Computational Thinking
|
|
- Derek Nicholson
- 6 years ago
- Views:
Transcription
1 Computational Thinking MASTER IN DIGITAL HUMANITIES 9 December 2016 Teresa Scantamburlo DAIS and ECLT Ca Foscari University
2 Thinking as a computational process Thinking can be usefully understood as a computational process What does this conjecture amount to? Not that the brain is something like an electronic computer Thinking as a form of symbol processing that can be carried out purely mechanically without having to know what the symbols stand for
3 All computation? So much of our thinking seems to have very little to do with calculations Example: I know my keys are in my coat pocket or on the fridge. That s where I always leave them. I felt in my coat pocket, and there s noting there. So my keys must be on the fridge, and that s where I should look This example seems to have nothing to do with numbers and calculation
4 Thinking and computation Thinking is about something (e.g., keys, coat pocket, refrigerator, etc.) Computation seems to be about nothing. It is the process of manipulating symbols in a mechanical way There is a conceptual gap between the two that need to be bridged This work has much to do with philosophy!
5 Gottfried Leibniz He envisioned a special alphabet (characteristica universalis), a system of symbols whose elements represented a welldefined idea His idea: Thought ca be reduced to a manipulation of symbols (calculus ratiocinatur) It will in fact suffice to take pen in hand and to say: let us calculate
6 Propositions vs. sentences What about ideas and objects of ordinary thought? Examples: My keys are in my coat pocket Dinosaurs were warm-blooded Life expectancy in the USA declined for the first time since 1993 Sentence à sequence of words Proposition à the idea expressed
7 Propositions What can be said about propositions? Propositions are considered to hold or to not hold. A sentence is true if the propositions it expresses holds, and false if that proposition does not hold Propositions are considered to be related to people in certain ways: people may or may not believe them, fear them, etc. (propositional attitude) Propositions are related to each other in certain ways: a proposition may imply or contradict another one, etc.
8 Example Consider this: The snark was a boojum (The hunting of the Snark, L. Caroll) If we assume that the sentence is true, even without knowing what the words snark and boojum mean, we can answer certain questions: What kind of think was the snark? It was a boojum Is it true that the snark was either a beejum or a boojum? Yes, because it was a boojum What is an example of something that was a boojum? The snark, of course
9 Other examples My keys are in my coat pocket or on the fridge Noting is in my coat pocket So: My keys are on the fridge Henry is in the basement or in the garden Nobody is in the basement So: Henry is in the garden The frumble is frimble or framble Nothing is frimble So: The frumble is framble Again, one does not need to know what frimble means! What does matter is the form of the sentence
10 Logical entailment A collection of sentences S 1, S 2,, S n logically entails another sentence S if the truth of S is implicit in the truth of S i sentences In determining if a collection of sentences logically entails another, it is not necessary to know what the terms in those sentences mean (certain keywords in sentence, such as and, do have specific functions) Example: The snark was a boojum Logically entails Something was a boojum My keys are in my coat pocket or on the fridge Noting is in my coat pocket Logically entails My keys are on the fridge
11 Using what is known Thinking seems to be much richer than logical entailments Positive aspects: with logical entailments we can extract knowledge from what is already known It might be helpful to visualize sentences as forming a kind of network with nodes for each terms and links between them according to the sentences in which they appear.
12 Example
13 Assignment 3 Try to represent the sentences in the example as a network (using at least 5 sentences) Consider the sentence George is a bachelor and find out some logical entailments
14 Web of belief
15 Some logical entailments George has never been the groom at wedding Mary has an unmarried son born in Boston No woman is the wife of any of Fred s children
16 Turing s model of computation Turing s model resulted from the analysis of the possible processes a human can go through while performing a calculation using paper and pencil applying rules from a given finite set. Note that the rules are followed blindly, without using insight or ingenuity
17 The analysis of a computation process By observing a human doing calculi, Turing ends up with a series of constrains: Only a finite number of symbols can be written down and used in any computation There is a fixed bound on the amount of scratch paper that a human can consider at a time At any time symbols can be written down or erased Current state of mind along with the last symbol written determine what to do next
18 Example 4231 x 77 = X 7 7 = =
19 Turing machine A Turing machine consists of: An unbounded tape divided into squares, each of which can hold exactly one symbol A tape head for reading and writing symbols from a given alphabet on the squares A controller which is in exactly one of finitely many states at any given time
20 Turing s main result From Turing analysis we derive a strong notion of computation: Anything computable by an algorithmic process can be computed by a Turing machine Therefore, if we prove that some task cannot be accomplished by a Turing machine, we can conclude that no algorithmic process can accomplish that task
21 Formulas in Turing machine Assume, for example, the statement: When the machine is in state R scanning the symbol a it will replace a by b, move one square to the right, and then shift into state S. Can be expressed as follows (quintuples): R a : b à S Analogous statements: R a : b ß S R a : b S
22 Testing a Turing machine Problem: Given natural number to see whether it is even or odd Number written on the tape (input): No limit to the amount of tape available Special symbol (blank square) for termination As the machine terminates squares will be all blank except for one (that will be 0 if the input was even, 1 it was odd) 4 states: Ø Q = starting state (from the leftmost square) Ø E, O = states resulting from scanning numbers, E (even), O (odd) Ø F = final state
23 Set of quintuples
24 Step 1
25 Step 2
26 Step 3
27 Step 4
28 Step 5
29 Step 6
30 Output Note that: Unlike physical devices, Turing machines being a mathematical abstraction has no limitations on the amount of tape they can use. But this might have some problems
31 Halting problem For example, consider the Turing machine consisting of these 2 quintuples: Q 1 : 1 à Q Q 2 : 2 ß Q Input: 1 2 Some Turing machines with some inputs eventually halt, others do not. This reasoning led Turing to the conclusion that some problems cannot be solved by Turing machines...
32 Some conclusions In summary for Some Turing machines with some inputs eventually halt, others do not. This reasoning led Turing to the conclusion that some problems cannot be solved by Turing machines and to the unsolvability of Hilbert s problem (decision problem)
33 Some references M. Davis, The Universal Computer: The Road from Leibniz to Turing, Norton & Company, 2000 H. Levesque, Thinking as Computation, 2012 IMT Press
Lecture notes on Turing machines
Lecture notes on Turing machines Ivano Ciardelli 1 Introduction Turing machines, introduced by Alan Turing in 1936, are one of the earliest and perhaps the best known model of computation. The importance
More informationComputability 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
More informationThe 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.
More informationThe universal computer
The universal computer The road from Leibniz to Turing Instructor: Viola Schiaffonati March, 19 th 2018 Thinking, calculating, programming 2 Processing information in an automatic way The birth of computer
More informationQ = Set of states, IE661: Scheduling Theory (Fall 2003) Primer to Complexity Theory Satyaki Ghosh Dastidar
IE661: Scheduling Theory (Fall 2003) Primer to Complexity Theory Satyaki Ghosh Dastidar Turing Machine A Turing machine is an abstract representation of a computing device. It consists of a read/write
More informationModels. 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
More informationLarge 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
More informationIV. Turing Machine. Yuxi Fu. BASICS, Shanghai Jiao Tong University
IV. Turing Machine Yuxi Fu BASICS, Shanghai Jiao Tong University Alan Turing Alan Turing (23Jun.1912-7Jun.1954), an English student of Church, introduced a machine model for effective calculation in On
More informationECS 120 Lesson 15 Turing Machines, Pt. 1
ECS 120 Lesson 15 Turing Machines, Pt. 1 Oliver Kreylos Wednesday, May 2nd, 2001 Before we can start investigating the really interesting problems in theoretical computer science, we have to introduce
More informationTuring Machines and the Church-Turing Thesis
CSE2001, Fall 2006 1 Turing Machines and the Church-Turing Thesis Today our goal is to show that Turing Machines are powerful enough to model digital computers, and to see discuss some evidence for the
More informationChapter 6: Turing Machines
Chapter 6: Turing Machines 6.1 The Turing Machine Definition A deterministic Turing machine (DTM) M is specified by a sextuple (Q, Σ, Γ, δ, s, f), where Q is a finite set of states; Σ is an alphabet of
More informationTuring machines Finite automaton no storage Pushdown automaton storage is a stack What if we give the automaton a more flexible storage?
Turing machines Finite automaton no storage Pushdown automaton storage is a stack What if we give the automaton a more flexible storage? What is the most powerful of automata? In this lecture we will introduce
More informationcse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska
cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 13 CHAPTER 4 TURING MACHINES 1. The definition of Turing machine 2. Computing with Turing machines 3. Extensions of Turing
More informationDecidability: 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
More informationIntroduction 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
More informationTURING MAHINES
15-453 TURING MAHINES TURING MACHINE FINITE STATE q 10 CONTROL AI N P U T INFINITE TAPE read write move 0 0, R, R q accept, R q reject 0 0, R 0 0, R, L read write move 0 0, R, R q accept, R 0 0, R 0 0,
More informationDecidable Languages - relationship with other classes.
CSE2001, Fall 2006 1 Last time we saw some examples of decidable languages (or, solvable problems). Today we will start by looking at the relationship between the decidable languages, and the regular and
More information1 Computational problems
80240233: Computational Complexity Lecture 1 ITCS, Tsinghua Univesity, Fall 2007 9 October 2007 Instructor: Andrej Bogdanov Notes by: Andrej Bogdanov The aim of computational complexity theory is to study
More informationLecture 13: Turing Machine
Lecture 13: Turing Machine Instructor: Ketan Mulmuley Scriber: Yuan Li February 19, 2015 Turing machine is an abstract machine which in principle can simulate any computation in nature. Church-Turing Thesis:
More informationUNIT-VIII COMPUTABILITY THEORY
CONTEXT SENSITIVE LANGUAGE UNIT-VIII COMPUTABILITY THEORY A Context Sensitive Grammar is a 4-tuple, G = (N, Σ P, S) where: N Set of non terminal symbols Σ Set of terminal symbols S Start symbol of the
More informationCS5371 Theory of Computation. Lecture 10: Computability Theory I (Turing Machine)
CS537 Theory of Computation Lecture : Computability Theory I (Turing Machine) Objectives Introduce the Turing Machine (TM)? Proposed by Alan Turing in 936 finite-state control + infinitely long tape A
More informationTuring Machines and Time Complexity
Turing Machines and Time Complexity Turing Machines Turing Machines (Infinitely long) Tape of 1 s and 0 s Turing Machines (Infinitely long) Tape of 1 s and 0 s Able to read and write the tape, and move
More informationThe Limit of Humanly Knowable Mathematical Truth
The Limit of Humanly Knowable Mathematical Truth Gödel s Incompleteness Theorems, and Artificial Intelligence Santa Rosa Junior College December 12, 2015 Another title for this talk could be... An Argument
More informationGottfried Wilhelm Leibniz (1666)
Euclid (c. -300) Euclid s GCD algorithm appeared in his Elements. Formulated geometrically: Find common measure for 2 lines. Used repeated subtraction of the shorter segment from the longer. Gottfried
More informationComputation. 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
More informationCS154, Lecture 10: Rice s Theorem, Oracle Machines
CS154, Lecture 10: Rice s Theorem, Oracle Machines Moral: Analyzing Programs is Really, Really Hard But can we more easily tell when some program analysis problem is undecidable? Problem 1 Undecidable
More informationTuring 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
More informationTuring Machines. The Language Hierarchy. Context-Free Languages. Regular Languages. Courtesy Costas Busch - RPI 1
Turing Machines a n b n c The anguage Hierarchy n? ww? Context-Free anguages a n b n egular anguages a * a *b* ww Courtesy Costas Busch - PI a n b n c n Turing Machines anguages accepted by Turing Machines
More informationChapter 8. Turing Machine (TMs)
Chapter 8 Turing Machine (TMs) Turing Machines (TMs) Accepts the languages that can be generated by unrestricted (phrase-structured) grammars No computational machine (i.e., computational language recognition
More information1 Acceptance, Rejection, and I/O for Turing Machines
1 Acceptance, Rejection, and I/O for Turing Machines Definition 1.1 (Initial Configuration) If M = (K,Σ,δ,s,H) is a Turing machine and w (Σ {, }) then the initial configuration of M on input w is (s, w).
More informationLogical Agents. Administrative. Thursday: Midterm 1, 7p-9p. Next Tuesday: DOW1013: Last name A-M DOW1017: Last name N-Z
Logical Agents Mary Herchenhahn, mary-h.com EECS 492 February 2 nd, 2010 Administrative Thursday: Midterm 1, 7p-9p DOW1013: Last name A-M DOW1017: Last name N-Z Next Tuesday: PS2 due PS3 distributed---
More informationLimits 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
More informationThe Church-Turing Thesis
The Church-Turing Thesis Huan Long Shanghai Jiao Tong University Acknowledgements Part of the slides comes from a similar course in Fudan University given by Prof. Yijia Chen. http://basics.sjtu.edu.cn/
More informationCS5371 Theory of Computation. Lecture 10: Computability Theory I (Turing Machine)
CS537 Theory of Computation Lecture : Computability Theory I (Turing Machine) Objectives Introduce the Turing Machine (TM) Proposed by Alan Turing in 936 finite-state control + infinitely long tape A stronger
More information(a) Definition of TMs. First Problem of URMs
Sec. 4: Turing Machines First Problem of URMs (a) Definition of the Turing Machine. (b) URM computable functions are Turing computable. (c) Undecidability of the Turing Halting Problem That incrementing
More informationMore Turing Machines. CS154 Chris Pollett Mar 15, 2006.
More Turing Machines CS154 Chris Pollett Mar 15, 2006. Outline Multitape Turing Machines Nondeterministic Turing Machines Enumerators Introduction There have been many different proposals for what it means
More informationMACHINE COMPUTING. the limitations
MACHINE COMPUTING the limitations human computing stealing brain cycles of the masses word recognition: to digitize all printed writing language education: to translate web content games with a purpose
More informationCST Part IB. Computation Theory. Andrew Pitts
Computation Theory, L 1 1/171 CST Part IB Computation Theory Andrew Pitts Corrections to the notes and extra material available from the course web page: www.cl.cam.ac.uk/teaching/0910/comptheory/ Introduction
More informationFinite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018
Finite Automata Theory and Formal Languages TMV027/DIT321 LP4 2018 Lecture 15 Ana Bove May 17th 2018 Recap: Context-free Languages Chomsky hierarchy: Regular languages are also context-free; Pumping lemma
More informationTuring Machines. Nicholas Geis. February 5, 2015
Turing Machines Nicholas Geis February 5, 2015 Disclaimer: This portion of the notes does not talk about Cellular Automata or Dynamical Systems, it talks about turing machines, however this will lay the
More informationIntroduction to Proofs
Introduction to Proofs Many times in economics we will need to prove theorems to show that our theories can be supported by speci c assumptions. While economics is an observational science, we use mathematics
More informationTuring 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
More informationTime-bounded computations
Lecture 18 Time-bounded computations We now begin the final part of the course, which is on complexity theory. We ll have time to only scratch the surface complexity theory is a rich subject, and many
More informationSpace Complexity. then the space required by M on input x is. w i u i. F ) on. September 27, i=1
Space Complexity Consider a k-string TM M with input x. Assume non- is never written over by. a The purpose is not to artificially reduce the space needs (see below). If M halts in configuration (H, w
More informationCS 361 Meeting 26 11/10/17
CS 361 Meeting 26 11/10/17 1. Homework 8 due Announcements A Recognizable, but Undecidable Language 1. Last class, I presented a brief, somewhat inscrutable proof that the language A BT M = { M w M is
More informationState Machines. Example FSM: Roboant
page 1 State Machines 1) State Machine Design 2) How can we improve on FSMs? 3) Turing Machines 4) Computability Oh genie, will you now tell me what it means to compute? Doctor, I think you ve built a
More informationHOW TO WRITE PROOFS. Dr. Min Ru, University of Houston
HOW TO WRITE PROOFS Dr. Min Ru, University of Houston One of the most difficult things you will attempt in this course is to write proofs. A proof is to give a legal (logical) argument or justification
More informationTuring Machines A Turing Machine is a 7-tuple, (Q, Σ, Γ, δ, q0, qaccept, qreject), where Q, Σ, Γ are all finite
The Church-Turing Thesis CS60001: Foundations of Computing Science Professor, Dept. of Computer Sc. & Engg., Turing Machines A Turing Machine is a 7-tuple, (Q, Σ, Γ, δ, q 0, q accept, q reject ), where
More informationCS4026 Formal Models of Computation
CS4026 Formal Models of Computation Turing Machines Turing Machines Abstract but accurate model of computers Proposed by Alan Turing in 1936 There weren t computers back then! Turing s motivation: find
More informationCS20a: Turing Machines (Oct 29, 2002)
CS20a: Turing Machines (Oct 29, 2002) So far: DFA = regular languages PDA = context-free languages Today: Computability 1 Church s thesis The computable functions are the same as the partial recursive
More informationAn Algebraic Characterization of the Halting Probability
CDMTCS Research Report Series An Algebraic Characterization of the Halting Probability Gregory Chaitin IBM T. J. Watson Research Center, USA CDMTCS-305 April 2007 Centre for Discrete Mathematics and Theoretical
More informationUndecidability 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
More informationChapter 3: The Church-Turing Thesis
Chapter 3: The Church-Turing Thesis 1 Turing Machine (TM) Control... Bi-direction Read/Write Turing machine is a much more powerful model, proposed by Alan Turing in 1936. 2 Church/Turing Thesis Anything
More informationChurch s undecidability result
Church s undecidability result Alan Turing Birth Centennial Talk at IIT Bombay, Mumbai Joachim Breitner April 21, 2011 Welcome, and thank you for the invitation to speak about Church s lambda calculus
More informationCardinality of Sets. P. Danziger
MTH 34-76 Cardinality of Sets P Danziger Cardinal vs Ordinal Numbers If we look closely at our notions of number we will see that in fact we have two different ways of conceiving of numbers The first is
More information240 Metaphysics. Frege s Puzzle. Chapter 26
240 Metaphysics Frege s Puzzle Frege s Puzzle 241 Frege s Puzzle In his 1879 Begriffsschrift (or Concept-Writing ), Gottlob Frege developed a propositional calculus to determine the truth values of propositions
More informationCSCC63 Worksheet Turing Machines
1 An Example CSCC63 Worksheet Turing Machines Goal. Design a turing machine, M that accepts only strings of the form {w#w w {0, 1} }. Idea. Describe in words how the machine would work. Read first symbol
More information15-251: Great Theoretical Ideas in Computer Science Fall 2016 Lecture 6 September 15, Turing & the Uncomputable
15-251: Great Theoretical Ideas in Computer Science Fall 2016 Lecture 6 September 15, 2016 Turing & the Uncomputable Comparing the cardinality of sets A B if there is an injection (one-to-one map) from
More informationMathematics-I Prof. S.K. Ray Department of Mathematics and Statistics Indian Institute of Technology, Kanpur. Lecture 1 Real Numbers
Mathematics-I Prof. S.K. Ray Department of Mathematics and Statistics Indian Institute of Technology, Kanpur Lecture 1 Real Numbers In these lectures, we are going to study a branch of mathematics called
More informationIntroduction to Languages and Computation
Introduction to Languages and Computation George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 400 George Voutsadakis (LSSU) Languages and Computation July 2014
More informationAutomata & 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
More informationAutomata Theory (2A) Young Won Lim 5/31/18
Automata Theory (2A) Copyright (c) 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More informationCISC-102 Winter 2016 Lecture 17
CISC-102 Winter 2016 Lecture 17 Logic and Propositional Calculus Propositional logic was eventually refined using symbolic logic. The 17th/18th century philosopher Gottfried Leibniz (an inventor of calculus)
More informationBusch Complexity Lectures: Turing Machines. Prof. Busch - LSU 1
Busch Complexity ectures: Turing Machines Prof. Busch - SU 1 The anguage Hierarchy a n b n c n? ww? Context-Free anguages n b n a ww egular anguages a* a *b* Prof. Busch - SU 2 a n b anguages accepted
More informationhighlights proof by contradiction what about the real numbers?
CSE 311: Foundations of Computing Fall 2013 Lecture 27: Turing machines and decidability highlights Cardinality A set S is countableiffwe can writeit as S={s 1, s 2, s 3,...} indexed by N Set of rationals
More informationFundamentals of Computer Science
Fundamentals of Computer Science Chapter 8: Turing machines Henrik Björklund Umeå University February 17, 2014 The power of automata Finite automata have only finite memory. They recognize the regular
More informationUnit 4: Computer as a logic machine
Unit 4: Computer as a logic machine Propositional logic Boolean algebra Logic gates Computer as a logic machine: symbol processor Development of computer The credo of early AI Reference copyright c 2013
More informationThe tape of M. Figure 3: Simulation of a Turing machine with doubly infinite tape
UG3 Computability and Intractability (2009-2010): Note 4 4. Bells and whistles. In defining a formal model of computation we inevitably make a number of essentially arbitrary design decisions. These decisions
More informationHalting and Equivalence of Program Schemes in Models of Arbitrary Theories
Halting and Equivalence of Program Schemes in Models of Arbitrary Theories Dexter Kozen Cornell University, Ithaca, New York 14853-7501, USA, kozen@cs.cornell.edu, http://www.cs.cornell.edu/~kozen In Honor
More informationFurther discussion of Turing machines
Further discussion of Turing machines In this lecture we will discuss various aspects of decidable and Turing-recognizable languages that were not mentioned in previous lectures. In particular, we will
More informationHarvard CS 121 and CSCI E-121 Lecture 14: Turing Machines and the Church Turing Thesis
Harvard CS 121 and CSCI E-121 Lecture 14: Turing Machines and the Church Turing Thesis Harry Lewis October 22, 2013 Reading: Sipser, 3.2, 3.3. The Basic Turing Machine The Basic Turing Machine a a b a
More informationUnderstanding The Law of Attraction
Psychic Insight 4-5-17 New Blog Below Understanding The Law of Attraction Everything in the universe is energy and vibrates at a distinct frequency. Understanding that these energies are attracted to similar
More informationCS 730/830: Intro AI. 2 handouts: slides, asst 5. Wheeler Ruml (UNH) Lecture 10, CS / 19. What is KR? Prop. Logic. Reasoning
CS 730/830: Intro AI 2 handouts: slides, asst 5 Wheeler Ruml (UNH) Lecture 10, CS 730 1 / 19 History of Logic Advice Taker The PSSH Introduction to Knowledge Representation and Wheeler Ruml (UNH) Lecture
More informationTuring Machines Part II
Turing Machines Part II COMP2600 Formal Methods for Software Engineering Katya Lebedeva Australian National University Semester 2, 2016 Slides created by Katya Lebedeva COMP 2600 Turing Machines 1 Why
More informationReasons for Rejecting a Counterfactual Analysis of Conclusive Reasons
Reasons for Rejecting a Counterfactual Analysis of Conclusive Reasons Abstract In a recent article [1], Charles Cross applies Lewisian counterfactual logic to explicate and evaluate Fred Dretske s conclusive
More informationBasic Logic and Proof Techniques
Chapter 3 Basic Logic and Proof Techniques Now that we have introduced a number of mathematical objects to study and have a few proof techniques at our disposal, we pause to look a little more closely
More informationThe Turing Machine. CSE 211 (Theory of Computation) The Turing Machine continued. Turing Machines
The Turing Machine Turing Machines Professor Department of Computer Science and Engineering Bangladesh University of Engineering and Technology Dhaka-1000, Bangladesh The Turing machine is essentially
More informationIST 4 Information and Logic
IST 4 Information and Logic mon tue wed thr fri sun T = today 3 M oh x= hw#x out oh M 7 oh oh 2 M2 oh oh x= hw#x due 24 oh oh 2 oh = office hours oh oh T M2 8 3 oh midterms oh oh Mx= MQx out 5 oh 3 4 oh
More informationTheoretical Computers and Diophantine Equations *
Theoretical Computers and Diophantine Equations * Alvin Chan Raffles Junior College & K. J. Mourad Nat. Univ. of Singapore l. Introduction This paper will discuss algorithms and effective computability,
More informationDescription Logics. Foundations of Propositional Logic. franconi. Enrico Franconi
(1/27) Description Logics Foundations of Propositional Logic Enrico Franconi franconi@cs.man.ac.uk http://www.cs.man.ac.uk/ franconi Department of Computer Science, University of Manchester (2/27) Knowledge
More informationFor all For every For each For any There exists at least one There exists There is Some
Section 1.3 Predicates and Quantifiers Assume universe of discourse is all the people who are participating in this course. Also let us assume that we know each person in the course. Consider the following
More informationThe Unsolvability of the Halting Problem. Chapter 19
The Unsolvability of the Halting Problem Chapter 19 Languages and Machines SD D Context-Free Languages Regular Languages reg exps FSMs cfgs PDAs unrestricted grammars Turing Machines D and SD A TM M with
More informationThe paradox of knowability, the knower, and the believer
The paradox of knowability, the knower, and the believer Last time, when discussing the surprise exam paradox, we discussed the possibility that some claims could be true, but not knowable by certain individuals
More informationCS187 - Science Gateway Seminar for CS and Math
CS187 - Science Gateway Seminar for CS and Math Fall 2013 Class 3 Sep. 10, 2013 What is (not) Computer Science? Network and system administration? Playing video games? Learning to use software packages?
More informationLinear-time Temporal Logic
Linear-time Temporal Logic Pedro Cabalar Department of Computer Science University of Corunna, SPAIN cabalar@udc.es 2015/2016 P. Cabalar ( Department Linear oftemporal Computer Logic Science University
More informationMost General computer?
Turing Machines Most General computer? DFAs are simple model of computation. Accept only the regular languages. Is there a kind of computer that can accept any language, or compute any function? Recall
More informationCSCI3390-Assignment 2 Solutions
CSCI3390-Assignment 2 Solutions due February 3, 2016 1 TMs for Deciding Languages Write the specification of a Turing machine recognizing one of the following three languages. Do one of these problems.
More informationProving Completeness for Nested Sequent Calculi 1
Proving Completeness for Nested Sequent Calculi 1 Melvin Fitting abstract. Proving the completeness of classical propositional logic by using maximal consistent sets is perhaps the most common method there
More informationPropositional Logic Truth-functionality Definitions Soundness Completeness Inferences. Modal Logic. Daniel Bonevac.
January 22, 2013 Modal logic is, among other things, the logic of possibility and necessity. Its history goes back at least to Aristotle s discussion of modal syllogisms in the Prior Analytics. But modern
More informationTuring machine recap. Universal Turing Machines and Undecidability. Alphabet size doesn t matter. Robustness of TM
Turing machine recap A Turing machine TM is a tuple M = (Γ, Q, δ) where Γ: set of symbols that TM s tapes can contain. Q: possible states TM can be in. qstart: the TM starts in this state qhalt: the TM
More informationInstitute 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
More informationCS151 Complexity Theory. Lecture 1 April 3, 2017
CS151 Complexity Theory Lecture 1 April 3, 2017 Complexity Theory Classify problems according to the computational resources required running time storage space parallelism randomness rounds of interaction,
More informationA non-turing-recognizable language
CS 360: Introduction to the Theory of Computing John Watrous, University of Waterloo A non-turing-recognizable language 1 OVERVIEW Thus far in the course we have seen many examples of decidable languages
More informationThe Converse of Deducibility: C.I. Lewis and the Origin of Modern AAL/ALC Modal 2011 Logic 1 / 26
The Converse of Deducibility: C.I. Lewis and the Origin of Modern Modal Logic Edwin Mares Victoria University of Wellington AAL/ALC 2011 The Converse of Deducibility: C.I. Lewis and the Origin of Modern
More informationPropositions as Types
Propositions as Types Martin Pfeifhofer & Felix Schett May 25, 2016 Contents 1 Introduction 2 2 Content 3 2.1 Getting Started............................ 3 2.2 Effective Computability And The Various Definitions.......
More information1 Propositional Logic
1 Propositional Logic Required reading: Foundations of Computation. Sections 1.1 and 1.2. 1. Introduction to Logic a. Logical consequences. If you know all humans are mortal, and you know that you are
More informationCPSC 421: Tutorial #1
CPSC 421: Tutorial #1 October 14, 2016 Set Theory. 1. Let A be an arbitrary set, and let B = {x A : x / x}. That is, B contains all sets in A that do not contain themselves: For all y, ( ) y B if and only
More informationCA320 - Computability & Complexity
CA320 - Computability & Complexity David Sinclair Overview In this module we are going to answer 2 important questions: Can all problems be solved by a computer? What problems be efficiently solved by
More informationSemantics and Generative Grammar. An Introduction to Intensional Semantics 1
An Introduction to Intensional Semantics 1 1. The Inadequacies of a Purely Extensional Semantics (1) Our Current System: A Purely Extensional Semantics The extension of a complex phrase is (always) derived
More informationComplexity Theory. Martin Ziegler
Complexity Theory Martin Ziegler Reminder: Asymptotics Landau: For f,g : Í Ñ write f=o(g) M n M: f(n) M g(n) f=ω(g) M n M: f(n) g(n)/m f=θ(g) f=o(g) f=ω(g) These notions neglect lower order terms and
More information