, with rich relations to both randomness and classical computability theory. We then apply these ideas to propose alternative foundations for further development in (intrinsic) generic computability.Toward these goals, we classify intrinsic density 0 as a new immunity property, specifying its position in the standard hierarchy from immune to cohesive for both general and ∆ 0 2 sets, and identify intrinsic density 1 2 as the stochasticity corresponding to permutation randomness. We also prove that Rice's Theorem extends to all intrinsic variations of generic computability, demonstrating in particular that no such notion considers ∅ ′ to be "computable".

The Computational Content of Intrinsic Density

“…(g −1 (A)) ≥ 1 − 1 n . Since this occurs for infinitely many n, we conclude that g samples A with upper density 1, and thus (by Lemma 4.2 of [1]) that A has absolute upper density 1. Take h(n) = n 2 (or, indeed, any computable superlinear function), and let f ≤ T A be as above.…”

“…In a previous paper [1], the author suggested that we instead consider sets to be thin if no computable process can sample the set at positive density infinitely often (i.e., with positive upper density); we say such a set has intrinsic density 0. More precisely: We can weaken this slightly, instead asking only that no sampling succeeds with positive density in the limit (i.e., with positive lower density).…”

“…However, since asymptotic density is not invariant under computable permutation, defining thinness in terms of density gives a notion that is Figure 1. [1] The graphs of the implications between the classical immunity properties; for Δ 0 2 sets, the implications are the same as in the general case, except that shh-immunity and hh-immunity become equivalent. All implications are strict, and any not shown (excepting those implied by transitivity) are false.…”

“…Figure 2. [1] The graph of implications between the classical immunity properties and intrinsic density 0. Again, for Δ 0 2 sets, shhimmunity and hh-immunity become equivalent; all other implications are as depicted.…”

Coarse Reducibility and Algorithmic Randomness

“…We will show in the next section that, as one might expect, the converse fails. The development of the theory of notions of robust information coding and related concepts have led to interactions with computability theory (as in Jockusch and Schupp [13]; Downey, Jockusch, and Schupp [4]; Downey, Jockusch, Mc-Nicholl, and Schupp [5]; and Hirschfeldt, Jockusch, McNicholl, and Schupp [10]), reverse mathematics (as in Dzhafarov and Igusa [7] and Hirschfeldt and Jockusch [9]), and algorithmic randomness (as in Astor [1]).…”