We say that A ≤LR B if every B-random number is A-random. Intuitively this means that if oracle A can identify some patterns on some real γ. In other words. B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e., restricted to the computably enumberable degrees) and their relationship with the Turing degrees. Among other results we show that whenever α in not GL₂ the LR degree of (...) α bounds $2^{\aleph _{0}}$ degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees. (shrink)

We say that A≤LRB if every B-random set is A-random with respect to Martin–Löf randomness. We study this relation and its interactions with Turing reducibility, classes, hyperimmunity and other recursion theoretic notions.

We show that the identity bounded Turing degrees of computably enumerable sets are not dense.

The strong weak truth table (sw) reducibility was suggested by Downey, Hirschfeldt, and LaForte as a measure of relative randomness, alternative to the Solovay reducibility. It also occurs naturally in proofs in classical computability theory as well as in the recent work of Soare, Nabutovsky, and Weinberger on applications of computability to differential geometry. We study the sw-degrees of c.e. reals and construct a c.e. real which has no random c.e. real (i.e., Ω number) sw-above it.

We prove a number of results in effective randomness, using methods in which Π⁰₁ classes play an essential role. The results proved include the fact that every PA Turing degree is the join of two random Turing degrees, and the existence of a minimal pair of LR degrees below the LR degree of the halting problem.

We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the sublattice (...) of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space. (shrink)

We show that there exists a real α such that, for all reals β, if α is linear reducible to β then β≤Tα. In fact, every random real satisfies this quasi-maximality property. As a corollary we may conclude that there exists no ℓ-complete Δ2 real. Upon realizing that quasi-maximality does not characterize the random reals–there exist reals which are not random but which are of quasi-maximal ℓ-degree–it is then natural to ask whether maximality could provide such a characterization. Such hopes, (...) however, are in vain since no real is of maximal ℓ-degree. (shrink)

The infinite time Turing machine analogue of Post's problem, the question whether there are semi-decidable supertask degrees between 0 and the supertask jump 0∇, has in a sense both positive and negative solutions. Namely, in the context of the reals there are no degrees between 0 and 0∇, but in the context of sets of reals, there are; indeed, there are incomparable semi-decidable supertask degrees. Both arguments employ a kind of transfinite-injury construction which generalizes canonically to oracles.

We show that given any (Turing) degree 0

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We show that there exists a properly Σ2 minimal degree b, and moreover that b can be chosen to join with 0′ to 0″ – so that b is a 0″ complement for every degree a such that 0′ ≤ a < 0″.

We show that there exists a properly Σ2 minimal degree b, and moreover that b can be chosen to join with 0′ to 0″ – so that b is a 0″ complement for every degree a such that 0′ ≤ a < 0″.

It is shown that for every (Turing) degree 0 < a < 0´ there is a minimal degree m < 0´ such that a ∨ m = 0´ (and therefore a ∧ m = 0).

We show that in the c.e. weak truth table degrees if b < c then there is an a which contains no hypersimple set and b < a < c. We also show that for every w < c in the c.e. wtt degrees such that w is hypersimple, there is a hypersimple a such that w < a < c. On the other hand, we know that there are intervals which contain no hypersimple set.

Charney's target article continues a critique of genetic blueprint models of development that suggests reconsideration of concepts of adaptation, inheritance, and environment, which can be well illustrated in current research on infant attachment. The concepts of development and adaptation are so heavily based on the model of genetics and inheritance forged in the modern synthesis that they will require reconsideration to accommodate epigenetic inheritance.

Del Giudice provides an extension of the life history theory of attachment that incorporates emerging data suggestive of sex differences in avoidant male and preoccupied female attachment patterns emerging in middle childhood. This commentary considers the place of disorganized attachment within this theory and why male children may be more prone to disorganized attachment by drawing on Trivers's parental investment theory.

Law and History contains a broad range of essays by prominent legal historians, which explore the ways in which history has been used by lawyers. Largely theoretical in focus, the volume covers a broad range of issues, including discussions of norms in medieval England, the works of Montesquieu, Maine, and Weber, and of the nature of legal argument in nineteenth-century England, and in twentieth- century war crimes trials.Readership: Scholars of law and history, social historians, legal historians.

It is shown that for every degree 0

Let us say that any (Turing) degree d > 0 satisfies the minimal complementation property (MCP) if for every degree 0 < a < d there exists a minimal degree b < d such that a ∨ b = d (and therefore a ∧ b = 0). We show that every degree d ≥ 0′ satisfies MCP. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim).

This is the fourth volume of a series entitled `Current Legal Issues' that are published each Summer as a sister volume to `Current Legal Problems'. The interaction of religious practice and the law raises a number of difficult and fascinating issues. What exactly do we mean by religious faith? To what extent are the Courts competent to pass judgement on disputes arising within religious organisations? Are some religious faiths more legitimate than others? Should the law grant special privileges to religious (...) believers? - for example exemption from provisions in human rights legislation which would otherwise restrict their activities. (shrink)