This article reports that some robustness of the notions of predicativity and of autonomous progression is broken down if as the given infinite total entity we choose some mathematical entities other than the traditionalω. Namely, the equivalence between normal transfinite recursion scheme and newdependent transfinite recursionscheme, which does hold in the context of subsystems of second order number theory, does not hold in the context of subsystems of second order set theory where the universeVof sets is treated as the given (...) totality. (shrink)

Let W be a recursively enumerable vector space over a recursive ordered field. We show the Turing equivalence of the following sets: the set of all tuples of vectors in W which are linearly dependent; the set of all tuples of vectors in W whose convex closures contain the zero vector; and the set of all pairs of tuples in W such that the convex closure of X intersects the convex closure of Y. We also form the analogous sets (...) consisting of tuples with given numbers of elements, and prove similar results on the Turing equivalence of these. (shrink)

In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure – Dedekind-complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if “computable” is replaced by “primitive recursive” , these definitions lead to a number of different concepts, which (...) we compare in this article. We summarize the known results and add new ones. In particular we show that there is a proper hierarchy among p. r. real numbers by nested interval representation, Cauchy representation, b -adic expansion representation, Dedekind cut representation, and continued fraction expansion representation. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers. (shrink)

In mathematics, various representations of real numbers have been investigated. All these representations are mathematically equivalent because they lead to the same real structure - Dedekind-complete ordered field. Even the effective versions of these representations are equivalent in the sense that they define the same notion of computable real numbers. Although the computable real numbers can be defined in various equivalent ways, if computable is replaced by primitive recursive (p. r., for short), these definitions lead to a number of (...) different concepts, which we compare in this article. We summarize the known results and add new ones. In particular we show that there is a proper hierarchy among p. r. real numbers by nested interval representation, Cauchy representation, b -adic expansion representation, Dedekind cut representation, and continued fraction expansion representation. Our goal is to clarify systematically how the primitive recursiveness depends on the representations of the real numbers. (shrink)

This article addresses three questions about well-being. First, is well-being future-sensitive? I.e., can present well-being depend on future events? Second, is well-being recursively dependent? I.e., can present well-being depend on itself? Third, can present and future well-being be interdependent? The third question combines the first two, in the sense that a yes to it is equivalent to yeses to both the first and second. To do justice to the diverse ways we contemplate well-being, I consider our thought and discourse about (...) well-being in three domains: everyday conversation, social science, and philosophy. This article’s main conclusion is that we must answer the third question with no. Present and future well-being cannot be interdependent. The reason, in short, is that a theory of well-being that countenances both future-sensitivity and recursive dependence would have us understand a person’s well-being at a time as so intricately tied to her well-being at other times that it would not make sense to consider her well-being an aspect of her state at particular times. It follows that we must reject either future-sensitivity or recursive dependence. I ultimately suggest, especially in light of arguments based on assumptions of empirical research on well-being, that the balance of reasons favors rejecting future-sensitivity. (shrink)

We use the multiple price list method and a recursive expected utility theory of smooth ambiguity to separate out attitude towards risk from that towards ambiguity. Based on this separation, we investigate if there are differences in agent behaviour under uncertainty over gain amounts vis-a-vis uncertainty over loss amounts. On an aggregate level, we find that (i) subjects are risk averse over gains and risk seeking over losses, displaying a “reflection effect” and (ii) they are ambiguity neutral over gains (...) and are mildly ambiguity seeking over losses. Further analysis shows that on an individual level, and with respect to both risky and ambiguous prospects, there is limited incidence of a reflection effect where subjects are risk/ambiguity averse (seeking) in gains and seeking (averse) in losses, though this incidence is higher for ambiguous prospects. A very high proportion of such cases of reflection exhibit risk (ambiguity) aversion in gains and risk (ambiguity) seeking in losses, with the reverse effect being significantly present in the case of risk but almost absent in case of ambiguity. Our results suggest that reflection across gains and losses is not a stable individual characteristic, but depends upon whether the form of uncertainty is precise or ambiguous, since we rarely find an individual who exhibits reflection in both risky and ambiguous prospects. We also find that correlations between attitudes towards risk and ambiguity were domain dependent. (shrink)

We show how two iterated products of selection functions can both be used in conjunction with systemTto interpret, via the dialectica interpretation and modified realizability, full classical analysis. We also show that one iterated product is equivalent over systemTto Spector’s bar recursion, whereas the other isT-equivalent to modified bar recursion. Modified bar recursion itself is shown to arise directly from the iteration of a different binary product of ‘skewed’ selection functions. Iterations of the dependent binary products are also considered but (...) in all cases are shown to beT-equivalent to the iteration of the simple products. (shrink)

This paper is concerned with the event-based state and fault estimation problem for a class of linear discrete systems with randomly occurring faults and missing measurements. Different from the static event-based transmission mechanism with a constant threshold, a dynamic event-based mechanism is exploited here to regulate the threshold parameter, thus further reducing the amount of data transmission. Some mutually independent Bernoulli random variables are used to characterize the phenomena of ROFs and missing measurements. In order to simultaneously estimate the system (...) state and the fault signals, the main attention of this paper is paid to the design of recursive filter; for example, for all DETM, ROFs, and missing measurements, an upper bound for the estimation error covariance is ensured and the relevant filter gain matrix is designed by minimizing the obtained upper bound. Moreover, the rigorous mathematical analysis is carried out for the exponential boundedness of the estimation error. It is clear that the developed algorithms are dependent on the threshold parameters and the upper bound together with the probabilities of missing measurements and ROFs. Finally, a numerical example is provided to indicate the effectiveness of the presented estimation schemes. (shrink)

The capacity to generate recursive sequences is a marker of rich, algorithmic cognition, and perhaps unique to humans. Yet, the precise processes driving recursive sequence generation remain mysterious. We investigated three potential cognitive mechanisms underlying recursive pattern processing: hierarchical reasoning, ordinal reasoning, and associative chaining. We developed a Bayesian mixture model to quantify the extent to which these three cognitive mechanisms contribute to adult humans’ performance in a sequence generation task. We further tested whether recursive rule (...) discovery depends upon relational information, either perceptual or semantic. We found that the presence of relational information facilitates hierarchical reasoning and drives the generation of recursive sequences across novel depths of center embedding. In the absence of relational information, the use of ordinal reasoning predominates. Our results suggest that hierarchical reasoning is an important cognitive mechanism underlying recursive pattern processing and can be deployed across embedding depths and relational domains. (shrink)

This paper provides a unifying axiomatic account of the interpretation of recursive types that incorporates both domain-theoretic and realizability models as concrete instances. Our approach is to view such models as full subcategories of categorical models of intuitionistic set theory. It is shown that the existence of solutions to recursive domain equations depends upon the strength of the set theory. We observe that the internal set theory of an elementary topos is not strong enough to guarantee their existence. (...) In contrast, as our first main result, we establish that solutions to recursive domain equations do exist when the category of sets is a model of full intuitionistic Zermelo–Fraenkel set theory. We then apply this result to obtain a denotational interpretation of FPC, a recursively typed lambda-calculus with call-by-value operational semantics. By exploiting the intuitionistic logic of the ambient model of intuitionistic set theory, we analyse the relationship between operational and denotational semantics. We first prove an “internal” computational adequacy theorem: the model always believes that the operational and denotational notions of termination agree. This allows us to identify, as our second main result, a necessary and sufficient condition for genuine “external” computational adequacy to hold, i.e. for the operational and denotational notions of termination to coincide in the real world. The condition is formulated as a simple property of the internal logic, related to the logical notion of 1-consistency. We provide useful sufficient conditions for establishing that the logical property holds in practice. Finally, we outline how the methods of the paper may be applied to concrete models of FPC. In doing so, we obtain computational adequacy results for an extensive range of realizability and domain-theoretic models. (shrink)

We introduce a new formulation of the axiom of dependent choice, which can be viewed as an abstract termination principle that in particular generalises recursive path orderings, the latter being fundamental tools used to establish termination of rewrite systems. We consider several variants of our termination principle, and relate them to general termination theorems in the literature.

Let KP- be the theory resulting from Kripke-Platek set theory by restricting Foundation to Set Foundation. Let G: V → V (V:= universe of sets) be a ▵0-definable set function, i.e. there is a ▵0-formula φ(x, y) such that φ(x, G(x)) is true for all sets x, and $V \models \forall x \exists!y\varphi (x, y)$ . In this paper we shall verify (by elementary proof-theoretic methods) that the collection of set functions primitive recursive in G coincides with the collection (...) of those functions which are Σ1-definable in KP- + Σ1-Foundation + ∀ x ∃!yφ (x, y). Moreover, we show that this is still true if one adds Π1-Foundation or a weak version of ▵0-Dependent Choices to the latter theory. (shrink)

This paper deals with a philosophical question that arises within the theory of computational complexity: how to understand the notion of INTRINSIC complexity or difficulty, as opposed to notions of difficulty that depend on the particular computational model used. The paper uses ideas from Blum's abstract approach to complexity theory to develop an extensional approach to this question. Among other things, it shows how such an approach gives detailed confirmation of the view that subrecursive hierarchies tend to rank functions in (...) terms of their intrinsic, and not just their model-dependent, difficulty, and it shows how the approach allows us to model the idea that intrinsic difficulty is a fuzzy concept. (shrink)

Let be a model of a theory T. Depending on wether is decidable or recursive, and on whether T is strongly minimal or -minimal, we find conditions on which guarantee that every infinite independent subset of is not recursively enumerable. For each of the same four cases we also find conditions on which guarantee that every infinite independent subset of has Turing degree 0'. More generally, let be a recursive -structure, R a relation symbol not in , ψ (...) a recursive infiniatary Π2 sentence in the language {R}, and let 0<α<ωCK1. We discuss conditions under which we can construct a recursive -structure such that ,Rψ for every Δ0α relation R on. (shrink)

Most of what is told in this paper has been told before by the same author, in a number of publications of various kinds, but this is the first time that all this material has been brought together and treated in a uniform way. Smaller errors in the earlier publications are corrected here without comment. It has been known since the 1920s that quadratic equations played a prominent role in Babylonian mathematics. See, most recently, Høyrup (Hist Sci 34:1–32, 1996, and (...) Lengths, widths, surfaces: a portrait of old Babylonian algebra and its kin. Springer, New York, 2002). What has not been known, however, is how quadratic equations came to play that role, since it is difficult to think of any practical use for quadratic equations in the life and work of a Babylonian scribe. One goal of the present paper is to show how the need to find solutions to quadratic equations actually arose in Mesopotamia not later than in the second half of the third millennium BC, and probably before that in connection with certain geometric division of property problems. This issue was brought up for the first time in Friberg (Cuneiform Digit Lib J 2009:3, 2009). In this connection, it is argued that the tool used for the first exact solution of a quadratic equation was either a clever use of the “conjugate rule” or a “completion of the square,” but that both methods ultimately depend on a certain division of a square, the same in both cases. Another, closely related goal of the paper is to discuss briefly certain of the most impressive achievements of anonymous Babylonian mathematicians in the first half of the second millennium BC, namely recursive geometric algorithms for the solution of various problems related to division of figures, more specifically trapezoidal fields. For an earlier, comprehensive (but less accessible) treatment of these issues, see Friberg (Amazing traces of a Babylonian origin in Greek mathematics. WorldScientific, Singapore 2007b, Ch. 11 and App. 1). (shrink)

We extend the earlier results on the equivalence between the Boolean and the multivalued dependencies in relational databases and fragments of the Boolean propositional logic. It is shown that these equivalences are still valid for the databases that store complex data elements obtained from the recursive nesting of record, list, set and multiset constructors. The major proof argument utilises properties of Brouwerian algebras.The equivalences have several consequences. Firstly, they provide new insights into databases that are not in first normal (...) form. Secondly, they characterise the implication of data dependencies in nested databases in purely logical terms. The database designer can take advantage of these equivalences to reduce database design problems to well-studied problems in Boolean propositional logic. Furthermore, relational database design solutions can be reused to solve problems for nested databases. (shrink)

We show that the length of a hierarchy of domains with totality, based on the standard domain for the natural numbers \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\Bbb N}$\end{document} and closed under dependent products of continuously parameterised families of domains will be the first ordinal not recursive in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $^3E$\end{document} and any real. As a part of the proof we show that the domains of the hierarchy (...) share important properties with the types of continuous functionals. The main result can also be viewed as a representation theorem for recursion in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $^3E$\end{document}. (shrink)

The component structures of two distinct neuropsychological systems are described. "System-Y" depends upon "system-X" which, on the other hand, can operate independently of system-Y. System-X provides a matrix upon which system-Y must operate, and, system-Y is transformed by the operations of system-X. In addition these neuropsychological structures reverberate in political history and in the cosmos. The most fundamental structure in the soul, in society, and in the cosmos, has the form of a conical spiral. It can be described mathematically as (...) an harmonic system and mythologically in terms of the birth, marriage and death of the divine-king. ;System-Y corresponds to the neocortex. System-X corresponds to the region below the cortex including the limbic brain. Many of the essential structures of system-Y are captured by Plato's image of the helmsman: orientation, locomotion, manual control and dexterity, visual guidance, verbal command, intention, and volition. System-X on the other hand has been systematically neglected by Western culture, beginning with Plato. ;The dissertation builds upon Yakovlev's distinction between "teleokinesis," "ectokinesis" and "endokinesis." "Teleokinesis" refers to goal-directed action in external space, and belongs to system-Y. Ectokinesis and endokinesis are components of system-X. "Endokinesis" refers to movements within the body. "Ectokinesis" refers to the emotions, which are expressions of endokinesis. The mechanics of ectokinesis is that of an harmonic or vibrational system, as is the mechanics of a musical instrument. ;Within a trance the structure of system-Y is temporarily altered, such that system-X enters the foreground of awareness. Because ectokinesis is analogous to music cultures inspired by trance experience understand the universe in terms of music. The structure of ectokinesis in a trance is that of a conical spiral. Plato inherited the mystical traditions of the ancient Near East but replaced the spiral of emotions with a spiral of ideas , by understanding the axis of the cone as the paradigmatic dimension, and the angular rotation of the spiral as the syntagmatic dimension, of language. Finally I explain how the mythology and political structure of theocratic society imitated the neuropsychological structures of trance experience. (shrink)

The aim of the paper is to propose a preferences representation model under risk where risk perception can be past experience dependent. A first step consists in considering a one period decision problem where individual preferences are no more defined only on decisions but on pairs (decision, past experience). The obtained criterion is used in the construction of a dynamic choice model under risk. The paper ends with an illustrative example concerning insurance demand. It appears that our model allows to (...) explain modifications in the insurance demand behavior over time observed on the insurance markets for catastrophic risk and difficult to justify with standard models. (shrink)

Pensons aux divers énoncés qui peuvent être composés à partir d'un ensemble fini ou dénombrable d'énoncés atomiques à l'aide de, disons, ‘˜’ et ‘&’; soit A n'importe lequel de ces énoncés; et soit l'ensemble SA des composantes atomiques de A. La valeur de vérité de A dépend évidemment des valeurs de vérité de certains membres de SA. En effet, si aux valeurs de vérité Vrai et Faux sont substitués les entiers 1 et 0, respectivement; la valeur de vérité VVV d'une (...) négation ˜A est 1–VV; et la valeur de vérité VV d'une conjonction A & B est min, VV), alors la valeur de vérité de A s'avère être une fonction numérique des valeurs de vérité de certains membres de SA. Et les fonctions de valeurs de vérité sont dites, à titre de résultat, récursivement définissables. (shrink)

The aim of this paper is to propose a preferences representation model where ambiguity attitudes can be exogenous events or past experience-dependent. We adapt the Recursive Smooth Ambiguity model proposed by Klibanoff (Journal of Economic Theory 144:930-976, 2009) by introducing past experience described by a sequence of neutral events occurring up to the moment of the decision. These neutral events do not provide any information on the true process, but are likely to strengthen or weaken the decision-maker’s ambiguity aversion (...) degree by modifying emotions. Our model can explain some observed behaviors and market inefficiencies. We propose two illustrations. First, we provide a behavioral explanation for the decrease in influenza vaccination in France that followed the H1N1 crisis in 2009–2010. Second, we contribute to the broad literature on the annuity puzzle by introducing the impact of emotions on ambiguity attitudes. (shrink)

This article generalises the refined A-translation method for extracting programs from classical proofs [U. Berger,W. Buchholz, H. Schwichtenberg, Refined program extraction from classical proofs, Annals of Pure and Applied Logic 114 3–25] to the scenario where additional assumptions such as choice principles are involved. In the case of choice principles, this is done by adding computational content to the ‘translated’ assumptions, an idea which goes back to [S. Berardi, M. Bezem, T. Coquand, On the computational content of the axiom of (...) choice, JSL 63 600–622] and has been further elaborated in [U. Berger, P. Oliva, Modified bar recursion and classical dependent choice, in: M. Baaz, S.D. Friedman, J. Kraijcek , Logic Colloquium ’01, in: Lecture Notes in Logic, vol. 20, Springer, Berlin, 2005, pp. 89–107]. We further investigate the applicability of this extension by means of a small but non-trivial example, and discuss the formalisation as well as some optimisations necessary in order to extract terminating programs. Both formalisation and term extraction have been implemented in the interactive proof assistant Minlog. (shrink)

A study on minimalist syntax develops an empirical argument for a crash-proof computational system. A crash-proof system is obtained if syntactic dependencies are strictly local (i.e. there is no long-distance Agree). Apparent long-distance dependencies turn out to be the outcome of a recursive chain on local complex dependencies. This framework allows for novel analyses of quirky subjects in Icelandic and Spanish, indefinite SE in Spanish and different types of expletive constructions in English, French, German, Spanish, Italian, and Icelandic.

Limited distribution phenomena related to negation and negative polarity are usually thought of in terms of affectivity where affective is understood as negative or downward entailing. In this paper I propose an analysis of affective contexts as nonveridical and treat negative polarity as a manifestation of the more general phenomenon of sensitivity to (non)veridicality (which is, I argue, what affective dependencies boil down to). Empirical support for this analysis will be provided by a detailed examination of affective dependencies in Greek, (...) but the distribution of any will also be shown to follow from (non)veridicality. (shrink)

This unique volume explores what ethics has to offer the practicing physician, nurse, and allied health care worker. The authors introduce the basic vocabulary of ethics and present and discuss the most commonly used ethical theories, using case studies to illustrate how ethics work within the context of health care. Newcomers to the field will learn what ethics is all about and how it relates to the pragmatic concerns of the health care professional. Those who already have a working knowledge (...) of the field will find a new approach to ethical theory--personalism--which the authors believe can make the practical use of ethics more effective and reliable. (shrink)

Experiments are generally thought of as actions or operations undertaken to test a scientific hypothesis in settings detached from the rest of society. In this paper a different notion of experiment will be discussed. It is an understanding that has been developed in the classical tradition of the Chicago School of Sociology since the 1890s, but has so far remained unexplored. This sociological understanding of experiment does not model itself strictly on the natural sciences. Rather, it implies a process of (...) societal self-experimentation without a fixed setting of a sociological experimenter. The paper discusses this notion of experiment in relation to the recursive dependency of the application and the production of sociological knowledge. It is contended that this concept of a self-experimental society offers theoretical insights that could well prove fruitful for a sociological concept of experiment beyond the realm of the laboratory. (shrink)

Talk of linguistic universals has given cognitive scientists the impression that languages are all built to a common pattern. In fact, there are vanishingly few universals of language in the direct sense that all languages exhibit them. Instead, diversity can be found at almost every level of linguistic organization. This fundamentally changes the object of enquiry from a cognitive science perspective. This target article summarizes decades of cross-linguistic work by typologists and descriptive linguists, showing just how few and unprofound the (...) universal characteristics of language are, once we honestly confront the diversity offered to us by the world's 6,000 to 8,000 languages. After surveying the various uses of “universal,” we illustrate the ways languages vary radically in sound, meaning, and syntactic organization, and then we examine in more detail the core grammatical machinery of recursion, constituency, and grammatical relations. Although there are significant recurrent patterns in organization, these are better explained as stable engineering solutions satisfying multiple design constraints, reflecting both cultural-historical factors and the constraints of human cognition.Linguistic diversity then becomes the crucial datum for cognitive science: we are the only species with a communication system that is fundamentally variable at all levels. Recognizing the true extent of structural diversity in human language opens up exciting new research directions for cognitive scientists, offering thousands of different natural experiments given by different languages, with new opportunities for dialogue with biological paradigms concerned with change and diversity, and confronting us with the extraordinary plasticity of the highest human skills. (shrink)

The paper proposes a novel solution to the problem of scope posed by natural language indefinites that captures both the difference in scopal freedom between indefinites and bona fide quantifiers and the syntactic sensitivity that the scope of indefinites does nevertheless exhibit. Following the main insight of choice functional approaches, we connect the special scopal properties of indefinites to the fact that their semantics can be stated in terms of choosing a suitable witness. This is in contrast to bona fide (...) quantifiers, the semantics of which crucially involves relations between sets of entities. We provide empirical arguments that this insight should not be captured by adding choice/Skolem functions to classical first-order logic, but in a semantics that follows Independence-Friendly Logic, in which scopal relations involving existentials are part of the recursive definition of truth and satisfaction. These scopal relations are resolved automatically as part of the interpretation of existentials. Additional support for this approach is provided by dependent indefinites, a cross-linguistically common class of special indefinites that can be straightforwardly analyzed in our semantic framework. (shrink)

A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitely many elements including h(x), f is a k(n)-enumerator if for every input x of length n, h(x) is among the first k(n) elements enumerated by f. If there is a k(n)-enumerator for h then h is called k(n)-enumerable. We also consider enumerators which are only A-recursive for some oracle A. We determine exactly how hard it is to enumerate the Kolmogorov (...) function, which assigns to each string x its Kolmogorov complexity: • For every underlying universal machine U, there is a constant a such that C is k(n)-enumerable only if k(n) ≥ n/a for almost all n. • For any given constant k, the Kolmogorov function is k-enumerable relative to an oracle A if and only if A is at least as hard as the halting problem. • There exists an r.e., Turing-incomplete set A such for every non-decreasing and unbounded recursive function k, the Kolmogorov function is k(n)-enumerable relative to A. The last result is obtained by using a relativizable construction for a nonrecursive set A relative to which the prefix-free Kolmogorov complexity differs only by a constant from the unrelativized prefix-free Kolmogorov complexity. Although every 2-enumerator for C is Turing hard for K, we show that reductions must depend on the specific choice of the 2-enumerator and there is no bound on the quantity of their queries. We show our negative results even for strong 2-enumerators as an oracle where the querying machine for any x gets directly an explicit list of all hypotheses of the enumerator for this input. The limitations are very general and we show them for any recursively bounded function g: • For every Turing reduction M and every non-recursive set B, there is a strong 2-enumerator f for g such that M does not Turing reduce B to f. • For every non-recursive set B, there is a strong 2-enumerator f for g such that B is not wtt-reducible to f. Furthermore, we deal with the resource-bounded case and give characterizations for the class ${\rm S}_{2}^{{\rm P}}$ introduced by Canetti and independently Russell and Sundaram and the classes PSPACE, EXP. • ${\rm S}_{2}^{{\rm P}}$ is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time tt-reduction which reduces A to every strong 2-enumerator for g. • PSPACE is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time Turing reduction which reduces A to every strong 2-enumerator for g. Interestingly, g can be taken to be the Kolmogorov function for the conditional space bounded Kolmogorov complexity. • EXP is the class of all sets A for which there is a polynomially bounded function g and a machine M which witnesses A ∈ PSPACEf for all strong 2-enumerators f for g. Finally, we show that any strong O(log n)-enumerator for the conditional space bounded Kolmogorov function must be PSPACE-hard if P = NP. (shrink)

Theories of meaning developed within the analytic tradition, starting with Gottlob Frege, and within continental philosophy, starting with Husserl, can be distinguished by their disagreement about the phenomenon of collapse or failure of meaning. Our text focuses on Frege’s legacy, taken up by Rudolph Carnap, which culminated in a view of the collapse of meaning defined first by a purely syntactic conception of categorial error and second, when Tarski entered the scene, by the paradoxes created by the conflict between the (...) use of the predicate truth and the concept of proof. As positivism’s conventionalism and late model-theoretic semantics became the paradigms derived from this position, the result, in our view, was a conception of meaning success and meaning failure dominated by dependence on a technological conception. We will argue that this dependence leads to the inability to evaluate the breaks in rational parameters and crises of meaning addressed by the phenomenological and transcendental perspectives. Husserl’s critique of logic as technology is the basis for the reflexive standpoint on the phenomena of meaning failure that leads to a theoretical and eidetic ideal approach to that phenomenon. It is then suggested that this phenomenological reflective standpoint on the problem of meaning failure is the origin of the doors that lead to the divergent paths between the analytic and continental traditions. Thus, this is an article with three argumentative phases connected into a larger unity. The first phase presents Husserl’s critique of a purely extensional, syntactic, and recursive conception of logical consequence and the concept of truth; the second presents Hussel’s larger conception of meaning-failure; the third—presented in an appendix— presents a lesson on the division between continental and analytic philosophy. (shrink)

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)

By compiling an experimental method combining both dialectic and rhetoric, 'Dialectic, Rhetoric and Contrast: The Infinite Middle of Meaning' demonstrates how singular meanings can be rendered in a spectrum of 12 repeating concepts that are in a continuum, gradated and symmetrical. The ability to arrange meaning into this pattern opens enquiry into its ontology, and presents meaning as closer to the sensation of colours or musical notes than the bivalent oppositions depicted in classical logic. However, the experiment does not assert (...) that this pattern suggests some sort of constant or absolute principle; instead, it theorises on the ways in which meaning can be considered to be recursive. To explain this, the book explores the concept of contrast itself. No exactitude on the precise existence of contrast can ever be struck because the answer varies infinitely depending upon the scale of measurement used to gauge the meeting point. This characteristic of contrast helps to define a whole new dimension from which sensation, meaning, cognition and consciousness can be analogised to the infinite forms between forms. At a time when the widest consensus in philosophy is the exhaustion of its central themes, the significance of such a hypothesis provides fresh impetus to revise some of the key meanings and concepts underpinning contemporary thought. To do this, the method explores the opposing themes of idealism and realism that run throughout western philosophy from Plato to the Speculative turn. This book will be of interest to professional academic audiences in the humanities and social sciences, from graduates to senior scholars. It will also be an interesting read to anyone wishing to keep abreast of developments in continental philosophy, epistemology, metaphysics, and the sociology of knowledge. (shrink)

The topics approached in the 52 papers included in this book are: neutrosophic sets; neutrosophic logic; generalized neutrosophic set; neutrosophic rough set; multigranulation neutrosophic rough set (MNRS); neutrosophic cubic sets; triangular fuzzy neutrosophic sets (TFNSs); probabilistic single-valued (interval) neutrosophic hesitant fuzzy set; neutro-homomorphism; neutrosophic computation; quantum computation; neutrosophic association rule; data mining; big data; oracle Turing machines; recursive enumerability; oracle computation; interval number; dependent degree; possibility degree; power aggregation operators; multi-criteria group decision-making (MCGDM); expert set; soft sets; LA-semihypergroups; single (...) valued trapezoidal neutrosophic number; inclusion relation; Q-linguistic neutrosophic variable set; vector similarity measure; fundamental neutro-homomorphism theorem; neutro-isomorphism theorem; quasi neutrosophic triplet loop; quasi neutrosophic triplet group; BE-algebra; cloud model; Maclaurin symmetric mean; pseudo-BCI algebra; hesitant fuzzy set; photovoltaic plan; decision-making trial and evaluation laboratory (DEMATEL); Choquet integral; fuzzy measure; clustering algorithm; and many more. (shrink)

Results in recursion-theoretic inductive inference have been criticized as depending on unrealistic self-referential examples. J. M. Bārzdiņš proposed a way of ruling out such examples, and conjectured that one of the earliest results of inductive inference theory would fall if his method were used. In this paper we refute Bārzdiņš' conjecture. We propose a new line of research examining robust separations; these are defined using a strengthening of Bārzdiņš' original idea. The preliminary results of the new line of research are (...) presented, and the most important open problem is stated as a conjecture. Finally, we discuss the extension of this work from function learning to formal language learning. (shrink)

My life as a quantum physicist / M. Nakahara -- A review on operator quantum error correction - Dedicated to Professor Mikio Nakahara on the occasion of his 60th birthday / C.-K. Li, Y.-T. Poon and N.-S. Sze -- Implementing measurement operators in linear optical and solid-state qubits / Y. Ota, S. Ashhab and F. Nori -- Fast and accurate simulation of quantum computing by multi-precision MPS: Recent development / A. Saitoh -- Entanglement properties of a quantum lattice-gas model on (...) square and triangular ladders / S. Tanaka, R. Tamura and H. Katsura -- On signal amplification from weak-value amplification / Y. Shikano -- Topological protection of quantum information / K. Fujii -- Quantum annealing with antiferromagnetic fluctuations for mean-field models / Y. Seki and H. Nishimori -- A method to change phase transition nature - Toward annealing methods / R. Tamura and S. Tanaka -- Computational analysis of the first stage of the photosynthetic system, the light-dependent reaction, by quantum chemical simulation method / M. Tada-Umezaki -- Two-qubit gate operation on selected nearest neighboring qubits in a neutral atom quantum computer / E. Hosseini Lapasar... [et al.] -- A simple operator quantum error correction scheme avoiding fully correlated errors / C. Bagnasco, Y. Kondo and M. Nakahara -- Black hole predictability, classical and quantum / A. Ishibashi -- Classical field simulation of finite-temperature Bose gases / T. Sato -- Atomic quantum simulations of lattice gauge theory: Effect of gauge symmetry breaking / K. Kasamatsu, I. Ichinose and T. Matsui -- Recursive construction of noiseless subsystem for qudits / U. Gungordu... [et al.] -- Composite quantum gates for precise quantum control / M. Bando... [et al.] -- New formulation of statistical mechanics using thermal pure quantum states / S. Sugiura and A. Shimizu -- Thermodynamics in unitary time evolution / T. N. Ikeda -- Second law of thermodynamics with QC-mutual information / T. Sagawa. (shrink)