Xizhong Zheng

Weakly Computable Real Numbers

A real number x is recursively approximable if it is a limit of a computable sequence of rational numbers. If, moreover, the sequence is increasing (decreasing or simply monotonic), then x is called left computable (right computable or semi-computable). x is called weakly computable if it is a difference of two left computable real numbers. We show that a real number is weakly computable if and only if there is a computable sequence (xs)s?N of rational numbers which converges to x weakly effectively, namely the sum of jumps of the sequence is bounded. It is also shown that the class of weakly computable real numbers extends properly the class of semi-computable real numbers and the class of recursively approximable real numbers extends properly the class of weakly computable real numbers.

Resource-Bounded Balanced Genericity, Stochasticity and Weak Randomness

We introduce balanced t(n)-genericity which is a refinement of the genericity concept of Ambos-Spies, Fleischhack and Huwig [2] and which in addition controls the frequency with which a condition is met. We show that this concept coincides with the resource-bounded version of Churchs stochasticity [6]. By uniformly describing these concepts and weaker notions of stochasticity introduced by Wilber [19] and Ko [11] in terms of prediction functions, we clarify the relations among these resource-bounded stochasticity concepts. Moreover, we give descriptions of these concepts in the framework of Lutzs resource-bounded measure theory [13] based on martingales: We show that t(n)-stochasticity coincides with a weak notion of t(n)-randomness based on so-called simple martingales but that it is strictly weaker than t(n)-randomness in the sense of Lutz.

Effectiveness of the global modulus of continuity on metric spaces

In the “δ—e” — definition of continuity of a function ƒ between metric spaces, the value of δ depends on χ, e and function ƒ as well. This kind of dependence can be described by a function, so called, “global modulus of continuity”, which maps the triple (ƒ, χ, e) to corresponding δ. By a recent result of Repovs and Semenov (Proc. Int. Conf. Topol. (Trieste, 1993), G. Gentili (Ed.), Rent. Ist. Mat. Univ. Trieste, vol. 25, 1993, pp. 441–446), there is a continuous global modulus of continuity for the function space C(X, Y) of all continuous functions from a locally compact metric space X to an arbitrary metric space Y. Based on Weihrauchs framework on computable metric spaces (K. Weihrauch, Theoret. Comput. Sci. 113 (1993) 191–210), we show that there is a computable global modulus of continuity for C(X, Y), if X is an “effectively locally compact” metric space and Y is a computable metric space. The proof is a direct construction not depending on the proof of Repovs and Semenov.

Resource Bounded Randomness and Weakly Complete Problems

We introduce and study resource bounded random sets based on Lutzs concept of resource bounded measure ([5, 6]). We concentrate on nc-randomness (c ≥ 2) which corresponds to the polynomial time bounded (p-) measure of Lutz, and which is adequate for studying the internal and quantative structure of E = DTIME(2lin). First we show that the class of nc-random sets has p-measure 1. This provides a new, simplified approach to p-measure 1-results. Next we compare randomness with genericity (in the sense of [1, 2]) and we show that nc+1-random sets are nc-generic, whereas the converse fails. From the former we conclude thatnc-random sets are not p-btt-complete for E. Our technical main results describe the distribution of the nc-random sets under p-m-reducibility. We show that every nc-random set in E has nk-random predecessors in E for any k ≥ 1, whereas the amount of randomness of the successors is bounded. We apply this result to answer a question raised by Lutz [8]: We show that the class of weakly complete sets has measure 1 in E and that there are weakly complete problems which are not p-btt-complete for E.

Closure Properties of Real Number Classes under Limits and Computable Operators

In effective analysis, various classes of real numbers are discussed. For example, the classes of computable, semi-computable, weakly computable, recursively approximable real numbers, etc. All these classes correspond to some kind of (weak) computability of the real numbers. In this paper we discuss mathematical closure properties of these classes under the limit, effective limit and computable function. Among others, we show that the class of weakly computable real numbers is not closed under effective limit and partial computable functions while the class of recursively approximable real numbers is closed under effective limit and partial computable functions.

Approaches to Effective Semi‐Continuity of Real Functions

For semi-continuous real functions we study different computability concepts defined via computability of epigraphs and hypographs. We call a real function f lower semi-computable of type one, if its open hypograph hypo(f) is recursively enumerably open in dom(f) × ℝ; we call f lower semi-computable of type two, if its closed epigraph Epi(f) is recursively enumerably closed in dom(f) × ℝ; we call f lower semi-computable of type three, if Epi(f) is recursively closed in dom(f) × ℝ. We show that type one and type two semi-computability are independent and that type three semi-computability plus effectively uniform continuity implies computability, which is false for type one and type two instead of type three. We show also that the integral of a type three semi-computable real function on a computable interval is not necessarily computable.

Computability on Continuou, Lower Semi-continuous and Upper Semi-continuous Real Functions

In this paper we extend computability theory to the spaces of continuous, upper and lower semi-continuous real functions. We apply the framework of TTE, Type-2 Theory of Effectivity, where not only computable objects but also computable functions on the spaces can be considered. First some basic facts about TTE are summarized. For each of the function spaces, we introduce several natural representations based on different intiuitive concepts of “effectivity” and prove their equivalence. Computability of several operations on the function spaces is investigated, among others limits, mappings to open sets, images of compact sets and preimages of open sets, maximum and minimum values. The positive results usually show computability in all arguments, negative results usually express non-continuity. Several of the problems have computable but not extensional solutions. Since computable functions map computable elements to computable elements, many previously known results on computability are obtained as simple corollaries.

A comparison of weak completeness notions

We compare the weak completeness notions for E in the sense of Lutzs resource-bounded measure theory (1992) with respect to the standard polynomial time reducibilities. Our results parallel results for classical completeness by Watanabe (1987) and others. We show that the weak completeness notions for 1-query reductions coincide: A set is weakly complete for E under 1-truth-table reducibility iff it is weakly complete for length-increasing one-one reducibility. For most of the other polynomial reducibilities, however, we obtain separations of the weak completeness notions where these reducibilities differ on E (Ladner et al. (1975)). In fact our separations simultaneously hold for the corresponding weak completeness notions for E and E/sub 2/, for the classical completeness notions, and for the weak completeness notions in the sense of the resource-bounded Baire category concepts of Ambos-Spies et al. (1988) and Ambos-Spies (1995).

Approaches to Effective Semi-continuity of Real Functions

By means of different effectivities of the epigraphs and hypographs of real functions we introduce several effectivizations of the semi-continuous real functions. We call a real function f lower semicomputable of type one if its hypograph hypo(f) := {(x, y) : f(x) > y & × ∈ dom(f)} is recursively enumerably open in dom(f) × IR; f is lower semi-computable of type two if its closed epigraph Epi(f) := {(x, y) : f(x) ≤ y & x ∈ dom(f)} is recursively enumerably closed in dom(f) × IR and f is lower semi-computable of type three if Epi(f) is recursively closed in dom(f) × IR. These semi-computabilities and computability of real functions are compared. We show that, type one and type two semi-computability are independent and that type three semicomputability plus effectively uniform continuity implies computability which is false for type one and type two instead of type three. We show also that the integral of a type three semi-computable real function on a computable interval is not necessarily computable.