Toward a Formal Philosophy of Hypercomputation
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Quantum Hypercomputation—Hype Or Computation?
Quantum Hypercomputation—Hype or Computation? Amit Hagar,∗ Alex Korolev† February 19, 2007 Abstract A recent attempt to compute a (recursion–theoretic) non–computable func- tion using the quantum adiabatic algorithm is criticized and found wanting. Quantum algorithms may outperform classical algorithms in some cases, but so far they retain the classical (recursion–theoretic) notion of computability. A speculation is then offered as to where the putative power of quantum computers may come from. ∗HPS Department, Indiana University, Bloomington, IN, 47405. [email protected] †Philosophy Department, University of BC, Vancouver, BC. [email protected] 1 1 Introduction Combining physics, mathematics and computer science, quantum computing has developed in the past two decades from a visionary idea (Feynman 1982) to one of the most exciting areas of quantum mechanics (Nielsen and Chuang 2000). The recent excitement in this lively and fashionable domain of research was triggered by Peter Shor (1994) who showed how a quantum computer could exponentially speed–up classical computation and factor numbers much more rapidly (at least in terms of the number of computational steps involved) than any known classical algorithm. Shor’s algorithm was soon followed by several other algorithms that aimed to solve combinatorial and algebraic problems, and in the last few years the- oretical study of quantum systems serving as computational devices has achieved tremendous progress. According to one authority in the field (Aharonov 1998, Abstract), we now have strong theoretical evidence that quantum computers, if built, might be used as powerful computational tool, capable of per- forming tasks which seem intractable for classical computers. -
A Probabilistic Anytime Algorithm for the Halting Problem
Computability 7 (2018) 259–271 259 DOI 10.3233/COM-170073 IOS Press A probabilistic anytime algorithm for the halting problem Cristian S. Calude Department of Computer Science, University of Auckland, Auckland, New Zealand [email protected] www.cs.auckland.ac.nz/~cristian Monica Dumitrescu Faculty of Mathematics and Computer Science, Bucharest University, Romania [email protected] http://goo.gl/txsqpU The first author dedicates his contribution to this paper to the memory of his collaborator and friend S. Barry Cooper (1943–2015). Abstract. TheHaltingProblem,the most (in)famous undecidableproblem,has important applications in theoretical andapplied computer scienceand beyond, hencethe interest in its approximate solutions. Experimental results reportedonvarious models of computationsuggest that haltingprogramsare not uniformly distributed– running times play an important role.A reason is thataprogram whicheventually stopsbut does not halt “quickly”, stops ata timewhich is algorithmically compressible. In this paperweworkwith running times to defineaclass of computable probability distributions on theset of haltingprograms in ordertoconstructananytimealgorithmfor theHaltingproblem withaprobabilisticevaluationofthe errorofthe decision. Keywords: HaltingProblem,anytimealgorithm, running timedistribution 1. Introduction The Halting Problem asks to decide, from a description of an arbitrary program and an input, whether the computation of the program on that input will eventually stop or continue forever. In 1936 Alonzo Church, and independently Alan Turing, proved that (in Turing’s formulation) an algorithm to solve the Halting Problem for all possible program-input pairs does not exist; two equivalent models have been used to describe computa- tion by algorithms (an informal notion), the lambda calculus by Church and Turing machines by Turing. The Halting Problem is historically the first proved undecidable problem; it has many applications in mathematics, logic and theoretical as well as applied computer science, mathematics, physics, biology, etc. -
Alan Turing, Hypercomputation, Adam Smith and Next Generation Intelligent Systems
J. Intell. Syst. 21 (2012), 325–330 DOI 10.1515/jisys-2012-0017 © de Gruyter 2012 Something Old, Something New, Something Borrowed, Something Blue Part 1: Alan Turing, Hypercomputation, Adam Smith and Next Generation Intelligent Systems Mark Dougherty Abstract. In this article intelligent systems are placed in the context of accelerated Turing machines. Although such machines are not currently a reality, the very real gains in computing power made over previous decades require us to continually reevaluate the potential of intelligent systems. The economic theories of Adam Smith provide us with a useful insight into this question. Keywords. Intelligent Systems, Accelerated Turing Machine, Specialization of Labor. 2010 Mathematics Subject Classification. Primary 01-02, secondary 01A60, 01A67. 1 Introduction The editor-in-chief of this journal has asked me to write a series of shorter articles in which I try to illuminate various challenges in intelligent systems. Given space requirements they cannot be regarded as comprehensive reviews, they are intended to provide an overview and to be a point of departure for discussion and debate. The title of the series implies that I will look at both very old and rather new concepts and I will not be afraid to borrow from other subjects. Nor will I be afraid to incorporate radical “blue skies” ideas or hypotheses. By challenges I mean aspects of our subject which cannot easily be tackled by either theoretical means or through empirical enquiry. That may be because the questions raised are not researchable or because no evidence has yet been ob- served. Or it may be that work is underway but that no firm conclusions have yet been reached by the research community. -
Self-Referential Basis of Undecidable Dynamics: from the Liar Paradox and the Halting Problem to the Edge of Chaos
Self-referential basis of undecidable dynamics: from The Liar Paradox and The Halting Problem to The Edge of Chaos Mikhail Prokopenko1, Michael Harre´1, Joseph Lizier1, Fabio Boschetti2, Pavlos Peppas3;4, Stuart Kauffman5 1Centre for Complex Systems, Faculty of Engineering and IT The University of Sydney, NSW 2006, Australia 2CSIRO Oceans and Atmosphere, Floreat, WA 6014, Australia 3Department of Business Administration, University of Patras, Patras 265 00, Greece 4University of Pennsylvania, Philadelphia, PA 19104, USA 5University of Pennsylvania, USA [email protected] Abstract In this paper we explore several fundamental relations between formal systems, algorithms, and dynamical sys- tems, focussing on the roles of undecidability, universality, diagonalization, and self-reference in each of these com- putational frameworks. Some of these interconnections are well-known, while some are clarified in this study as a result of a fine-grained comparison between recursive formal systems, Turing machines, and Cellular Automata (CAs). In particular, we elaborate on the diagonalization argument applied to distributed computation carried out by CAs, illustrating the key elements of Godel’s¨ proof for CAs. The comparative analysis emphasizes three factors which underlie the capacity to generate undecidable dynamics within the examined computational frameworks: (i) the program-data duality; (ii) the potential to access an infinite computational medium; and (iii) the ability to im- plement negation. The considered adaptations of Godel’s¨ proof distinguish between computational universality and undecidability, and show how the diagonalization argument exploits, on several levels, the self-referential basis of undecidability. 1 Introduction It is well-known that there are deep connections between dynamical systems, algorithms, and formal systems. -
Biological Computation: a Road
918 Biological Computation: A Road to Complex Engineered Systems Nelson Alfonso Gómez-Cruz Modeling and Simulation Laboratory Universidad del Rosario Bogotá, Colombia [email protected] Carlos Eduardo Maldonado Research Professor Universidad del Rosario Bogotá, Colombia [email protected] Provided that there is no theoretical frame for complex engineered systems (CES) as yet, this paper claims that bio-inspired engineering can help provide such a frame. Within CES bio- inspired systems play a key role. The disclosure from bio-inspired systems and biological computation has not been sufficiently worked out, however. Biological computation is to be taken as the processing of information by living systems that is carried out in polynomial time, i.e., efficiently; such processing however is grasped by current science and research as an intractable problem (for instance, the protein folding problem). A remark is needed here: P versus NP problems should be well defined and delimited but biological computation problems are not. The shift from conventional engineering to bio-inspired engineering needs bring the subject (or problem) of computability to a new level. Within the frame of computation, so far, the prevailing paradigm is still the Turing-Church thesis. In other words, conventional 919 engineering is still ruled by the Church-Turing thesis (CTt). However, CES is ruled by CTt, too. Contrarily to the above, we shall argue here that biological computation demands a more careful thinking that leads us towards hypercomputation. Bio-inspired engineering and CES thereafter, must turn its regard toward biological computation. Thus, biological computation can and should be taken as the ground for engineering complex non-linear systems. -
State of the Art of Information Technology Computing Models for Autonomic Cloud Computing †
Proceedings State of the Art of Information Technology Computing Models for Autonomic Cloud Computing † Eugene Eberbach Department of Computer Science and Robotics Engineering Program, Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609-2280, USA; [email protected]; Tel.: +1-508-831-4937 † Presented at the IS4SI 2017 Summit DIGITALISATION FOR A SUSTAINABLE SOCIETY, Gothenburg, Sweden, 12–16 June 2017. Published: 9 June 2017 Abstract: In the paper we present several models of computation for autonomic cloud computing. In particular, we justify that for autonomic cloud computing if we require perfect self-reflection, we need models of computation going beyond Turing machines. To approximate self-reflection, models below Turing machines are sufficient. The above claims are illustrated using as an example the DIME Network Architecture. Keywords: autonomic cloud computing; Turing machine; incompleteness; self-reflection; superTuring machines; DIME network architecture 1. Introduction A distributed system is a collection of independent computers that appears to its users as a single coherent system with basic characteristics: 1. concurrency of components, lack of a global clock, scalability and independent failures of components, 2. underlying transparency (inherited by cloud computing), hiding unnecessary details of management, networking, multi-computer implementation or services, 3. providing a uniform way of looking at the whole distributed system. We will consider two subclasses of distributed systems represented by cloud and autonomic computing. Cloud Computing is a relatively new concept in computing that emerged in the late 1990s and the 2000s (but having roots from 1960s—IBM VMs for System360/370/390), first under the name software as a service—the details of computing are hidden transparently in the networking “cloud” 1. -
Halting Problems: a Historical Reading of a Paradigmatic Computer Science Problem
PROGRAMme Autumn workshop 2018, Bertinoro L. De Mol Halting problems: a historical reading of a paradigmatic computer science problem Liesbeth De Mol1 and Edgar G. Daylight2 1 CNRS, UMR 8163 Savoirs, Textes, Langage, Universit´e de Lille, [email protected] 2 Siegen University, [email protected] 1 PROGRAMme Autumn workshop 2018, Bertinoro L. De Mol and E.G. Daylight Introduction (1) \The improbable symbolism of Peano, Russel, and Whitehead, the analysis of proofs by flowcharts spearheaded by Gentzen, the definition of computability by Church and Turing, all inventions motivated by the purest of mathematics, mark the beginning of the computer revolution. Once more, we find a confirmation of the sentence Leonardo jotted despondently on one of those rambling sheets where he confided his innermost thoughts: `Theory is the captain, and application the soldier.' " (Metropolis, Howlett and Rota, 1980) Introduction 2 PROGRAMme Autumn workshop 2018, Bertinoro L. De Mol and E.G. Daylight Introduction (2) Why is this `improbable' symbolism considered relevant in comput- ing? ) Different non-excluding answers... 1. (the socio-historical answers) studying social and institutional developments in computing to understand why logic, or, theory, was/is considered to be the captain (or not) e.g. need for logic framed in CS's struggle for disciplinary identity and independence (cf (Tedre 2015)) 2. (the philosophico-historical answers) studying history of computing on a more technical level to understand why and how logic (or theory) are in- troduced in the computing practices { question: is there something to com- puting which makes logic epistemologically relevant to it? ) significance of combining the different answers (and the respective approaches they result in) ) In this talk: focus on paradigmatic \problem" of (theoretical) computer science { the halting problem Introduction 3 PROGRAMme Autumn workshop 2018, Bertinoro L. -
Undecidable Problems: a Sampler (.Pdf)
UNDECIDABLE PROBLEMS: A SAMPLER BJORN POONEN Abstract. After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathemat- ics. 1. Introduction The goal of this survey article is to demonstrate that undecidable decision problems arise naturally in many branches of mathematics. The criterion for selection of a problem in this survey is simply that the author finds it entertaining! We do not pretend that our list of undecidable problems is complete in any sense. And some of the problems we consider turn out to be decidable or to have unknown decidability status. For another survey of undecidable problems, see [Dav77]. 2. Two notions of undecidability There are two common settings in which one speaks of undecidability: 1. Independence from axioms: A single statement is called undecidable if neither it nor its negation can be deduced using the rules of logic from the set of axioms being used. (Example: The continuum hypothesis, that there is no cardinal number @0 strictly between @0 and 2 , is undecidable in the ZFC axiom system, assuming that ZFC itself is consistent [G¨od40,Coh63, Coh64].) The first examples of statements independent of a \natural" axiom system were constructed by K. G¨odel[G¨od31]. 2. Decision problem: A family of problems with YES/NO answers is called unde- cidable if there is no algorithm that terminates with the correct answer for every problem in the family. (Example: Hilbert's tenth problem, to decide whether a mul- tivariable polynomial equation with integer coefficients has a solution in integers, is undecidable [Mat70].) Remark 2.1. -
Lambda Calculus and Computation 6.037 – Structure and Interpretation of Computer Programs
Lambda Calculus and Computation 6.037 { Structure and Interpretation of Computer Programs Benjamin Barenblat [email protected] Massachusetts Institute of Technology With material from Mike Phillips, Nelson Elhage, and Chelsea Voss January 30, 2019 Benjamin Barenblat 6.037 Lambda Calculus and Computation : Build a calculating machine that gives a yes/no answer to all mathematical questions. Figure: Alonzo Church Figure: Alan Turing (1912-1954), (1903-1995), lambda calculus Turing machines Theorem (Church, Turing, 1936): These models of computation can't solve every problem. Proof: next! Limits to Computation David Hilbert's Entscheidungsproblem (1928) Benjamin Barenblat 6.037 Lambda Calculus and Computation Figure: Alonzo Church Figure: Alan Turing (1912-1954), (1903-1995), lambda calculus Turing machines Theorem (Church, Turing, 1936): These models of computation can't solve every problem. Proof: next! Limits to Computation David Hilbert's Entscheidungsproblem (1928): Build a calculating machine that gives a yes/no answer to all mathematical questions. Benjamin Barenblat 6.037 Lambda Calculus and Computation Theorem (Church, Turing, 1936): These models of computation can't solve every problem. Proof: next! Limits to Computation David Hilbert's Entscheidungsproblem (1928): Build a calculating machine that gives a yes/no answer to all mathematical questions. Figure: Alonzo Church Figure: Alan Turing (1912-1954), (1903-1995), lambda calculus Turing machines Benjamin Barenblat 6.037 Lambda Calculus and Computation Proof: next! Limits to Computation David Hilbert's Entscheidungsproblem (1928): Build a calculating machine that gives a yes/no answer to all mathematical questions. Figure: Alonzo Church Figure: Alan Turing (1912-1954), (1903-1995), lambda calculus Turing machines Theorem (Church, Turing, 1936): These models of computation can't solve every problem. -
Halting Problem
Automata Theory CS411-2015S-FR2 Final Review David Galles Department of Computer Science University of San Francisco FR2-0: Halting Problem Halting Machine takes as input an encoding of a Turing Machine e(M) and an encoding of an input string e(w), and returns “yes” if M halts on w, and “no” if M does not halt on w. Like writing a Java program that parses a Java function, and determines if that function halts on a specific input e(M) Halting yes e(w) Machine no FR2-1: Halting Problem Halting Machine takes as input an encoding of a Turing Machine e(M) and an encoding of an input string e(w), and returns “yes” if M halts on w, and “no” if M does not halt on w. Like writing a Java program that parses a Java function, and determines if that function halts on a specific input How might the Java version work? Check for loops while (<test>) <body> Use program verification techniques to see if test can ever be false, etc. FR2-2: Halting Problem The Halting Problem is Undecidable There exists no Turing Machine that decides it There is no Turing Machine that halts on all inputs, and always says “yes” if M halts on w, and always says “no” if M does not halt on w Prove Halting Problem is Undecidable by Contradiction: FR2-3: Halting Problem Prove Halting Problem is Undecidable by Contradiction: Assume that there is some Turing Machine that solves the halting problem. e(M) Halting yes e(w) Machine no We can use this machine to create a new machine Q: Q e(M) runs forever e(M) Halting yes e(M) Machine no yes FR2-4: Halting Problem Q e(M) runs forever e(M) Halting yes e(M) Machine no yes R yes MDUPLICATE MHALT no yes FR2-5: Halting Problem Machine Q takes as input a Turing Machine M, and either halts, or runs forever. -
The Halting Paradox
The Halting Paradox Bill Stoddart 20 December 2017 Abstract The halting problem is considered to be an essential part of the theoretical background to computing. That halting is not in general computable has been “proved” in many text books and taught on many computer science courses, and is supposed to illustrate the limits of computation. However, Eric Hehner has a dissenting view, in which the specification of the halting problem is called into question. Keywords: halting problem, proof, paradox 1 Introduction In his invited paper [1] at The First International Conference on Unifying Theories of Programming, Eric Hehner dedicates a section to the proof of the halting problem, claiming that it entails an unstated assumption. He agrees that the halt test program cannot exist, but concludes that this is due to an inconsistency in its specification. Hehner has republished his arguments arXiv:1906.05340v1 [cs.LO] 11 Jun 2019 using less formal notation in [2]. The halting problem is considered to be an essential part of the theoretical background to computing. That halting is not in general computable has been “proved” in many text books and taught on many computer science courses to illustrate the limits of computation. Hehner’s claim is therefore extraordinary. Nevertheless he is an eminent and highly creative computer scientist whose opinions reward careful attention. In judging Hehner’s thesis we take the view that to illustrate the limits of computation we need a consistent specification which cannot be implemented. For our discussion we use a guarded command language for sequential state based programs. Our language includes named procedures and named en- quiries and tests. -
Equivalence of the Frame and Halting Problems
algorithms Article Equivalence of the Frame and Halting Problems Eric Dietrich and Chris Fields *,† Department of Philosophy, Binghamton University, Binghamton, NY 13902, USA; [email protected] * Correspondence: fi[email protected]; Tel.: +33-6-44-20-68-69 † Current address: 23 Rue des Lavandières, 11160 Caunes Minervois, France. Received: 1 July 2020; Accepted: 16 July 2020; Published: 20 July 2020 Abstract: The open-domain Frame Problem is the problem of determining what features of an open task environment need to be updated following an action. Here we prove that the open-domain Frame Problem is equivalent to the Halting Problem and is therefore undecidable. We discuss two other open-domain problems closely related to the Frame Problem, the system identification problem and the symbol-grounding problem, and show that they are similarly undecidable. We then reformulate the Frame Problem as a quantum decision problem, and show that it is undecidable by any finite quantum computer. Keywords: artificial intelligence; entanglement; heuristic search; Multiprover Interactive Proof*(MIP); robotics; separability; system identification 1. Introduction The Frame Problem (FP) was introduced by McCarthy and Hayes [1] as the problem of circumscribing the set of axioms that must be deployed in a first-order logic representation of a changing environment. From an operational perspective, it is the problem of circumscribing the set of (object, property) pairs that need to be updated following an action. In a fully-circumscribed domain comprising a finite set of objects, each with a finite set of properties, the FP is clearly solvable by enumeration; in such domains, it is merely the problem of making the enumeration efficient.