In this paper we answer the following well-known open question in computable model theory. Does there exist a computable not ‮א‬₀-categorical saturated structure with a unique computable isomorphism type? Our answer is affirmative and uses a construction based on Kolmogorov complexity. With a variation of this construction, we also provide an example of an ‮א‬₁-categorical but not ‮א‬₀-categorical saturated $\Sigma _{1}^{0}$ -structure with a unique computable isomorphism type. In addition, using the construction we give an example of an ‮א‬₁-categorical but (...) not ‮א‬₁-categorical theory whose only non-computable model is the prime one. (shrink)

In this paper we investigate computable models of -categorical theories and Ehrenfeucht theories. For instance, we give an example of an -categorical but not -categorical theory such that all the countable models of except its prime model have computable presentations. We also show that there exists an -categorical but not -categorical theory such that all the countable models of except the saturated model, have computable presentations.

Bridging the gap between the world of science and the realm of the spiritual, B. Alan Wallace introduces a natural theory of human consciousness that has its roots in contemporary physics and Buddhism. Wallace's "special theory of ontological relativity" suggests that mental phenomena are _conditioned_ by the brain, but do not _emerge_ from it. Rather, the entire natural world of mind and matter, subjects and objects, arises from a unitary dimension of reality that is more fundamental than these dualities, as (...) proposed by Wolfgang Pauli and Carl Jung. To test his hypothesis, Wallace employs the Buddhist meditative practice of _samatha_, refining one's attention and metacognition, to create a kind of telescope to examine the space of the mind. Drawing on the work of the physicist John Wheeler, he then proposes a more general theory in which the participatory nature of reality is envisioned as a self-excited circuit. In comparing these ideas to the Buddhist theory known as the Middle Way philosophy, Wallace explores further aspects of his "general theory of ontological relativity," which can be investigated by means of _vipasyana_, or insight, meditation. Wallace then focuses on the theme of symmetry in reference to quantum cosmology and the "problem of frozen time," relating these issues to the theory and practices of the Great Perfection school of Tibetan Buddhism. He concludes with a discussion of the general theme of complementarity as it relates to science and religion. The theories of relativity and quantum mechanics were major achievements in the physical sciences, and the theory of evolution has had an equally deep impact on the life sciences. However, rigorous scientific methods do not yet exist to observe mental phenomena, and naturalism has its limits for shedding light on the workings of the mind. A pioneer of modern consciousness research, Wallace offers a practical and revolutionary method for exploring the mind that combines the keenest insights of contemporary physicists and philosophers with the time-honored meditative traditions of Buddhism. (shrink)

Bridging the gap between the world of science and the realm of the spiritual, B. Alan Wallace introduces a natural theory of human consciousness that has its roots in contemporary physics and Buddhism. Wallace's "special theory of ontological relativity" suggests that mental phenomena are _conditioned_ by the brain, but do not _emerge_ from it. Rather, the entire natural world of mind and matter, subjects and objects, arises from a unitary dimension of reality that is more fundamental than these dualities, as (...) proposed by Wolfgang Pauli and Carl Jung. To test his hypothesis, Wallace employs the Buddhist meditative practice of _samatha_, refining one's attention and metacognition, to create a kind of telescope to examine the space of the mind. Drawing on the work of the physicist John Wheeler, he then proposes a more general theory in which the participatory nature of reality is envisioned as a self-excited circuit. In comparing these ideas to the Buddhist theory known as the Middle Way philosophy, Wallace explores further aspects of his "general theory of ontological relativity," which can be investigated by means of _vipasyana_, or insight, meditation. Wallace then focuses on the theme of symmetry in reference to quantum cosmology and the "problem of frozen time," relating these issues to the theory and practices of the Great Perfection school of Tibetan Buddhism. He concludes with a discussion of the general theme of complementarity as it relates to science and religion. The theories of relativity and quantum mechanics were major achievements in the physical sciences, and the theory of evolution has had an equally deep impact on the life sciences. However, rigorous scientific methods do not yet exist to observe mental phenomena, and naturalism has its limits for shedding light on the workings of the mind. A pioneer of modern consciousness research, Wallace offers a practical and revolutionary method for exploring the mind that combines the keenest insights of contemporary physicists and philosophers with the time-honored meditative traditions of Buddhism. (shrink)

The spectrum of a relation on a computable structure is the set of Turing degrees of the image of R under all isomorphisms between and any other computable structure . The relation is intrinsically computably enumerable if its image under all such isomorphisms is c.e. We prove that any computable partially ordered set is isomorphic to the spectrum of an intrinsically c.e. relation on a computable structure. Moreover, the isomorphism can be constructed in such a way that the image of (...) the minimum element of the partially ordered set is computable. This solves the spectrum problem. The theorem and modifications of its proof produce computably categorical structures whose expansions by finite number of constants are not computably categorical and, indeed, ones whose expansions can have any finite number of computable isomorphism types. They also provide examples of computably categorical structures that remain computably categorical under expansions by constants but have no Scott family. (shrink)

Whenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the given (...) class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings ; integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories. (shrink)

We build an א₁-categorical but not א₀-categorical theory whose only computably presentable model is the saturated one. As a tool, we introduce a notion related to limitwise monotonic functions.

We investigate dependence of recursively enumerable graphs on the equality relation given by a specific r.e. equivalence relation on ω. In particular we compare r.e. equivalence relations in terms of graphs they permit to represent. This defines partially ordered sets that depend on classes of graphs under consideration. We investigate some algebraic properties of these partially ordered sets. For instance, we show that some of these partial ordered sets possess atoms, minimal and maximal elements. We also fully describe the isomorphism (...) types of some of these partial orders. (shrink)

LetEbe a computably enumerable equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized (...) byE. One can also define the following pre-order$ \le _{lo} $on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2 $if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$ \le _{lo} $. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees. (shrink)

In this paper we introduce effectiveness into model theory of intuitionistic logic. The main result shows that any computable theory T of intuitionistic predicate logic has a Kripke model with decidable forcing such that for any sentence φ, φ is forced in the model if and only if φ is intuitionistically deducible from T.

In this paper, the notions of Fα-categorical and Gα-categorical structures are introduced by choosing the isomorphism such that the function itself or its graph sits on the α-th level of the Ershov hierarchy, respectively. Separations obtained by natural graphs which are the disjoint unions of countably many finite graphs. Furthermore, for size-bounded graphs, an easy criterion is given to say when it is computable-categorical and when it is only G2-categorical; in the latter case it is not Fα-categorical for any recursive (...) ordinal α. (shrink)

In this paper we investigate computable isomorphisms of Boolean algebras with operators (BAOs). We prove that there are examples of polymodal Boolean algebras with finitely many computable isomorphism types. We provide an example of a polymodal BAO such that it has exactly one computable isomorphism type but whose expansions by a constant have more than one computable isomorphism type. We also prove a general result showing that BAOs are complete with respect to the degree spectra of structures, computable dimensions, expansions (...) by constants, and the degree spectra of relations. (shrink)

Finite automata have been recently used as alternative, discrete models in theoretical physics, especially in problems related to the dichotomy between endophysical/intrinsic and exophysical/ extrinsic perception . These studies deal with Moore experiments; the main result states that it is impossible to determine the initial state of an automaton, and, consequently, a discrete model of Heisenberg uncertainty has been suggested. For this aim the classical theory of finite automata — which considers automata with initial states — is not adequate, and (...) a new approach is necessary. A study of finite deterministic automata without initial states is exactly the aim of this paper. We will define and investigate the complexity of various types of simulations between automata. Minimal automata will be constructed and proven to be unique up to an isomorphism. We will build our results on an extension of Myhill-Nerode technique; all constructions will make use of “automata responses” to simple experiments only, i.e., no information about the internal machinery will be considered available. (shrink)

Cholak, Goncharov, Khoussainov, and Shore [1] showed that for each k > 0 there is a computably categorical structure whose expansion by a constant has computable dimension k. We show that the same is true with k replaced by ω. Our proof uses a version of Goncharov's method of left and right operations.

We construct a computable ℵ0-categorical structure whose first order theory is computably equivalent to the true first order theory of arithmetic.

We define a new type of two player game occurring on a tree. The tree may have no root and may have arbitrary degrees of nodes. These games extend the class of games considered by Gurevich-Harrington in [5]. We prove that in the game one of the players has a winning strategy which depends on finite bounded information about the past part of a play and on future of each play that is isomorphism types of tree nodes. This result extends (...) further the Gurevich-Harrington determinacy theorem from [5]. (shrink)

The aim of this paper is to shed light on our understanding of large scale properties of infinite strings. We say that one string $\alpha $ has weaker large scale geometry than that of $\beta $ if there is color preserving bi-Lipschitz map from $\alpha $ into $\beta $ with small distortion. This definition allows us to define a partially ordered set of large scale geometries on the classes of all infinite strings. This partial order compares large scale geometries of (...) infinite strings. As such, it presents an algebraic tool for classification of global patterns. We study properties of this partial order. We prove, for instance, that this partial order has a greatest element and also possess infinite chains and antichains. We also investigate the sets of large scale geometries of strings accepted by finite state machines such as Büchi automata. We provide an algorithm that describes large scale geometries of strings accepted by Büchi automata. This connects the work with the complexity theory. We also prove that the quasi-isometry problem is a $\Sigma _2^0$ -complete set, thus providing a bridge with computability theory. Finally, we build algebraic structures that are invariants of large scale geometries. We invoke asymptotic cones, a key concept in geometric group theory, defined via model-theoretic notion of ultra-product. Partly, we study asymptotic cones of algorithmically random strings, thus connecting the topic with algorithmic randomness. (shrink)

In this paper, we initiate the study of Ehrenfeucht–Fraïssé games for some standard finite structures. Examples of such standard structures are equivalence relations, trees, unary relation structures, Boolean algebras, and some of their natural expansions. The paper concerns the following question that we call the Ehrenfeucht–Fraïssé problem. Given nω as a parameter, and two relational structures and from one of the classes of structures mentioned above, how efficient is it to decide if Duplicator wins the n-round EF game ? We (...) provide algorithms for solving the Ehrenfeucht–Fraïssé problem for the mentioned classes of structures. The running times of all the algorithms are bounded by constants. We obtain the values of these constants as functions of n. (shrink)

In this paper we show that for any set X ⊆ ω there exists a structure [MATHEMATICAL SCRIPT CAPITAL A] that has no presentation computable in X such that [MATHEMATICAL SCRIPT CAPITAL A]2 has a computable presentation. We also show that there exists a structure [MATHEMATICAL SCRIPT CAPITAL A] with infinitely many computable isomorphism types such that [MATHEMATICAL SCRIPT CAPITAL A]2 has exactly one computable isomorphism type.

In this paper we study the question as to which computable algebras are isomorphic to non-computable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi_{1}^{0}$$\end{document}-algebras. We show that many known algebras such as the standard model of arithmetic, term algebras, fields, vector spaces and torsion-free abelian groups have non-computable\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi_{1}^{0}$$\end{document}-presentations. On the other hand, many of this structures fail to have non-computable \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma_{1}^{0}$$\end{document}-presentation.

In this paper we study the question as to which computable algebras are isomorphic to non-computable $\Pi_{1}^{0}$ -algebras. We show that many known algebras such as the standard model of arithmetic, term algebras, fields, vector spaces and torsion-free abelian groups have non-computable $\Pi_{1}^{0}$ -presentations. On the other hand, many of this structures fail to have non-computable $\Sigma_{1}^{0}$ -presentation.

This paper shows how the notion of randomness defines, in a natural way, an algebra. It turns out that the algebra is computably enumerable and finitely generated. The paper investigates algebraic and effective properties of this algebra.

A unary algebra is an algebraic system A = , where ƒ 0 ,…,ƒ n are unary operations on A and n ∈ ω. In the paper we develop the theory of effective unary algebras. We investigate well-known questions of constructive model theory with respect to the class of unary algebras. In the paper we construct unary algebras with a finite number of recursive isomorphism types. We give the notions of program, uniform, and algebraic dimensions of models, and then we (...) investigate these notions on unary algebras. We find connections between algebraic and effective properties of r.e. representable unary algebras. We also deal with finitely generated r.e. unary algebras. We show the connections between trees and unary algebras. Our interests also concern recursive automorphisms groups, r.e. subalgebra and congruence lattices of effective unary algebras. (shrink)

The _Platform Sutra_ records the teachings of Hui-neng, the Sixth Patriarch, who is revered as one of the two great figures in the founding of Ch'an (Zen) Buddhism. This translation is the definitive English version of the eighth-century Ch'an classic. Phillip B. Yampolsky has based his translation on the Tun-huang manuscript, the earliest extant version of the work. A critical edition of the Chinese text is given at the end of the volume. Dr. Yampolsky also furnishes a lengthy and detailed (...) historical introduction which contains much information hitherto unavailable even to scholars, and provides the context essential to an understanding of Hui-neng's work. He gives an account of the history and legends of Ch'an Buddhism, with particular attention to the traditions associated with Hui-neng, quoting or summarizing the most important narratives. He then discusses the various texts of the _Platform Sutra_, and analyzes its contents. (shrink)

The late Jacob Klein’s important book is, remarkably, a lucid presentation of esoteric argument. Dealing with the famed Platonic triad, Theaetetus, Sophist, and Statesman, Klein settles the dispute about the missing dialogue, "The Philosopher," by first denying that it is missing and second showing that it is unnecessary. He argues, in short, that the triad is a dyad. That argument is reinforced by the distinction Klein strongly implies between the Socratic Theaetetus and the Eleatic Sophist and Statesman. "We can now (...) understand why the conversations of the ‘trilogy’ occur in two, not three, days and why the Sophist, the Statesman, and the Philosopher are dealt with in two, not three, dialogues". (shrink)

'The problem of the twentieth-century is the problem of the color-line.' Originally published in 1903, The Souls of Black Folk is a classic study of race, culture, and education at the turn of the twentieth century. With its singular combination of essays, memoir, and fiction, this book vaulted W. E. B. Du Bois to the forefront of American political commentary and civil rights activism. The Souls of Black Folk is an impassioned, at times searing account of the situation of African (...) Americans in the United States. Du Bois makes a forceful case for the access of African Americans to higher education, memorably extols the achievements of black culture (above all the spirituals or 'sorrow songs'), and advances the provocative and influential argument that due to the inequalities and pressures of the 'race problem', African American identity is characterized by 'double consciousness'. This edition includes a valuable appendix of other writing by Du Bois, which sheds light on his attitudes and intentions. ABOUT THE SERIES: For over 100 years Oxford World's Classics has made available the widest range of literature from around the globe. Each affordable volume reflects Oxford's commitment to scholarship, providing the most accurate text plus a wealth of other valuable features, including expert introductions by leading authorities, helpful notes to clarify the text, up-to-date bibliographies for further study, and much more. (shrink)

While the question of free will does not figure as prominently in Buddhist writings as it does in western theology, philosophy, and psychology, it is a topic that was addressed in the earliest Buddhist writings. According to these accounts, for pragmatic and ethical reasons, the Buddha rejected both determinism and indeterminism as understood at that time. Rather than asking the metaphysical question of whether already humans have free will, Buddhist tradition takes a more pragmatic approach, exploring ways in which we (...) can acquire greater freedom to make wise choices that are truly conducive to our own and others' genuine well-being. One key to achieving such freedom is the cultivation of attentional skills so that one can deliberately focus one's attention with continuity and clarity on one's chosen object. A second theme is the cultivation of insight into the manner in which our own attitudes shape experience, allowing for the possibility of altering not only the way we experience events in the present, but also how we are influenced by our memories of the past. Finally, the Great Perfection school of Tibetan Buddhism emphasizes the realization of the deepest dimension of consciousness -- pristine awareness -- which transcends the nexus of causality. This is regarded as the ultimate source of freedom and the ultimate nature of human identity. (shrink)

Knowledge of mechanisms is critical for causal reasoning. We contrasted two possible organizations of causal knowledge—an interconnected causal network, where events are causally connected without any boundaries delineating discrete mechanisms; or a set of disparate mechanisms—causal islands—such that events in different mechanisms are not thought to be related even when they belong to the same causal chain. To distinguish these possibilities, we tested whether people make transitive judgments about causal chains by inferring, given A causes B and B causes C, (...) that A causes C. Specifically, causal chains schematized as one chunk or mechanism in semantic memory led to transitive causal judgments. On the other hand, chains schematized as multiple chunks led to intransitive judgments despite strong intermediate links. Normative accounts of causal intransitivity could not explain these intransitive judgments. (shrink)

Transformations in the biological, medical and legal processes of infertility, substantial modifications in family structure and the advancement of methods and techniques of reproductive technology will affect the next step in both legal and medical terms to address the regulation of bioethics and law in Kosovo. There is a need to establish perspectives in both ethical and professional terms, since the Republic of Kosovo is in the process of drafting a Civil Code. Many of these issues have been raised and (...) addressed during the review and evaluation of family law in the context of harmonisation and inclusion of this law in the Civil Code of the Republic of Kosovo. During the several meetings of official members with different interest groups regarding family law, the need has been raised to regulate family law to be included in the Civil Code for motherhood and fatherhood in the case of reproduction with biomedical assistance, as well as for the birth contract as a donation for another person (so-called surrogate motherhood). These bioethical and legal issues indicate the urgent need for legal harmonisation of a multidimensional platform specifically based on the principles of public health and universal human rights Conclusion. These bio-ethical and legal interferences indicate the urgent need for legal harmonization of a multidimensional platform specifically designed based on the principles of public health and universal human rights. (shrink)

A guide for scientists who would like to contribute to the professional development of science teachers for elementary schools. Based on information from over 180 programs, describes what activities work and why, and suggests how to identify programs teachers have found to be effective and take the initial steps to become involved. Also provides vignettes illustrating the daily work of science teachers. Annotation copyright by Book News, Inc., Portland, OR.

The author has made this treatise on Madhva’s realistic school of Ved?nta philosophy convincing to the modern mind by employing western logical apparatus in substantiating Madhva’s ideas. Following the Indian classical tradition, the author has examined the validity of Advaitaved?nta, the Absolute Monism of ?a?kara and Bh?skara as P?rvapak?a, and logically proved its inconsistencies. He has then established the Dvaitasiddh?nta i.e., the Monotheistic Dualism of Madhva. He has successfully brought out the nuances of the realistic school of Indian philosophical thought (...) in this work. A Modern Introduction to M?dhva Philosophy is Prof B H Kotabagi’s posthumous publication. (shrink)

On September 1, 2023, Texas made important revisions to it its decades‐old statute granting legal safe harbor immunity to physicians who withhold or withdraw life‐sustaining treatment over the objection of critically ill patients’ surrogate decision‐makers. However, lawmakers left untouched glaring flaws in a key safeguard for patients—the transfer option. The transfer option is ethically important because, when no hospital is willing to accept the patient in transfer, that fact is taken as strong evidence that the surrogates’ treatment requests fall outside (...) accepted medical practice. But there are serious shortcomings in how the transfer option is carried out in Texas and many other states, which undermines the ethical usefulness of the process. We identify these shortcomings and recommend revisions to state statutes and professional guidelines to overcome them. (shrink)