Church s s Thesis and Hume s s Problem:
|
|
|
- David Marsh
- 5 years ago
- Views:
Transcription
1 Church s s Thesis and Hume s s Problem: Justification as Performance Kevin T. Kelly Department of Philosophy Carnegie Mellon University kk3n@andrew.cmu.edu
2 Further Reading (With C. Glymour) Why You'll Never Know Whether Roger Penrose is a Computer, Behavioral and Brain Sciences, 13: The Logic of Reliable Inquiry, Oxford: Oxford University Press, (with O. Schulte) Church's Thesis and Hume's Problem, in Logic and Scientific Methods, M. L. Dalla Chiara, et al., eds. Dordrecht: Kluwer, (with O. Schulte and C. Juhl) Learning Theory and the Philosophy of Science, Philosophy of Science 64: The Logic of Success, British Journal for the Philosophy of Science, 51:2000. Uncomputability: The Problem of Induction Internalized, Theoretical Computer Science 317: Computability and Learnibility, Textbook manuscript, 2005.
3 Fateful First Cut Relations of Ideas Analytic Certain A Priori Philosophy of Math Proofs and algorithms Truth-finding Computability Matters of fact Synthetic Uncertain A Posteriori Philosophy of Science Confirmation Rationality Probability
4 Painful Progress Relations of Ideas Analytic Certain A Priori Philosophy of Math Proofs and algorithms Truth-finding Computability Matters of fact Synthetic Uncertain A Posteriori Philosophy of Science Confirmation Rationality Probability
5 Slight Rearrangement Philosophy of Math Philosophy of Science Certain A Priori Uncertain A Posteriori Proofs and algorithms Truth-finding Computability Confirmation Rationality Probability
6 Philosophy of Science 101 Formal Questions Scientific Questions Unverifiable Verifiable The sun will never stop rising The given number is even
7 Wrong Twice! Formal Questions Unverifiable The given computation will never halt Verifiable Scientific Questions Unverifiable The sun will never stop rising Verifiable The given number is even The next emerald is green
8 Classical Excuse Relations of Ideas Matters of fact Completed analysis Rational Animal Unbounded experience Rational??? Always black? Reason Observation
9 But Maybe Relations of Ideas Matters of fact Unbounded analysis Unbounded experience?? Rational??? Always Never white? black? Reason Observation
10 Kant Synthetic A Priori A Posteriori Unbounded experience temporal intuition 3+2 = 5??? Always Never white? black? Intuition Observation
11 Oops! Synthetic A Priori A Posteriori spatial intuition Unbounded experience?? Never cross??? Always Never white? black? Intuition Observation
12 After Turing Formal Problems Empirical Problems Unbounded computation Input n Unbounded reality???? Never halts? Always Never white? black? Computer Observation
13 Bad First Cut Formal Questions Unverifiable The given computation will never halt Verifiable Scientific Questions Unverifiable The sun will never stop rising Verifiable The given number is even The next emerald is green
14 Good First Cut Formal Questions Unverifiable The given computation will never halt Verifiable Scientific Questions Unverifiable The sun will never stop rising Verifiable The given number is even The next emerald is green
15 Computational Epistemology For formal and empirical questions: Justification = truth-finding performance Find the truth in the best possible sense And then as efficiently as possible.
16 Verification as Support 1.0 Deductive verification.5 Inductive verification
17 Paradigms Evolve Objective support by evidence.
18 Paradigms Evolve Objective support by evidence. Coherent personal opinion. Rational update.
19 Paradigms Evolve Objective support by evidence. Coherent personal opinion. Rational update.
20 But Leading Metaphors Matter Myopic Bedrock = rationality intuitions Success = being rational Truth-finding performance deferred or ignored Problem complexity ignored Little resemblance to computability
21 Verification as Halting Conclusion Data
22 Verification as Halting Conclusion Data Yes!
23 Verification as Halting Conclusion Data
24 Verification as Halting Conclusion Data
25 Verification as Halting Conclusion Data Bang!
26 Verification as Halting Conclusion Data Can t take it back.
27 Verifiability Conclusion true Yes! Conclusion false
28 Laws are Not Verifiable Always green?
29 Laws are Not Verifiable Always green?
30 Laws are Not Verifiable Always green?
31 Laws are Not Verifiable Always green?
32 Laws are Not Verifiable Always green? You must halt with yes if it s true!
33 Laws are Not Verifiable Always green? You must halt with yes if it s true!
34 Laws are Not Verifiable Always green? Yes!
35 Laws are Not Verifiable Always green?
36 Laws are Not Verifiable Always green?
37 A Purely Formal Problem Never shoots?
38 Analogy? I ain t a shootin, yeller belly! Never shoots?
39 Apparently Not Game s up, J.R.! Yer program s showin! Never shoots? 666
40 Apparently Not Now shaddup! I m calculatin a-preeorey-like what yer gonna do. Never shoots? 666
41 Apparently Not?!!! Never shoots?
42 But Then Again I got it! Two kin play that opry oary game! Never shoots?
43 But Then Again I won t shoot til I see him shoot in this here opry oary 666 sim-yooo-laytion! Never shoots?
44 But Then Again I ain t a shootin Never shoots?
45 But Then Again I still ain t a shootin Never shoots?
46 But Then Again You check that there opry oary all you want Never shoots?
47 But Then Again An all you ll see is I ain t a-shootin Never shoots?
48 But Then Again Aw, shucks, J.R. I reckon You ain t never gonna shoot Aw, shucks, J.R. I reckon you ain t never gonna shoot. Never shoots?
49 But Then Again Never shoots? 666 Bang! 23455
50 But Then Again Never shoots?
51 But Then Again See ya in the grand ol opry oary, sucker! Never shoots?
52 But Then Again Take that! Never shoots? Bang!
53 But Then Again Never shoots?
54 But Then Again ROBO SHERIFF Fooled A priori Never shoots?
55 Rice-Shapiro Theorem Whatever a computable cognitive scientist could verify about an arbitrary computer s input-output behavior by formally analyzing its program Q QDoc
56 Rice-Shapiro Theorem could also have been verified by a behaviorist computer who empirically studies only the arbitrary computer s inputoutput behavior! Q QDoc
57 Uncomputable Induction H N Yes! Will see a non-halting machine Verifiable by ideal agent.
58 Uncomputable Induction H N? Will see a non-halting machine Q Not verifiable by computable agent.
59 Similarity Halting Problem Unverifiable Demon can fool computable agent Answer runs beyond formal experience Universal Law Unverifiable Demon can fool ideal agent Answer runs beyond empirical experience
60 A Difference Halting Problem Input ends Can handle some examples a priori Universal Law Input never ends Problem of induction Internal problem of induction eventually catches up!
61 Give and Take Formal input Jaw Breaker Fits in mouth but may never melt May never swallow Empirical input Noodle May never fit in mouth May never swallow...
62 Another Difference Formal Science Incompleteness Empirical Science Problem of induction
63 Not Necessarily Formal Science Incompleteness Add more powerful axiom Who knows if new axiom is consistent with old? Assume it is, keep checking, and hope Empirical Science Problem of induction Conjecture a theory Who knows if theory is consistent with data? Assume it is, keep checking, and hope
64 Thesis The problem of induction and the problem of uncomputability are essentially similar. So the two should be understood similarly. From the ground upward.
65 Unverifiability Confirmation Support Coherence Rational update, etc. A House Divided Uncomputability. Suck it up Logic vs. engineering Try to approximate, etc.?!
66 Unified Approach: Gun Control Find the truth in the best feasible sense. So drop the halting requirement if infeasible. Q
67 Start with Problems,, Not Methods Partition Q over set K of inputs A B C D Finite input Infinite input
68 Convergence Convergence with halting????a Convergence in limit (in limit) No more revisions??? A?? B C? C C... revisions
69 Success Converge to right answer A B C D Don t converge to wrong answer
70 Special Cases Verify A Refute A A B A B Decide A A B
71 Special Cases Compute partial φ φ φ = 0 φ = 1 φ = 2... Dom(φ) Theory selection Μ 0 Μ 1 Μ 2...
72 Solutions and Optimality Solution: method that succeeds on each input. Optimal solution: solves in best possible sense Solvable problem: has solution. Problem complexity: best sense of solvability
73 A B Verifiability convergence with 1 revision ending with no. Refutability convergence with 1 revision ending with yes. Decidability convergence with 0 revisions.
74 A B n-refutation = n+1 rev ending with -A n-verification = n+1 rev ending with A n-decision = n rev
75 ... Underdetermination and open Lim-ver Lim-ref Lim-dec... 1-ver 1-ref 1-dec 0-ver 0-ref 0-dec clopen or closed
76 ... Lim-ver Lim-ref Uncomputability and r.e. Lim-dec... 1-ver 1-ref 1-dec 0-ver 0-ref 0-dec decidable or Co-r.e.
77 ... Uncomputability + Underdetermination and r.e. Lim-ver Lim-ref Lim-dec... 1-ver 1-ref 1-dec 0-ver 0-ref 0-dec decidable or Co-r.e.
78 Ideal empirical Effective empirical Effective formal
79 Topological invariants Recursive invariants Recursive invariants
80 Lower Bound Combination Lower Bound
81 Empirical Borel Mixed effective Borel Formal arithmetical
82 Empirical Borel Mixed effective Borel Formal arithmetical
83 Infinitely divisible Finitely divisible Matter
84 Lim-ref Infinitely divisible Finitely divisible Lim-ver I say inf when you let me cut fin fin inf fin
85 Lim-ref -Lim-ver Infinitely divisible Finitely divisible Lim-ver -Lim-ref I let you cut when you say fin inf fin inf inf
86 Lim-ref -Lim-ver Infinite domain Finite domain Lim-ver -Lim-ref
87 Lim-ref -Lim-ver Infinite domain Finite domain Lim-ver -Lim-ref I halt on another input each time he says fin in my opry oary simulation of him ganderin at my program
88 Finite divisibility Finite divisibility Finite domain... Ideal empirical Effective empirical Effective formal
89 Computable Uncomputable???. Human subject Cognitive scientist
90 Computable Uncomputable You can verify human computability in the limit UCUCCCCCCCC Human subject Cognitive scientist
91 Computable Uncomputable But only if you aren t computable! UCUCUCCUCCU Human subject Cognitive scientist
92 Ideal Lim-ver Lim-ref Lim-dec
93 We can verify our computability in the limit only if we are not computable! Computable Lim-ver Lim-ref Lim-dec
94 Even assuming computability, you can converge to the true computable behavior only if you are not computable! Total computable functions φ 0 φ 32 φ
95 (with Oliver Schulte) T -T There exists T such that T makes a unique prediction at each stage; The predictions are very uncomputable
96 (with Oliver Schulte) T -T There exists T such that T makes a unique prediction at each stage; The predictions are very uncomputable But T is refutable by a computable method!
97 Ideal rationality No consistent computable convergent method. Exists inconsistent computable refuting method. Exists consistent computable non-convergent method. So a foolish consistency precludes truth-finding!
98 The very idea of insulating deduction from empirical data restricts the truth-finding power of effective science. Empirical data n Yes Theorem Prover? Q
99 The very idea of insulating deduction from empirical data restricts the truth-finding power of effective science. Empirical data Theorem Prover n n Q Yes
100 Q Justification = truth-finding performance. Performance depends on problem complexity. Formal and empirical complexity are similar. Computational epistemology is unified Standard epistemology is divided. Insistence on division weakens truth-finding performance of effective science.
101 Q Standard epistemology Q Computational epistemology
102 I m rational Q Standard epistemology Q Computational epistemology
103 I m rational Q Standard epistemology Q Computational epistemology
104
Ockham Efficiency Theorem for Randomized Scientific Methods
Ockham Efficiency Theorem for Randomized Scientific Methods Conor Mayo-Wilson and Kevin T. Kelly Department of Philosophy Carnegie Mellon University Formal Epistemology Workshop (FEW) June 19th, 2009 1
Belief Revision and Truth-Finding
Belief Revision and Truth-Finding Kevin T. Kelly Department of Philosophy Carnegie Mellon University kk3n@andrew.cmu.edu Further Reading (with O. Schulte and V. Hendricks) Reliable Belief Revision, in
Epistemological and Computational Constraints of Simulation Support for OR Questions
Epistemological and Computational Constraints of Simulation Support for OR Questions Andreas Tolk, PhD Approved for Public Release; Distribution Unlimited. Case Number 16-3321 2 M&S as a Discipline Foundations
Gödel s Incompleteness Theorem. Overview. Computability and Logic
Gödel s Incompleteness Theorem Overview Computability and Logic Recap Remember what we set out to do in this course: Trying to find a systematic method (algorithm, procedure) which we can use to decide,
Formal Epistemology: Lecture Notes. Horacio Arló-Costa Carnegie Mellon University
Formal Epistemology: Lecture Notes Horacio Arló-Costa Carnegie Mellon University hcosta@andrew.cmu.edu Bayesian Epistemology Radical probabilism doesn t insists that probabilities be based on certainties;
Lecture 14 Rosser s Theorem, the length of proofs, Robinson s Arithmetic, and Church s theorem. Michael Beeson
Lecture 14 Rosser s Theorem, the length of proofs, Robinson s Arithmetic, and Church s theorem Michael Beeson The hypotheses needed to prove incompleteness The question immediate arises whether the incompleteness
Cogito ergo sum non machina!
Cogito ergo sum non machina! About Gödel s First Incompleteness Theorem and Turing machines. Ricardo Pereira Tassinari 1 Philosophy Department of State University of São Paulo - UNESP - Campus Marília
What are the recursion theoretic properties of a set of axioms? Understanding a paper by William Craig Armando B. Matos
What are the recursion theoretic properties of a set of axioms? Understanding a paper by William Craig Armando B. Matos armandobcm@yahoo.com February 5, 2014 Abstract This note is for personal use. It
Creative Objectivism, a powerful alternative to Constructivism
Creative Objectivism, a powerful alternative to Constructivism Copyright c 2002 Paul P. Budnik Jr. Mountain Math Software All rights reserved Abstract It is problematic to allow reasoning about infinite
Gödel s Incompleteness Theorem. Overview. Computability and Logic
Gödel s Incompleteness Theorem Overview Computability and Logic Recap Remember what we set out to do in this course: Trying to find a systematic method (algorithm, procedure) which we can use to decide,
2 KEVIN T. KELLY AND OLIVER SCHULTE to an arbitrarily small extent. :::The dierence from our point of view between the single and compound symbols is
KEVIN T. KELLY AND OLIVER SCHULTE CHURCH'S THESIS AND HUME'S PROBLEM ABSTRACT. We argue that uncomputability and classical scepticism are both reections of inductive underdetermination, so that Church's
HOW TO DO THINGS WITH AN INFINITE REGRESS
HOW TO DO THINGS WITH AN INFINITE REGRESS Kevin T. Kelly Department of Philosophy Carnegie Mellon University August 14, 2000 Abstract Scientific method may be viewed either as an argument justifying a
The Arithmetical Hierarchy
Chapter 11 The Arithmetical Hierarchy Think of K as posing the problem of induction for computational devices, for it is impossible to tell for sure whether a given computation will never halt. Thus, K
Decidability: Church-Turing Thesis
Decidability: Church-Turing Thesis While there are a countably infinite number of languages that are described by TMs over some alphabet Σ, there are an uncountably infinite number that are not Are there
Agency and Interaction in Formal Epistemology
Agency and Interaction in Formal Epistemology Vincent F. Hendricks Department of Philosophy / MEF University of Copenhagen Denmark Department of Philosophy Columbia University New York / USA CPH / August
On some Metatheorems about FOL
On some Metatheorems about FOL February 25, 2014 Here I sketch a number of results and their proofs as a kind of abstract of the same items that are scattered in chapters 5 and 6 in the textbook. You notice
2 KEVIN KELLY, OLIVER SCHULTE, VINCENT HENDRICKS In this paper, we provide a learning theoretic analysis of the eects of the AGM axioms on reliabile i
KEVIN KELLY, OLIVER SCHULTE, VINCENT HENDRICKS RELIABLE BELIEF REVISION ABSTRACT. Philosophical logicians proposing theories of rational belief revision have had little to say about whether their proposals
09. Kant and Logical Positivism Topics:
09. Kant and Logical Positivism Topics: I. Two Types of Skepticism II. Kant s Proposal III. Logical Positivism I. Two Types of Skepticism Metaphysical skepticism = Skepticism about claims that go beyond
Undecidability. Andreas Klappenecker. [based on slides by Prof. Welch]
![Undecidability. Andreas Klappenecker. [based on slides by Prof. Welch]](/thumbs/81/83378912.jpg "Undecidability. Andreas Klappenecker. [based on slides by Prof. Welch]")
Undecidability Andreas Klappenecker [based on slides by Prof. Welch] 1 Sources Theory of Computing, A Gentle Introduction, by E. Kinber and C. Smith, Prentice-Hall, 2001 Automata Theory, Languages and
Automata & languages. A primer on the Theory of Computation. Laurent Vanbever. ETH Zürich (D-ITET) October,
Automata & languages A primer on the Theory of Computation Laurent Vanbever www.vanbever.eu ETH Zürich (D-ITET) October, 19 2017 Part 5 out of 5 Last week was all about Context-Free Languages Context-Free
Turing Machines. Lecture 8
Turing Machines Lecture 8 1 Course Trajectory We will see algorithms, what can be done. But what cannot be done? 2 Computation Problem: To compute a function F that maps each input (a string) to an output
Standing on the Shoulders of a Giant
Standing on the Shoulders of a Giant One Person s Experience of Turing s Impact David Harel The Weizmann Institute of Science If I have seen further, it is because I stand on the shoulders of giants (Isaac
Acknowledgments 2. Part 0: Overview 17
Contents Acknowledgments 2 Preface for instructors 11 Which theory course are we talking about?.... 12 The features that might make this book appealing. 13 What s in and what s out............... 14 Possible
M.Burgin Department of Mathematics University of California, Los Angeles Los Angeles, CA 90095
Proceedings of the High Performance Computing Symposium, San Diego, 1999, pp. 224-228 SUPER-RECURSIVE ALGORITHMS AS A TOOL FOR HIGH PERFORMANCE COMPUTING M.Burgin Department of Mathematics University of
Computability Theory. CS215, Lecture 6,
Computability Theory CS215, Lecture 6, 2000 1 The Birth of Turing Machines At the end of the 19th century, Gottlob Frege conjectured that mathematics could be built from fundamental logic In 1900 David
Institute for Applied Information Processing and Communications (IAIK) Secure & Correct Systems. Decidability
Decidability and the Undecidability of Predicate Logic IAIK Graz University of Technology georg.hofferek@iaik.tugraz.at 1 Fork of ways Brainteaser: Labyrinth Guards One to salvation One to perdition Two
6.825 Techniques in Artificial Intelligence. Logic Miscellanea. Completeness and Incompleteness Equality Paramodulation
6.825 Techniques in Artificial Intelligence Logic Miscellanea Completeness and Incompleteness Equality Paramodulation Lecture 9 1 Logic is a huge subject. It includes esoteric mathematical and philosophical
15 Skepticism of quantum computing
15 Skepticism of quantum computing Last chapter, we talked about whether quantum states should be thought of as exponentially long vectors, and I brought up class BQP/qpoly and concepts like quantum advice.
Topology and experimental distinguishability
Topology and experimental distinguishability Gabriele Carcassi Christine A Aidala David J Baker Mark J Greenfield University of Michigan Motivation for this work This work has its root in an effort to
Large Numbers, Busy Beavers, Noncomputability and Incompleteness
Large Numbers, Busy Beavers, Noncomputability and Incompleteness Food For Thought November 1, 2007 Sam Buss Department of Mathematics U.C. San Diego PART I Large Numbers, Busy Beavers, and Undecidability
CSE 105 THEORY OF COMPUTATION
CSE 105 THEORY OF COMPUTATION Spring 2017 http://cseweb.ucsd.edu/classes/sp17/cse105-ab/ Today's learning goals Sipser Ch 5.1 Define and explain core examples of decision problems: A DFA, E DFA, EQ DFA,
Lecture 11: Measuring the Complexity of Proofs
IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Advanced Course on Computational Complexity Lecture 11: Measuring the Complexity of Proofs David Mix Barrington and Alexis Maciel July
The roots of computability theory. September 5, 2016
The roots of computability theory September 5, 2016 Algorithms An algorithm for a task or problem is a procedure that, if followed step by step and without any ingenuity, leads to the desired result/solution.
Philosophy 220. Truth-Functional Equivalence and Consistency
Philosophy 220 Truth-Functional Equivalence and Consistency Review Logical equivalency: The members of a pair of sentences [of natural language] are logically equivalent if and only if it is not [logically]
CSE 4111/5111/6111 Computability Jeff Edmonds Assignment 3: Diagonalization & Halting Problem Due: One week after shown in slides
CSE 4111/5111/6111 Computability Jeff Edmonds Assignment 3: Diagonalization & Halting Problem Due: One week after shown in slides First Person: Second Person: Family Name: Family Name: Given Name: Given
Obsevations, Truth and Logic
Obsevations, Truth and Logic Jaap van Oosten Department of Mathematics Utrecht University Axioma, Groningen May 9, 2018 Art and Science have the same basic subject matter: Observations. Art (visual art):
6-1 Computational Complexity
6-1 Computational Complexity 6. Computational Complexity Computational models Turing Machines Time complexity Non-determinism, witnesses, and short proofs. Complexity classes: P, NP, conp Polynomial-time
Learning Goals of CS245 Logic and Computation
Learning Goals of CS245 Logic and Computation Alice Gao April 27, 2018 Contents 1 Propositional Logic 2 2 Predicate Logic 4 3 Program Verification 6 4 Undecidability 7 1 1 Propositional Logic Introduction
Turing Machines (TM) The Turing machine is the ultimate model of computation.
TURING MACHINES Turing Machines (TM) The Turing machine is the ultimate model of computation. Alan Turing (92 954), British mathematician/engineer and one of the most influential scientists of the last
Last Time. Inference Rules
Last Time When program S executes it switches to a different state We need to express assertions on the states of the program S before and after its execution We can do it using a Hoare triple written
A Strong Relevant Logic Model of Epistemic Processes in Scientific Discovery
A Strong Relevant Logic Model of Epistemic Processes in Scientific Discovery (Extended Abstract) Jingde Cheng Department of Computer Science and Communication Engineering Kyushu University, 6-10-1 Hakozaki,
Preptests 55 Answers and Explanations (By Ivy Global) Section 4 Logic Games
Section 4 Logic Games Questions 1 6 There aren t too many deductions we can make in this game, and it s best to just note how the rules interact and save your time for answering the questions. 1. Type
Limits of Computation
The real danger is not that computers will begin to think like men, but that men will begin to think like computers Limits of Computation - Sydney J. Harris What makes you believe now that I am just talking
Disproof MAT231. Fall Transition to Higher Mathematics. MAT231 (Transition to Higher Math) Disproof Fall / 16
Disproof MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Disproof Fall 2014 1 / 16 Outline 1 s 2 Disproving Universal Statements: Counterexamples 3 Disproving Existence
Great Theoretical Ideas
15-251 Great Theoretical Ideas in Computer Science Gödel s Legacy: Proofs and Their Limitations Lecture 25 (November 16, 2010) The Halting Problem A Quick Recap of the Previous Lecture Is there a program
Old Evidence, Logical Omniscience & Bayesianism
Old Evidence, Logical Omniscience & Bayesianism Branden Fitelson Department of Philosophy Group in Logic and the Methodology of Science & Institute for Cognitive and Brain Sciences University of California
Arithmetical Hierarchy
Arithmetical Hierarchy 1 The Turing Jump Klaus Sutner Carnegie Mellon University Arithmetical Hierarchy 60-arith-hier 2017/12/15 23:18 Definability Formal Systems Recall: Oracles 3 The Use Principle 4
Complexity, Undecidability and Tilings. Chaim Goodman-Strauss Univ Arkansas
Complexity, Undecidability and Tilings Chaim Goodman-Strauss Univ Arkansas strauss@uark.edu Why are tiling puzzles difficult? Why are tiling puzzles difficult? (And how difficult are they anyway?) There
arxiv: v1 [cs.lo] 16 Jul 2017
![arxiv: v1 [cs.lo] 16 Jul 2017](/thumbs/94/119554062.jpg "arxiv: v1 [cs.lo] 16 Jul 2017")
SOME IMPROVEMENTS IN FUZZY TURING MACHINES HADI FARAHANI arxiv:1707.05311v1 [cs.lo] 16 Jul 2017 Department of Computer Science, Shahid Beheshti University, G.C, Tehran, Iran h farahani@sbu.ac.ir Abstract.
Inductive vs. Deductive Statistical Inference
Inductive vs. Deductive Statistical Inference Konstantin Genin Kevin T. Kelly February 28, 2018 Abstract The distinction between deductive (infallible, monotonic) and inductive (fallible, non-monotonic)
Arithmetical Hierarchy
Arithmetical Hierarchy Klaus Sutner Carnegie Mellon University 60-arith-hier 2017/12/15 23:18 1 The Turing Jump Arithmetical Hierarchy Definability Formal Systems Recall: Oracles 3 We can attach an orcale
Lecture 11: Gödel s Second Incompleteness Theorem, and Tarski s Theorem
Lecture 11: Gödel s Second Incompleteness Theorem, and Tarski s Theorem Valentine Kabanets October 27, 2016 1 Gödel s Second Incompleteness Theorem 1.1 Consistency We say that a proof system P is consistent
CSE 311 Lecture 28: Undecidability of the Halting Problem. Emina Torlak and Kevin Zatloukal
CSE 311 Lecture 28: Undecidability of the Halting Problem Emina Torlak and Kevin Zatloukal 1 Topics Final exam Logistics, format, and topics. Countability and uncomputability A quick recap of Lecture 27.
Propositional natural deduction
Propositional natural deduction COMP2600 / COMP6260 Dirk Pattinson Australian National University Semester 2, 2016 Major proof techniques 1 / 25 Three major styles of proof in logic and mathematics Model
Epistemology, Reliable Inquiry and Topology
Epistemology, Reliable Inquiry and Topology Oliver Schulte Department of Philosophy Carnegie Mellon University Pittsburgh, PA 15213 USA E-mail: schulte+@cmu.edu Cory Juhl Department of Philosophy University
Review of unsolvability
Review of unsolvability L L H To prove unsolvability: show a reduction. To prove solvability: show an algorithm. Unsolvable problems (main insight) Turing machine (algorithm) properties Pattern matching
Comment on Leitgeb s Stability Theory of Belief
Comment on Leitgeb s Stability Theory of Belief Hanti Lin Kevin T. Kelly Carnegie Mellon University {hantil, kk3n}@andrew.cmu.edu Hannes Leitgeb s stability theory of belief provides three synchronic constraints
Introduction to Turing Machines
Introduction to Turing Machines Deepak D Souza Department of Computer Science and Automation Indian Institute of Science, Bangalore. 12 November 2015 Outline 1 Turing Machines 2 Formal definitions 3 Computability
Artificial Intelligence. 3 Problem Complexity. Prof. Dr. Jana Koehler Fall 2016 HSLU - JK
Artificial Intelligence 3 Problem Complexity Prof. Dr. Jana Koehler Fall 2016 Agenda Computability and Turing Machines Tractable and Intractable Problems P vs. NP Decision Problems Optimization problems
Undecidability and Rice s Theorem. Lecture 26, December 3 CS 374, Fall 2015
Undecidability and Rice s Theorem Lecture 26, December 3 CS 374, Fall 2015 UNDECIDABLE EXP NP P R E RECURSIVE Recap: Universal TM U We saw a TM U such that L(U) = { (z,w) M z accepts w} Thus, U is a stored-program
Complexity Theory Part I
Complexity Theory Part I Outline for Today Recap from Last Time Reviewing Verifiers Nondeterministic Turing Machines What does nondeterminism mean in the context of TMs? And just how powerful are NTMs?
PHI Searle against Turing 1
1 2 3 4 5 6 PHI2391: Confirmation Review Session Date & Time :2014-12-03 SMD 226 12:00-13:00 ME 14.0 General problems with the DN-model! The DN-model has a fundamental problem that it shares with Hume!
Ch 3.2: Direct proofs
Math 299 Lectures 8 and 9: Chapter 3 0. Ch3.1 A trivial proof and a vacuous proof (Reading assignment) 1. Ch3.2 Direct proofs 2. Ch3.3 Proof by contrapositive 3. Ch3.4 Proof by cases 4. Ch3.5 Proof evaluations
Poincaré s Thesis CONTENTS. Peter Fekete
Poincaré s Thesis CONTENTS Peter Fekete COPYWRITE PETER FEKETE 6 TH OCT. 2011 NO PART OF THIS PAPER MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY,
Lecture 4: Constructing the Integers, Rationals and Reals
Math/CS 20: Intro. to Math Professor: Padraic Bartlett Lecture 4: Constructing the Integers, Rationals and Reals Week 5 UCSB 204 The Integers Normally, using the natural numbers, you can easily define
Decidability and Undecidability
Decidability and Undecidability Major Ideas from Last Time Every TM can be converted into a string representation of itself. The encoding of M is denoted M. The universal Turing machine U TM accepts an
We prove that the creator is infinite Turing machine or infinite Cellular-automaton.
Do people leave in Matrix? Information, entropy, time and cellular-automata The paper proves that we leave in Matrix. We show that Matrix was built by the creator. By this we solve the question how everything
Sequence of Topics. 1. The Nature of Truth. 2. Biography of Alan Turing ( ) 3. The Church-Turing Thesis. 4. Interactive Computing
1 Sequence of Topics 1. The Nature of Truth 2. Biography of Alan Turing (1912-1954) 3. The Church-Turing Thesis 4. Interactive Computing 5. Rationalism Versus Empiricism 6. Can Machines Think? 7. Stephen
CISC 876: Kolmogorov Complexity
March 27, 2007 Outline 1 Introduction 2 Definition Incompressibility and Randomness 3 Prefix Complexity Resource-Bounded K-Complexity 4 Incompressibility Method Gödel s Incompleteness Theorem 5 Outline
Confirmation and Chaos *
Confirmation and Chaos * Maralee Harrell Department of Philosophy, Colorado College Clark Glymour Institute for Human and Machine Cognition, University of West Florida and Department of Philosophy, Carnegie
17.1 The Halting Problem
CS125 Lecture 17 Fall 2016 17.1 The Halting Problem Consider the HALTING PROBLEM (HALT TM ): Given a TM M and w, does M halt on input w? Theorem 17.1 HALT TM is undecidable. Suppose HALT TM = { M,w : M
Propositional Logic: Syntax
Logic Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about time (and programs) epistemic
Philosophy of Science: Models in Science
Philosophy of Science: Models in Science Kristina Rolin 2012 Questions What is a scientific theory and how does it relate to the world? What is a model? How do models differ from theories and how do they
258 Handbook of Discrete and Combinatorial Mathematics
258 Handbook of Discrete and Combinatorial Mathematics 16.3 COMPUTABILITY Most of the material presented here is presented in far more detail in the texts of Rogers [R], Odifreddi [O], and Soare [S]. In
Steeple #3: Goodstein s Theorem (glimpse only!)
Steeple #3: Goodstein s Theorem (glimpse only!) Selmer Bringsjord (with Naveen Sundar G.) Are Humans Rational? v of 12717 RPI Troy NY USA Back to the beginning Back to the beginning Main Claim Back to
STRUCTURES WITH MINIMAL TURING DEGREES AND GAMES, A RESEARCH PROPOSAL
STRUCTURES WITH MINIMAL TURING DEGREES AND GAMES, A RESEARCH PROPOSAL ZHENHAO LI 1. Structures whose degree spectrum contain minimal elements/turing degrees 1.1. Background. Richter showed in [Ric81] that
Hoare Logic: Part II
Hoare Logic: Part II COMP2600 Formal Methods for Software Engineering Jinbo Huang Australian National University COMP 2600 Hoare Logic II 1 Factorial {n 0} fact := 1; i := n; while (i >0) do fact := fact
Opleiding Informatica
Opleiding Informatica Tape-quantifying Turing machines in the arithmetical hierarchy Simon Heijungs Supervisors: H.J. Hoogeboom & R. van Vliet BACHELOR THESIS Leiden Institute of Advanced Computer Science
Computation. Some history...
Computation Motivating questions: What does computation mean? What are the similarities and differences between computation in computers and in natural systems? What are the limits of computation? Are
Logic. Propositional Logic: Syntax. Wffs
Logic Propositional Logic: Syntax Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about
Week 2. Week 1 Recap. Week 2
Week 2 Week 1 Recap In our first session, we outlined our big-picture goals for this course. We are going to spend these first couple weeks looking at mathematical proof, then we will take a couple weeks
GS03/4023: Validation and Verification Predicate Logic Jonathan P. Bowen Anthony Hall
GS03/4023: Validation and Verification Predicate Logic Jonathan P. Bowen www.cs.ucl.ac.uk/staff/j.bowen/gs03 Anthony Hall GS03 W1 L3 Predicate Logic 12 January 2007 1 Overview The need for extra structure
Gödel s Argument for Cantor s Cardinals Matthew W. Parker Centre for Philosophy of Natural and Social Science
www.logicnest.com Gödel s Argument for Cantor s Cardinals Matthew W. Parker Centre for Philosophy of Natural and Social Science The Hume Cantor Principle: If there is a 1-1 correspondence between two collections,
Recap (1) 1. Automata, grammars and other formalisms can be considered as mechanical devices to solve mathematical problems
Computability 1 Recap (1) 1. Automata, grammars and other formalisms can be considered as mechanical devices to solve mathematical problems Mathematical problems are often the formalization of some practical
Poincaré, Heisenberg, Gödel. Some limits of scientific knowledge. Fernando Sols Universidad Complutense de Madrid
Poincaré, Heisenberg, Gödel. Some limits of scientific knowledge. Fernando Sols Universidad Complutense de Madrid Henry Poincaré (1854-1912) nonlinear dynamics Werner Heisenberg (1901-1976) uncertainty
Examples: P: it is not the case that P. P Q: P or Q P Q: P implies Q (if P then Q) Typical formula:
Logic: The Big Picture Logic is a tool for formalizing reasoning. There are lots of different logics: probabilistic logic: for reasoning about probability temporal logic: for reasoning about time (and
Feyerabend: How to be a good empiricist
Feyerabend: How to be a good empiricist Critique of Nagel s model of reduction within a larger critique of the logical empiricist view of scientific progress. Logical empiricism: associated with figures
Recursion Theory. Joost J. Joosten
Recursion Theory Joost J. Joosten Institute for Logic Language and Computation University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam Room P 3.26, +31 20 5256095 jjoosten@phil.uu.nl www.phil.uu.nl/
Lecture Notes on Invariants for Arbitrary Loops
15-414: Bug Catching: Automated Program Verification Lecture Notes on Invariants for Arbitrary Loops Matt Fredrikson Ruben Martins Carnegie Mellon University Lecture 5 1 Introduction The previous lecture
Sequential Decision Problems
Sequential Decision Problems Michael A. Goodrich November 10, 2006 If I make changes to these notes after they are posted and if these changes are important (beyond cosmetic), the changes will highlighted
Computational Tasks and Models
1 Computational Tasks and Models Overview: We assume that the reader is familiar with computing devices but may associate the notion of computation with specific incarnations of it. Our first goal is to
Before We Start. Turing Machines. Languages. Now our picture looks like. Theory Hall of Fame. The Turing Machine. Any questions? The $64,000 Question
Before We Start s Any questions? Languages The $64,000 Question What is a language? What is a class of languages? Now our picture looks like Context Free Languages Deterministic Context Free Languages
Experimental logics, mechanism and knowable consistency (SELLC 2010 in Guangzhou, China)
Experimental logics, mechanism and knowable consistency (SELLC 2010 in Guangzhou, China) Martin Kaså, University of Gothenburg, Sweden Overview In a paper published in 1975, Robert Jeroslow introduced
Turing Centenary Lecture
Turing Centenary Lecture P.D.Welch University of Bristol Visiting Research Fellow, Isaac Newton Institute Early Life King s College 1931 King s College 1931 Hardy Eddington He attended Eddington s lectures
Notes on the Point-Set Topology of R Northwestern University, Fall 2014
Notes on the Point-Set Topology of R Northwestern University, Fall 2014 These notes give an introduction to the notions of open and closed subsets of R, which belong to the subject known as point-set topology.
PROOF-THEORETIC REDUCTION AS A PHILOSOPHER S TOOL
THOMAS HOFWEBER PROOF-THEORETIC REDUCTION AS A PHILOSOPHER S TOOL 1. PROOF-THEORETIC REDUCTION AND HILBERT S PROGRAM Hilbert s program in the philosophy of mathematics comes in two parts. One part is a
Induction 1 = 1(1+1) = 2(2+1) = 3(3+1) 2
Induction 0-8-08 Induction is used to prove a sequence of statements P(), P(), P(3),... There may be finitely many statements, but often there are infinitely many. For example, consider the statement ++3+
Gödel s Incompleteness Theorems
15-251: Great Theoretical Ideas in Computer Science Spring 2016, Lecture 16 Gödel s Incompleteness Theorems Don t stress, Kurt, it s easy! Proving the famous Gödel Incompleteness Theorems is easy if you
Notes on Complexity Theory Last updated: December, Lecture 2
Notes on Complexity Theory Last updated: December, 2011 Jonathan Katz Lecture 2 1 Review The running time of a Turing machine M on input x is the number of steps M takes before it halts. Machine M is said
Models. Models of Computation, Turing Machines, and the Limits of Turing Computation. Effective Calculability. Motivation for Models of Computation
Turing Computation /0/ Models of Computation, Turing Machines, and the Limits of Turing Computation Bruce MacLennan Models A model is a tool intended to address a class of questions about some domain of