6.2 Presenting Formal and Informal Mathematics
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Why Machines Don't
KI - Künstliche Intelligenz https://doi.org/10.1007/s13218-019-00599-w SURVEY Why Machines Don’t (yet) Reason Like People Sangeet Khemlani1 · P. N. Johnson‑Laird2,3 Received: 15 February 2019 / Accepted: 19 June 2019 © This is a U.S. government work and not under copyright protection in the U.S.; foreign copyright protection may apply 2019 Abstract AI has never come to grips with how human beings reason in daily life. Many automated theorem-proving technologies exist, but they cannot serve as a foundation for automated reasoning systems. In this paper, we trace their limitations back to two historical developments in AI: the motivation to establish automated theorem-provers for systems of mathematical logic, and the formulation of nonmonotonic systems of reasoning. We then describe why human reasoning cannot be simulated by current machine reasoning or deep learning methodologies. People can generate inferences on their own instead of just evaluating them. They use strategies and fallible shortcuts when they reason. The discovery of an inconsistency does not result in an explosion of inferences—instead, it often prompts reasoners to abandon a premise. And the connectives they use in natural language have diferent meanings than those in classical logic. Only recently have cognitive scientists begun to implement automated reasoning systems that refect these human patterns of reasoning. A key constraint of these recent implementations is that they compute, not proofs or truth values, but possibilities. Keywords Reasoning · Mental models · Cognitive models 1 Introduction problems for mathematicians). Their other uses include the verifcation of computer programs and the design of com- The great commercial success of deep learning has pushed puter chips. -
Artificial Consciousness and the Consciousness-Attention Dissociation
Consciousness and Cognition 45 (2016) 210–225 Contents lists available at ScienceDirect Consciousness and Cognition journal homepage: www.elsevier.com/locate/concog Review article Artificial consciousness and the consciousness-attention dissociation ⇑ Harry Haroutioun Haladjian a, , Carlos Montemayor b a Laboratoire Psychologie de la Perception, CNRS (UMR 8242), Université Paris Descartes, Centre Biomédical des Saints-Pères, 45 rue des Saints-Pères, 75006 Paris, France b San Francisco State University, Philosophy Department, 1600 Holloway Avenue, San Francisco, CA 94132 USA article info abstract Article history: Artificial Intelligence is at a turning point, with a substantial increase in projects aiming to Received 6 July 2016 implement sophisticated forms of human intelligence in machines. This research attempts Accepted 12 August 2016 to model specific forms of intelligence through brute-force search heuristics and also reproduce features of human perception and cognition, including emotions. Such goals have implications for artificial consciousness, with some arguing that it will be achievable Keywords: once we overcome short-term engineering challenges. We believe, however, that phenom- Artificial intelligence enal consciousness cannot be implemented in machines. This becomes clear when consid- Artificial consciousness ering emotions and examining the dissociation between consciousness and attention in Consciousness Visual attention humans. While we may be able to program ethical behavior based on rules and machine Phenomenology learning, we will never be able to reproduce emotions or empathy by programming such Emotions control systems—these will be merely simulations. Arguments in favor of this claim include Empathy considerations about evolution, the neuropsychological aspects of emotions, and the disso- ciation between attention and consciousness found in humans. -
Computability and Complexity
Computability and Complexity Lecture Notes Herbert Jaeger, Jacobs University Bremen Version history Jan 30, 2018: created as copy of CC lecture notes of Spring 2017 Feb 16, 2018: cleaned up font conversion hickups up to Section 5 Feb 23, 2018: cleaned up font conversion hickups up to Section 6 Mar 15, 2018: cleaned up font conversion hickups up to Section 7 Apr 5, 2018: cleaned up font conversion hickups up to the end of these lecture notes 1 1 Introduction 1.1 Motivation This lecture will introduce you to the theory of computation and the theory of computational complexity. The theory of computation offers full answers to the questions, • what problems can in principle be solved by computer programs? • what functions can in principle be computed by computer programs? • what formal languages can in principle be decided by computer programs? Full answers to these questions have been found in the last 70 years or so, and we will learn about them. (And it turns out that these questions are all the same question). The theory of computation is well-established, transparent, and basically simple (you might disagree at first). The theory of complexity offers many insights to questions like • for a given problem / function / language that has to be solved / computed / decided by a computer program, how long does the fastest program actually run? • how much memory space has to be used at least? • can you speed up computations by using different computer architectures or different programming approaches? The theory of complexity is historically younger than the theory of computation – the first surge of results came in the 60ties of last century. -
An Overview of Automated Reasoning
An overview of automated reasoning John Harrison Intel Corporation 3rd February 2014 (10:30{11:30) Talk overview I What is automated reasoning? I Early history and taxonomy I Automation, its scope and limits I Interactive theorem proving I Hobbes (1651): \Reason . is nothing but reckoning (that is, adding and subtracting) of the consequences of general names agreed upon, for the marking and signifying of our thoughts." I Leibniz (1685) \When there are disputes among persons, we can simply say: Let us calculate [calculemus], without further ado, to see who is right." Nowadays, by `automatic and algorithmic' we mean `using a computer program'. What is automated reasoning? Attempting to perform logical reasoning in an automatic and algorithmic way. An old dream! I Leibniz (1685) \When there are disputes among persons, we can simply say: Let us calculate [calculemus], without further ado, to see who is right." Nowadays, by `automatic and algorithmic' we mean `using a computer program'. What is automated reasoning? Attempting to perform logical reasoning in an automatic and algorithmic way. An old dream! I Hobbes (1651): \Reason . is nothing but reckoning (that is, adding and subtracting) of the consequences of general names agreed upon, for the marking and signifying of our thoughts." Nowadays, by `automatic and algorithmic' we mean `using a computer program'. What is automated reasoning? Attempting to perform logical reasoning in an automatic and algorithmic way. An old dream! I Hobbes (1651): \Reason . is nothing but reckoning (that is, adding and subtracting) of the consequences of general names agreed upon, for the marking and signifying of our thoughts." I Leibniz (1685) \When there are disputes among persons, we can simply say: Let us calculate [calculemus], without further ado, to see who is right." What is automated reasoning? Attempting to perform logical reasoning in an automatic and algorithmic way. -
Inventing Computational Rhetoric
INVENTING COMPUTATIONAL RHETORIC By Michael W. Wojcik A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Digital Rhetoric and Professional Writing — Master of Arts 2013 ABSTRACT INVENTING COMPUTATIONAL RHETORIC by Michael W. Wojcik Many disciplines in the humanities are developing “computational” branches which make use of information technology to process large amounts of data algorithmically. The field of computational rhetoric is in its infancy, but we are already seeing interesting results from applying the ideas and goals of rhetoric to text processing and related areas. After considering what computational rhetoric might be, three approaches to inventing computational rhetorics are presented: a structural schema, a review of extant work, and a theoretical exploration. Copyright by MICHAEL W. WOJCIK 2013 For Malea iv ACKNOWLEDGEMENTS Above all else I must thank my beloved wife, Malea Powell, without whose prompting this thesis would have remained forever incomplete. I am also grateful for the presence in my life of my terrific stepdaughter, Audrey Swartz, and wonderful granddaughter Lucille. My thesis committee, Dean Rehberger, Bill Hart-Davidson, and John Monberg, pro- vided me with generous guidance and inspiration. Other faculty members at Michigan State who helped me explore relevant ideas include Rochelle Harris, Mike McLeod, Joyce Chai, Danielle Devoss, and Bump Halbritter. My previous academic program at Miami University did not result in a degree, but faculty there also contributed greatly to my the- oretical understanding, particularly Susan Morgan, Mary-Jean Corbett, Brit Harwood, J. Edgar Tidwell, Lori Merish, Vicki Smith, Alice Adams, Fran Dolan, and Keith Tuma. -
Computability Theory
CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2019 Computability Theory This section is partly inspired by the material in \A Course in Mathematical Logic" by Bell and Machover, Chap 6, sections 1-10. Other references: \Introduction to the theory of computation" by Michael Sipser, and \Com- putability, Complexity, and Languages" by M. Davis and E. Weyuker. Our first goal is to give a formal definition for what it means for a function on N to be com- putable by an algorithm. Historically the first convincing such definition was given by Alan Turing in 1936, in his paper which introduced what we now call Turing machines. Slightly before Turing, Alonzo Church gave a definition based on his lambda calculus. About the same time G¨odel,Herbrand, and Kleene developed definitions based on recursion schemes. Fortunately all of these definitions are equivalent, and each of many other definitions pro- posed later are also equivalent to Turing's definition. This has lead to the general belief that these definitions have got it right, and this assertion is roughly what we now call \Church's Thesis". A natural definition of computable function f on N allows for the possibility that f(x) may not be defined for all x 2 N, because algorithms do not always halt. Thus we will use the symbol 1 to mean “undefined". Definition: A partial function is a function n f :(N [ f1g) ! N [ f1g; n ≥ 0 such that f(c1; :::; cn) = 1 if some ci = 1. In the context of computability theory, whenever we refer to a function on N, we mean a partial function in the above sense. -
Automated Reasoning for Explainable Artificial Intelligence∗
EPiC Series in Computing Volume 51, 2017, Pages 24–28 ARCADE 2017. 1st International Workshop on Automated Reasoning: Challenges, Appli- cations, Directions, Exemplary Achievements Automated Reasoning for Explainable Artificial Intelligence∗ Maria Paola Bonacina1 Dipartimento di Informatica Universit`adegli Studi di Verona Strada Le Grazie 15 I-37134 Verona, Italy, EU [email protected] Abstract Reasoning and learning have been considered fundamental features of intelligence ever since the dawn of the field of artificial intelligence, leading to the development of the research areas of automated reasoning and machine learning. This paper discusses the relationship between automated reasoning and machine learning, and more generally be- tween automated reasoning and artificial intelligence. We suggest that the emergence of the new paradigm of XAI, that stands for eXplainable Artificial Intelligence, is an oppor- tunity for rethinking these relationships, and that XAI may offer a grand challenge for future research on automated reasoning. 1 Artificial Intelligence, Automated Reasoning, and Ma- chine Learning Ever since the beginning of artificial intelligence, scientists concurred that the capability to learn and the capability to reason about problems and solve them are fundamental pillars of intelligent behavior. Similar to many subfields of artificial intelligence, automated reasoning and machine learning have grown into highly specialized research areas. Both have experienced the dichotomy between the ideal of imitating human intelligence, whereby the threshold of success would be the indistinguishability of human and machine, as in the Turing’s test [12], and the ideal that machines can and should learn and reason in their own way. According to the latter ideal, the measure of success is independent of similarity to human behavior. -
Introduction to the Theory of Computation Computability, Complexity, and the Lambda Calculus Some Notes for CIS262
Introduction to the Theory of Computation Computability, Complexity, And the Lambda Calculus Some Notes for CIS262 Jean Gallier and Jocelyn Quaintance Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA e-mail: [email protected] c Jean Gallier Please, do not reproduce without permission of the author April 28, 2020 2 Contents Contents 3 1 RAM Programs, Turing Machines 7 1.1 Partial Functions and RAM Programs . 10 1.2 Definition of a Turing Machine . 15 1.3 Computations of Turing Machines . 17 1.4 Equivalence of RAM programs And Turing Machines . 20 1.5 Listable Languages and Computable Languages . 21 1.6 A Simple Function Not Known to be Computable . 22 1.7 The Primitive Recursive Functions . 25 1.8 Primitive Recursive Predicates . 33 1.9 The Partial Computable Functions . 35 2 Universal RAM Programs and the Halting Problem 41 2.1 Pairing Functions . 41 2.2 Equivalence of Alphabets . 48 2.3 Coding of RAM Programs; The Halting Problem . 50 2.4 Universal RAM Programs . 54 2.5 Indexing of RAM Programs . 59 2.6 Kleene's T -Predicate . 60 2.7 A Non-Computable Function; Busy Beavers . 62 3 Elementary Recursive Function Theory 67 3.1 Acceptable Indexings . 67 3.2 Undecidable Problems . 70 3.3 Reducibility and Rice's Theorem . 73 3.4 Listable (Recursively Enumerable) Sets . 76 3.5 Reducibility and Complete Sets . 82 4 The Lambda-Calculus 87 4.1 Syntax of the Lambda-Calculus . 89 4.2 β-Reduction and β-Conversion; the Church{Rosser Theorem . 94 4.3 Some Useful Combinators . -
The Ratiolog Project: Rational Extensions of Logical Reasoning
The RatioLog Project: Rational Extensions of Logical Reasoning Ulrich Furbach · Claudia Schon · Frieder Stolzenburg · Karl-Heinz Weis · Claus-Peter Wirth Abstract Higher-level cognition includes logical reasoning 1 Rational Reasoning and Question Answering and the ability of question answering with common sense. The RatioLog project addresses the problem of rational rea- The development of formal logic played a big role in the soning in deep question answering by methods from auto- field of automated reasoning, which led to the development mated deduction and cognitive computing. In a first phase, of the field of artificial intelligence (AI). Applications of au- we combine techniques from information retrieval and ma- tomated deduction in mathematics have been investigated chine learning to find appropriate answer candidates from from the early years on. Nowadays automated deduction the huge amount of text in the German version of the free techniques are successfully applied in hard- and software encyclopedia “Wikipedia”. In a second phase, an automated verification and many other areas (for an overview see [2]). theorem prover tries to verify the answer candidates on In contrast to formal logical reasoning, however, human the basis of their logical representations. In a third phase reasoning does not strictly follow the rules of classical logic. — because the knowledge may be incomplete and inconsis- Reasons may be incomplete knowledge, incorrect beliefs, tent —, we consider extensions of logical reasoning to im- and inconsistent norms. From the very beginning of AI re- prove the results. In this context, we work toward the appli- search, there has been a strong emphasis on incorporating cation of techniques from human reasoning: We employ de- mechanisms for rationality, such as abductive or defeasible feasible reasoning to compare the answers w.r.t. -
Automated Reasoning on the Web
Automated Reasoning on the Web Slim Abdennadher1, Jos´eJ´ulioAlves Alferes2, Grigoris Antoniou3, Uwe Aßmann4, Rolf Backofen5, Cristina Baroglio6, Piero A. Bonatti7, Fran¸coisBry8, W lodzimierz Drabent9, Norbert Eisinger8, Norbert E. Fuchs10, Tim Geisler11, Nicola Henze12, Jan Ma luszy´nski4, Massimo Marchiori13, Alberto Martelli14, Sara Carro Mart´ınez15, Wolfgang May16, Hans J¨urgenOhlbach8, Sebastian Schaffert8, Michael Schr¨oder17, Klaus U. Schulz8, Uta Schwertel8, and Gerd Wagner18 1 German University in Cairo (Egypt), 2 New University of Lisbon (Portugal), 3 Institute of Computer Science, Foundation for Research and Technology – Hel- las (Greece), 4 University of Link¨oping (Sweden), 5 University of Jena (Germany), 6 University of Turin (Italy), 7 University of Naples (Italy), 8 University of Mu- nich (Germany), 9 Polish Academy of Sciences (Poland), 10 University of Zurich (Switzerland), 11 webXcerpt (Germany), 12 University of Hannover (Germany), 13 University of Venice (Italy), 14 University of Turin (Italy), 15 Telef´onica Investi- gaci´ony Desarollo (Spain), 16 University of G¨ottingen(Germany), 17 University of Dresden (Germany), 18 University of Eindhoven (The Netherlands) Abstract Automated reasoning is becoming an essential issue in many Web systems and applications, especially in emerging Semantic Web applications. This article first discusses reasons for this evolution. Then, it presents research issues currently investigated towards automated reasoning on the Web and it introduces into selected applications demonstrating the practical impact of the approach. Finally, it introduces a research endeavor called REWERSE (cf. http://rewerse.net) recently launched by the authors of this article which is concerned with developing automated reasoning methods and tools for the Web as well as demonstrator applications. -
Programming with Recursion
Purdue University Purdue e-Pubs Department of Computer Science Technical Reports Department of Computer Science 1980 Programming With Recursion Dirk Siefkes Report Number: 80-331 Siefkes, Dirk, "Programming With Recursion" (1980). Department of Computer Science Technical Reports. Paper 260. https://docs.lib.purdue.edu/cstech/260 This document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact [email protected] for additional information. '. \ PI~Q(:[V\!'<lMTNr. 11'1'1'11 nr:CUI~SrON l1irk Siefke$* Tcchnische UniveTsit~t Berlin Inst. Soft\~are Theor. Informatik 1 Berlin 10, West Germany February, 1980 Purdue University Tech. Report CSD-TR 331 Abstract: Data structures defined as term algebras and programs built from recursive definitions complement each other. Computing in such surroundings guides us into Nriting simple programs Nith a clear semantics and performing a rigorous cost analysis on appropriate data structures. In this paper, we present a programming language so perceived, investigate some basic principles of cost analysis through it, and reflect on the meaning of programs and computing. -;;----- - -~~ Visiting at. the Computer Science Department of Purdue University with the help of a grant of the Stiftung Volksl.... agem.... erk. I am grateful for the support of both, ViV-Stiftung and Purdue. '1 1 CONTENTS O. Introduction I. 'llle formnlislIl REe Syntax and semantics of REC-programs; the formalisms REC(B) and RECS; recursive functions. REC l~ith VLI-r-t'iblcs. -2. Extensions and variations Abstract languages and compilers. REC over words as data structure; the formalisms REC\\'(q). REe for functions with variable I/O-dimension; the forma l:i sms VREC (B) . -
Computer Algorithms As Persuasive Agents: the Rhetoricity of Algorithmic Surveillance Within the Built Ecological Network
COMPUTER ALGORITHMS AS PERSUASIVE AGENTS: THE RHETORICITY OF ALGORITHMIC SURVEILLANCE WITHIN THE BUILT ECOLOGICAL NETWORK Estee N. Beck A Dissertation Submitted to the Graduate College of Bowling Green State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY May 2015 Committee: Kristine Blair, Advisor Patrick Pauken, Graduate Faculty Representative Sue Carter Wood Lee Nickoson © 2015 Estee N. Beck All Rights Reserved iii ABSTRACT Kristine Blair, Advisor Each time our students and colleagues participate online, they face invisible tracking technologies that harvest metadata for web customization, targeted advertising, and even state surveillance activities. This practice may be of concern for computers and writing and digital humanities specialists, who examine the ideological forces in computer-mediated writing spaces to address power inequality, as well as the role ideology plays in shaping human consciousness. However, the materiality of technology—the non-human objects that surrounds us—is of concern to those within rhetoric and composition as well. This project shifts attention to the materiality of non-human objects, specifically computer algorithms and computer code. I argue that these technologies are powerful non-human objects that have rhetorical agency and persuasive abilities, and as a result shape the everyday practices and behaviors of writers/composers on the web as well as other non-human objects. Through rhetorical inquiry, I examine literature from rhetoric and composition, surveillance studies, media and software studies, sociology, anthropology, and philosophy. I offer a “built ecological network” theory that is the manufactured and natural rhetorical rhizomatic network full of spatial, material, social, linguistic, and dynamic energy layers that enter into reflexive and reciprocal relations to support my claim that computer algorithms have agency and persuasive abilities.