It is known that every non-universal self-full degree in the structure of the degrees of computably enumerable equivalence relations (ceers) under computable reducibility has exactly one strong minimal cover. This leaves little room for embedding wide partial orders as initial segments using self-full degrees. We show that considerably more can be done by staying entirely inside the collection of non-self-full degrees. We show that the poset can be embedded as an initial segment of the degrees of ceers (...) with infinitely many classes. A further refinement of the proof shows that one can also embed the free distributive lattice generated by the lower semilattice as an initial segment of the degrees of ceers with infinitely many classes. (shrink)

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\scr{D}_{h}$ . In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable, locally finite lattice) is isomorphic to an initial segment of $\scr{D}_{h}$ . Corollaries include the decidability of the two quantifier theory of $\scr{D}_{h}$ and the undecidability of its three quantifier theory. The key tool in (...) the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $\omega _{1}^{CK}$ . Somewhat surprisingly, the set theoretic analog of this forcing does not preserve ω₁. On the other hand, we construct countable lattices that are not isomorphic to any initial segment of $\scr{D}_{h}$. (shrink)

Using properties of${\cal K}$-pairs of sets, we show that every nonzero enumeration degreeabounds a nontrivial initial segment of enumeration degrees whose nonzero elements have all the same jump asa. Some consequences of this fact are derived, that hold in the local structure of the enumeration degrees, including: There is an initial segment of enumeration degrees, whose nonzero elements are all high; there is a nonsplitting high enumeration degree; every noncappable enumeration degree is high; every nonzero low (...) enumeration degree can be capped by degrees of any possible local jump ; every enumeration degree that bounds a nonzero element of strictly smaller jump, is bounding; every low enumeration degree below a non low enumeration degreeacan be capped belowa. (shrink)

Recent results on initial segments of the Turing degrees are presented, and some conjectures about initial segments that have implications for the existence of nontrivial automorphisms of the Turing degrees are indicated.

Given a model \ of set theory, and a nontrivial automorphism j of \, let \\) be the submodel of \ whose universe consists of elements m of \ such that \=x\) for every x in the transitive closure of m ). Here we study the class \ of structures of the form \\), where the ambient model \ satisfies a frugal yet robust fragment of \ known as \, and \=m\) whenever m is a finite ordinal in the sense (...) of \ Our main achievement is the calculation of the theory of \ as precisely \-\. The following theorems encapsulate our principal results:Theorem A. Every structure in \ satisfies \-\.Theorem B. Each of the following three conditions is sufficient for a countable structure \ to be in \: \ is a transitive model of \-\. \ is a recursively saturated model of \-\. \ is a model of \.Theorem C. Suppose \ is a countable recursively saturated model of \ and I is a proper initial segment of \ that is closed under exponentiation and contains \. There is a group embedding \ from \\) into \\) such that I is the longest initial segment of \ that is pointwise fixed by \ for every nontrivial \.\)In Theorem C, \\) is the group of automorphisms of the structure X, and \ is the ordered set of rationals. (shrink)

A 2-affable ultrafilter has only finitely many predecessors in the Rudin-Keisler ordering of isomorphism classes of ultrafilters over the natural numbers. If the continuum hypothesis is true, then there is an ℵ 1 -sequence of ultrafilters D α such that the strict Rudin-Keisler predecessors of D α are precisely the isomorphs of the D β 's for $\beta.

MOTIVATION The constructivization of ; => D(_£^') poses several problems. For some of these the tools of MD can be modified; for others new methods will need to be established. What must we do to make a full approximation to ; * D? We ...

We show that in the lattice E Π of Π 0 1 classes there are initial segments [ $\emptyset$ , P] = L(P) which are not Boolean algebras, but which have a decidable theory. In fact, we will construct for any finite distributive lattice L which satisfies the dual of the usual reduction property a Π 0 1 class P such that L is isomorphic to the lattice L(P)*, which is L(P), modulo finite differences. For the 2-element lattice, we (...) obtain a minimal class, first constructed by Cenzer, Downey, Jockusch and Shore in 1993. For the simplest new Π 0 1 class P constructed, P has a single, non-computable limit point and L(P)* has three elements, corresponding to $\emptyset$ , P and a minimal class P $_0 \subset$ P. The element corresponding to P 0 has no complement in the lattice. On the other hand, the theory of L(P) is shown to be decidable. A Π 0 1 class P is said to be decidable if it is the set of paths through a computable tree with no dead ends. We show that if P is decidable and has only finitely many limit points, then L(P)* is always a Boolean algebra. We show that if P is a decidable Π 0 1 class and L(P) is not a Boolean algebra, then the theory of L(P)interprets the theory of arithmetic and is therefore undecidable. (shrink)

We combine two techniques of set theory relating to minimal degrees of constructibility. Jensen constructed a minimal real which is additionally a Π 1 2 singleton. Groszek built an initial segment of order type 1 + α * , for any ordinal α. This paper shows how to force a Π 1 2 singleton such that the c-degrees beneath it, all represented by reals, are of type 1 + α * , for many ordinals α. We also examine (...) the definability α needs to be so represented by a real. (shrink)

A set S of degrees is said to be an initial segment if c ≤ d ∈ S→-c∈S. Shoenfield has shown that if P is the lattice of all subsets of a finite set then there is an initial segment of degrees isomorphic to P. Rosenstein [2] (independently) proved the same to hold of the lattice of all finite subsets of a countable set. We shall show that “countable set” may be replaced by “set of cardinality (...) at most that of the continuum.” This result is also an extension of [3, Corollary 2 to Theorem 15], which states that there is a sublattice of degrees isomorphic to the lattice of all finite subsets of 2N. (A sublattice of degrees is a subset closed under ∪ and ∩; an initial segment closed under ∪ is necessarily a sublattice, but not conversely.)It seems worth noting that the proof of the present result was preceded in time by our proof [4] of the analogous theorem for hyperdegrees, and is in fact an adaptation of that proof. Thus the present work has been influenced much more directly by the Gandy-Sacks forcing construction of a minimal hyperdegree [1] than by previous work on initial segments of degrees. (shrink)

A standard result about hyperdegrees is proved for allΔ2n+11-degrees.

Suppose that 〈xk〉k∈ℕ is a countable sequence of real numbers. Working in the usual subsystems for reverse mathematics, RCA0 suffices to prove the existence of a sequence of reals 〈uk〉k∈ℕ such that for each k, uk is the minimum of {x0, x1, …, xk}. However, if we wish to prove the existence of a sequence of integer indices of minima of initial segments of 〈xk〉k∈ℕ, the stronger subsystem WKL0 is required. Following the presentation of these reverse mathematics results, we (...) will derive computability theoretic corollaries and use them to illustrate a distinction between computable analysis and constructive analysis. (shrink)

By an "inverse iteration" of Sacks forcing over a model of V = L, we produce a model in which the degrees of constructibility of nonconstructible reals have order type ω 1 *.

The WEAVER ++ model discussed by Levelt et al. assumes incremental encoding and articulation following complete encoding. However, many of the response latency results can also be accounted for by assuming incremental articulation. Another temporal variable, initial segment duration, can distinguish WEAVER ++'s incremental encoding account from the incremental articulation account.

A generalization is given of McAloon's result on initial segments ofmodels of GlΔ0, the fragment of Peano Arithmetic where the induction scheme is restricted to formulas with bounded quantifiers.

In any nonstandard model of Peano arithmetic, the standard part is not first-order definable. But we show that in some model the standard part is definable as the unique solution of a formula , where P is a unary predicate variable.

If $M \models PA (= Peano Arithmetic)$ , we set $A^M = \{N \subset_e M: N \models PA\}$ and study this family.

In this paper we describe the well-founded initial segment of the free Heyting algebra ������α on finitely many, α, generators. We give a complete classification of initial sublattices of ������₂ isomorphic to ������₁ (called 'low ladders'), and prove that for 2 < α < ω, the height of the well-founded initial segment of ������α.

For any countable nonstandard modelM of a sufficiently strong fragment of arithmeticT, and any nonstandard numbersa, c εM, M⊨c≦a, there is a modelK ofT which agrees withM up toa and such that inK there is a proof of contradiction inT with Gödel number $ \leqq 2^{a^c } $.

Where AR is the set of arithmetic Turing degrees, 0 (ω ) is the least member of { $\mathbf{\alpha}^{(2)}|\mathbf{a}$ is an upper bound on AR}. This situation is quite different if we examine HYP, the set of hyperarithmetic degrees. We shall prove (Corollary 1) that there is an a, an upper bound on HYP, whose hyperjump is the degree of Kleene's O. This paper generalizes this example, using an iteration of the jump operation into the transfinite which is based on (...) results of Jensen and is detailed in [3] and [4]. In $\S1$ we review the basic definitions from [3] which are needed to state the general results. (shrink)

The relatively simple central nervous system (CNS) of the Drosophila embryo provides a useful model system for investigating the mechanisms that generate and pattern complex nervous systems. Central to the generation of different types of neurons by precursor neuroblasts is the initial specification of neuroblast identity and the Drosophila segment polarity genes, genes that specify regions within a segment or repeating unit of the Drosophila embryo, have emerged recently as significant players in this process. During neurogenesis the (...) segment polarity genes are expressed in the neuroectodermal cells from which neuroblasts delaminate and they continue to be expressed in neuroblasts and their progeny. Loss-of-function mutations in these genes lead to a failure in the formation of neuroblasts and/or specification of neuroblast identity. Results from several recent studies suggest that regulatory interactions between segment polarity genes during neurogenesis lead to an increase in the number of neuroblasts and specification of different identities to neuroblasts within a population of cells. BioEssays 21:472–485, 1999. © 1999 John Wiley & Sons, Inc. (shrink)

chical reinforcement learning that does not rely on a pri ori hierarchical structures Thus the approach deals with a more di cult problem compared with existing work It in volves learning to segment sequences to create hierarchical structures based on reinforcement received during task ex ecution with di erent levels of control communicating with each other through sharing reinforcement estimates obtained by each others The algorithm segments sequences to re duce non Markovian temporal dependencies to facilitate the learning of (...) the overall task Initial experiments demon strated the basic promise of the approach.. (shrink)

The formation of segments in the Drosophila early embryo is understood in greater detail than any other complex developmental process. Now, by studying other types of insect embryo, we can hope to deduce something of the ancestral mechanism of segmentation and the ways in which it has been modified in evolution. The parasitic wasp, Copidosoma floridanum, is spectacularly atypical of insects in that the small egg cell divides extensively, with no initial syncytial phase, and forms eventually some 2000 embryos(1). (...) This process raises intriguing questions about the control of embryonic polarity and segmentation. (shrink)

Acoustic speech output results from coordinated articulation of dozens of muscles, bones and cartilages of the vocal mechanism. While we commonly take the fluency and speed of our speech productions for granted, the neural mechanisms facilitating the requisite muscular control are not completely understood. Previous neuroimaging and electrophysiology studies of speech sensorimotor control has typically concentrated on speech sounds (i.e., phonemes, syllables and words) in isolation; sentence-length investigations have largely been used to inform coincident linguistic processing. In this study, we (...) examined the neural representations of segmental features (place and manner of articulation, and voicing status) in the context of fluent, continuous speech production. We used recordings from the cortical surface (electrocorticography (ECoG)) to simultaneously evaluate the spatial topography and temporal dynamics of the neural correlates of speech articulation that may mediate the generation of hypothesized gestural or articulatory scores. We found that the representation of place of articulation involved broad networks of brain regions during all phases of speech production: preparation, execution and monitoring. In contrast, manner of articulation and voicing status were dominated by auditory cortical responses after speech had been initiated. These results provide a new insight into the articulatory and auditory processes underlying speech production in terms of their motor requirements and acoustic correlates. (shrink)

By drawing on service encounter data in Japanese, this paper analyzes a previously undocumented request action initiated by a service provider to a client as a necessary step to provide the service. The service provider and their client, both exercising respective deontic rights, collaboratively construct a request turn in particular ways. In this case, due to the Japanese SOV word order, the service provider takes advantage of segmenting their request turn to allow the recipient clients to begin compliance, who thereby (...) acquiesce to the service provider’s deontic authority at the earliest point. By bringing the request turn to completion after the client’s compliance while reflexively showing their deontic and beneficiary stances in the turn final component, the service provider displays commitment to balancing out their relative deontic status to the client. (shrink)

In the perceptual learning of lexical tones, an automatic and robust attention-to-phonology system enables native tonal listeners to adapt to acoustically non-optimal speech, such as phonetic conflicts in daily communications. Previous tone research reveals that non-native listeners who do not linguistically employ lexical tones in their mother tongue may find it challenging to attend to the tonal dimension or integrate it with the segmental features. However, it is unknown whether the attentional interference initially caused by a maternal attentional system would (...) continue influencing the non-optimal tone perception for simultaneous bilingual teenagers. From an endpoint in the age of language acquisition, we investigate whether the tone-specific attention mechanism developed by the Urdu-Cantonese simultaneous bilinguals is automatic enough to assist them in adapting to a phonetically-conflicting environment. Three groups of teenagers engaged in a four-condition ABX task: Urdu-Cantonese simultaneous bilinguals, Cantonese native listeners, and Urdu-speaking, late learners of Cantonese. The results showed that although the simultaneous bilinguals could phonologically process Cantonese tones in a Cantonese-like way under a conflict-free listening condition, they still failed in adapting to the phonetic conflicts, especially the segment-induced ones. It thus demonstrated that the simultaneous exposure and years of regular education in Hong Kong local schools still could not automatically guarantee simultaneous bilingual processing of Cantonese tones. In interpreting the findings, it hypothesized that, except for simultaneous exposure, the development of a tone-specific attention mechanism is also likely to be L1-inhibitory, tone experience-driven, and language-specific for simultaneous bilinguals. (shrink)

We present a number of results on the structure of initial segments of models of Peano arithmetic with the arithmetic operations of addition, subtraction, multiplication, division, exponentiation and logarithm. Each of the binary operations introduced is defined in two dual ways, often with quite different results, and we attempt to systematise the issues and show how various calculations may be carried out. To understand the behaviour of addition and subtraction we introduce a notion of derivative on cuts, analogous to (...) differentiation in the calculus. Multiplication, division and other operations are described by higher order versions of derivative. The work here is presented as important preliminary work related to a nonstandard measure theory of non‐definable bounded subsets of a model of Peano arithmetic. (shrink)

Instead of attempting to reduce rituals of healing to so‐called primitive psychotherapy this essay raises the question of whether contemporary psychotherapies might not fruitfully be viewed as ritual processes through which a small segment of modern society receives ritual leadership in times of crisis. Selected phenomena in contemporary psychotherapeutic practice are analyzed in an attempt to discern the ritual processes which are manifest in them. It is concluded that most modalities of contemporary psychotherapy manifest elements of ritualized submission, containment, (...) and enactment. (shrink)

Let U be an initial segment of $^*{\mathbb N}$ closed under addition (such U is called a cut) with uncountable cofinality and A be a subset of U, which is the intersection of U and an internal subset of $^*{\mathbb N}$ . Suppose A has lower U-density α strictly between 0 and 3/5. We show that either there exists a standard real $\epsilon$ > 0 and there are sufficiently large x in A such that | (A+A) ∩ [0, (...) 2x]| > (10/3+ $\epsilon$ ) | A ∩ [0, x]| or A is a large subset of an arithmetic progression of difference greater than 1 or A is a large subset of the union of two arithmetic progressions with the same difference greater than 2 or A is a large subset of the union of three arithmetic progressions with the same difference greater than 4. (shrink)

The initial segment of the latest volume of the _Ignatian Philosophical Yearbook _is centered on the discourse surrounding the presence of cultural heritage in the cyber domain. It explores the dissemination of this heritage by cultural institutions such as museums, libraries, and archives, as well as initiatives undertaken by Internet users. The authors scrutinize a comprehensive array of topics concerning the digital presentation of cultural artifacts, employing carefully selected examples in their discourse.

We investigate the initial segments of the Medvedev lattice as Brouwer algebras, and study the propositional logics connected to them.