Computability

Here, by introducing a version of “Unexpected hanging paradox” first we try to open a new way and a new explanation for paradoxes, similar to liar paradox. Also, we will show that we have a semantic situation which no syntactical logical system could support it. Finally, we propose a claim in Theory of Computation about the consistency of this Theory. One of the major claim is:Theory of Computation and Classical Logic leads us to a contradiction.

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)

Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”,first we attempt to open a new explanation for some paradoxes. In the second step, by applying this paradox, it (...) is demonstrated that any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction. We conclude that our mathematical frame work is inappropriate for Theory of Computation. Furthermore, the result provides us a reason that many problems in Complexity Theory resist to be solved.(This work is completed in 2017 -5- 2, it is in vixra in 2017-5-14, presented in Unilog 2018, Vichy). (shrink)

We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability.

Bedau's influential (1997) account analyzes weak emergence in terms of the non-derivability of a system’s macrostates from its microstates except by simulation. I offer an improved version of Bedau’s account of weak emergence in light of insights from information theory. Non-derivability alone does not guarantee that a system’s macrostates are weakly emergent. Rather, it is non-derivability plus the algorithmic compressibility of the system’s macrostates that makes them weakly emergent. I argue that the resulting information-theoretic picture provides a metaphysical account of (...) weak emergence rather than a merely epistemic one. (shrink)

At its core this book is concerned with logic and computation with respect to the mathematical characterization of sentient biophysical structure and its behavior. -/- Three related theories are presented: The first of these provides an explanation of how sentient individuals come to be in the world. The second describes how these individuals operate. And the third proposes a method for reasoning about the behavior of individuals in groups. -/- These theories are based upon a new explanation of experience in (...) nature, the construction of senses, and motile behavior. This new approach is developed from first principles to enable a rigorous and systematic explanation of the variety of associated intelligent behaviors. -/- Alongside this development is a further account that focuses upon the nature of our work. It discusses the existential aspects of scientific inquiry, its epistemology and logic. It seeks to clarify the nature of the mathematical characterization and computation of natural behaviors, dealing with questions in the foundations of logic. It explores methodological issues related to reduction and the refinement of ideas from intuition to formal logical structure. -/- In support of this inquiry we work toward the development of a calculus for biophysical construction and its dynamics. If successful this mechanics mathematically characterizes sensory and motile behavior. -/- Upon this foundation we propose a model of apprehension and explore how its products are processed by the organism. Finally, we develop a probabilistic theory that enables us to reason about inaccessible factors in group behavior. -/- The mechanics we propose suggests the design and physical realization of a new model of computation; one in which structure and the concurrency of action are a first-order consideration. -/- We identify opportunities for experimental verification of the theory and we suggest a proof of our results in practice by the identification of this mechanism, allowing the construction of machines that experience. (shrink)

Review by: Johanna N. Y. Franklin The Bulletin of Symbolic Logic, Volume 19, Issue 1, Page 115-118, March 2013.

Elvira Mayordomo and Wolfgang Merkle The Bulletin of Symbolic Logic, Volume 19, Issue 1, Page 135-136, March 2013.

Infinite time Turing machines represent a model of computability that extends the operations of Turing machines to transfinite ordinal time by defining the content of each cell at limit steps to be the lim sup of the sequences of previous contents of that cell. In this paper, we study a computational model obtained by replacing the lim sup rule with an ‘eventually constant’ rule: at each limit step, the value of each cell is defined if and only if the content (...) of that cell has stabilized before that limit step and is then equal to this constant value. We call these machines weak infinite time Turing machines. We study different variants of wITTMs adding multiple tapes, heads, or bidimensional tapes. We show that some of these models are equivalent to each other concerning their computational strength. We show that wITTMs decide exactly the arithmetic relations on natural numbers. (shrink)

We show that there is a minimal pair in the nonuniform generic degrees, and hence also in the uniform generic degrees. This fact contrasts with Igusa’s result that there are no minimal pairs for relative generic computability and answers a basic structural question mentioned in several papers in the area.

We study the logical and computational properties of basic theorems of uncountable mathematics, in particular Pincherle's theorem, published in 1882. This theorem states that a locally bounded function is bounded on certain domains, i.e. one of the first ‘local-to-global’ principles. It is well-known that such principles in analysis are intimately connected to (open-cover) compactness, but we nonetheless exhibit fundamental differences between compactness and Pincherle's theorem. For instance, the main question of Reverse Mathematics, namely which set existence axioms are necessary to (...) prove Pincherle's theorem, does not have an unique or unambiguous answer, in contrast to compactness. We establish similar differences for the computational properties of compactness and Pincherle's theorem. We establish the same differences for other local-to-global principles, even going back to Weierstrass. We also greatly sharpen the known computational power of compactness, for the most shared with Pincherle's theorem however. Finally, countable choice plays an important role in the previous, we therefore study this axiom together with the intimately related Lindelöf lemma. (shrink)

Yuri Matiyasevich's theorem states that the set of all Diophantine equations which have a solution in non-negative integers is not recursive. Craig Smoryński's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. Let R be a subring of Q with or without 1. By H_{10}(R), we denote the problem of whether there exists an algorithm which for any given Diophantine equation with integer coefficients, can decide whether (...) or not the equation has a solution in R. We prove that a positive solution to H_{10}(R) implies that the set of all Diophantine equations with a finite number of solutions in R is recursively enumerable. We show the converse implication for every infinite set R \subseteq Q such that there exist computable functions \tau_1,\tau_2:N \to Z which satisfy (\forall n \in N \tau_2(n) \neq 0) \wedge ({\frac{\tau_1(n)}{\tau_2(n)}: n \in N}=R). This implication for R=N guarantees that Smoryński's theorem follows from Matiyasevich's theorem. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have a rational solution is not recursive. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have only finitely many rational solutions is not recursively enumerable. These conjectures are equivalent by our results for R=Q. (shrink)

I give a detailed review of 'The Outer Limits of Reason' by Noson Yanofsky from a unified perspective of Wittgenstein and evolutionary psychology. I indicate that the difficulty with such issues as paradox in language and math, incompleteness, undecidability, computability, the brain and the universe as computers etc., all arise from the failure to look carefully at our use of language in the appropriate context and hence the failure to separate issues of scientific fact from issues of how language works. (...) I discuss Wittgenstein's views on incompleteness, paraconsistency and undecidability and the work of Wolpert on the limits to computation. To sum it up: The Universe According to Brooklyn---Good Science, Not So Good Philosophy. -/- Those wishing a comprehensive up to date framework for human behavior from the modern two systems view may consult my book ‘The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle’ 2nd ed (2019). Those interested in more of my writings may see ‘Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019) and Suicidal Utopian Delusions in the 21st Century 4th ed (2019) . (shrink)

Eu li muitas discussões recentes sobre os limites da computação e do universo como computador, na esperança de encontrar alguns comentários sobre o trabalho surpreendente do físico polimatemático e teórico da decisão David Wolpert, mas não encontrei uma única citação e assim que eu apresento este muito breve Resumo. Wolpert provou alguma impossibilidade impressionante ou teoremas da incompletude (1992 a 2008-Veja arxiv dot org) nos limites à inferência (computação) que são tão gerais que são independentes do dispositivo que faz a (...) computação, e mesmo independente das leis da física, para que eles se apliquem em computadores, física e comportamento humano. Eles fazem uso da diagonalização de cantor, o paradoxo mentiroso e worldlines (linhas do mundo) para fornecer o que pode ser o teorema final na teoria da máquina de Turing, e, aparentemente, fornecer insights sobre a impossibilidade, incompletude, os limites da computação, e do universo como computador, em todos os universos possíveis e todos os seres ou mecanismos, gerando, entre outras coisas, um princípio de incerteza mecânica não quântica e uma prova de monoteísmo. Há umas conexões óbvias ao trabalho clássico de Chaitin, Solomonoff, Komolgarov e Wittgenstein e à noção que nenhum programa (e assim nenhum dispositivo) pode gerar uma seqüência (ou dispositivo) com maior complexidade do que possui. Pode-se dizer que este corpo de trabalho implica ateísmo, uma vez que não pode haver qualquer entidade mais complexa do que o universo físico e do ponto de vista Wittgensteiniano, ' mais complexo ' é sem sentido (não tem condições de satisfação, ou seja, criador de verdade ou teste). Mesmo um ' Deus ' (ou seja, um ' dispositivo ' com tempo ilimitado/espaço e energia) não pode determinar se um determinado ' número ' é ' aleatório ', nem encontrar uma determinada maneira de mostrar que uma determinada ' fórmula ', ' teorema ' ou ' sentença ' ou ' Device ' (todos estes sendo linguagem complexa jogos) faz parte de um determinado «sistema». Aqueles que desejam um quadro até à data detalhado para o comportamento humano da opinião moderna dos dois sistemas consultar meu livros Falando Macacos 3ª Ed (2019), A Estrutura Lógica da Filosofia, Psicologia, Mente e Linguagem em Ludwig Wittgenstein e John Searle 2a Ed (2019), Suicídio Pela Democracia,4aEd(2019), Entendendo as Conexões entre Ciência, Filosofia, Psicologia, Religião, Política e Economia Artigos e Análises 2006-2019 (2019), Ilusões Utópicas Suicidas no 21St século 5a Ed (2019), A Estrutura Lógica do Comportamento Humano (2019), e A Estrutura Lógica da Consciência (2019) y outras. (shrink)

I develop a theory of counterfactuals about relative computability, i.e. counterfactuals such as 'If the validity problem were algorithmically decidable, then the halting problem would also be algorithmically decidable,' which is true, and 'If the validity problem were algorithmically decidable, then arithmetical truth would also be algorithmically decidable,' which is false. These counterfactuals are counterpossibles, i.e. they have metaphysically impossible antecedents. They thus pose a challenge to the orthodoxy about counterfactuals, which would treat them as uniformly true. What’s more, I (...) argue that these counterpossibles don’t just appear in the periphery of relative computability theory but instead they play an ineliminable role in the development of the theory. Finally, I present and discuss a model theory for these counterfactuals that is a straightforward extension of the familiar comparative similarity models. (shrink)

In this paper I address the question of whether the incompleteness theorems imply that “the mind cannot be mechanized,” where this is understood in the specific sense that “the mathematical outputs of the idealized human mind do not coincide with the mathematical outputs of any idealized finite machine.” Gödel argued that his incompleteness theorems implied a weaker, disjunctive conclusion to the effect that either “the mind cannot be mechanized” or “mathematical truth outstrips the idealized human mind.” Others, most notably, Lucas (...) and Penrose, have claimed more—they have claimed that the incompleteness theorems actually imply the first disjunct. I will show that by sharpening the fundamental concepts involved and articulating the background assumptions governing them, one can prove Gödel’s disjunction, one can show that the arguments of Lucas and Penrose fail, and one can see what likely led proponents of the first disjunct astray. (shrink)

Review of Dowek, Gilles, Computation, Proof, Machine, Cambridge University Press, Cambridge, 2015. Translation of Les Métamorphoses du calcul, Le Pommier, Paris, 2007. Translation from the French by Pierre Guillot and Marion Roman.

Inspired from a joint work by A. Beckmann, S. Buss and S. Friedman, we propose a class of set-theoretic functions, predicatively computable set functions. Each function in this class is polynomial time computable when we restrict to finite binary strings.

A review of Computability: Turing, Gödel, Church, and Beyond by Copeland, Posy and Shagrir.

A primary goal of quantum computer science is to find an explanation for the fact that quantum computers are more powerful than classical computers. In this paper I argue that to answer this question is to compare algorithmic processes of various kinds and to describe the possibility spaces associated with these processes. By doing this, we explain how it is possible for one process to outperform its rival. Further, in this and similar examples little is gained in subsequently asking a (...) how-actually question. Once one has explained how-possibly, there is little left to do. (shrink)

A function is diagonally non-computable if it diagonalizes against the universal partial computable function. D.n.c. functions play a central role in algorithmic randomness and reverse mathematics. Flood and Towsner asked for which functions h, the principle stating the existence of an h-bounded d.n.c. function implies Ramsey-type weak König’s lemma. In this paper, we prove that for every computable order h, there exists an ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega}$$\end{document} -model of h-DNR which is not a not (...) model of the Ramsey-type graph coloring principle for two colors and therefore not a model of RWKL. The proof combines bushy tree forcing and a technique introduced by Lerman, Solomon and Towsner to transform a computable non-reducibility into a separation over ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\omega}$$\end{document} -models. (shrink)

Given a graph G, we say that a subset D of the vertex set V is a dominating set if it is near all the vertices, in that every vertex outside of D is adjacent to a vertex in D. A domatic k-partition of G is a partition of V into k dominating sets. In this paper, we will consider issues of computability related to domatic partitions of computable graphs. Our investigation will center on answering two types of questions for (...) the case when k = 3. First, if domatic 3-partitions exist in a computable graph, how complicated can they be? Second, a decision problem: given a graph, how difficult is it to decide whether it has a domatic 3-partition? We will completely classify this decision problem for highly computable graphs, locally finite computable graphs, and computable graphs in general. Specifically, we show the decision problems for these kinds of graphs to be ${\Pi^{0}_{1}}$ -, ${\Pi^{0}_{2}}$ -, and ${\Sigma^{1}_{1}}$ -complete, respectively. (shrink)

A simple inductive argument shows natural languages to have infinitly many sentences, but workers in the field have uncovered clear evidence of a diverse group of ‘exceptional’ languages from Proto-Uralic to Dyirbal and most recently, Pirahã, that appear to lack recursive devices entirely. We argue that in an information-theoretic setting non-recursive natural languages appear neither exceptional nor functionally inferior to the recursive majority.

Several extensions of Gödel's system TT with new forms of recursion have been designed for the purpose of giving a computational interpretation to classical analysis. One can organise many of these extensions into two groups: those based on bar recursion , which include Spector's original bar recursion, modified bar recursion and the more recent products of selections functions, or those based on open recursion which in particular include the symmetric Berardi–Bezem–Coquand functional. We relate these two groups by showing that both (...) open recursion and the BBC functional are primitive recursively equivalent to a variant of modified bar recursion. Our results, in combination with existing research, essentially complete the classification up to primitive recursive equivalence of those extensions of system TT used to give a direct computational interpretation to choice principles. (shrink)

Kolmogorov complexity was originally defined for finitely-representable objects. Later, the definition was extended to real numbers based on the asymptotic behaviour of the sequence of the Kolmogorov complexities of the finitely-representable objects—such as rational numbers—used to approximate them.This idea will be taken further here by extending the definition to continuous functions over real numbers, based on the fact that every continuous real function can be represented as the limit of a sequence of finitely-representable enclosures, such as polynomials with rational coefficients.Based (...) on this definition, we will prove that for any growth rate imaginable, there are real functions whose Kolmogorov complexities have higher growth rates. In fact, using the concept of prevalence, we will prove that ‘almost every’ continuous real function has such a high-growth Kolmogorov complexity. An asymptotic bound on the Kolmogorov complexities of total single-valued computable real functions will be presented as well. (shrink)

For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect to this class. These two computability notions are natural generalizations of certain notions introduced in a previous paper co-authored by Andreas Weiermann and in another previous paper by the same authors, respectively. Under certain weak assumptions about the class in question, (...) we show that conditional computability is preserved by substitution, that all conditionally computable real functions are locally uniformly computable, and that the ones with compact domains are uniformly computable. The introduced notions have some similarity with the uniform computability and its non-uniform extension considered by Katrin Tent and Martin Ziegler, however, there are also essential differences between the conditional computability and the non-uniform computability in question. (shrink)

An important link between model theory and proof theory is to construe a deductive disproof of S as an attempted construction of a countermodel to it. In the function logic outlined here, this idea is implemented in such a way that different kinds of individuals can be introduced into the countermodel in any order whatsoever. This imposes connections between the length of the branches of the tree that a disproof is and their number. If there are already n individuals in (...) the countermodel that is being constructed, the next individual has to be considered in its relations to each of the n old ones, creating 2^n different cases and accordingly at least 2^n different branches. Hence a disproof procedure of a polynomial length is normally not equivalent with an exponential one. Because every computation can be represented as a deduction with the same number of constant terms, the same holds for nondeterministic computations. Apparent exceptions seem to come about if a branch created by a new individual i is redundant. But when the disproof is a shortest one (contains the minimum number of different constant terms) then not introducing that idle term at all would result in an even shorter disproof, violating the shortness assumption. (shrink)

Sufficient conditions are given for the computation of an arc that accesses a point on the boundary of an open subset of the plane from a point within the set. The existence of a not-computably-accessible but computable point on a computably compact arc is also demonstrated.

The paper investigates how the mathematical languages used to describe and to observe automatic computations influence the accuracy of the obtained results. In particular, we focus our attention on Single and Multi-tape Turing machines which are described and observed through the lens of a new mathematical language which is strongly based on three methodological ideas borrowed from Physics and applied to Mathematics, namely: the distinction between the object (we speak here about a mathematical object) of an observation and the instrument (...) used for this observation; interrelations holding between the object and the tool used for the observation; the accuracy of the observation determined by the tool. Results of the observation executed by the traditional and new languages are compared and discussed. (shrink)

Results of R. Miller in 2009 proved several theorems about algebraic fields and computable categoricity. Also in 2009, A. Frolov, I. Kalimullin, and R. Miller proved some results about the degree spectrum of an algebraic field when viewed as a subfield of its algebraic closure. Here, we show that the same computable categoricity results also hold for finite-branching trees under the predecessor function and for connected, finite-valence, pointed graphs, and we show that the degree spectrum results do not hold for (...) these trees and graphs. We also offer an explanation for why the degree spectrum results distinguish these classes of structures: although all three structures are algebraic structures, the fields are what we call effectively algebraic. (shrink)

We analyze the pointwise convergence of a sequence of computable elements of L1 in terms of algorithmic randomness. We consider two ways of expressing the dominated convergence theorem and show that, over the base theory RCA0, each is equivalent to the assertion that every Gδ subset of Cantor space with positive measure has an element. This last statement is, in turn, equivalent to weak weak Königʼs lemma relativized to the Turing jump of any set. It is also equivalent to the (...) conjunction of the statement asserting the existence of a 2-random relative to any given set and the principle of Σ2 collection. (shrink)

This paper commences from the critical observation that the Turing Test (TT) might not be best read as providing a definition or a genuine test of intelligence by proxy of a simulation of conversational behaviour. Firstly, the idea of a machine producing likenesses of this kind served a different purpose in Turing, namely providing a demonstrative simulation to elucidate the force and scope of his computational method, whose primary theoretical import lies within the realm of mathematics rather than cognitive modelling. (...) Secondly, it is argued that a certain bias in Turing’s computational reasoning towards formalism and methodological individualism contributed to systematically unwarranted interpretations of the role of the TT as a simulation of cognitive processes. On the basis of the conceptual distinction in biology between structural homology vs. functional analogy, a view towards alternate versions of the TT is presented that could function as investigative simulations into the emergence of communicative patterns oriented towards shared goals. Unlike the original TT, the purpose of these alternate versions would be co-ordinative rather than deceptive. On this level, genuine functional analogies between human and machine behaviour could arise in quasi-evolutionary fashion. (shrink)