S. Barry Cooper

The d.r.e. degrees are not dense

Abstract By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n -r.e. degrees ( n ⩾2) and of the ω-r.e. degrees.

On minimal pairs of enumeration degrees

For sets of natural numbers A and B, A is enumeration reducible to B if there is some effective algorithm which when given any enumeration of B will produce an enumeration of A . Gutteridge [5] has shown that in the upper semilattice of the enumeration degrees there are no minimal degrees (see Cooper [3]), and in this paper we study those pairs of degrees with gib 0 . Case [1] constructed a minimal pair. This minimal pair construction can be relativised to any gib, and following a suggestion of Jockusch we can also fix one of the degrees and still construct the pair. These methods yield an easier proof of Cases exact pair theorem for countable ideals. 0″ is an upper bound for the minimal pair constructed in §1, and in §2 we improve this bound to any Σ 2 -high Δ 2 degree. In contrast to this we show that every low degree c bounds a degree a which is not in any minimal pair bounded by c . The structure of the co-r.e. e-degrees is isomorphic to that of the r.e. Turing degrees, and Gutteridge has constructed co-r.e. degrees which form a minimal pair in the e-degrees. In §3 we show that if a, b is any minimal pair of co-r.e. degrees such that a is low then a, b is a minimal pair in the e-degrees (and so Gutteridges result follows). As a corollary of this we can embed any countable distributive lattice and the two nondistributive five-element lattices in the e-degrees below 0′. However the lowness assumption is necessary, as we also prove that there is a minimal pair of (high) r.e. degrees which is not a minimal pair in the e-degrees (under the isomorphism). In §4 we present more concise proofs of some unpublished work of Lagemann on bounding incomparable pairs and embedding partial orderings. As usual, { W i } i ∈ ω is the standard listing of the recursively enumerable sets, D u is the finite set with canonical index u and {‹ m, n ›} m, n ∈ ω is a recursive, one-to-one coding of the pairs of numbers onto the numbers. Capital italic letters will be variables over sets of natural numbers, and lower case boldface letters from the beginning of the alphabet will vary over degrees.

Incomputability in Nature

To what extent is incomputability relevant to the material Universe? We look at ways in which this question might be answered, and the extent to which the theory of computability, which grew out of the work of Godel, Church, Kleene and Turing, can contribute to a clear resolution of the current confusion. It is hoped that the presentation will be accessible to the non-specialist reader.

Weak density and cupping in the d-r.e. degrees

Consider the Turing degrees of differences of recursively enumerable sets (the d-r.e. degrees). We show that there is a properly d-r.e. degree (a d-r.e. degree that is not r.e.) between any two comparable r.e. degrees, and that given a high r.e. degreeh, every nonrecursive d-r.e. degree ≦h cups toh by a low d-r.e. degree.

Cupping and noncupping in the enumeration degrees of ∑20 sets

Abstract We prove the following three theorems on the enumeration degrees of ∑ 2 0 sets. Theorem A: There exists a nonzero noncuppable ∑ 2 0 enumeration degree. Theorem B: Every nonzero Δ 2 0 enumeration degree is cuppable to 0′ e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low Δ 2 0 enumeration degree with the anticupping property .

How the World Computes

Partial Cylindrical Algebraic Decomposition I: The Lifting Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560 Grant Olney Passmore and Paul B. Jackson Multi-valued Functions in Computability Theory . . . . . . . . . . . . . . . . . . . . . 571 Arno Pauly Relative Randomness for Martin-Löf Random Sets . . . . . . . . . . . . . . . . . . . 581 NingNing Peng, Kojiro Higuchi, Takeshi Yamazaki, and Kazuyuki Tanaka Table ofThe idea that the Universe is a program in a giant quantum computer is both fascinating and suffers from various problems. Nonetheless, it can provide a unified picture of physics and this can be very useful for the problem of Quantum Gravity where such a unification is necessary. In previous work we proposed Quantum Graphity, a simple way to model a dynamical spacetime as a quantum computation. In this paper, we give an easily readable introduction to the idea of the universe as a quantum computation, the problem of quantum gravity, and the graphity models.

First Steps into Metapredicativity in Explicit Mathematics

Metapredicativity is a new general term in proof theory which describes the analysis and study of formal systems whose proof-theoretic strength is beyond the Feferman-Schutte ordinal Γ0 but which are nevertheless amenable to purely predicative methods. Typical examples of formal systems which are apt for scaling the initial part of metapredicativity are the transfinitely iterated fixed point theories IDα whose detailed proof-theoretic analysis is given by Jager, Kahle, Setzer and Strahm in [18]. In this paper we assume familiarity with [18]. For natural extensions of Friedman’s ATR that can be measured against transfinitely iterated fixed point theories the reader is referred to Jager and Strahm [20]. In the mid seventies, Feferman [3, 4] introduced systems of explicit mathematics in order to provide an alternative foundation of constructive mathematics. More precisely, it was the origin of Feferman’s program to give a logical account of Bishop-style constructive mathematics. Right from the beginning, systems of explicit mathematics turned out to be of general interest for proof theory, mainly in connection with the proof-theoretic analysis of subsystems of first and second order arithmetic and set theory, cf. e.g. Jager [15] and Jager and Pohlers [19]. More recently, systems of explicit mathematics have been used to develop a general logical framework for functional programming and type theory, where it is possible to derive correctness and termination properties of functional programs. Important references in this connection are Feferman [6, 7, 9] and Jager [17]. Universes are a frequently studied concept in constructive mathematics at least since the work of Martin-Lof, cf. e.g. Martin-Lof [23] or Palmgren [27] for

Turing's Titanic machine?

Embodied and disembodied computing at the Turing Centenary.

A splitting theorem for the n-r.e. degrees

We prove a splitting theorem for the n-r.e. degrees, of which the Sacks Splitting Theorem [9] for the r.e. (=1-r.e.) degrees is a special case. For background terminology and notation see [4] and [11]

There is No Low Maximal D.C.E. Degree

AbstractWe show that for any computably enumerable (c.e.) degree a andany low n–c.e. degree l (n ≥ 1), if l< a, then there are n–c.e. degreesa 0 ,a 1 such that l< a 0 ,a 1 < a and a 0 ∨a 1 = a. In particular, thereis no low maximal d.c.e. degree. 1 Introduction A set A is n-c.e. if there is a computable function f(x,s) such that for everyx f(x,0) = 0 , lim s f(x,s) = A(x), and |{s : f(x,s) 6= f(x,s+1)}| ≤ n.So, in particular, the 1-c.e. sets are precisely the c.e. sets. The 2-c.e. setsare also known as the d.c.e. sets as they are the differences of c.e. sets, i.e.the ones of the form B − C with both B and C c.e. Similarly the n-c.e.sets are those obtained by starting with c.e. sets and alternating the Booleanoperations of difference and union.A (Turing) degree is called n–c.e. if it contains an n–c.e. set. An(n + 1)–c.e. degree is a properly (n + 1)–c.e. degree if it contains no n–c.e. set. As for sets, a 1–c.e. degree is a c.e. degree, and a 2–c.e. de-gree is called a d.c.e. degree. Let D