The principle ADS asserts that every linear order on ω has an infinite ascending or descending sequence. This has been studied extensively in the reverse mathematics literature, beginning with the work of Hirschfeldt and Shore [16]. We introduce the principle ADC, which asserts that every such linear order has an infinite ascending or descending chain. The two are easily seen to be equivalent over the base system RCA 0 of second order arithmetic; they are even computably equivalent. However, we prove that ADC is strictly weaker than ADS under Weihrauch (uniform) reducibility. In fact, we show that even the principle SADS, which is the restriction of ADS to linear orders of type ω + ω * , is not Weihrauch reducible to ADC. In this connection, we define a more natural stable form of ADS that we call General-SADS, which is the restriction of ADS to linear orders of type k + ω, ω + ω * , or ω + k, where k is a finite number. We define General-SADC analogously. We prove that General-SADC is not Weihrauch reducible to SADS, and so in particular, each of SADS and SADC is strictly weaker under Weihrauch reducibility than its general version. Finally, we turn to the principle CAC, which asserts that every partial order on ω has an infinite chain or antichain. This has two previously studied stable variants, SCAC and WSCAC, which were introduced by Hirschfeldt and Jockusch [16], and by Jockusch, Kastermans, Lempp, Lerman, and Solomon [18], respectively, and which are known to be equivalent over RCA 0 . Here, we show that SCAC is strictly weaker than WSCAC under even computable reducibility.

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".

This paper concerns algorithms that give correct answers with (asymptotic) density 1. A dense description of a function g : ω → ω is a partial function f on ω such that {n : f (n) = g(n)} has density 1. We define g to be densely computable if it has a partial computable dense description f . Several previous authors have studied the stronger notions of generic computability and coarse computability, which correspond respectively to requiring in addition that g and f agree on the domain of f , and to requiring that f be total. Strengthening these two notions, call a function g effectively densely computable if it has a partial computable dense description f such that the domain of f is a computable set and f and g agree on the domain of f . We compare these notions as well as asymptotic approximations to them that require for each ε > 0 the existence of an appropriate description that is correct on a set of lower density of at least 1−ε. We determine which implications hold among these various notions of approximate computability and show that any Boolean combination of these notions is satisfied by a c.e. set unless it is ruled out by these implications. We define reducibilities corresponding to dense and effectively dense reducibility and show that their uniform and nonuniform versions are different. We show that there are natural embeddings of the Turing degrees into the corresponding degree structures, and that these embeddings are not surjective and indeed that sufficiently random sets have quasiminimal degree. We show that nontrivial upper cones in the generic, dense, and effective dense degrees are of measure 0 and use this fact to show that there are minimal pairs in the dense degrees.

In a previous paper, the author introduced the idea of intrinsic density -a restriction of asymptotic density to sets whose density is invariant under computable permutation. We prove that sets with well-defined intrinsic density (and particularly intrinsic density 0) exist only in Turing degrees that are either high (a ′ ≥ T ∅ ′′ ) or compute a diagonally non-computable function. By contrast, a classic construction of an immune set in every noncomputable degree actually yields a set with intrinsic lower density 0 in every non-computable degree.We also show that the former result holds in the sense of reverse mathematics, in that (over RCA 0 ) the existence of a dominating or diagonally non-computable function is equivalent to the existence of a set with intrinsic density 0.

We define the notion of a determined Borel code in reverse math, and consider the principle DPB, which states that every determined Borel set has the property of Baire. We show that this principle is strictly weaker than ATR 0 . Any ω-model of DPB must be closed under hyperarithmetic reduction, but DPB is not a theory of hyperarithmetic analysis. We show that whenever M ⊆ 2 ω is the second-order part of an ω-model of DPB, then for every Z ∈ M , there is a G ∈ M such that G is ∆ 1 1 -generic relative to Z. Stephen G. Simpson. Subsystems of second order arithmetic. Perspectives in Logic.