Let A be a subset of the natural numbers, and let F A n (x 1 ; : : : ; xn ) = hØA (x 1 ); : : : ; ØA (xn )i; where ØA is the characteristic function of A. An oracle Turing machine with oracle A could certainly compute F A n with n queries to A. There are some sets A (e.g., the halting set) for which F A n can be computed with substantially fewer than n queries. One key reason for this is that the questions asked to the oracle can depend on previous answers, i.e., the questions are adaptive. We examine when it is possible to save queries. A set A is terse if the computation of F A n from A requires n queries. A set A is superterse if the computation of F A n from any set requires n queries. A set A is verbose if F A 2 n \Gamma1 can be computed with n queries to A. The range of possible query savings is limited by the following theorem: F A n cannot be computed with only blog nc queries to a set X unless A is recursive. In addition we produce the following: (1) a verbose ...

A query-bounded Turing machine is an oracle machine which computes its output function from a bounded number of queries to its oracle. In this paper we investigate the behavior of nondeterministic query-bounded Turing machines. In particular we study how easily such machines can compute the function F A n (x 1 , . . . , x n ) from A, where A # N and F A n (x 1 , . . . , x n ) = ##A (x 1 ), . . . , #A (x n )#. We show that each truth-table degree contains a set A such that, F A n can be nondeterministically computed from A by asking at most one question per nondeterministic branch; and that every set of the form A # also has this property. On the other hand, we show that if A is a 1-generic set then F A n cannot be nondeterministically computed from A in less that n queries to A; and that each non-zero r.e. Turing degree contains an r.e. set A with the same property. If the machines involved can only make queries that are part of their input then all sets such that F A n ca...

Blum and Blum (1975) showed that a class B of suitable recursive approximations to the halting problem K is reliably EX-learnable but left it open whether or not B is in NUM . By showing B to be not in NUM we resolve this old problem. Moreover, variants of this problem obtained by approximating any given recursively enumerable set A instead of the halting problem K are studied. All corresponding function classes U(A) are still EX-inferable but may fail to be reliably EX-learnable, for example if A is non-high and hypersimple. Blum and Blum (1975) considered only approximations to K defined by monotone complexity functions. We prove this condition to be necessary for making learnability independent of the underlying complexity measure. The class ~ B of all recursive approximations to K generated by all total complexity functions is shown to be not even behaviorally correct learnable for a class of natural complexity measures. On the other hand, there are complexity measures such that ~ B is EX -learnable. A similar result is obtained for all classes ~ U(A). For natural complexity measures, B is shown to be not robustly learnable, but again there are complexity measures such that B and, more generally, every class U(A) is robustly EX-learnable. This result extends the criticism of Jain, Smith and Wiehagen (1998), since the classes defined by artificial complexity measures turn out to be robustly learnable while those defined by natural complexity measures are not robustly learnable. 1 Supported by the Deutsche Forschungsgemeinschaft (DFG) under Heisenberg grant no. Ste 967/1--1. 2 Supported by the Grant-in-Aid for Scientific Research in Fundamental Areas from the Japanese Ministry of Education, Science, Sports, and Culture under grant no. 10558047. Part of th...

We study connections between strong reducibilities and properties of computably enumerable sets such as simplicity. We call a class S of computably enumerable sets bounded iff there is an m-in- complete computably enumerable set A such that every set in S is m-reducible to A. For example, we show that the class of eectively simple sets is bounded; but the class of maximal sets is not. Furthermore, the class of computably enumerable sets Turing reducible to a computably enumerable set B is bounded iff B is low 2 . For r = bwtt, tt, wtt and T , there is a bounded class intersecting every computably enumerable r-degree; for r = c, d and p, no such class exists. AMS Classication: 03D30; 03D25 Keywords: Computably enumerable sets (= Recursively enumerable sets); Simple sets; m-reducibility; Strong reducibilities; 3 classes; Ideals; Exact pairs 1 Introduction With a typical priority argument, one can show that for any simple set A, there is a simple set B such that B m A. Carl ...

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This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to know the dynamical properties of a c.e. set We , not merely whether an element x is enumerated in We but when, relative to its appearance in other c.e. sets. We present here