Knowledge Representation and Reasoning Prolog and Ailog Practical 2012 – 2013 Exercises
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Graph-Based Reasoning in Collaborative Knowledge Management for Industrial Maintenance Bernard Kamsu-Foguem, Daniel Noyes
Graph-based reasoning in collaborative knowledge management for industrial maintenance Bernard Kamsu-Foguem, Daniel Noyes To cite this version: Bernard Kamsu-Foguem, Daniel Noyes. Graph-based reasoning in collaborative knowledge manage- ment for industrial maintenance. Computers in Industry, Elsevier, 2013, Vol. 64, pp. 998-1013. 10.1016/j.compind.2013.06.013. hal-00881052 HAL Id: hal-00881052 https://hal.archives-ouvertes.fr/hal-00881052 Submitted on 7 Nov 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Open Archive Toulouse Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in: http://oatao.univ-toulouse.fr/ Eprints ID: 9587 To link to this article: doi.org/10.1016/j.compind.2013.06.013 http://www.sciencedirect.com/science/article/pii/S0166361513001279 To cite this version: Kamsu Foguem, Bernard and Noyes, Daniel Graph-based reasoning in collaborative knowledge management for industrial -
UNIT-I Mathematical Logic Statements and Notations
UNIT-I Mathematical Logic Statements and notations: A proposition or statement is a declarative sentence that is either true or false (but not both). For instance, the following are propositions: “Paris is in France” (true), “London is in Denmark” (false), “2 < 4” (true), “4 = 7 (false)”. However the following are not propositions: “what is your name?” (this is a question), “do your homework” (this is a command), “this sentence is false” (neither true nor false), “x is an even number” (it depends on what x represents), “Socrates” (it is not even a sentence). The truth or falsehood of a proposition is called its truth value. Connectives: Connectives are used for making compound propositions. The main ones are the following (p and q represent given propositions): Name Represented Meaning Negation ¬p “not p” Conjunction p q “p and q” Disjunction p q “p or q (or both)” Exclusive Or p q “either p or q, but not both” Implication p ⊕ q “if p then q” Biconditional p q “p if and only if q” Truth Tables: Logical identity Logical identity is an operation on one logical value, typically the value of a proposition that produces a value of true if its operand is true and a value of false if its operand is false. The truth table for the logical identity operator is as follows: Logical Identity p p T T F F Logical negation Logical negation is an operation on one logical value, typically the value of a proposition that produces a value of true if its operand is false and a value of false if its operand is true. -
Introduction to Logic
IntroductionI Introduction to Logic Propositional calculus (or logic) is the study of the logical Christopher M. Bourke relationship between objects called propositions and forms the basis of all mathematical reasoning. Definition A proposition is a statement that is either true or false, but not both (we usually denote a proposition by letters; p, q, r, s, . .). Computer Science & Engineering 235 Introduction to Discrete Mathematics [email protected] IntroductionII ExamplesI Example (Propositions) Definition I 2 + 2 = 4 I The derivative of sin x is cos x. The value of a proposition is called its truth value; denoted by T or I 6 has 2 factors 1 if it is true and F or 0 if it is false. Opinions, interrogative and imperative sentences are not Example (Not Propositions) propositions. I C++ is the best language. I When is the pretest? I Do your homework. ExamplesII Logical Connectives Connectives are used to create a compound proposition from two Example (Propositions?) or more other propositions. I Negation (denoted or !) ¬ I 2 + 2 = 5 I And (denoted ) or Logical Conjunction ∧ I Every integer is divisible by 12. I Or (denoted ) or Logical Disjunction ∨ I Microsoft is an excellent company. I Exclusive Or (XOR, denoted ) ⊕ I Implication (denoted ) → I Biconditional; “if and only if” (denoted ) ↔ Negation Logical And A proposition can be negated. This is also a proposition. We usually denote the negation of a proposition p by p. ¬ The logical connective And is true only if both of the propositions are true. It is also referred to as a conjunction. Example (Negated Propositions) Example (Logical Connective: And) I Today is not Monday. -
Chapter 1 Logic and Set Theory
Chapter 1 Logic and Set Theory To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. – Ian Stewart Does God play dice? The mathematics of chaos In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. That is, a proof is a logical argument, not an empir- ical one. One must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics. An unproven proposition for which there is some sort of empirical evidence is known as a conjecture. Mathematical logic is the framework upon which rigorous proofs are built. It is the study of the principles and criteria of valid inference and demonstrations. Logicians have analyzed set theory in great details, formulating a collection of axioms that affords a broad enough and strong enough foundation to mathematical reasoning. The standard form of axiomatic set theory is denoted ZFC and it consists of the Zermelo-Fraenkel (ZF) axioms combined with the axiom of choice (C). Each of the axioms included in this theory expresses a property of sets that is widely accepted by mathematicians. It is unfortunately true that careless use of set theory can lead to contradictions. Avoiding such contradictions was one of the original motivations for the axiomatization of set theory. 1 2 CHAPTER 1. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. -
Next Generation Logic Programming Systems Gopal Gupta
Next Generation Logic Programming Systems Gopal Gupta In the last 30 years logic programming (LP), with Prolog as the most representative logic programming language, has emerged as a powerful paradigm for intelligent reasoning and deductive problem solving. In the last 15 years several powerful and highly successful extensions of logic programming have been proposed and researched. These include constraint logic programming (CLP), tabled logic programming, inductive logic programming, concurrent/parallel logic programming, Andorra Prolog and adaptations for non- monotonic reasoning such as answer set programming. These extensions have led to very powerful applications in reasoning, for example, CLP has been used in industry for intelligently and efficiently solving large scale scheduling and resource allocation problems, tabled logic programming has been used for intelligently solving large verification problems as well as non- monotonic reasoning problems, inductive logic programming for new drug discoveries, and answer set programming for practical planning problems, etc. However, all these powerful extensions of logic programming have been developed by extending standard logic programming (Prolog) systems, and each one has been developed in isolation from others. While each extension has resulted in very powerful applications in reasoning, considerably more powerful applications will become possible if all these extensions were combined into a single system. This power will come about due to search space being dramatically pruned, reducing the time taken to find a solution given a reasoning problem coded as a logic program. Given the current state-of-the-art, even two extensions are not available together in a single system. Realizing multiple or all extensions in a single framework has been rendered difficult by the enormous complexity of implementing reasoning systems in general and logic programming systems in particular. -
Logic-Based Technologies for Intelligent Systems: State of the Art and Perspectives
information Article Logic-Based Technologies for Intelligent Systems: State of the Art and Perspectives Roberta Calegari 1,* , Giovanni Ciatto 2 , Enrico Denti 3 and Andrea Omicini 2 1 Alma AI—Alma Mater Research Institute for Human-Centered Artificial Intelligence, Alma Mater Studiorum–Università di Bologna, 40121 Bologna, Italy 2 Dipartimento di Informatica–Scienza e Ingegneria (DISI), Alma Mater Studiorum–Università di Bologna, 47522 Cesena, Italy; [email protected] (G.C.); [email protected] (A.O.) 3 Dipartimento di Informatica–Scienza e Ingegneria (DISI), Alma Mater Studiorum–Università di Bologna, 40136 Bologna, Italy; [email protected] * Correspondence: [email protected] Received: 25 February 2020; Accepted: 18 March 2020; Published: 22 March 2020 Abstract: Together with the disruptive development of modern sub-symbolic approaches to artificial intelligence (AI), symbolic approaches to classical AI are re-gaining momentum, as more and more researchers exploit their potential to make AI more comprehensible, explainable, and therefore trustworthy. Since logic-based approaches lay at the core of symbolic AI, summarizing their state of the art is of paramount importance now more than ever, in order to identify trends, benefits, key features, gaps, and limitations of the techniques proposed so far, as well as to identify promising research perspectives. Along this line, this paper provides an overview of logic-based approaches and technologies by sketching their evolution and pointing out their main application areas. Future perspectives for exploitation of logic-based technologies are discussed as well, in order to identify those research fields that deserve more attention, considering the areas that already exploit logic-based approaches as well as those that are more likely to adopt logic-based approaches in the future. -
Compiling a Default Reasoning System Into Prolog
Compiling A Default Reasoning System into Prolog David Poole Department of Computer Science University of British Columbia Vancouver BC Canada VT W p o olecsub cca July Abstract Articial intelligence researchers have b een designing representa tion systems for default and ab ductive reasoning Logic Programming researchers have b een working on techniques to improve the eciency of Horn Clause deduction systems This pap er describ es how one such default and ab ductive reasoning system namely Theorist can b e translated into Horn clauses with negation as failure so that we can use the clarity of ab ductive reasoning systems and the eciency of Horn clause deduction systems Wethus showhowadvances in expressivepower that articial intelligence workers are working on can directly utilise advances in eciency that logic programming re searchers are working on Actual co de from a running system is given Intro duction Many p eople in Articial Intelligence have b een working on default reasoning and ab ductive diagnosis systems The systems implemented so far eg are only prototyp es or have b een develop ed in A Theorist to Prolog Compiler ways that cannot take full advantage in the advances of logic programming implementation technology Many p eople are working on making logic programming systems more ecient These systems however usually assume that the input is in the form of Horn clauses with negation as failure This pap er shows howto implement the default reasoning system Theorist by compiling its input into Horn clauses -
Logical Vs. Natural Language Conjunctions in Czech: a Comparative Study
ITAT 2016 Proceedings, CEUR Workshop Proceedings Vol. 1649, pp. 68–73 http://ceur-ws.org/Vol-1649, Series ISSN 1613-0073, c 2016 K. Prikrylová,ˇ V. Kubon,ˇ K. Veselovská Logical vs. Natural Language Conjunctions in Czech: A Comparative Study Katrin Prikrylová,ˇ Vladislav Kubon,ˇ and Katerinaˇ Veselovská Charles University in Prague, Faculty of Mathematics and Physics Czech Republic {prikrylova,vk,veselovska}@ufal.mff.cuni.cz Abstract: This paper studies the relationship between con- ceptions and irregularities which do not abide the rules as junctions in a natural language (Czech) and their logical strictly as it is the case in logic. counterparts. It shows that the process of transformation Primarily due to this difference, the transformation of of a natural language expression into its logical representa- natural language sentences into their logical representation tion is not straightforward. The paper concentrates on the constitutes a complex issue. As we are going to show in most frequently used logical conjunctions, and , and it the subsequent sections, there are no simple rules which analyzes the natural language phenomena which∧ influence∨ would allow automation of the process – the majority of their transformation into logical conjunction and disjunc- problematic cases requires an individual approach. tion. The phenomena discussed in the paper are temporal In the following text we are going to restrict our ob- sequence, expressions describing mutual relationship and servations to the two most frequently used conjunctions, the consequences of using plural. namely a (and) and nebo (or). 1 Introduction and motivation 2 Sentences containing the conjunction a (and) The endeavor to express natural language sentences in the form of logical expressions is probably as old as logic it- The initial assumption about complex sentences contain- self. -
Introduction to Deductive Logic
Introduction to Deductive Logic Part I: Logic & Language There is no royal road to logic, and really valuable ideas can only be had at the price of close attention. Charles Saunders Peirce INTRODUCTION TO DEDUCTIVE LOGIC PART I: LOGIC AND LANGUAGE I. INTRODUCTION What is Logic? . 4 What is an argument? . 5 Three Criteria for Evaluating Arguments . 6 II. ARGUMENT ANALYSIS – PART 1 Preliminary Issues . 9 Statements and Truth Value Premises and Conclusions . 9 Statements vs Sentences 10 Simple vs Compound Statements 12 Recognizing premises and conclusions . 13 Indicator Language. 15 TWO TYPES OF ARGUMENTS Inductive Reasoning/Argument . 18 Deductive Reasoning/Argument . 19 III. ANALYSIS – PART II FORMALIZING ARGUMENT Preliminary Issue: Formal vs. Informal Argument . 24 Simple and Compound Statements . 25 Types of Compound Statements . 26 Formation Rules for Sentential . 26 DEFINING LOGICAL OPERATORS . 29 Negation . 29 Conjunction . 30 Disjunction . 31 Conditional/ Material Implication . 32 Bi-conditional . 33 APPLICATION: TRANSLATING USING LOGICAL OPERATORS . 35 TRUTH TABLE RULES/SUMMARY OF LOGICAL OPERATORS . 38 TRUTH TABLE APPLICATIONS . 39 Application I: Substitute and Calculate Method . 39 Application II: Calculating with Unknown Values . 41 Application III: Full Truth Table Method . 42 A Short Cut Technique for statements. 44 Truth Table Test for Logical Equivalence . 45 Application IV: Using Truth Tables to Facilitate Reasoning . 47 THE NEED FOR PREDICATE . 50 Using Predicate to Represent Categorical Statements . 52 Preliminary Issues– Grammar . 53 Formation Rules for Predicate . 55 Expressing Quantity . 56 APPLICATION: Symbolizing Categorical Statements using Predicate . 56 Negating Standard Form Categorical Statements. 57 Summary Chart for Standard Form Categorical Statements . 59 Extending Predicate . 61 A Universe of Discourse . -
Hardware Abstract the Logic Gates References Results Transistors Through the Years Acknowledgements
The Practical Applications of Logic Gates in Computer Science Courses Presenters: Arash Mahmoudian, Ashley Moser Sponsored by Prof. Heda Samimi ABSTRACT THE LOGIC GATES Logic gates are binary operators used to simulate electronic gates for design of circuits virtually before building them with-real components. These gates are used as an instrumental foundation for digital computers; They help the user control a computer or similar device by controlling the decision making for the hardware. A gate takes in OR GATE AND GATE NOT GATE an input, then it produces an algorithm as to how The OR gate is a logic gate with at least two An AND gate is a consists of at least two A NOT gate, also known as an inverter, has to handle the output. This process prevents the inputs and only one output that performs what inputs and one output that performs what is just a single input with rather simple behavior. user from having to include a microprocessor for is known as logical disjunction, meaning that known as logical conjunction, meaning that A NOT gate performs what is known as logical negation, which means that if its input is true, decision this making. Six of the logic gates used the output of this gate is true when any of its the output of this gate is false if one or more of inputs are true. If all the inputs are false, the an AND gate's inputs are false. Otherwise, if then the output will be false. Likewise, are: the OR gate, AND gate, NOT gate, XOR gate, output of the gate will also be false. -
Mathematical Reasoning with Diagrams Mateja Jamnik
Mathematical Reasoning with Diagrams Mateja Jamnik ISBN: 1-57586-324-3 Copyright notice: Excerpted from Mathematical Reasoning with Diagrams by Mateja Jamnik, published by CSLI Publications. ©2001 by CSLI Publications. All rights reserved. This text may be used and shared in accordance with the fair-use provisions of U.S. copyright law, and it may be archived and redistributed in electronic form, provided that this entire notice, including copyright information, is carried and provided that CSLI Publications is notified and no fee is charged for access. Archiving, redistribution, or republication of this text on other terms, in any medium, requires the consent of CSLI Publications. Contents Foreword vii Preface xi 1 Introduction 1 2 The History of Diagrammatic Systems 11 3 Diagrammatic Theorems and the Problem Domain 27 4 The Constructive ω-Rule and Schematic Proofs 49 5 Designing a Diagrammatic Reasoning System 71 6 Diagrammatic Operations 89 7 The Construction of Schematic Proofs 103 8 The Verification of Schematic Proofs 123 9 Diamond in Action 149 10 Complete Automation 163 Appendix A: More Examples of Diagrammatic Theorems 175 Appendix B: The ω-Rule 181 Glossary 185 References 190 Index 199 v Foreword The advent of the modern computer in the nineteen-fifties immediately suggested a new research challenge: to seek ways of programming these versatile machines which would make them behave as much like intel- ligent human beings as possible. After fifty years or so, this quest has produced some intriguing results, but until now progress has been dis- appointingly slow. This book is a welcome and encouraging sign that things may at last be about to change. -
Set Notation and Concepts
Appendix Set Notation and Concepts “In mathematics you don’t understand things. You just get used to them.” John von Neumann (1903–1957) This appendix is primarily a brief run-through of basic concepts from set theory, but it also in Sect. A.4 mentions set equations, which are not always covered when introducing set theory. A.1 Basic Concepts and Notation A set is a collection of items. You can write a set by listing its elements (the items it contains) inside curly braces. For example, the set that contains the numbers 1, 2 and 3 can be written as {1, 2, 3}. The order of elements do not matter in a set, so the same set can be written as {2, 1, 3}, {2, 3, 1} or using any permutation of the elements. The number of occurrences also does not matter, so we could also write the set as {2, 1, 2, 3, 1, 1} or in an infinity of other ways. All of these describe the same set. We will normally write sets without repetition, but the fact that repetitions do not matter is important to understand the operations on sets. We will typically use uppercase letters to denote sets and lowercase letters to denote elements in a set, so we could write M ={2, 1, 3} and x = 2 as an element of M. The empty set can be written either as an empty list of elements ({})orusing the special symbol ∅. The latter is more common in mathematical texts. A.1.1 Operations and Predicates We will often need to check if an element belongs to a set or select an element from a set.