Solovay reducibility and continuity

The objective of this study is a better understandingof the relationships between reduction and continuity. Solovay reduction is a variation of Turing reduction based on the distance of two real numbers. We characterize Solovay reduction by the existence of a certain real function that is computable (in the sense of computable analysis) and Lipschitz continuous. We ask whether thereexists a reducibility concept that corresponds to H¨older continuity. The answer is affirmative. We introduce quasi Solovay reduction and characterize this new reduction via H¨older continuity. In addition, we separate it from Solovay reduction and Turing reduction and investigate the relationships between complete sets and partial randomness.

CONSERVATIVITY OF ULTRAFILTERS OVER SUBSYSTEMS OF SECOND ORDER ARITHMETIC

We extend the usual language of second order arithmetic to one in which we can discuss an ultrafilter over of the sets of a given model. The semantics are based on fixing a subclass of the sets in a structure for the basic language that corresponds to the intended ultrafilter. In this language we state axioms that express the notion that the subclass is an ultrafilter and additional ones that say it is idempotent or Ramsey. The axioms for idempotent ultrafilters prove, for example, Hindman’s theorem and its generalizations such as the Galvin--Glazer theorem and iterated versions of these theorems (IHT and IGG). We prove that adding these axioms to IHT produce conservative extensions of ACA0+IHT,${\rm{ACA}}_{\rm{0}}^ +$, ATR0,${\rm{\Pi }}_2^1$-CA0, and${\rm{\Pi }}_2^1$-CA0for all sentences of second order arithmetic and for full Z2for the class of${\rm{\Pi }}_4^1$sentences. We also generalize and strengthen a metamathematical result of Wang (1984) to show, for example, that any${\rm{\Pi }}_2^1$theorem ∀X∃YΘ(X,Y) provable in ACA0or${\rm{ACA}}_{\rm{0}}^ +$there aree,k∈ ℕ such that ACA0or${\rm{ACA}}_{\rm{0}}^ +$proves that ∀X(Θ(X, Φe(J(k)(X))) where Φeis theeth Turing reduction andJ(k)is thekth iterate of the Turing or Arithmetic jump, respectively. (A similar result is derived for${\rm{\Pi }}_3^1$theorems of${\rm{\Pi }}_1^1$-CA0and the hyperjump.)

The logic of interactive turing reduction

The paper gives a soundness and completeness proof for the implicative fragment of intuitionistic calculus with respect to the semantics of computability logic, which understands intuitionistic implication as interactive algorithmic reduction. This concept — more precisely, the associated concept of reducibility — is a generalization of Turing reducibility from the traditional, input/output sorts of problems to computational tasks of arbitrary degrees of interactivity.