Church's Thesis and the Conceptual Analysis of Computability
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“The Church-Turing “Thesis” As a Special Corollary of Gödel's
“The Church-Turing “Thesis” as a Special Corollary of Gödel’s Completeness Theorem,” in Computability: Turing, Gödel, Church, and Beyond, B. J. Copeland, C. Posy, and O. Shagrir (eds.), MIT Press (Cambridge), 2013, pp. 77-104. Saul A. Kripke This is the published version of the book chapter indicated above, which can be obtained from the publisher at https://mitpress.mit.edu/books/computability. It is reproduced here by permission of the publisher who holds the copyright. © The MIT Press The Church-Turing “ Thesis ” as a Special Corollary of G ö del ’ s 4 Completeness Theorem 1 Saul A. Kripke Traditionally, many writers, following Kleene (1952) , thought of the Church-Turing thesis as unprovable by its nature but having various strong arguments in its favor, including Turing ’ s analysis of human computation. More recently, the beauty, power, and obvious fundamental importance of this analysis — what Turing (1936) calls “ argument I ” — has led some writers to give an almost exclusive emphasis on this argument as the unique justification for the Church-Turing thesis. In this chapter I advocate an alternative justification, essentially presupposed by Turing himself in what he calls “ argument II. ” The idea is that computation is a special form of math- ematical deduction. Assuming the steps of the deduction can be stated in a first- order language, the Church-Turing thesis follows as a special case of G ö del ’ s completeness theorem (first-order algorithm theorem). I propose this idea as an alternative foundation for the Church-Turing thesis, both for human and machine computation. Clearly the relevant assumptions are justified for computations pres- ently known. -
A Compositional Analysis for Subset Comparatives∗ Helena APARICIO TERRASA–University of Chicago
A Compositional Analysis for Subset Comparatives∗ Helena APARICIO TERRASA–University of Chicago Abstract. Subset comparatives (Grant 2013) are amount comparatives in which there exists a set membership relation between the target and the standard of comparison. This paper argues that subset comparatives should be treated as regular phrasal comparatives with an added presupposi- tional component. More specifically, subset comparatives presuppose that: a) the standard has the property denoted by the target; and b) the standard has the property denoted by the matrix predi- cate. In the account developed below, the presuppositions of subset comparatives result from the compositional principles independently required to interpret those phrasal comparatives in which the standard is syntactically contained inside the target. Presuppositions are usually taken to be li- censed by certain lexical items (presupposition triggers). However, subset comparatives show that presuppositions can also arise as a result of semantic composition. This finding suggests that the grammar possesses more than one way of licensing these inferences. Further research will have to determine how productive this latter strategy is in natural languages. Keywords: Subset comparatives, presuppositions, amount comparatives, degrees. 1. Introduction Amount comparatives are usually discussed with respect to their degree or amount interpretation. This reading is exemplified in the comparative in (1), where the elements being compared are the cardinalities corresponding to the sets of books read by John and Mary respectively: (1) John read more books than Mary. |{x : books(x) ∧ John read x}| ≻ |{y : books(y) ∧ Mary read y}| In this paper, I discuss subset comparatives (Grant (to appear); Grant (2013)), a much less studied type of amount comparative illustrated in the Spanish1 example in (2):2 (2) Juan ha leído más libros que El Quijote. -
Interpretations and Mathematical Logic: a Tutorial
Interpretations and Mathematical Logic: a Tutorial Ali Enayat Dept. of Philosophy, Linguistics, and Theory of Science University of Gothenburg Olof Wijksgatan 6 PO Box 200, SE 405 30 Gothenburg, Sweden e-mail: [email protected] 1. INTRODUCTION Interpretations are about `seeing something as something else', an elusive yet intuitive notion that has been around since time immemorial. It was sys- tematized and elaborated by astrologers, mystics, and theologians, and later extensively employed and adapted by artists, poets, and scientists. At first ap- proximation, an interpretation is a structure-preserving mapping from a realm X to another realm Y , a mapping that is meant to bring hidden features of X and/or Y into view. For example, in the context of Freud's psychoanalytical theory [7], X = the realm of dreams, and Y = the realm of the unconscious; in Khayaam's quatrains [11], X = the metaphysical realm of theologians, and Y = the physical realm of human experience; and in von Neumann's foundations for Quantum Mechanics [14], X = the realm of Quantum Mechanics, and Y = the realm of Hilbert Space Theory. Turning to the domain of mathematics, familiar geometrical examples of in- terpretations include: Khayyam's interpretation of algebraic problems in terms of geometric ones (which was the remarkably innovative element in his solution of cubic equations), Descartes' reduction of Geometry to Algebra (which moved in a direction opposite to that of Khayyam's aforementioned interpretation), the Beltrami-Poincar´einterpretation of hyperbolic geometry within euclidean geom- etry [13]. Well-known examples in mathematical analysis include: Dedekind's interpretation of the linear continuum in terms of \cuts" of the rational line, Hamilton's interpertation of complex numbers as points in the Euclidean plane, and Cauchy's interpretation of real-valued integrals via complex-valued ones using the so-called contour method of integration in complex analysis. -
Enumerations of the Kolmogorov Function
Enumerations of the Kolmogorov Function Richard Beigela Harry Buhrmanb Peter Fejerc Lance Fortnowd Piotr Grabowskie Luc Longpr´ef Andrej Muchnikg Frank Stephanh Leen Torenvlieti Abstract A recursive enumerator for a function h is an algorithm f which enu- merates for an input x finitely many elements including h(x). f is a aEmail: [email protected]. Department of Computer and Information Sciences, Temple University, 1805 North Broad Street, Philadelphia PA 19122, USA. Research per- formed in part at NEC and the Institute for Advanced Study. Supported in part by a State of New Jersey grant and by the National Science Foundation under grants CCR-0049019 and CCR-9877150. bEmail: [email protected]. CWI, Kruislaan 413, 1098SJ Amsterdam, The Netherlands. Partially supported by the EU through the 5th framework program FET. cEmail: [email protected]. Department of Computer Science, University of Mas- sachusetts Boston, Boston, MA 02125, USA. dEmail: [email protected]. Department of Computer Science, University of Chicago, 1100 East 58th Street, Chicago, IL 60637, USA. Research performed in part at NEC Research Institute. eEmail: [email protected]. Institut f¨ur Informatik, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany. fEmail: [email protected]. Computer Science Department, UTEP, El Paso, TX 79968, USA. gEmail: [email protected]. Institute of New Techologies, Nizhnyaya Radi- shevskaya, 10, Moscow, 109004, Russia. The work was partially supported by Russian Foundation for Basic Research (grants N 04-01-00427, N 02-01-22001) and Council on Grants for Scientific Schools. hEmail: [email protected]. School of Computing and Department of Mathe- matics, National University of Singapore, 3 Science Drive 2, Singapore 117543, Republic of Singapore. -
Bundled Fragments of First-Order Modal Logic: (Un)Decidability
Bundled Fragments of First-Order Modal Logic: (Un)Decidability Anantha Padmanabha Institute of Mathematical Sciences, HBNI, Chennai, India [email protected] https://orcid.org/0000-0002-4265-5772 R Ramanujam Institute of Mathematical Sciences, HBNI, Chennai, India [email protected] Yanjing Wang Department of Philosophy, Peking University, Beijing, China [email protected] Abstract Quantified modal logic is notorious for being undecidable, with very few known decidable frag- ments such as the monodic ones. For instance, even the two-variable fragment over unary predic- ates is undecidable. In this paper, we study a particular fragment, namely the bundled fragment, where a first-order quantifier is always followed by a modality when occurring in the formula, inspired by the proposal of [15] in the context of non-standard epistemic logics of know-what, know-how, know-why, and so on. As always with quantified modal logics, it makes a significant difference whether the domain stays the same across possible worlds. In particular, we show that the predicate logic with the bundle ∀ alone is undecidable over constant domain interpretations, even with only monadic predicates, whereas having the ∃ bundle instead gives us a decidable logic. On the other hand, over increasing domain interpretations, we get decidability with both ∀ and ∃ bundles with unrestricted predicates, where we obtain tableau based procedures that run in PSPACE. We further show that the ∃ bundle cannot distinguish between constant domain and variable domain interpretations. 2012 ACM Subject Classification Theory of computation → Modal and temporal logics Keywords and phrases First-order modal logic, decidability, bundled fragments Digital Object Identifier 10.4230/LIPIcs.FSTTCS.2018.43 Acknowledgements We thank the reviewers for the insightful and helpful comments. -
How to Build an Interpretation and Use Argument
Timothy S. Brophy Professor and Director, Institutional Assessment How to Build an University of Florida Email: [email protected] Interpretation and Phone: 352‐273‐4476 Use Argument To review the concept of validity as it applies to higher education Today’s Goals Provide a framework for developing an interpretation and use argument for assessments Validity What it is, and how we examine it • Validity refers to the degree to which evidence and theory support the interpretations of the test scores for proposed uses of tests. (p. 11) defined Source: • American Educational Research Association (AERA), American Psychological Association (APA), & National Council on Measurement in Education (NCME). (2014). Standards for educational and psychological testing. Validity Washington, DC: AERA The Importance of Validity The process of validation Validity is, therefore, the most involves accumulating relevant fundamental consideration in evidence to provide a sound developing tests and scientific basis for the proposed assessments. score and assessment results interpretations. (p. 11) Source: AERA, APA, & NCME. (2014). Standards for educational and psychological testing. Washington, DC: AERA • Most often this is qualitative; colleagues are a good resource • Review the evidence How Do We • Common sources of validity evidence Examine • Test Content Validity in • Construct (the idea or theory that supports the Higher assessment) • The validity coefficient Education? Important distinction It is not the test or assessment itself that is validated, but the inferences one makes from the measure based on the context of its use. Therefore it is not appropriate to refer to the ‘validity of the test or assessment’ ‐ instead, we refer to the validity of the interpretation of the results for the test or assessment’s intended purpose. -
Theory of Computation
A Universal Program (4) Theory of Computation Prof. Michael Mascagni Florida State University Department of Computer Science 1 / 33 Recursively Enumerable Sets (4.4) A Universal Program (4) The Parameter Theorem (4.5) Diagonalization, Reducibility, and Rice's Theorem (4.6, 4.7) Enumeration Theorem Definition. We write Wn = fx 2 N j Φ(x; n) #g: Then we have Theorem 4.6. A set B is r.e. if and only if there is an n for which B = Wn. Proof. This is simply by the definition ofΦ( x; n). 2 Note that W0; W1; W2;::: is an enumeration of all r.e. sets. 2 / 33 Recursively Enumerable Sets (4.4) A Universal Program (4) The Parameter Theorem (4.5) Diagonalization, Reducibility, and Rice's Theorem (4.6, 4.7) The Set K Let K = fn 2 N j n 2 Wng: Now n 2 K , Φ(n; n) #, HALT(n; n) This, K is the set of all numbers n such that program number n eventually halts on input n. 3 / 33 Recursively Enumerable Sets (4.4) A Universal Program (4) The Parameter Theorem (4.5) Diagonalization, Reducibility, and Rice's Theorem (4.6, 4.7) K Is r.e. but Not Recursive Theorem 4.7. K is r.e. but not recursive. Proof. By the universality theorem, Φ(n; n) is partially computable, hence K is r.e. If K¯ were also r.e., then by the enumeration theorem, K¯ = Wi for some i. We then arrive at i 2 K , i 2 Wi , i 2 K¯ which is a contradiction. -
Interpreting Arguments
Informal Logic IX.1, Winter 1987 Interpreting Arguments JONATHAN BERG University of Haifa We often speak of "the argument in" Two notions which would merit ex a particular piece of discourse. Although tensive analysis in a broader context will such talk may not be essential to only get a bit of clarification here. By philosophy in the philosophical sense, 'claims' I mean what are variously call it is surely a practical requirement for ed 'propositions', 'statements', or 'con most work in philosophy, whether in the tents'. I remain neutral with regard to interpretation of classical texts or in the their ontological status and shall not pre discussion of contemporary work, and tend to know exactly how to individuate it arises as well in everyday contexts them, especially in relation to sentences. wherever we care about reasons given I shall assume that for every claim there in support of a claim. (I assume this is is at least one literal, declarative why developing the ability to identify sentence typically used for making that arguments in discourse is a major goal claim, and often I shall not distinguish in many courses in informal logic or between the claim, proper, and such a critical thinking-the major goal when I sentence. teach the subject.) Thus arises the ques What I mean when I say that one tion, just how do we get from text to claim follows from other claims is much argument? harder to specify than what I do not I shall propose a set of principles for mean. I certainly do not have in mind the extrication of argu ments from any mere psychological or causal con argumentative prose. -
Why We Should Abandon the Semantic Subset Principle
LANGUAGE LEARNING AND DEVELOPMENT https://doi.org/10.1080/15475441.2018.1499517 Why We Should Abandon the Semantic Subset Principle Julien Musolinoa, Kelsey Laity d’Agostinob, and Steve Piantadosic aDepartment of Psychology and Center for Cognitive Science, Rutgers University, Piscataway, NJ, USA; bDepartment of Philosophy, Rutgers University, New Brunswick, USA; cDepartment of Brain and Cognitive Sciences, University of Rochester, Rochester, USA ABSTRACT In a recent article published in this journal, Moscati and Crain (M&C) show- case the explanatory power of a learnability constraint called the Semantic Subset Principle (SSP) (Crain et al. 1994). If correct, M&C’s argument would represent a compelling demonstration of the operation of an innate, domain specific, learning principle. However, in trying to make the case for the SSP, M&C fail to clearly define their hypothesis, omit discussion of key facts, and contradict the theory itself. Moreover, their presentation does not address the core arguments and alternatives against their account available in the literature and misrepresents the position of their critics. Once these shortcomings are understood, a very different conclusion emerges: its historical significance notwithstanding, the SSP represents a flawed and obsolete account that should be abandoned. We discuss the implications of the demise of the SSP and describe more powerful and plausible alternatives that provide a better foundation for ongoing work in language acquisition. Introduction A common assumption within generative models of language acquisition is that learners are guided by a built-in constraint called the Subset Principle (SP) (Baker, 1979; Pinker, 1979; Dell, 1981; Berwick, 1985; Manzini & Wexler, 1987; among others). -
Soundness of Workflow Nets: Classification
DOI 10.1007/s00165-010-0161-4 The Author(s) © 2010. This article is published with open access at Springerlink.com Formal Aspects Formal Aspects of Computing (2011) 23: 333–363 of Computing Soundness of workflow nets: classification, decidability, and analysis W. M. P. van der Aalst1,2,K.M.vanHee1, A. H. M. ter Hofstede1,2,N.Sidorova1, H. M. W. Verbeek1, M. Voorhoeve1 andM.T.Wynn2 1 Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: [email protected] 2 Queensland University of Technology, Brisbane, Australia Abstract. Workflow nets, a particular class of Petri nets, have become one of the standard ways to model and analyze workflows. Typically, they are used as an abstraction of the workflow that is used to check the so-called soundness property. This property guarantees the absence of livelocks, deadlocks, and other anomalies that can be detected without domain knowledge. Several authors have proposed alternative notions of soundness and have suggested to use more expressive languages, e.g., models with cancellations or priorities. This paper provides an overview of the different notions of soundness and investigates these in the presence of different extensions of workflow nets. We will show that the eight soundness notions described in the literature are decidable for workflow nets. However, most extensions will make all of these notions undecidable. These new results show the theoretical limits of workflow verification. Moreover, we discuss some of the analysis approaches described in the literature. Keywords: Petri nets, Decidability, Workflow nets, Reset nets, Soundness, Verification 1. -
Undecidable Problems: a Sampler
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. -
The Interplay Between Computability and Incomputability Draft 619.Tex
The Interplay Between Computability and Incomputability Draft 619.tex Robert I. Soare∗ January 7, 2008 Contents 1 Calculus, Continuity, and Computability 3 1.1 When to Introduce Relative Computability? . 4 1.2 Between Computability and Relative Computability? . 5 1.3 The Development of Relative Computability . 5 1.4 Turing Introduces Relative Computability . 6 1.5 Post Develops Relative Computability . 6 1.6 Relative Computability in Real World Computing . 6 2 Origins of Computability and Incomputability 6 2.1 G¨odel’s Incompleteness Theorem . 8 2.2 Incomputability and Undecidability . 9 2.3 Alonzo Church . 9 2.4 Herbrand-G¨odel Recursive Functions . 10 2.5 Stalemate at Princeton Over Church’s Thesis . 11 2.6 G¨odel’s Thoughts on Church’s Thesis . 11 ∗Parts of this paper were delivered in an address to the conference, Computation and Logic in the Real World, at Siena, Italy, June 18–23, 2007. Keywords: Turing ma- chine, automatic machine, a-machine, Turing oracle machine, o-machine, Alonzo Church, Stephen C. Kleene,klee Alan Turing, Kurt G¨odel, Emil Post, computability, incomputabil- ity, undecidability, Church-Turing Thesis (CTT), Post-Church Second Thesis on relative computability, computable approximations, Limit Lemma, effectively continuous func- tions, computability in analysis, strong reducibilities. 1 3 Turing Breaks the Stalemate 12 3.1 Turing’s Machines and Turing’s Thesis . 12 3.2 G¨odel’s Opinion of Turing’s Work . 13 3.3 Kleene Said About Turing . 14 3.4 Church Said About Turing . 15 3.5 Naming the Church-Turing Thesis . 15 4 Turing Defines Relative Computability 17 4.1 Turing’s Oracle Machines .