A set theory model of reality, representation and language based on the relation of completeness and incompleteness is explored. The problem of completeness of mathematics is linked to its counterpart in quantum mechanics. That model includes two Peano arithmetics or Turing machines independent of each other. The complex Hilbert space underlying quantum mechanics as the base of its mathematical formalism is interpreted as a generalization of Peano arithmetic: It is a doubled infinite set of doubled Peano arithmetics (...) having a remarkable symmetry to the axiom of choice. The quantity of information is interpreted as the number of elementary choices (bits). Quantum information is seen as the generalization of information to infinite sets or series. The equivalence of that model to a quantum computer is demonstrated. The condition for the Turing machines to be independent of each other is reduced to the state of Nash equilibrium between them. Two relative models of language as game in the sense of game theory and as ontology of metaphors (all mappings, which are not one-to-one, i.e. not representations of reality in a formal sense) are deduced. (shrink)

A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent (...) Peano arithmetics discussed in Section I. Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them. Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines. The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV. Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations. (shrink)

Any computer can create a model of reality. The hypothesis that quantum computer can generate such a model designated as quantum, which coincides with the modeled reality, is discussed. Its reasons are the theorems about the absence of “hidden variables” in quantum mechanics. The quantum modeling requires the axiom of choice. The following conclusions are deduced from the hypothesis. A quantum model unlike a classical model can coincide with reality. Reality can be interpreted as a (...) quantum computer. The physical processes represent computations of the quantum computer. Quantum information is the real fundament of the world. The conception of quantum computer unifies physics and mathematics and thus the material and the ideal world. Quantum computer is a non-Turing machine in principle. Any quantum computing can be interpreted as an infinite classical computational process of a Turing machine. Quantum computer introduces the notion of “actually infinite computational process”. The discussed hypothesis is consistent with all quantum mechanics. The conclusions address a form of neo-Pythagoreanism: Unifying the mathematical and physical, quantum computer is situated in an intermediate domain of their mutual transformation. (shrink)

Quantum complementarity is interpreted in terms of duality and opposition. Any two conjugates are considered both as dual and opposite. Thus quantum mechanics introduces a mathematical model of them in an exact and experimental science. It is based on the complex Hilbert space, which coincides with the dual one. The two dual Hilbert spaces model both duality and opposition to resolve unifying the quantum and smooth motions. The model involves necessarily infinity even in any finitely dimensional subspace (...) of the complex Hilbert space being due to the complex basis. Furthermore, infinity is what unifies duality and opposition, universality and openness, completeness and incompleteness in it. The deduced core of quantum complementarity in terms of infinity, duality and opposition allows of resolving a series of various problems in different branches of philosophy: the common structure of incompleteness in Gödel’s (1931) theorems and Enstein, Podolsky, and Rosen’s argument (1935); infinity as both complete and incomplete; grounding and self-grounding, metaphor and representation between language and reality, choice and information, the totality and an observer, the basic idea of philosophical phenomenology. The main conclusion is: Quantum complementarity unifies duality and opposition in a consistent way underlying the physical world. (shrink)

We describe a proof by a computer program of the unsolvability of the halting problem. The halting problem is posed in a constructive, formal language. The computational paradigm formalized is Pure LISP, not Turing machines. The machine was led to the proof by the authors, who suggested certain function definitions and stated certain intermediate lemmas. The machine checked that every suggested definition was admissible and the machine proved the main theorem and every lemma. We (...) believe this is the first instance of a machine checking that a given problem is not solvable by machine. (shrink)

Non-commuting quantities and hidden parameters – Wave-corpuscular dualism and hidden parameters – Local or nonlocal hidden parameters – Phase space in quantum mechanics – Weyl, Wigner, and Moyal – Von Neumann’s theorem about the absence of hidden parameters in quantum mechanics and Hermann – Bell’s objection – Quantum-mechanical and mathematical incommeasurability – Kochen – Specker’s idea about their equivalence – The notion of partial algebra – Embeddability of a qubit into a bit – Quantum computer is (...) not Turing machine – Is continuality universal? – Diffeomorphism and velocity – Einstein’s general principle of relativity – „Mach’s principle“ – The Skolemian relativity of the discrete and the continuous – The counterexample in § 6 of their paper – About the classical tautology which is untrue being replaced by the statements about commeasurable quantum-mechanical quantities – Logical hidden parameters – The undecidability of the hypothesis about hidden parameters – Wigner’s work and и Weyl’s previous one – Lie groups, representations, and psi-function – From a qualitative to a quantitative expression of relativity − psi-function, or the discrete by the random – Bartlett’s approach − psi-function as the characteristic function of random quantity – Discrete and/ or continual description – Quantity and its “digitalized projection“ – The idea of „velocity−probability“ – The notion of probability and the light speed postulate – Generalized probability and its physical interpretation – A quantum description of macro-world – The period of the as-sociated de Broglie wave and the length of now – Causality equivalently replaced by chance – The philosophy of quantum information and religion – Einstein’s thesis about “the consubstantiality of inertia ant weight“ – Again about the interpretation of complex velocity – The speed of time – Newton’s law of inertia and Lagrange’s formulation of mechanics – Force and effect – The theory of tachyons and general relativity – Riesz’s representation theorem – The notion of covariant world line – Encoding a world line by psi-function – Spacetime and qubit − psi-function by qubits – About the physical interpretation of both the complex axes of a qubit – The interpretation of the self-adjoint operators components – The world line of an arbitrary quantity – The invariance of the physical laws towards quantum object and apparatus – Hilbert space and that of Minkowski – The relationship between the coefficients of -function and the qubits – World line = psi-function + self-adjoint operator – Reality and description – Does a „curved“ Hilbert space exist? – The axiom of choice, or when is possible a flattening of Hilbert space? – But why not to flatten also pseudo-Riemannian space? – The commutator of conjugate quantities – Relative mass – The strokes of self-movement and its philosophical interpretation – The self-perfection of the universe – The generalization of quantity in quantum physics – An analogy of the Feynman formalism – Feynman and many-world interpretation – The psi-function of various objects – Countable and uncountable basis – Generalized continuum and arithmetization – Field and entanglement – Function as coding – The idea of „curved“ Descartes product – The environment of a function – Another view to the notion of velocity-probability – Reality and description – Hilbert space as a model both of object and description – The notion of holistic logic – Physical quantity as the information about it – Cross-temporal correlations – The forecasting of future – Description in separable and inseparable Hilbert space – „Forces“ or „miracles“ – Velocity or time – The notion of non-finite set – Dasein or Dazeit – The trajectory of the whole – Ontological and onto-theological difference – An analogy of the Feynman and many-world interpretation − psi-function as physical quantity – Things in the world and instances in time – The generation of the physi-cal by mathematical – The generalized notion of observer – Subjective or objective probability – Energy as the change of probability per the unite of time – The generalized principle of least action from a new view-point – The exception of two dimensions and Fermat’s last theorem. (shrink)

The recent development of the research field of Computing and Philosophy has triggered investigations into the theoretical foundations of computing and information. This thesis consists of two parts which are the result of studies in two areas of Philosophy of Computing and Philosophy of Information regarding the production of meaning and the value system with applications. The first part develops a unified dual-aspect theory of information and computation, in which information is characterized as structure, and computation is the information dynamics. (...) This enables naturalization of epistemology, based on interactive information representation and communication. In the study of systems modeling, meaning, truth and agency are discussed within the framework of the PI/PC unification. The second part of the thesis addresses the necessity of ethical judgment in rational agency illustrated by the problem of information privacy and surveillance in the networked society. The value grounds and socio-technological solutions for securing trustworthiness of computing are analyzed. Privacy issues clearly show the need for computing professionals to contribute to understanding of the technological mechanisms of Information and Communication Technology. The main original contribution of this thesis is the unified dual-aspect theory of computation/information. Semantics of information is seen as a part of the data-information-knowledge structuring, in which complex structures are self-organized by the computational processing of information. Within the unified model, complexity is a result of computational processes on informational structures. The thesis argues for the necessity of computing beyond the Turing-Church limit, motivated by natural computation, and wider by pancomputationalism and paninformationalism, seen as two complementary views of the same physical reality. Moreover, it follows that pancomputationalism does not depend on the assumption that the physical world on some basic level is digital. Contrary to many believes it is entirely compatible with dual quantum-mechanical computing. (shrink)

This book provides an interdisciplinary approach to one of the most fascinating and important open questions in science: What is quantum mechanics really talking about? In the last decades quantum mechanics has given rise to a new quantum technological era, a revolution taking place today especially within the field of quantum information processing; which goes from quantum teleportation and cryptography to quantum computation. Quantum theory is probably our best confirmed physical theory. However, in (...) spite of its great empirical effectiveness it stands today still without a universally accepted physical representation that allows us to understand its relation to the world and reality. The novelty of the book comes from the multiple perspectives put forward by top researchers in quantum mechanics, from Europe as well as North and South America, discussing the meaning and structure of the theory of quanta. The book comprises in a balanced manner physical, philosophical, logical and mathematical approaches to quantum mechanics and quantum information. Going from quantum superpositions and entanglement to dynamics and the problem of identity; from quantum logic, computation and quasi-set theory to the category approach and teleportation; from realism and empiricism to operationalism and instrumentalism; the book considers from different angles some of the most intriguing questions in the field. From Buenos Aires to Brussels and Cagliari, from Florence to Florianópolis, the interaction between different groups is reflected in the many different articles. This book is interesting not only to the specialists but also to the general public attempting to get a grasp on some of the most fundamental questions of present quantum physics. (shrink)

The paper addresses Leon Hen.kin's proposition as a " lighthouse", which can elucidate a vast territory of knowledge uniformly: logic, set theory, information theory, and quantum mechanics: Two strategies to infinity are equally relevant for it is as universal and t hus complete as open and thus incomplete. Henkin's, Godel's, Robert Jeroslow's, and Hartley Rogers' proposition are reformulated so that both completeness and incompleteness to be unified and thus reduced as a joint property of infinity and of all infinite (...) sets. However, only Henkin's proposition equivalent to an internal position to infinity is consistent . This can be retraced back to set theory and its axioms, where that of choice is a key. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that some essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, these phenomena can be elucidated as both complete and incomplete, after which choice is the border between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or " observer'' , or "user" to infinity implied by Henkin's proposition as the only consistent ones as to infinity. An essential area of contemporary knowledge may be synthesized from a single viewpoint. (shrink)

Making Sense of Inner Sense 'Terra cognita' is terra incognita. It is difficult to find someone not taken abackand fascinated by the incomprehensible but indisputable fact: there are material systems which are aware of themselves. Consciousness is self-cognizing code. During homo sapiens's relentness and often frustrated search for self-understanding various theories of consciousness have been and continue to be proposed. However, it remains unclear whether and at what level the problems of consciousness and intelligent thought can be resolved. Science's greatest (...) challenge is to answer the fundamental question: what precisely does a cognitive state amount to in physical terms? Albert Einstein insisted that the fundamental ideas of science are essentially simple and can be expressed in a language comprehensible to everyone. When one thinks about the complexities which present themselves in modern physics and even more so in the physics of life, one may wonder whether Einstein really meant what he said. Are we to consider the fundamental problem of the mind, whose understanding seems to lie outside the limits of the mind, to be essentially simple too? Knowledge is neither automatic nor universally deductive. Great new ideas are typically counterintuitive and outrageous, and connecting them by simple logical steps to existing knowledge is often a hard undertaking. The notion of a tensor was needed to provide the general theory of relativity; the notion of entropy had to be developed before we could get full insight into the laws of thermodynamics; the notice of information bit is crucial for communication theory, just as the concept of a Turing machine is instrumental in the deep understanding of a computer. To understand something, consciousness must reach an adequate intellectual level, even more so in order to understand itself. Reality is full of unending mysteries, the true explanation of which requires very technical knowledge, often involving notions not given directly to intuition. Even though the entire content and the results of this study are contained in the eight pages of the mathematical abstract, it would be unrealistic and impractical to suggest that anyone can gain full insight into the theory that presented here after just reading abstract. In our quest for knowledge we are exploring the remotest areas of the macrocosm and probing the invisible particles of the microcosm, from tiny neutrinos and strange quarks to black holes and the Big Bang. But the greatest mystery is very close to home: the greatest mystery is human consciousness. The question before us is whether the logical brain has evolved to a conceptual level where it is able to understand itself. (shrink)

We argue that the intractable part of the measurement problem -- the 'big' measurement problem -- is a pseudo-problem that depends for its legitimacy on the acceptance of two dogmas. The first dogma is John Bell's assertion that measurement should never be introduced as a primitive process in a fundamental mechanical theory like classical or quantum mechanics, but should always be open to a complete analysis, in principle, of how the individual outcomes come about dynamically. The second dogma is (...) the view that the quantum state has an ontological significance analogous to the significance of the classical state as the 'truthmaker' for propositions about the occurrence and non-occurrence of events, i.e., that the quantum state is a representation of physical reality. We show how both dogmas can be rejected in a realist information-theoretic interpretation of quantum mechanics as an alternative to the Everett interpretation. The Everettian, too, regards the 'big' measurement problem as a pseudo-problem, because the Everettian rejects the assumption that measurements have definite outcomes, in the sense that one particular outcome, as opposed to other possible outcomes, actually occurs in a quantum measurement process. By contrast with the Everettians, we accept that measurements have definite outcomes. By contrast with the Bohmians and the GRW 'collapse' theorists who add structure to the theory and propose dynamical solutions to the 'big' measurement problem, we take the problem to arise from the failure to see the significance of Hilbert space as a new kinematic framework for the physics of an indeterministic universe, in the sense that Hilbert space imposes kinematic objective probabilistic constraints on correlations between events. (shrink)

Quantum Information Theory and the Foundations of Quantum Mechanics is a conceptual analysis of one of the most prominent and exciting new areas of physics, providing the first full-length philosophical treatment of quantum information theory and the questions it raises for our understanding of the quantum world. -/- Beginning from a careful, revisionary, analysis of the concepts of information in the everyday and classical information-theory settings, Christopher G. Timpson argues for an ontologically deflationary account of the (...) nature of quantum information. Against what many have supposed, quantum information can be clearly defined (it is not a primitive or vague notion) but it is not part of the material contents of the world. Timpson's account sheds light on the nature of nonlocality and information flow in the presence of entanglement and, in particular, dissolves puzzles surrounding the remarkable process of quantum teleportation. In addition it permits a clear view of what the ontological and methodological lessons provided by quantum information theory are; lessons which bear on the gripping question of what role a concept like information has to play in fundamental physics. Topics discussed include the slogan 'Information is Physical', the prospects for an informational immaterialism (the view that information rather than matter might fundamentally constitute the world), and the status of the Church-Turing hypothesis in light of quantum computation. -/- With a clear grasp of the concept of information in hand, Timpson turns his attention to the pressing question of whether advances in quantum information theory pave the way for the resolution of the traditional conceptual problems of quantum mechanics: the deep problems which loom over measurement, nonlocality and the general nature of quantum ontology. He marks out a number of common pitfalls to be avoided before analysing in detail some concrete proposals, including the radical quantum Bayesian programme of Caves, Fuchs, and Schack. One central moral which is drawn is that, for all the interest that the quantum information-inspired approaches hold, no cheap resolutions to the traditional problems of quantum mechanics are to be had. (shrink)

Two strategies to infinity are equally relevant for it is as universal and thus complete as open and thus incomplete. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, thеse phenomena can be elucidated as both complete and incomplete, after which choice is the (...) border between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or “observer”, or “user” to infinity implied by Henkin’s proposition as the only consistent ones as to infinity. (shrink)

Summary Proposed by Einstein, Podolsky, and Rosen (EPR) in 1935, the entangled state has played a central part in exploring the foundation of quantum mechanics. At the end of the twentieth century, however, some physicists and mathematicians set aside the epistemological debates associated with EPR and turned it from a philosophical puzzle into practical resources for information processing. This paper examines the origin of what is known as quantum information. Scientists had considered making quantum computers and employing (...) entanglement in communications for a long time. But the real breakthrough only occurred in the 1980s when they shifted focus from general-purpose systems such as Turing machines to algorithms and protocols that solved particular problems, including quantum factorization, quantum search, superdense code, and teleportation. Key to their development was two groups of mathematical manipulations and deformations of entanglement—quantum parallelism and ‘feedback EPR’—that served as conceptual templates. The early success of quantum parallelism and feedback EPR was owed to the idealized formalism of entanglement researchers had prepared for philosophical discussions. Yet, such idealization is difficult to hold when the physical implementation of quantum information processors is at stake. A major challenge for today's quantum information scientists and engineers is thus to move from Einstein et al.'s well-defined scenarios into realistic models. (shrink)

As a compact representation of joint probability distributions over a dependence graph of random variables, and a tool for modelling and reasoning in the presence of uncertainty, Bayesian networks are of great importance for artificial intelligence to combine domain knowledge, capture causal relationships, or learn from incomplete datasets. Known as a NP-hard problem in a classical setting, Bayesian inference pops up as a class of algorithms worth to explore in a quantum framework. This paper explores such a research direction (...) and improves on previous proposals by a judicious use of the utility function in an entangled configuration. It proposes a completely quantum mechanical decision-making process with a proven computational advantage. A prototype implementation in Qiskit (a Python-based program development kit for the IBM Q machine) is discussed as a proof-of-concept. (shrink)

Accelerating Turing machines have attracted much attention in the last decade or so. They have been described as “the work-horse of hypercomputation” (Potgieter and Rosinger 2010: 853). But do they really compute beyond the “Turing limit”—e.g., compute the halting function? We argue that the answer depends on what you mean by an accelerating Turing machine, on what you mean by computation, and even on what you mean by a Turing machine. We show first (...) that in the current literature the term “accelerating Turing machine” is used to refer to two very different species of accelerating machine, which we call end-stage-in and end-stage-out machines, respectively. We argue that end-stage-in accelerating machines are not Turing machines at all. We then present two differing conceptions of computation, the internal and the external, and introduce the notion of an epistemic embedding of a computation. We argue that no accelerating Turing machine computes the halting function in the internal sense. Finally, we distinguish between two very different conceptions of the Turing machine, the purist conception and the realist conception; and we argue that Turing himself was no subscriber to the purist conception. We conclude that under the realist conception, but not under the purist conception, an accelerating Turing machine is able to compute the halting function in the external sense. We adopt a relatively informal approach throughout, since we take the key issues to be philosophical rather than mathematical. (shrink)

Accelerating Turing machines are Turing machines of a sort able to perform tasks that are commonly regarded as impossible for Turing machines. For example, they can determine whether or not the decimal representation of contains n consecutive 7s, for any n; solve the Turing-machine halting problem; and decide the predicate calculus. Are accelerating Turing machines, then, logically impossible devices? I argue that they are not. There are implications concerning the nature of effective procedures (...) and the theoretical limits of computability. Contrary to a recent paper by Bringsjord, Bello and Ferrucci, however, the concept of an accelerating Turing machine cannot be used to shove up Searle's Chinese room argument. (shrink)

The papers collected in this volume exemplify some of the trends in current approaches to logic, language and computation. Written by authors with varied academic backgrounds, the contributions are intended for an interdisciplinary audience. The first part of this volume addresses issues relevant for multi-agent systems: reasoning with incomplete information, reasoning about knowledge and beliefs, and reasoning about games. Proofs as formal objects form the subject of Part II. Topics covered include: contributions on logical frameworks, linear logic, and different approaches (...) to formalized reasoning. Part III focuses on representations and formal methods in linguistic theory, addressing the areas of comparative and temporal expressions, modal subordination, and compositionality. (shrink)

Arthur Clark and Michael Kube–McDowell (“The Triger”, 2000) suggested the sci-fi idea about the direct transformation from a chemical substance to another by the action of a newly physical, “Trigger” field. Karl Brohier, a Nobel Prize winner, who is a dramatic persona in the novel, elaborates a new theory, re-reading and re-writing Pauling’s “The Nature of the Chemical Bond”; according to Brohier: “Information organizes and differentiates energy. It regularizes and stabilizes matter. Information propagates through matter-energy and mediates the interactions of (...) matter-energy.” Dr Horton, his collaborator in the novel replies: “If the universe consists of energy and information, then the Trigger somehow alters the information envelope of certain substances –“. “Alters it, scrambles it, overwhelms it, destabilizes it” Brohier adds. There is a scientific debate whether or how far chemistry is fundamentally reducible to quantum mechanics. Nevertheless, the fact that many essential chemical properties and reactions are at least partly representable in terms of quantum mechanics is doubtless. For the quantum mechanics itself has been reformulated as a theory of a special kind of information, quantum information, chemistry might be in turn interpreted in the same terms. Wave function, the fundamental concept of quantum mechanics, can be equivalently defined as a series of qubits, eventually infinite. A qubit, being defined as the normed superposition of the two orthogonal subspaces of the complex Hilbert space, can be interpreted as a generalization of the standard bit of information as to infinite sets or series. All “forces” in the Standard model, which are furthermore essential for chemical transformations, are groups [U(1),SU(2),SU(3)] of the transformations of the complex Hilbert space and thus, of series of qubits. One can suggest that any chemical substances and changes are fundamentally representable as quantum information and its transformations. If entanglement is interpreted as a physical field, though any group above seems to be unattachable to it, it might be identified as the “Triger field”. It might cause a direct transformation of any chemical substance by from a remote distance. Is this possible in principle? (shrink)

A central problem in the interpretation of non-relativistic quantum mechanics is to relate the conceptual structure of the theory to the classical idea of the state of a physical system. This paper approaches the problem by presenting an analysis of the notion of an elementary physical proposition. The notion is shown to be realized in standard formulations of the theory and to illuminate the significance of proofs of the impossibility of hidden variable extensions. In the interpretation of quantum (...) mechanics that emerges from this analysis, the philosophically distinctive features of the theory derive from the fact that it seeks to represent a reality of which complete knowledge is essentially unattainable. (shrink)

The nature of quantum computation is discussed. It is argued that, in terms of the amount of information manipulated in a given time, quantum and classical computation are equally efficient. Quantum superposition does not permit quantum computers to ''perform many computations simultaneously'' except in a highly qualified and to some extent misleading sense. Quantum computation is therefore not well described by interpretations of quantum mechanics which invoke the concept of vast numbers of parallel universes. (...) Rather, entanglement makes available types of computation processes which, while not exponentially larger than classical ones, are unavailable to classical systems. The essence of quantum computation is that it uses entanglement to generate and manipulate a physical representation of the correlations between logical entities, without the need to completely represent the logical entities themselves. (shrink)

The bookModeling Reality covers a wide range of fascinating subjects, accessible to anyone who wants to learn about the use of computer modeling to solve a diverse range of problems, but who does not possess a specialized training in mathematics or computer science. The material presented is pitched at the level of high-school graduates, even though it covers some advanced topics (cellular automata, Shannon's measure of information, deterministic chaos, fractals, game theory, neural networks, genetic algorithms, and Turing machines). These (...) advanced topics are explained in terms of well known simple concepts: Cellular automata - Game of Life, Shannon's formula - Game of twenty questions, Game theory - Television quiz, etc. The book is unique in explaining in a straightforward, yet complete, fashion many important ideas, related to various models of reality and their applications. Twenty-five programs, written especially for this book, are provided on an accompanying CD. They greatly enhance its pedagogical value and make learning of even the more complex topics an enjoyable pleasure. (shrink)

This book provides a general survey of the main concepts, questions and results that have been developed in the recent interactions between quantum information, quantum computation and logic. Divided into 10 chapters, the books starts with an introduction of the main concepts of the quantum-theoretic formalism used in quantum information. It then gives a synthetic presentation of the main “mathematical characters” of the quantum computational game: qubits, quregisters, mixtures of quregisters, quantum logical gates. Next, (...) the book investigates the puzzling entanglement-phenomena and logically analyses the Einstein–Podolsky–Rosen paradox and introduces the reader to quantum computational logics, and new forms of quantum logic. The middle chapters investigate the possibility of a quantum computational semantics for a language that can express sentences like “Alice knows that everybody knows that she is pretty”, explore the mathematical concept of quantum Turing machine, and illustrate some characteristic examples that arise in the framework of musical languages. The book concludes with an analysis of recent discussions, and contains a Mathematical Appendix which is a survey of the definitions of all main mathematical concepts used in the book. (shrink)

The old Bohr–Einstein debate about the completeness of quantum mechanics (QM) was held on an ontological ground. The completeness problem becomes more tractable, however, if it is preliminarily discussed from a semantic viewpoint. Indeed every physical theory adopts, explicitly or not, a truth theory for its observative language, in terms of which the notions of semantic objectivity and semantic completeness of the physical theory can be introduced and inquired. In particular, standard QM adopts a verificationist theory of truth that (...) implies its semantic nonobjectivity; moreover, we show in this paper that standard QM is semantically complete, which matches Bohr's thesis. On the other hand, one of the authors has provided a Semantic Realism (or SR) interpretation of QM that adopts a Tarskian theory of truth as correspondence for the observative language of QM (which was previously mantained to be impossible); according to this interpretation QM is semantically objective, yet incomplete, which matches EPR's thesis. Thus, standard QM and the SR interpretation of QM come to opposite conclusions. These can be reconciled within an integrationist perspective that interpretes non-Tarskian theories of truth as theories of metalinguistic concepts different from truth. (shrink)

Some regularities enjoy only an attenuated existence in a body of training data. These are regularities whose statistical visibility depends on some systematic recoding of the data. The space of possible recodings is, however, infinitely large – it is the space of applicable Turing machines. As a result, mappings that pivot on such attenuated regularities cannot, in general, be found by brute-force search. The class of problems that present such mappings we call the class of “type-2 problems.” Type-1 problems, (...) by contrast, present tractable problems of search insofar as the relevant regularities can be found by sampling the input data as originally coded. Type-2 problems, we suggest, present neither rare nor pathological cases. They are rife in biologically realistic settings and in domains ranging from simple animat behaviors to language acquisition. Not only are such problems rife – they are standardly solved! This presents a puzzle. How, given the statistical intractability of these type-2 cases, does nature turn the trick? One answer, which we do not pursue, is to suppose that evolution gifts us with exactly the right set of recoding biases so as to reduce specific type-2 problems to type-1 mappings. Such a heavy-duty nativism is no doubt sometimes plausible. But we believe there are other, more general mechanisms also at work. Such mechanisms provide general strategies for managing problems of type-2 complexity. Several such mechanisms are investigated. At the heart of each is a fundamental ploy – namely, the maximal exploitation of states of representation already achieved by prior, simpler learning so as to reduce the amount of subsequent computational search. Such exploitation both characterizes and helps make unitary sense of a diverse range of mechanisms. These include simple incremental learning, modular connectionism, and the developmental hypothesis of “representational redescription”. In addition, the most distinctive features of human cognition – language and culture – may themselves be viewed as adaptations enabling this representation/computation trade-off to be pursued on an even grander scale. (shrink)

The usual representation of quantum algorithms, limited to the process of solving the problem, is physically incomplete. We complete it in three steps: extending the representation to the process of setting the problem, relativizing the extended representation to the problem solver to whom the problem setting must be concealed, and symmetrizing the relativized representation for time reversal to represent the reversibility of the underlying physical process. The third steps projects the input state of the representation, where the problem solver (...) is completely ignorant of the setting and thus the solution of the problem, on one where she knows half solution. Completing the physical representation shows that the number of computation steps required to solve any oracle problem in an optimal quantum way should be that of a classical algorithm endowed with the advanced knowledge of half solution. (shrink)

This paper argues that ordinary object languages for fundamental physics are incomplete, essentially because they are extensional, and consequently lack any adequate formal representation of contingency. It is shown that it is impossible to formulate adequate deduction systems for general transformations in such languages. This is argued in detail for the time reversal transformation. Two important controversies about the application of time reversal in quantum mechanics are summarized at the start, to provide the context of this problem, and show (...) its serious implications, but the aim here is not to solve these problems. The flaw is not special to quantum mechanics: it is a general feature traced to extensionality, and demonstrated through a simple example in classical physics. It is proposed that this defect can be overcome by extending to an intensional semantics, but this involves extending the usual formalism of physics. A detailed proposal for such an extension is given in Part 2. (shrink)

The purpose of this book is to initiate a new discipline, namely a formalized epistemological method drawn from the cognitive strategies practised in the most effective among the modern scientific disciplines, as well as from general philosophical thinking. Indeed, what is lacking in order to improve our knowledge and our domination of the modes which nowadays are available for the generation and communication of knowledge, thoroughly and rapidly and with precision and detail? It is a systematic explication of the epistemological (...) essence encrypted in the specialized languages and algorithms of the major modern scientific approaches, a systematic cross-referencing of the explicated results, and a final elaboration of a new coherent whole. Quantum mechanics, like a diver, can take us down to the level of the very first actions of our conceptualization of reality. And starting from there, it can induce an explicit understanding of certain fundamental features of the new scientific thinking. A formalized epistemology should not be mistaken for a crossdisciplinary or a multidisciplinary project. The latter projects are designed to offer to nonspecialists access to information, to results obtained inside specialized disciplines, as well as a certain understanding of these results; whereas a formalized epistemology should equip anyone with a framework for conceptualizing himself in whatever domain and direction he or she might choose. A formalized epistemology should not be mistaken either for an approach belonging to the modern cognitive sciences. These try to establish as neutrally as possible descriptions of how the human body-and-mind work spontaneously when knowledge is generated; whereas a method of conceptualization should establish what conceptual-operational deliberate procedures have to be applied in order to represent and to achieve processes of generation of knowledge optimized accordingly to any definite aims. This book addresses philosophers of science, physicists, mathematicians, logicians, computer scientists, researchers in cognitive sciences, and biologists, as well as any intellectual who is interested in scientific and philosophical thinking. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Emergent Spacetime, the Megastructure Problem, and the Metaphysics of the SelfSusan Schneider (bio)The aim of this article is to introduce new thoughts on some pressing topics relating to my book, Artificial You, ranging from the fundamental nature of reality to quantum theory and emergence in large language models (LLM) like GPT-4. Since Artificial You was published, the innovations in the domain of AI chatbots like GPT-4 have been (...) rapid-fire, and at times dizzying. The thoughtful discussion in this symposium offers an important opportunity for cross-cultural engagement on these issues and more. I am grateful for the opportunity to learn from the insights of these authors and editors. [End Page 314]The order of my comments will be as follows: in the context of considering Christian Coseru's intriguing response (section 1), I begin by discussing the nature of the self, which I use as a springboard into larger, vexing issues about the nature of fundamental reality, given work on emergent spacetime (ES) (section 2). I offer a "Prototime Interpretation" of quantum mechanics in which spacetime emerges from a prototemporal arena, and use this idea to explore the nature of the self and mind.Using the assumption of ES, I devise an approach to metaphysically fundamental objects (which I call "quantum objects") based on quantum entanglement and the bundle theory of substance. I illustrate how the self or mind, insofar as it is a substance that supervenes on quantum objects, could be partly or fully determined by the properties of the quantum objects on which it supervenes. Intriguingly, this would mean that the self or mind is not located in any singular point in space at a given time. I also explain how the Prototime Interpretation of quantum mechanics is compatible with change and causation at the fundamental level.In section 3 I turn to Inês Hipólito's interesting article, discussing the import of not underestimating AI autonomy in light of the increasingly sophisticated large-language models (LLMs).1 I raise the "AI Megastructure Problem," a new problem of AI control arising in the context of the incipient AGI systems that are now populating the AI ecosystem. The problem, which arose from a collaboration with Kyle Kilian, considers the possibility of emergent properties from the interaction of the LLMs and other AI's in the AI ecosystem (Schneider and Kilian 2023). Then, turning to Noam Cook's intriguing discussion of Japanese philosophy, I consider the pressing issue of testing for machine consciousness, devising a modified version of my AI consciousness test (ACT), originally developed with astrophysicist Edwin Turner. I add an "interpretability component" to the ACT to deal with the case of LLMs (section 4).I. Patternism and the Biological Process ViewChristian Coseru's intriguing piece focuses on the chapters on personal identity in Artificial You, proposing that when it comes to the nature of the self, one should adopt "a biologically informed process-ontology perspective on personal identity," which "means recognizing that organic identity is different in kind: it is an identity of form rather than of matter."His position is akin to the patternist views that I argue against in Artificial You in the sense that it embraces the idea that form, rather than the particular matter that comprises one's brain and body, is key to one's survival over time. But there are important differences as well. For transhumanists tend to believe that the self can survive radical changes in one's substrate, transitioning from thinking with a biological brain to computing with a network of microchips, for instance. Coseru departs from this, writing: [End Page 315]As organisms, persons endure by undergoing constant change on a fundamentally cellular level, including the change in the neural pathways that realize the cognitive processes of thinking and expressing these very thoughts right now.Furthermore:[I]f persons are processes rather than things, then they cannot be cognitively enhanced by augmenting, replacing or transferring their functioning parts (e.g., brain cells).I will call this position the "Biological Process View" (BPV). I have several questions about it. First, how does this position solve one of the key problems I raised for... (shrink)

In his article “On Mechanical Recognition” R. J. Nelson brings to bear a branch of mathematical logic called automata theory on problems of artificial intelligence. Specifically he attacks the anti-mechanist claim that “[i]nasmuch as human recognition to a very great extent relies on context and on the ability to grasp wholes with some independence of the quality of the parts, even to fill in the missing parts on the basis of expectations, it follows that computers cannot in principle be programmed (...) to recognize or learn to recognize all patterns”, p. 24). Nelson proposes, contrary to this claim, that “gestalt recognition is not beyond digital automata”. in particular, he claims, he will establish by what he calls “Turing machine arguments” that the following four theses are true: automata can recognize different pattern types in one and the same set of instances;automata can recognize the “same” pattern in different sets of instances;automata can recognize incomplete, degraded patterns having missing or indeterminate parts;automata can recognize family resemblances. (shrink)

There exists a growing literature on the so-called physical Church-Turing thesis in a relativistic spacetime setting. The physical Church-Turing thesis is the conjecture that no computing device that is physically realizable can exceed the computational barriers of a Turing machine. By suggesting a concrete implementation of a beyond-Turing computer in a spacetime setting, Istvan Nemeti and Gyula David have shown how an appreciation of the physical Church-Turing thesis necessitates the confluence of mathematical, computational, physical, (...) and indeed cosmological ideas. In this essay, I will honour Istvan's seventieth birthday, as well as his longstanding interest in, and his seminal contributions to, this field going back to as early as 1987 by modestly proposing how the concrete implementation in Nemeti and D\'avid might be complemented by a quantum-information-theoretic communication protocol between the computing device and the logician who sets the beyond-Turing computer a task such as determining the consistency of Zermelo-Fraenkel set theory. This suggests that even the foundations of quantum theory and, ultimately, quantum gravity may play an important role in determining the validity of the physical Church-Turing thesis. (shrink)

Alan Turing anticipated many areas of current research incomputer and cognitive science. This article outlines his contributionsto Artificial Intelligence, connectionism, hypercomputation, andArtificial Life, and also describes Turing's pioneering role in thedevelopment of electronic stored-program digital computers. It locatesthe origins of Artificial Intelligence in postwar Britain. It examinesthe intellectual connections between the work of Turing and ofWittgenstein in respect of their views on cognition, on machineintelligence, and on the relation between provability and truth. Wecriticise widespread and influential misunderstandings (...) of theChurch–Turing thesis and of the halting theorem. We also explore theidea of hypercomputation, outlining a number of notional machines thatcompute the uncomputable. (shrink)

Throughout what is now the more than 50-year history of the computer many theories have been advanced regarding the contribution this machine would make to changes both in the structure of society and in ways of thinking. Like other theories regarding the future, these should also be taken with a pinch of salt. The history of the development of computer technology contains many predictions which have failed to come true and many applications that have not been foreseen. While we (...) must reserve judgment as to the question of the impact on the structure of society and human thought, there is no reason to wait for history when it comes to the question: what are the properties that could give the computer such far-reaching importance? The present book is intended as an answer to this question. The fact that this is a theoretical analysis is due to the nature of the subject. No other possibilities are available because such a description of the properties of the computer must be valid for any kind of application. An additional demand is that the description should be capable of providing an account of the properties which permit and limit these possible applications, just as it must make it possible to characterize a computer as distinct from a) other machines whether clocks, steam engines, thermostats, or mechanical and automatic calculating machines, b) other symbolic media whether printed, mechanical, or electronic and c) other symbolic languages whether ordinary languages, spoken or written or formal languages. This triple limitation, however, (with regard to other machines, symbolic media and symbolic languages) raises a theoretical question as it implies a meeting between concepts of mechanical-deterministic systems, which stem from mathematical physics, and concepts of symbolic systems which stem from the description of symbolic activities common to the humanities. The relationship between science and the humanities has traditionally been seen from a dualistic perspective, as a relationship between two clearly separate subject areas, each studied on its own set of premises and using its own methods. In the present case, however, this perspective cannot be maintained since there is both a common subject area and a new - and specific - kind of interaction between physical and symbolic processes. (shrink)

In this paper we explore the possibility of giving a justification of the “semantic information” content and measure, in the framework of the recent coalgebraic approach to quantum systems and quantum computation, extended to QFT systems. In QFT, indeed, any quantum system has to be considered as an “open” system, because it is always interacting with the background fluctuations of the quantum vacuum. Namely, the Hamiltonian in QFT always includes the quantum system and its inseparable (...) thermal bath, formally “entangled” like an algebra with its coalgebra, according to the principle of the “doubling” of the degrees of freedom between them. This is the core of the representation theory of cognitive neuroscience based on QFT. Moreover, in QFT, the probabilities of the quantum states follow a Wigner distribution, based on the notion and measure of quasiprobability, where regions integrated under given expectation values do not represent mutually exclusive states. This means that a computing agent, either natural or artificial, in QFT, against the quantum Turing machine paradigm, is able to change dynamically the representation space of its computations. This depends on the possibility of interpreting QFT system computations within the framework of category theory logic and its principle of duality between opposed categories, such as the algebra and coalgebra categories of QFT. This allows us to justify and not only to suppose, like in the “theory of strong semantic information” of L. Floridi, the definition of modal “local truth” and the notion of semantic information as a measure of it, despite both measures being defined on quasiprobability distributions. (shrink)

The cognition of quantum processes raises a series of questions about ordering and information connecting the states of one and the same system before and after measurement: Quantum measurement, quantum in-variance and the non-locality of quantum information are considered in the paper from an epistemological viewpoint. The adequate generalization of ‘measurement’ is discussed to involve the discrepancy, due to the fundamental Planck constant, between any quantum coherent state and its statistical representation as a statistical ensemble (...) after measurement. Quantum in-variance designates the relation of any quantum coherent state to the corresponding statistical ensemble of measured results. A set-theory corollary is the curious in-variance to the axiom of choice: Any coherent state excludes any well-ordering and thus excludes also the axiom of choice. However the above equivalence requires it to be equated to a well-ordered set after measurement and thus requires the axiom of choice for it to be able to be obtained. Quantum in-variance underlies quantum information and reveals it as the relation of an unordered quantum “much” (i.e. a coherent state) and a well-ordered “many” of the measured results (i.e. a statistical ensemble). It opens up to a new horizon, in which all physical processes and phenomena can be interpreted as quantum computations realizing relevant operations and algorithms on quantum information. All phenomena of entanglement can be described in terms of the so defined quantum information. Quantum in-variance elucidates the link between general relativity and quantum mechanics and thus, the problem of quantum gravity. The non-locality of quantum information unifies the exact position of any space-time point of a smooth trajectory and the common possibility of all space-time points due to a quantum leap. This is deduced from quantum in-variance. Epistemology involves the relation of ordering and thus a generalized kind of information, quantum one, to explain the special features of the cognition in quantum mechanics. (shrink)

One of the most tantalizing questions about the interpretation of Quantum Theory is the objective vs. subjective meaning of quantum states. Here, by focusing on a typical EPR experiment upon which a selection procedure is performed on one side, we will confront the fully epistemic view of quantum states with its results. Our statement is that such a view cannot be considered complete, although the opposite attitude would also pose well-known problems of interpretation.

We explore the possibility of using quantum mechanical principles for hypercomputation through the consideration of a quantum algorithm for computing the Turing halting problem. The mathematical noncomputability is compensated by the measurability of the values of quantum observables and of the probability distributions for these values. Some previous no-go claims against quantum hypercomputation are then reviewed in the light of this new positive proposal.

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and show why it is helpful to (...) consider the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. (shrink)

This conference was devoted to the 80 years of the Copenhagen Interpretation, and to the question of the relevance of the Copenhagen interpretation for the present understanding of quantum mechanics. It is in this framework that fundamental questions raised by quantum mechanics, especially in information theory, were discussed throughout the conference. As has become customary in our series of conference in Växjö, we were glad to welcome a fruitful assembly of theoretical physicists, experimentalists, mathematicians and even philosophers interested (...) in the foundations of probability and physics. The nature of quantum fluctuations---in the form of Stochastic Electrodynamics or in other approaches to stochastic quantum mechanics---was also a central topic discussed during the conference, especially during debates. We should also mention talks on the completeness or incompleteness of quantum mechanics; on macroscopic quantum systems; on Bell's inequality, entanglement and experiments on quantum nonlocality (and locality); on Bohmian mechanics; on the connection between quantum mechanics and general relativity; on quantum probability; on quantum computing, quantum teleportation and quantum cryptography technologies; and more generally on the mathematical formalism of quantum mechanics and on the philosophical problems raised by its interpretations. (shrink)

It has been argued that neural networks and other forms of analog computation may transcend the limits of Turing-machine computation; proofs have been offered on both sides, subject to differing assumptions. In this article I argue that the important comparisons between the two models of computation are not so much mathematical as epistemological. The Turing-machine model makes assumptions about information representation and processing that are badly matched to the realities of natural computation (information representation and processing (...) in or inspired by natural systems). This points to the need for new models of computation addressing issues orthogonal to those that have occupied the traditional theory of computation. (shrink)

Aiming to unravel the mystery of quantum mechanics, this book is concerned with questions about action-at-a-distance, holism, and whether quantum mechanics gives a complete account of microphysical reality. With rigorous arguments and clear thinking, the author provides an introduction to the philosophy of physics.

In 1935/1936 Kurt Gödel wrote three notebooks on the foundations of quantum mechanics, which have now been entirely transcribed for the first time. Whereas a lot of the material is rather technical in character, many of Gödel's remarks have a philosophical background and concentrate on Leibnizian monadology as well as on vitalism. Obviously influenced by the vitalistic writings of Hans Driesch and his ‘proofs’ for the existence of an entelechy in every living organism, Gödel briefly develops the idea of (...) a computing machine which closely resembles Turing's groundbreaking conception. After introducing the notebooks on quantum mechanics, this article describes Gödel's vitalistic Weltbild and the ideas leading to the development of his computing machine. It investigates a notion of lawlike sequence which closely resembles Turing's concept of a computable number and which Gödel himself calls ‘problematic’, and compares it to the opposed concept of randomness, drawing upon the notion of program-size complexity. Finally, Gödel's machine is implemented in a dialect of the Lisp programing language. (shrink)

Information is a basic structure of the world, while computation is a process of the dynamic change of information. This book provides a cutting-edge view of world's leading authorities in fields where information and computation play a central role. It sketches the contours of the future landscape for the development of our understanding of information and computation, their mutual relationship and the role in cognition, informatics, biology, artificial intelligence, and information technology. -/- This book is an utterly enjoyable and engaging (...) read which gives readers an opportunity to understand and relate phenomena seemingly unrelated in a completely new light — especially the connections between information, computation, cognition and life. -/- Contents: Cybersemiotics and the Question of Knowledge (Søren Brier) Information Dynamics in a Categorical Setting (Mark Burgin) Mathematics as a Biological Process (G J Chaitin) Information, Causation and Computation (John Collier) From Descartes to Turing: The Computational Content of Supervenience (S Barry Cooper) A Dialogue Concerning Two World Systems: Info-Computational vs Mechanistic (Gordana Dodig-Crnkovic & Vincent C Müller) Does Computing Embrace Self-Organization? (Wolfgang Hofkirchner) Analysis of Information and Computation in Physics Explains Cognitive Paradigms: From Full Cognition to Laplace Determinism to Statistical Determinism to Modern Approach (Vladik Kreinovich, Roberto Araiza & Juan Ferret) Bodies — Both Informed and Transformed Embodied Computation and Information Processing (Bruce J MacLennan) Computation on Information, Meaning and Representations: An Evolutionary Approach (Christophe Menant) Interior Grounding, Reflection, and Self-Consciousness (Marvin Minsky) A Molecular Dynamic Network: Minimal Properties and Evolutionary Implications (Walter Riofrio) Super-recursive Features of Evolutionary Processes and the Models for Computational Evolution (Darko Roglic) Towards a Modeling View of Computing (Oron Shagrir) What's Information, for an Organism or Intelligent Machine? How can a Machine or Organism Mean? (Aaron Sloman) Inconsistent Knowledge as a Natural Phenomenon: The Ranking of Reasonable Inferences as a Computational Approach to Naturally Inconsistent (Legal) Theories (Kees (CNJ) de Vey Mestdagh & Jaap Henk (JH) Hoepman) On the Algorithmic Nature of the World (Hector Zenil & Jean-Paul Delahaye) . (shrink)

The usual representation of quantum algorithms is limited to the process of solving the problem. We extend it to the process of setting the problem. Bob, the problem setter, selects a problem-setting by the initial measurement. Alice, the problem solver, unitarily computes the corresponding solution and reads it by the final measurement. This simple extension creates a new perspective from which to see the quantum algorithm. First, it highlights the relevance of time-symmetric quantum mechanics to quantum (...) computation: the problem-setting and problem solution, in their quantum version, constitute pre- and post-selection, hence the process as a whole is bound to be affected by both boundary conditions. Second, it forces us to enter into relational quantum mechanics. There must be a representation of the quantum algorithm with respect to Bob, and another one with respect to Alice, from whom the outcome of the initial measurement, specifying the setting and thus the solution of the problem, must be concealed. Time-symmetrizing the quantum algorithm to take into account both boundary conditions leaves the representation to Bob unaltered. It shows that the representation to Alice is a sum over histories in each of which she remains shielded from the information coming to her from the initial measurement, not from that coming to her backwards in time from the final measurement. In retrospect, all is as if she knew in advance, before performing her problem-solving action, half of the information that specifies the solution of the problem she will read in the future and could use this information to reach the solution with fewer computation steps. This elucidates the quantum computational speedup in all the quantum algorithms examined. (shrink)

Does the quantum state represent reality or our knowledge of reality? In making this distinction precise, we are led to a novel classification of hidden variable models of quantum theory. We show that representatives of each class can be found among existing constructions for two-dimensional Hilbert spaces. Our approach also provides a fruitful new perspective on arguments for the nonlocality and incompleteness of quantum theory. Specifically, we show that for models wherein the quantum state has the (...) status of something real, the failure of locality can be established through an argument considerably more straightforward than Bell’s theorem. The historical significance of this result becomes evident when one recognizes that the same reasoning is present in Einstein’s preferred argument for incompleteness, which dates back to 1935. This fact suggests that Einstein was seeking not just any completion of quantum theory, but one wherein quantum states are solely representative of our knowledge. Our hypothesis is supported by an analysis of Einstein’s attempts to clarify his views on quantum theory and the circumstance of his otherwise puzzling abandonment of an even simpler argument for incompleteness from 1927. (shrink)

Bob hides a ball in one of four drawers. Alice is to locate it. Classically she has to open up to three drawers, quantally just one. The fundamental reason for this quantum speedup is not known. The usual representation of the quantum algorithm is limited to the process of solving the problem. We extend it to the process of setting the problem. The number of the drawer with the ball becomes a unitary transformation of the random outcome of (...) the preparation measurement. This extended, time-symmetric, representation brings in relational quantum mechanics. It is with respect to Bob and any external observer and cannot be with respect to Alice. It would tell her the number of the drawer with the ball before she opens any drawer. To Alice, the projection of the quantum state due to the preparation measurement should be retarded at the end of her search; in the input state of the search, the drawer number is determined to Bob and undetermined to Alice. We show that, mathematically, one can ascribe any part of the selection of the random outcome of the preparation measurement to the final Alice’s measurement. Ascribing half of it explains the speedup of the present algorithm. This leaves the input state to Bob unaltered and projects that to Alice on a state of lower entropy where she knows half of the number of the drawer with the ball in advance. The quantum algorithm turns out to be a sum over histories in each of which Alice knows in advance that the ball is in a pair of drawers and locates it by opening one of the two. In the sample of quantum algorithms examined, the part of the random outcome of the initial measurement selected by the final measurement is one half or slightly above it. Conversely, given an oracle problem, the assumption it is one half always corresponds to an existing quantum algorithm and gives the order of magnitude of the number of oracle queries required by the optimal one. (shrink)

Many researchers determine the question “Why anything rather than nothing?” as the most ancient and fundamental philosophical problem. Furthermore, it is very close to the idea of Creation shared by religion, science, and philosophy, e.g. as the “Big Bang”, the doctrine of “first cause” or “causa sui”, the Creation in six days in the Bible, etc. Thus, the solution of quantum mechanics, being scientific in fact, can be interpreted also philosophically, and even religiously. However, only the philosophical interpretation is (...) the topic of the text. The essence of the answer of quantum mechanics is: 1. The creation is necessary in a rigorous mathematical sense. Thus, it does not need any choice, free will, subject, God, etc. to appear. The world exists in virtue of mathematical necessity, e.g. as any mathematical truth such as 2+2=4. 2. The being is less than nothing rather than more than nothing. So, the creation is not an increase of nothing, but the decrease of nothing: it is a deficiency in relation of nothing. Time and its “arrow” are the way of that diminishing or incompleteness to nothing. (shrink)

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and show why it is helpful to (...) consider the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. These, provides a new interpretation of Quantum Mechanics.More exactly what we see here is "Particle-Fuzzy time" interpretation of quantum Mechanics, in contrast to some other interpretations of Quantum Mechanics like " Wave-Particle" interpretation. At the end, we propound a question about the possible solution of a paradox in Physics, the contradiction between General Relativity and Quantum Mechanics. (shrink)

The central motivating idea behind the development of this work is the concept of prespace, a hypothetical structure that is postulated by some physicists to underlie the fabric of space or space-time. I consider how such a structure could relate to space and space-time, and the rest of reality as we know it, and the implications of the existence of this structure for quantum theory. Understanding how this structure could relate to space and to the rest of reality requires, (...) I believe, that we consider how space itself relates to reality, and how other so-called "spaces" used in physics relate to reality. In chapter 2, I compare space and space-time to other spaces used in physics, such as configuration space, phase space and Hilbert space. I support what is known as the "property view" of space, opposing both the traditional views of space and space-time, substantivalism and relationism. I argue that all these spaces are property spaces. After examining the relationships of these spaces to causality, I argue that configuration space has, due to its role in quantum mechanics, a special status in the microscopic world similar to the status of position space in the macroscopic world. In chapter 3, prespace itself is considered. One way of approaching this structure is through the comparison of the prespace structure with a computational system, in particular to a cellular automaton, in which space or space-time and all other physical quantities are broken down into discrete units. I suggest that one way open for a prespace metaphysics can be found if physics is made fully discrete in this way. I suggest as a heuristic principle that the physical laws of our world are such that the computational cost of implementing those laws on an arbitrary computational system is minimized, adapting a heuristic principle of this type proposed by Feynman. In chapter 4, some of the ideas of the previous chapters are applied in an examination of the physics and metaphysics of quantum theory. I first discuss the "measurement problem" of quantum mechanics: this problem and its proposed solution are the primary subjects of chapter 4. It turns out that considering how quantum theory could be made fully discrete leads naturally to a suggestion of how standard linear quantum mechanics could be modified to give rise to a solution to the measurement problem. The computational heuristic principle reinforces the same solution. I call the modified quantum mechanics Critical Complexity Quantum Mechanics (CCQM). I compare CCQM with some of the other proposed solutions to the measurement problem, in particular the spontaneous localization model of Ghirardi, Rimini and Weber. Finally, in chapters 5 and 6, I argue that the measure of complexity of quantum mechanical states I introduce in CCQM also provides a new definition of entropy for quantum mechanics, and suggests a solution to the problem of providing an objective foundation for statistical mechanics, thermodynamics, and the arrow of time. (shrink)

I propose to consider the question, "Can machines think?" This should begin with definitions of the meaning of the terms "machine" and "think." The definitions might be framed so as to reflect so far as possible the normal use of the words, but this attitude is dangerous, If the meaning of the words "machine" and "think" are to be found by examining how they are commonly used it is difficult to escape the conclusion that the meaning and the (...) answer to the question, "Can machines think?" is to be sought in a statistical survey such as a Gallup poll. But this is absurd. Instead of attempting such a definition I shall replace the question by another, which is closely related to it and is expressed in relatively unambiguous words. The new form of the problem can be described in terms of a game which we call the 'imitation game." It is played with three people, a man (A), a woman (B), and an interrogator (C) who may be of either sex. The interrogator stays in a room apart front the other two. The object of the game for the interrogator is to determine which of the other two is the man and which is the woman. He knows them by labels X and Y, and at the end of the game he says either "X is A and Y is B" or "X is B and Y is A." The interrogator is allowed to put questions to A and B. We now ask the question, "What will happen when a machine takes the part of A in this game?" Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman? These questions replace our original, "Can machines think?". (shrink)