Computability in structures representing a Scott set

Continuing work begun in [10], we utilize a notion of forcing for which the generic objects are structures and which allows us to determine whether these “generic” structures compute certain sets and enumerations. The forcing conditions are bounded complexity types which are consistent with a given theory and are elements of a given Scott set. These generic structures will “represent” this given Scott set, in the sense that the structure has a certain weak saturation property with respect to bounded complexity types in the Scott set. For example, if ? is a nonstandard model of PA, then ? represents the Scott set ? = n∈ω | ?⊧“the nth prime divides a” | a∈?. The notion of forcing yields two main results. The first characterizes the sets of natural numbers computable in all models of a given theory representing a given Scott set. We show that the characteristic function of such a set must be enumeration reducible to a complete existential type which is consistent with the given theory and is an element of the given Scott set. The second provides a sufficient condition for the existence of a structure ? such that ? represents a countable jump ideal and ? does not compute an enumeration of a given family of sets ?. This second result is of particular interest when the family of sets which cannot be enumerated is ? = Rep[Th(?)]. Under this additional assumption, the second result generalizes a result on TA [6] and on certain other completions of PA [10]. For example, we show that there also exist models of completions of ZF from which one cannot enumerate the family of sets represented by the theory. Received: 8 October 1997 / Published online: 25 January 2001     

The computable Lipschitz degrees of computably enumerable sets are not dense

The computable Lipschitz reducibility was introduced by Downey, Hirschfeldt and LaForte under the name of strong weak truth-table reducibility (Downey et al. (2004) [6]). This reducibility measures both the relative randomness and the relative computational power of real numbers. This paper proves that the computable Lipschitz degrees of computably enumerable sets are not dense. An immediate corollary is that the Solovay degrees of strongly c.e. reals are not dense. There are similarities to Barmpalias and Lewis’ proof that the identity bounded Turing degrees of c.e. sets are not dense (George Barmpalias, Andrew E.M. Lewis (2006) [2]), however the problem for the computable Lipschitz degrees is more complex.     

The Geometry of Standard Deontic Logic

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “n-opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m” (m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid).      

No random reals in countable support iterations

We prove a preservation theorem for limit steps of countable support iterations of proper forcing notions whose particular cases are preservations of the following properties on limit steps: “no random reals are added”, “μ(Random(V))≠1”, “no dominating reals are added”, “Cohen(V) is not comeager”. Consequently, countable support iterations of σ-centered forcing notions do not add random reals. The work was supported by BRF of Israel Academy of Sciences and by grant GA SAV 365 of Slovak Academy of Sciences.     

The distribution of ITRM-recognizable reals

Infinite Time Register Machines (ITRM's) are a well-established machine model for infinitary computations. Their computational strength relative to oracles is understood, see e.g. ,  and . We consider the notion of recognizability, which was first formulated for Infinite Time Turing Machines in [6] and applied to ITRM's in [3]. A real x is ITRM-recognizable iff there is an ITRM-program P   such that P^y$P^y$ stops with output 1 iff y=x$y=x$, and otherwise stops with output 0. In [3], it is shown that the recognizable reals are not contained in the ITRM-computable reals. Here, we investigate in detail how the ITRM  -recognizable reals are distributed along the canonical well-ordering _<L${<}_L$ of Gödel's constructible hierarchy L  . In particular, we prove that the recognizable reals have gaps in _<L${<}_L$, that there is no universal ITRM in terms of recognizability and consider a relativized notion of recognizability.     

Relativized Schnorr tests with universal behavior

A Schnorr test relative to some oracle A may informally be called “universal” if it covers all Schnorr tests. Since no true universal Schnorr test exists, such an A cannot be computable. We prove that the sets with this property are exactly those with high Turing degree. Our method is closely related to the proof of Terwijn and Zambella’s characterization of the oracles which are low for Schnorr tests. We also consider the oracles which compute relativized Schnorr tests with the weaker property of covering all computable reals. The degrees of these oracles strictly include the hyperimmune degrees and are strictly included in the degrees not computably traceable.     

On an obstruction for perfect matchings

Steffens [3] introduced a substructure (called below a “compressed set”) which prevents a graph from having a perfect matching, and proved that a countable graph possesses a perfect matching if and only if it does not contain such a substructure. In this paper we study some properties of compressed sets. Dedicated to Paul Erdős on his seventieth birthday     

Extending and interpreting Post’s programme

Computability theory concerns information with a causal-typically algorithmic-structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals which produced richness of local computability-theoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Post’s programme was the genesis of a more far reaching research project.In this article we will firstly review the history of Post’s programme, and look at two interesting developments of Post’s approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computability-theoretic infrastructure. We will conclude by viewing Post’s programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world.     

On one-sided versus two-sided classification

One-sided classifiers are computable devices which read the characteristic function of a set and output a sequence of guesses which converges to 1 iff the set on the input belongs to the gven class. Such a classifier istwo-sided if the sequence of its output in addition converges to 0 on setsnot belonging to the class. The present work obtains the below mentionedresults for one-sided classes (= Σ⁰ ₂ classes) with respect to four areas: Turing complexity, 1-reductions, index sets and measure. There are one-sided classes which are not two-sided. This can have two reasons: (1) the class has only high Turing complexity. Then there are some oracles which allow to construct noncomputale two-sided classifiers. (2) The class is difficult because of some topological constraints and then there are also no nonrecursive two-sided classifiers. For case (1), several results are obtainedto localize the Turing complexity of certain types of one-sided classes. The concepts of 1-reduction, 1-completeness and simple sets is transferred to one-sided classes: There are 1-complete classes and simple classes, but no class is at the same time 1-complete nd simple. The one-sided classes have a natural numbering. Most of the common index sets relative to this numbering have the high complexity Π¹ ₁: the index set of the class {0,1}^∞, the index set of the equality problem and the index set of all two-sided classes. On the other side the index set of the empty class has complexity Π⁰ ₂; Π⁰ ₂ and Σ⁰ ₂ are the least complexities any nontrivial index set can have. Lusin showed that any one-sided class is measurable. Concerning the effectiveness of this measure, it is shown that a one-sided class has recursive measure 0 if it has measure 0, but that thre are one-sided classes having measure 1 without having measure 1 effectively. The measure of a two-sided class can be computed in the limit. Received: 2 December 1999 / Revised version: 28 February 2000 / Published online: 15 June 2001     

A version of the Jensen–Johnsbråten coding at arbitrary level n≥ 3

We generalize, on higher projective levels, a construction of “incompatible” generic Δ¹ ₃ real singletons given by Jensen and Johnsbr?ten. Received: 3 November 1998 / Revised version: 23 April 2000 / Published online: 3 October 2001     

Generics for computable Mathias forcing

We study the complexity of generic reals for computable Mathias forcing in the context of computability theory. The n-generics and weak n-generics form a strict hierarchy under Turing reducibility, as in the case of Cohen forcing. We analyze the complexity of the Mathias forcing relation, and show that if G is any n  -generic with n≥2$n\ge 2$ then it satisfies the jump property G⁽ⁿ⁻¹⁾_≡TG^′⊕∅⁽ⁿ⁾$G^{(n-1)}{\equiv }_T{G}^{\prime }\oplus {\varnothing }^{(n)}$. We prove that every such G has generalized high Turing degree, and so cannot have even Cohen 1-generic degree. On the other hand, we show that every Mathias n-generic real computes a Cohen n-generic real.     

On the filter of computably enumerable supersets of an r-maximal set

We study the filter ℒ^*(A) of computably enumerable supersets (modulo finite sets) of an r-maximal set A and show that, for some such set A, the property of being cofinite in ℒ^*(A) is still Σ⁰ ₃-complete. This implies that for this A, there is no uniformly computably enumerable “tower” of sets exhausting exactly the coinfinite sets in ℒ^*(A). Received: 6 November 1999 / Revised version: 10 March 2000 /?Published online: 18 May 2001     

Baire property and axiom of choice

We show that without using inaccessible cardinals it is possible to get models of “ZF+all sets of reals have the Baire property +DC(ω₁)” and “ZFC+all projective sets have the Baire property+the union of less than ω₂ many meager sets is meager”, answering two well-known open questions of Woodin and Judah, respectively. The authors would like to thank the Israel Academy of Sciences BSF for partial support. The second author would like to thank the Landau Center for Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Lebesgue numbers and Atsuji spaces in subsystems of second-order arithmetic

We study Lebesgue and Atsuji spaces within subsystems of second order arithmetic. The former spaces are those such that every open covering has a Lebesgue number, while the latter are those such that every continuous function defined on them is uniformly continuous. The main results we obtain are the following: the statement “every compact space is Lebesgue” is equivalent to ; the statements “every perfect Lebesgue space is compact” and “every perfect Atsuji space is compact” are equivalent to ; the statement “every Lebesgue space is Atsuji” is provable in ; the statement “every Atsuji space is Lebesgue” is provable in . We also prove that the statement “the distance from a closed set is a continuous function” is equivalent to . Received: February 2, 1996     

Spaces of orders and their Turing degree spectra

We investigate computability theoretic and topological properties of spaces of orders on computable orderable groups. A left order on a group G is a linear order of the domain of G, which is left-invariant under the group operation. Right orders and bi-orders are defined similarly. In particular, we study groups for which the spaces of left orders are homeomorphic to the Cantor set, and their Turing degree spectra contain certain upper cones of degrees. Our approach unifies and extends Sikora’s (2004) [28] investigation of orders on groups in topology and Solomon’s (2002) [31] investigation of these orders in computable algebra. Furthermore, we establish that a computable free group F_n of rank n>1 has a bi-order in every Turing degree.     

On the constructive Dedekind reals

In order to build the collection of Cauchy reals as a set in constructive set theory, the only power set-like principle needed is exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, Constructive Zermelo–Fraenkel set theory with subset collection replaced by exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class.     

Left Determined Model Structures for Locally Presentable Categories

We extend a result of Cisinski on the construction of cofibrantly generated model structures from (Grothendieck) toposes to locally presentable categories and from monomorphism to more general cofibrations. As in the original case, under additional conditions, the resulting model structures are “left determined” in the sense of Rosicky and Tholen.     

Selective covering properties of product spaces

We study the preservation of selective covering properties, including classic ones introduced by Menger, Hurewicz, Rothberger, Gerlits and Nagy, and others, under products with some major families of concentrated sets of reals.     

Truth‐table Schnorr randomness and truth‐table reducible randomness

Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth‐table Schnorr randomness (defined in 6 too only by martingales) and truth‐table reducible randomness, for which we prove that van Lambalgen's Theorem holds. We also show that the classes of truth‐table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van Lambalgen's Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth‐table Schnorr randomness. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim