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 class of r.e. degrees, called the array nonrecursive degrees, previously studied by the authors in connection with multiple permitting arguments relative to r.e. sets, is extended to the degrees in general. This class contains all degrees which satisfy a (i.e. a 2 GL 2 ) but in addition there exist low r.e. degrees which are array nonrecursive (a.n.r.).

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Abstract. We answer a question of Jockusch by constructing a hyperimmunefree minimal degree below a 1-generic one. To do this we introduce a new forcing notion called arithmetical Sacks forcing. Some other applications are presented. 1. introduction Two fundamental construction techniques in set theory and computability theory are forcing with finite strings as conditions resulting in various forms of Cohen genericity, and forcing with perfect trees, resulting in various forms of minimality. Whilst these constructions are clearly incompatible, this paper was motivated by the general question of “How can minimality and (Cohen) genericity interact?”. Jockusch [5] showed that for n ≥ 2, no n-generic degree can bound a minimal degree, and Haught [4] extended earlier work of Chong and Jockusch to show that that every nonzero Turing degree below a 1-generic degree below 0 ′ was itself 1generic. Thus, it seemed that these forcing notions were so incompatible that perhaps no minimal degree could even be comparable with a 1-generic one. However, this conjecture was shown to fail independently by Chong and Downey [1] and by Kumabe [7]. In each of those papers, a minimal degree below m < 0 ′ and a 1-generic a < 0 ′ ′ are constructed with m < a. The specific question motivating the present paper is one of Jockusch who asked whether a hyperimmune-free (minimal) degree could be below a 1-generic one. The point here is that the construction of a hyperimmune-free degree by and large directly uses forcing with perfect trees, and is a much more “pure ” form of Spector-Sacks forcing [10] and [9]. This means that it is not usually possible to use tricks such as full approximation or forcing with partial computable trees, which are available to us when we only wish to construct (for instance) minimal degrees. For instance, minimal degrees can be below computably enumerable ones, whereas no degree below 0 ′ can be hyperimmune-free. Moreover, the results of Jockusch [5], in fact prove that for n ≥ 2, if 0 < a ≤ b and b is n-generic, then a bounds a n-generic degrees and, in particular, certainly is not hyperimmune free. This contrasts quite strongly with the main result below. In this paper we will answer Jockusch’s question, proving the following result.

Algorithmic and resource-bounded Baire category and corresponding genericity concepts introduced in computability theory and computational complexity theory, respectively, have become elegant and powerful tools in these settings. Here we introduce some new genericity notions based on extension functions computable by nite automata which are tailored for capturing diagonalizations over regular sets and functions. We show that the generic sets obtained either by the partial regular extension functions of any given xed constant length or by all total regular extension of constant length are just the sets with saturated (also called disjunctive) characteristic sequences. Here a sequence is saturated if every string occurs in as a substring. We also show that these automatic generic sets are not regular but may be context free. Furthermore, we introduce stronger automatic genericity notions based on regular extension functions of nonconstant length and we show that the corresponding generic sets are bi-immune for the classes of regular and context free languages.

Abstract. A real is called properly n-generic if it is n-generic but not n + 1generic. We show that every 1-generic real computes a properly 1-generic real. On the other hand, if m> n � 2 then an m-generic real cannot compute a properly n-generic real.

We say that a real X is relatively r.e. if there exists a real Y such that X is r.e. (Y) and X ̸≤T Y. We say X is relatively REA if there exists such a Y ≤T X. We define A ≤e1 B if there exists a Σ1 set C such that n ∈ A if and only if there is a finite E ⊆ B with (n, E) ∈ C. In this paper we show that a real X is relatively r.e. if and only if X ̸≤e1 X. We prove that every nonempty Π 0 1 class contains a real which is not relatively r.e. We also construct a real which is relatively r.e. but not relatively REA. We say that a real X is relatively simple and above if there exists a real Y such that X is r.e. (Y) and there is no infinite Z ⊆ X such that Z is r.e. (Y). We prove that every 1-generic real is relatively simple and above. 1

Abstract. We study low level nondefinability in the Turing degrees. We prove a variety of results, including for example, that being array nonrecursive is not definable by a Σ1 or Π1 formula in the language (≤,REA) where REA stands for the “r.e. in and above ” predicate. In contrast, this property is definable by a Π2 formula in this language. We also show that the Σ1-theory of (D, ≤,REA) is decidable. 1.