On Characterizing Spector Classes

We study in this paper characterizations of various interesting classes of relations arising in recursion theory. We first determine which Spector classes on the structure of arithmetic arise from recursion in normal type 2 objects, giving a partial answer to a problem raised by Moschovakis [8], where the notion of Spector class was first essentially introduced. Our result here was independently discovered by S. G. Simpson (see [3]). We conclude our study of Spector classes by examining two simple relations between them and a natural hierarchy to which they give rise

On symmetric differences of NP-hard sets with weakly P-selective sets

AbstractThe symmetric differences of NP-hard sets with weakly-P-selective sets are investigated. We show that if there exist a weakly-P-selective set A and an NP-⩽Pm-hard set H such that H - AϵPbtt(sparse) and A — HϵPm(sparse) then P = NP. So no NP-⩽Pm-hard set has sparse symmetric difference with any weakly-P-selective set unless P = NP. The proof of our main result is an interesting application of the tree prunning techniques (Fortune 1979; Mahaney 1982). In addition, we show that there exists a P-selective set which has exponentially dense symmetric difference with every set in Pbtt(sparse)

A dichotomy result for a pointwise summable sequence of operators

AbstractLet X be a separable Banach space and Q be a coanalytic subset of XN×X. We prove that the set of sequences (ei)i∈N in X which are weakly convergent to some e∈X and Q((ei)i∈N,e) is a coanalytic subset of XN. The proof applies methods of effective descriptive set theory to Banach space theory. Using Silver’s Theorem [J. Silver, Every analytic set is Ramsey, J. Symbolic Logic 35 (1970) 60–64], this result leads to the following dichotomy theorem: if X is a Banach space, (aij)i,j∈N is a regular method of summability and (ei)i∈N is a bounded sequence in X, then there exists a subsequence (ei)i∈L such that either (I) there exists e∈X such that every subsequence (ei)i∈H of (ei)i∈L is weakly summable w.r.t. (aij)i,j∈N to e and Q((ei)i∈H,e); or (II) for every subsequence (ei)i∈H of (ei)i∈L and every e∈X with Q((ei)i∈H,e)the sequence (ei)i∈H is not weakly summable to e w.r.t. (aij)i,j∈N. This is a version for weak convergence of an Erdös–Magidor result, see [P. Erdös, M. Magidor, A note on Regular Methods of Summability, Proc. Amer. Math. Soc. 59 (2) (1976) 232–234]. Both theorems obtain some considerable generalizations

P-Selectivity, Immunity, and the Power of One Bit

We prove that P-sel, the class of all P-selective sets, is EXP-immune, but is not EXP/1-immune. That is, we prove that some infinite P-selective set has no infinite EXP-time subset, but we also prove that every infinite P-selective set has some infinite subset in EXP/1. Informally put, the immunity of P-sel is so fragile that it is pierced by a single bit of information. The above claims follow from broader results that we obtain about the immunity of the P-selective sets. In particular, we prove that for every recursive function f, P-sel is DTIME(f)-immune. Yet we also prove that P-sel is not \Pi_2^p/1-immune

On the extension of computable real functions

International audienceWe investigate interrelationships among different notions from mathematical analysis, effective topology, and classical computability theory. Our main object of study is the class of computable functions defined over an interval with the boundary being a left-c.e. real number. We investigate necessary and sufficient conditions under which such functions can be computably extended. It turns out that this depends on the behavior of the function near the boundary as well as on the class of left-c.e. real numbers to which the boundary belongs, that is, how it can be constructed. Of particular interest a class of functions is investigated: sawtooth functions constructed from computable enumerations of c.e. sets