Solovay Reducibility Research Articles

Solovay reducibility via Lipschitz functions and signed-digit representation

We explore Solovay reducibility in the context of computably approximable reals, extending its natural characterization for left-c.e. reals via computable Lipschitz functions. Our paper offers two distinct characterizations: the first employs Lipschitz functions, while the second utilizes Turing reductions with bounded use with respect to signed-digit representation. Additionally, we examine multiple related reducibilities and establish separations among them. These results contribute to a refined perspective of the relationship between Solovay reducibility and computable Lipschitz functions.

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.

Randomness and Solovay degrees

We consider the behaviour of Schnorr randomness, a randomness notion weaker than Martin-L o f's, for left-r.e. reals under Solovay reducibility. Contrasting with results on Martin-L o f-randomenss, we show that Schnorr randomness is not upward closed in the Solovay degrees. Next, some left-r.e. Schnorr random α is the sum of two left-r.e. reals that are far from random. We also show that the left-r.e. reals of effective dimension > r , for some rational r , form a filter in the Solovay degrees.

L1-Computability, Layerwise Computability and Solovay Reducibility

We propose a hierarchy of classes of functions that corresponds to the hierarchy of randomness notions. Each class of functions converges at the corresponding random points. Wegive various characterizations of the classes, that is, characterizations via integral tests, L 1 -computability and layerwise computability. Furthermore, the relation among these classes is formulated

Degrees of monotone complexity

AbstractLevin and Schnorr (independently) introduced the monotone complexity, Km (α), of a binary string α. We use monotone complexity to define the relative complexity (or relative randomness) of reals. We define a partial ordering ≤Km on 2ω by α ≤Km β iff there is a constant c such that Km(α | n) ≤ Km(β | n)+ c for all n. The monotone degree of α is the set of all β such that α Km β and β Km α. We show the monotone degrees contain an antichain of size , a countable dense linear ordering (of degrees of cardinality ), and a minimal pair.Downey, Hirschfeldt, LaForte, Nies and others have studied a similar structure, the K-degrees, where K is the prefix-free Kolmogorov complexity. A minimal pair of K-degrees was constructed by Csima and Montalban. Of particular interest are the noncomputable trivial reals, first constructed by Solovay. We defineareal to be (Km,K)-trivial if for some constant c, Km(α | n) ≤ K(n) + c for all n. It is not known whether there is a Km-minimal real, but we show that any such real must be (Km,K)-trivial.Finally, we consider the monotone degrees of the computably enumerable (c.e.) and strongly computably enumerable (s.c.e.) reals. We show there is no minimal c.e. monotone degree and that Solovay reducibility does not imply monotone reducibility on the c.e. reals. We also show the s.c.e. monotone degrees contain an infinite antichain and a countable dense linear ordering.

A C.E. Real That Cannot Be SW-Computed by Any Ω Number

The strong weak truth table (sw) reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky, and Weinberger on applications of computability to differential geometry. We study the sw-degrees of c.e. reals and construct a c.e. real which has no random c.e. real (i.e., Ω number) sw-above it.

Randomness and reducibility

We study reducibilities that act as measures of relative randomness on reals, concentrating particularly on their behavior on the computably enumerable reals. One such reducibility, called domination or Solovay reducibility, has already proved to be a powerful tool in the study of randomness of effectively presented reals. Motivated by certain shortcomings of Solovay reducibility, we introduce two new measures of relative randomness and investigate their properties and the relationships between them and Solovay reducibility.

Randomness, Computability, and Density

We study effectively given positive reals (more specifically, computably enumerable reals) under a measure of relative randomness introduced by Solovay [manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 1975] and studied by Calude, Hertling, Khoussainov, and Wang [Theoret. Comput. Sci., 255 (2001), pp. 125--149], Calude [Theoret. Comput. Sci., 271 (2002), pp. 3--14], Kucera and Slaman [SIAM J. Comput., 31 (2002), pp. 199--211], and Downey, Hirschfeldt, and LaForte [Mathematical Foundations of Computer Science 2001, Springer-Verlag, Berlin, 2001, pp. 316--327], among others. This measure is called domination or Solovay reducibility and is defined by saying that $\alpha$ dominates $\beta$ if there are a constant c and a partial computable function $\varphi$ such that for all positive rationals $q < \alpha$ we have $\varphi(q)\!\downarrow < \beta$ and $\beta- \varphi(q) \leqslant c(\alpha- q)$. The intuition is that an approximating sequence for $\alpha$generates one for $\beta$ whose rate of convergence is not much slower than that of the original sequence. It is not hard to show that if $\alpha$ dominates $\beta$, then the initial segment complexity of $\alpha$ is at least that of $\beta$. In this paper we are concerned with structural properties of the degree structure generated by Solovay reducibility. We answer a natural question in this area of investigation by proving the density of the Solovay degrees. We also provide a new characterization of the random computably enumerable reals in terms of splittings in the Solovay degrees. Specifically, we show that the Solovay degrees of computably enumerable reals are dense, that any incomplete Solovay degree splits over any lesser degree, and that the join of any two incomplete Solovay degrees is incomplete, so that the complete Solovay degree does not split at all. The methodology is of some technical interest, since it includes a priority argument in which the injuries are themselves controlled by randomness considerations.