Computability Theory

This book is about methods and techniques popularly employed in the field of computational intelligence (CI). According to [1], CI can be broadly defined as ??the branch of science studying problems for which there are no effective computational algorithms??. These include problems such as extracting meaning from perception, understanding language, solving illdefined vision tasks, for which formulating exact, analytical algorithms is either not possible or computationally very demanding. Seeing from this viewpoint, it is instructive to define CI based on the problems studied, rather than the types of method used for solutions.

The received view is that computational states are individuated at least in part by their semantic properties. I offer an alternative, according to which computational states are individuated by their functional properties. Functional properties are specified by a mechanistic explanation without appealing to any semantic properties. The primary purpose of this paper is to formulate the alternative view of computational individuation, point out that it supports a robust notion of computational explanation, and defend it on the grounds of how computational states are individuated within computability theory and computer science. A secondary purpose is to show that existing arguments for the semantic view are defective.

The following publications involve content that is directly relevant to the topic of this thesis. For more information. P. van Loock, C. Weedbrook, and M. Gu. Building gaussian cluster states by linear optics. Phys. Rev. A, 76(3):032321, 2007. S. Sridharan, M. Gu, and M. James. Gate complexity using dynamic programming. Phys. Rev. A, 78(5):052327, 2008. Nicolas C. Menicucci, Peter van Loock, Mile Gu, C. Weedbrook, T. Ralph, and M. Nielsen. Universal quantum computation with continuous-variable cluster states. Phys.

The aim of this paper is to take a look at Péter's talk Rekursivität und Konstruktivität delivered at the Constructivity in Mathematics Col-loquium in 1957, where she challenged Church's Thesis from a constructive point of view. The discussion of her argument and motivations is then connected to her earlier work on recursion theory as well as her later work on theoretical computer science.

Agli albori del pensiero moderno, nella quinta parte del Discours de la Méthode, René Descartes aveva immaginato che un giorno sarebbe stato possibile costruire un automa dotato degli stessi organi e dello stesso aspetto di una scimmia, o di qualche altro animale. Nel caso di questi esseri, privi della ragione, Descartes pensava che sarebbe stato difficile distinguere una copia dall'originale; nel caso in cui l'automa fosse stato invece la copia di un essere umano, il trucco sarebbe stato scoperto senza alcun imbarazzo, imponendo all'automa di usare il linguaggio in modo appropriato o di eseguire i compiti che un essere intelligente è normalmente in grado di fare.

In this multidisciplinary investigation we show how an evidence-based perspective of quantification-in terms of algorithmic verifiability and algorithmic computability-admits evidence-based definitions of well-definedness and effective com-putability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA-over the structure N of the natural numbers-that are complementary, not contradictory. The first yields the weak, standard, interpretation I_P A(N, SV) of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under I_P A(N, SV). The second yields a strong, finitary, interpretation I_P A(N, SC) of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under I_P A(N, SC). We situate our investigation within a broad analysis of quantification visávisá vis: • Hilbert's ε-calculus • Gödel's ω-consistency • The Law of the Excluded Middle • Hilbert's ω-Rule • An Algorithmic ω-Rule • Gentzen's Rule of Infinite Induction • Rosser's Rule C • Markov's Principle • The Church-Turing Thesis • Aristotle's particularisation (defined as the postulation that from the provability of a formula [(∀x)F (x)] in a formal theory L, we may conclude the existence of some unspecified object a, in the domain of any well-defined interpretation I of L, for which the interpretation F * (a) of [F (a)] is true) • Wittgenstein's perspective of constructive mathematics • An evidence-based perspective of quantification. By showing how these are formally interrelated , we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's ε-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines.

Esempi di algoritmi possono essere tratti dalle matematiche elementari: sono algoritmi, ad esempio, gli insiemi di regole che consentono di eseguire le quattro operazioni, come pure è un algoritmo il procedimento euclideo per la ricerca del massimo comun divisore di due numeri naturali. In logica, il metodo delle tavole di verità è un algoritmo che permette di stabilire se una formula del calcolo proposizionale è o meno una tautologia.

We present a general computational theory of cancer and its developmental dynamics. The theory is based on a theory of the architecture and function of developmental control networks which guide the formation of multicellular organisms. Cancer networks are special cases of developmental control networks. Cancer results from transformations of normal developmental networks. Our theory generates a natural classification of all possible cancers based on their network architecture. Each cancer network has a unique topology and semantics and developmental dynamics that result in distinct clinical tumor phenotypes. We apply this new theory with a series of proof of concept cases for all the basic cancer types. These cases have been computationally modeled, their behavior simulated and mathematically described using a multicellular systems biology approach. There are fascinating correspondences between the dynamic developmental phenotype of computationally modeled {\em in silico} cancers a...

Este trabajo se centra en la discusión sobre la disyunción de Gödel, así como las posiciones antimecanicistas que surgen de los teoremas de Gödel, en el escenario actual de la pluralidad de la lógica. El trabajo está motivado por dos ideas, en primer lugar, que lo que prueban las posiciones antimecanicistas de Lucas y Penrose no es la afirmación de que la mente humana no puede ser capturada por ninguna máquina, sino que no puede ser capturada por el tipo de cálculo que lleva una máquina de Turing . Segundo, aunque los resultados limitantes de Gödel son de gran importancia en el campo de la aritmética clásica, cuando pensamos en la mente matemática humana, las paradojas e inconsistencias entran en escena y nos motivan a cambiar la lógica. Por lo tanto, sostengo que la ruta paraconsistente es aceptable en esta discusión.

Gli algoritmi sono metodi per la soluzione di problemi. Possiamo caratterizzare un problema mediante i dati di cui si dispone all'inizio e dei risultati che si vogliono ottenere: risolvere un problema significa ottenere in uscita i risultati desiderati a partire da un certo insieme di dati presi in ingresso. I dati in ingresso vengono anche detti (valori in) input ei risultati in uscita (valori in) output. Possiamo assumere che ciascun problema consista di un insieme di casi particolari, o istanze.

Este es el primer ensayo, de una serie de dos, en los que se quiere presentar visiones panorámicas de la noción de computabilidad de dos nuevos paradigmas de computación. A saber, la computación cuántica y la computación basada en ADN o molecular, así como de los circuitos integrados tridimensionales que constituyen una alternativa para continuar con la miniaturización de procesadores hasta más allá del orden del nanómetro. Aquí se parte de la misma noción de computación, de sus limitaciones lógicas y se delinea el problema de decidir si acaso el determinismo y el no-determinismo coinciden en dispositivos de complejidades polinomiales en el tiempo. En cuanto a la computación cuántica, se presenta a sus elementos matemáticos básicos y a las ventajas que implican en comunicaciones y criptografía. Finalmente se menciona a algunas de las alternativas actuales para implementarlas. Esta presentación es una reseña del estado actual del área, y tan sólo es novedosa en cuanto a sus ejemplos; está escrita originalmente en lengua castellana, y está dirigida a un público extenso, conformado tanto por profesionistas como estudiantes de nivel universitario.

In this thesis we introduce the reader to the program of reverse mathematics. This will be done in the usual context of second-order arithmetic and classical logic. However, instead of the common base theory RCA 0 (Recursive Comprehension Axiom), here we will take the much less explored path of using the weaker subsystem RCA * 0 as the base theory. This will enable us to analyze the reverse mathematics of some of the theorems that are provable in RCA 0 already. In particular, we will see that the Baire Category Theorem is equivalent to Σ 0 1 induction, over the base theory RCA * 0 . As an expository thesis, our main objective is not only to gather a few relevant articles into one coherent work, but also to give more explanations to make the material more easily accessible.

«Adam Olszewski (1999) propone que la Tesis de Church-Turing puede ser usada para refutar el platonismo matemático1. Para ello postula una máquina que al lanzar una moneda define una función que ―computa el valor (0 o 1) para la moneda n que se haya lanzado y dice que dicha función es efectivamente computable pero no Turing-computable.

Entonces, dado que según el platonismo matemático (PM), toda función de enteros positivos a enteros positivos ya existe, existiría una función efectivamente computable pero no Turing-computable (computable por una máquina de Turing).

Esto contradice a la Tesis de Church-Turing (CT) que dice que toda función efectivamente computable es Turing-computable. Por contraposición, si CT es verdadera, entonces PM es falso. Rafał Urbaniak (2011) critica este argumento desafiando la afirmación de que lo que esta máquina de hecho realiza sea una computación. Revisaré dicha crítica y detallaré algunos puntos de la misma. Además introduciré la noción de Tesis de Church-Turing Física.»

Despite its significance in neuroscience and computation, McCulloch and Pitts's celebrated 1943 paper has received little historical and philosophical attention. In 1943 there already existed a lively community of biophysicists doing mathematical work on neural networks. What was novel in McCulloch and Pitts's paper was their use of logic and computation to understand neural, and thus mental, activity. McCulloch and Pitts's contributions included (i) a formalism whose refinement and generalization led to the notion of finite automata (an important formalism in computability theory), (ii) a technique that inspired the notion of logic design (a fundamental part of modern computer design), (iii) the first use of computation to address the mind-body problem, and (iv) the first modern computational theory of mind and brain.

This paper reviews and evaluates the potential use of modern visualization techniques in archaeology. It suggests the need to apply and develop such techniques as a central part of any modern archaeological investigation. The use of these methods is associated with wider questions about data representations, in particular, their integration with archaeological theory and their role in facilitating analysis and shaping interpretation. Concern for these questions and with the overall potential that information systems provide to capture, represent, analyze, and model archaeological information suggests the need for a new interdisciplinary focus, Archaeological Information Science. For such a focus to prosper, archaeologists need to develop additional skills that go beyond mere technical ones. They need to become more active in the design and creation of future information archaeological systems. To this end, archaeologists are urged to view this task as a way to extend archaeology in ...

As part of an interdisciplinary project to develop a computational cognitive model of a reader of narrative text, we are developing a computational theory of how natural-language-understanding systems can automatically acquire new vocabulary by determining from context the meaning of words that are unknown, misunderstood, or used in a new sense. 'Context' includes surrounding text, grammatical information, and background knowledge, but no external sources. Our thesis is that the meaning of such a word can be determined from context, can be revised upon further encounters with the word, "converges" to a dictionary-like definition if enough context has been provided and there have been enough exposures to the word, and eventually "settles down" to a "steady state" that is always subject to revision upon further encounters with the word. The system is being implemented in the SNePS knowledgerepresentation and reasoning system.

In order to understand the Universe completely and achieve Theory of Everything we must understand how our consciousness experiences and understands all objects in the Universe in general. The aim of the project is to synthesize philosophy, mathematics, physics, information theory, language and cognitive science into a single architectonic framework or system of our reason itself and to model the original synthetic unity of apperception -the framework within which all our thoughts, knowledge and experience is produced.

Solomonoff induction is a brute-force method of predicting future bits of a binary string given past bits. It does so by comparing the past bits to the output of every possible input program to a universal Turing machine. It uses the programs of the matching output to predict future bits, weighted by the length of the program. Because it uses every possible input to a universal Turing machine, some of the programs do not halt. Therefore this semi-measure is incomputable. It does however have some nice properties; it is enumerable and multiplicatively dominates all other enumerable semi-measures.

En 1986 Lynn Steen señaló muy certeramente que: the computer revolution is a mathematical revolution. En este ensayo se analiza el papel que el ordenador, como intruso en el ecosistema matemático, ha jugado, juega y jugará en ese hacer. 1. En 1976 Appel, Haken y Koch hacen pública la demostración de la conjetura de los cuatro colores: bastan cuatro colores para colorear el plano con la condición de que dos regiones que poseen frontera común-no limitada a un punto-tengan colores diferentes. La conjetura de los 4C se convirtió así, desde 1976, en el teorema de los 4C. No tendría más importancia esta noticia-salvo la confirmación de una conjetura que llevaba como tal muchos años y cuya validez depende de la topología de la superficie a colorear-si no fuera porque, en esa demostración, se había empleado el ordenador. Algo que venía siendo corriente en temas próximos a la Teoría de números o en el cómputo de cifras decimales del desarrollo de números trascendentes como Ò, en test de primalidad, incluso en Análisis en el manejo de ecuaciones en diferencias finitas... Sin embargo, en la demostración del teorema 4C el ordenador parecía revestir un papel específico en el sentido de que se mostraba como factor constructivo al sugerir posibles estrategias al matemático y se hacía imprescindible en la verificación de 1482 configuraciones, verificación que no podía sustituirse por procesòhumano' alguno al exigir miles de páginas de programas de computación no publicadas y, de publicarse, no revisables o verificables paso a paso por algún matemático, revisión sólo factible por otro ordenador. En la misma línea, con las mismas características antes apuntadas, en 1986 se realizan demostraciones de teoremas en campos que parecían marginados a la faceta

Deliberate contextual vocabulary acquisition (CVA) is a reader's ability to figure out a (not the) meaning of an unknown word from its "context", without external sources of help such as dictionaries or people. The appropriate context for such CVA is the "belief-revised integration" of the reader's prior knowledge with the reader's "internalization" of the text. We discuss unwarranted assumptions behind some classic objections to CVA, and present and defend a computational theory of CVA that we have adapted to a new classroom curriculum designed to help students use CVA to improve their reading comprehension.

1) An account of the relation between constructive and recursive functions, including the status of Church's Thesis, with a correspondence between Yannis Moschovakis and Alonzo Church;
2) A formalization of Kripke's Schema as a closed principle, using function generality, rather than second order quantification in Martin-Löf's Constructive Type Theory, and a classical verification-object for this principle;
3) Relative  to a verification-object for Kripke's Principle, a meaning explanation is given for the Kreisel connective for Brouwer's Creating Subject and a demonstration is given for each of of his axioms CS1-CS3 for the ensuing  Theory of The Creating Subject.
4) On pp. 19-21 I offer a novel analysis of Fitch's paradox and explain why the "knowability axiom" is unacceptable

On pense généralement que l'impossibilité, l'incomplétdulité, la paracohérence, l'indécidabilité, le hasard, la calcul, le paradoxe, l'incertitude et les limites de la raison sont des questions scientifiques physiques ou mathématiques disparates ayant peu ou rien dans terrain d'entente. Je suggère qu'ils sont en grande partie des problèmes philosophiques standard (c.-à-d., jeux de langue) qui ont été la plupart du temps résolus par Wittgenstein plus de 80 ans.
Je fournis un bref résumé de quelques-unes des principales conclusions de deux des plus éminents étudiants du comportement des temps modernes, Ludwig Wittgenstein et John Searle, sur la structure logique de l'intentionnalité (esprit, langue, comportement), en prenant comme point de départ La découverte fondamentale de Wittgenstein, à savoir que tous les problèmes véritablement « philosophiques » sont les mêmes, les confusions sur la façon d'utiliser la langue dans un contexte particulier, et donc toutes les solutions sont les mêmes— en regardant comment la langue peut être utilisée dans le contexte en cause afin que sa vérité (Conditions de satisfaction ou COS) sont claires. Le problème fondamental est que l'on peut dire n'importe quoi, mais on ne peut pas signifier (état clair COS pour) toute déclaration arbitraire et le sens n'est possible que dans un contexte très spécifique.
Je dissé que quelques écrits de quelques-uns des principaux commentateurs sur ces questions d'un point de vue wittgensteinien dans le cadre de la perspective moderne des deux systèmes de pensée (popularisé comme «penser vite, penser lentement»), en utilisant une nouvelle table de intentionnalité et la nomenclature de nouveaux systèmes doubles. Je montre qu'il s'agit d'un puissant heuristique pour décrire la vraie nature de ces questions scientifiques, physiques ou mathématiques putatives qui sont vraiment mieux abordés comme des problèmes philosophiques standard de la façon dont la langue doit être utilisée (jeux de langue dans Wittgenstein terminologie).

Some philosophers have conflated functionalism and computationalism. I reconstruct how this came about and uncover two assumptions that made the conflation possible. They are the assumptions that (i) psychological functional analyses are computational descriptions and (ii) everything may be described as performing computations. I argue that, if we want to improve our understanding of both the metaphysics of mental states and the functional relations between them, we should reject these assumptions. #

In this paper we propose an approach to quantum λ-calculi. The "quantum data-classical control" paradigm is considered. Starting from a measurement-free untyped quantum λ-calculus called Q, we will study standard properties such as confluence and subject reduction and some good quantum properties. We will focus on the expressive power, analyzing the relationship with other quantum computational models. Successively, we will add an explicit measurement operator to Q. On the resulting calculus, called Q * , we will propose a complete study of reduction sequences regardless of their finiteness, proving confluence results. Moreover, since the stronger motivation behind quantum computing is the research of new results in computational complexity, we will also propose a calculus which captures the three classes of quantum polytime complexity, showing an ICC-like approach in the quantum setting.

Many scientific questions are considered solved to the best possible degree when we have a method for computing a solution. This is especially true in mathematics and those areas of science in which phenomena can be described mathematically: one only has to think of the methods of symbolic algebra in order to solve equations, or laws of physics which allow one to calculate unknown quantities from known measurements. The crowning achievement of mathematics would thus be a systematic way to compute the solution to any mathematical problem. The hope that this was possible was perhaps first articulated by the 18th century mathematician-philosopher G. W. Leibniz. Advances in the foundations of mathematics in the early 20th century made it possible in the 1920s to first formulate the question of whether there is such a systematic way to find a solution to every mathematical problem. This became known as the decision problem, and it was considered a major open problem in the 1920s and 1930s. Alan Turing solved it in his first, groundbreaking paper "On computable numbers" (1936). In order to show that there cannot be a systematic computational procedure that solves every mathematical question, Turing had to provide a convincing analysis of what a computational procedure is. His abstract, mathematical model of computability is that of a Turing Machine. He showed that no Turing machine, and hence no computational procedure at all, could solve the Entscheidungsproblem.

There is an intensive discussion nowadays about the meaning of effective computability, with implications to the status and provability of the Church–Turing Thesis (CTT). I begin by reviewing what has become the dominant account of the way Turing and Church viewed, in 1936, effective computability. According to this account, to which I refer as the Gandy–Sieg account, Turing and Church aimed to characterize the functions that can be computed by a human computer. In addition, Turing provided a highly convincing argument for CTT by analyzing the processes carried out by a human computer. I then contend that if the Gandy–Sieg account is correct, then the notion of effective computability has changed after 1936. Today computer scientists view effective computability in terms of finite machine computation. My contention is supported by the current formulations of CTT, which always refer to machine computation, and by the current argumentation for CTT, which is different from the main arguments advanced by Turing and Church. I finally turn to discuss Robin Gandy's characterization of machine computation. I suggest that there is an ambiguity regarding the types of machines Gandy was postulating. I offer three interpretations, which differ in their scope and limitations, and conclude that none provides the basis for claiming that Gandy characterized finite machine computation.

In this investigation we differentiate distinctly between human reasoning and mechanistic reasoning by showing that Tarski's classic definitions admit finitary evidence-based definitions of the satisfaction and truth of the atomic formulas of the first-order Peano Arithmetic PA over the domain N of the natural numbers in two, hitherto unsuspected and essentially different, ways:

(1) in terms of classical algorithmic verifiabilty; and

(2) in terms of finitary algorithmic computability.

We then show that the two definitions correspond to two distinctly different assignments of satisfaction and truth to the compound formulas of PA over N---PA(N, SV) and PA(N, SC).

We further show that the PA axioms are true over N, and that the PA rules of inference preserve truth over N, under both PA(N, SV) and PA(N, SC); and:

(a) that if we assume the satisfaction and truth of the compound formulas of PA are always non-finitarily decidable under PA(N, SV), then this assignment corresponds to the classical non-finitary standard interpretation PA(N, S) of PA over the domain N (accepted as consistent by Gentzen's argument); and

(b) that the satisfaction and truth of the compound formulas of PA are always finitarily decidable under the assignment PA(N, SC), from which we may finitarily conclude that PA is consistent.

We further conclude that the appropriate inference to be drawn from Goedel's 1931 paper on undecidable arithmetical propositions is that we can define PA formulas which---under interpretation---are algorithmically verifiable as always true over N, but not algorithmically computable as always true over N.

We conclude from this that Lucas' Goedelian argument is validated if the assignment PA(N, SV) can be treated as circumscribing the ambit of human reasoning about 'true' arithmetical propositions, and the assignment PA(N, SC) as circumscribing the ambit of mechanistic reasoning about 'true' arithmetical propositions.

In his famous paper, An Unsolvable Problem of Elementary Number Theory , Alonzo Church (1936) identified the intuitive notion of effective calculability with the mathematically precise notion of recursiveness. This proposal, known as Church's Thesis, has been widely accepted. Only a few papers have been written against it. One of these is László Kalmár's An Argument Against the Plausibility of Church's Thesis from 1959. The aim of this paper is to present Kalmár's argument and to fill in missing details based on his general philosophical thoughts on mathematics.

This paper deals with the question: what are the key requirements for a physical system to perform digital computation? Oftentimes, cognitive scientists are quick to employ the notion of computation simpliciter when asserting basically that cognitive activities are computational. They employ this notion as if there is a consensus on just what it takes for a physical system to compute. Some cognitive scientists in referring to digital computation simply adhere to Turing computability. But if cognition is indeed computational, then it is concrete computation that is required for explaining cognition as an embodied phenomenon. Three accounts of computation are examined here: 1. Formal Symbol Manipulation. 2. Physical Symbol Systems and 3. The Mechanistic account. I argue that the differing requirements implied by these accounts justify the demand that one commits to a particular account when employing the notion of digital computation in regard to physical systems, rather than use these accounts interchangeably.

Computation and its Limits is an innovative cross-disciplinary investigation of the relationship between computing and physical reality. It begins by exploring the mystery of why mathematics is so effective in science and seeks to explain this in terms of the modelling of one part of physical reality by another. Going from the origins of counting to the most blue-skies proposals for novel methods of computation, the authors investigate the extent to which the laws of nature and of logic constrain what we can compute. In the ...

In  this  article,  we  will  show  that  uncomputability  is  a  relative property  not  only  of  oracle  Turing  machines,  but  also  of  subrecursive classes. We will define the concept of a Turing submachine, and a recursive relative  version  for  the  Busy Beaver  function which we  will  call  Busy Beaver Plus function. Therefore, we will prove that the computable Busy Beaver Plus function defined on any Turing submachine is not computable by any program running on this submachine. We will thereby demonstrate the existence of a “paradox” of computability a la Skolem: a function is computable when “seen from the outside” the subsystem, but uncomputable when “seen from within” the same subsystem. Finally, we will raise the possibility of defining universal submachines,  and a hierarchy of negative Turing degrees.

Semantics connected to some information based metaphor are well-known in logic literature: a paradigmatic example is Kripke semantic for Intuitionistic Logic. In this paper we start from the concrete problem of providing suitable logic-algebraic models for the calculus of attribute dependencies in Formal Contexts with information gaps and we obtain an intuitive model based on the notion of "passage of information" showing that Kleene algebras, semi-simple Nelson algebras, three-valued Łukasiewicz algebras and Post algebras of order three are, in a sense, "naturally" and directly connected to partially defined information systems. In this way we can provide for these logic-algebraic structures a "raison d'être" different from the original motivations concerning, for instance, computability theory.

A standing challenge for the science of mind is to account for the datum that every mind faces in the most immediate -that is, unmediated -fashion: its phenomenal experience. The complementary tasks of explaining what it means for a system to give rise to experience and what constitutes the content of experience (qualia) in computational terms are particularly challenging, given the multiple realizability of computation. In this paper, we identify a set of conditions that a computational theory must satisfy for it to constitute not just a sufficient but a necessary, and therefore naturalistic and intrinsic, explanation of qualia. We show that a common assumption behind many neurocomputational theories of the mind, according to which mind states can be formalized solely in terms of instantaneous vectors of activities of representational units such as neurons, does not meet the requisite conditions, in part because it relies on inactive units to shape presently experienced qualia and implies a homogeneous representation space, which is devoid of intrinsic structure. We then sketch a naturalistic computational theory of qualia, which posits that experience is realized by dynamical activity-space trajectories (rather than points) and that its richness is measured by the representational capacity of the trajectory space in which it unfolds.