Lecture 13 of 41. More Propositional and Predicate Logic


 Alexander Garrett
 2 years ago
 Views:
Transcription
1 Lecture 13 of 41 More Propositional and Predicate Logic Monday, 20 September 2004 William H. Hsu, KSU Reading: Sections , Russell and Norvig 2e Review: Chapter 6, R&N 2e
2 Lecture Outline Today s Reading Chapter 8, Russell and Norvig Recommended references: Nilsson and Genesereth (excerpt of Chapter 5 online) Next Week s Reading: Chapters 910, R&N Previously: Propositional and FirstOrder Logic Last Wednesday (15 Sep 2004) Logical agent framework Logic in general: tools for KR, inference, problem solving Propositional logic: normal forms, sequent rules (modus ponens, resolution) Firstorder logic (FOL): predicates, functions, quantifiers Last Friday (17 Sep 2004) FOL agents, issues: frame, ramification, qualification problems Solutions: situation calculus, circumscription by successor state axioms Today: FOL Knowledge Bases Next Week: Resolution Theorem Proving, Logic Programming Basics
3 Validity and Satisfiability
4 Proof Methods
5 Logical Agents: Taking Stock
6 FOL: Atomic Sentences (Atomic WellFormed Formulae)
7 FOL: Complex Sentences (WellFormed Formulae)
8 Truth in FOL
9 Models for FOL: Example
10 Universal Quantification
11 Existential Quantification
12 Quantifier Properties
13 Taking Stock: FOL Inference Previously: Logical Agents and Calculi Review: FOL in Practice Agent toy world: Wumpus World in FOL Situation calculus Frame problem and variants (see R&N sidebar) Representational vs. inferential frame problems Qualification problem: what if? Ramification problem: what else? (side effects) Successorstate axioms FOL Knowledge Bases FOL Inference Proofs Patternmatching: unification Theoremproving as search Generalized Modus Ponens (GMP) Forward Chaining and Backward Chaining
14 Automated Deduction (Chapters R&N)
15 Example Proof??? Apply Sequent Rules to Generate New Assertions Modus Ponens And Introduction Universal Elimination
16 Search with Primitive Inference Rules
17 A Brief History of Reasoning: Chapter 8 End Notes, R&N
18 Knowledge Engineering KE: Process of Choosing logical language (basis of KR) Building KB Implementing proof theory Inferring new facts Analogy: Programming Languages / Software Engineering Choosing programming language (basis of software engineering) Writing program Choosing / writing compiler Running program Example Domains Electronic circuits (Section 8.3 R&N) Exercise Look up, read about protocol analysis Find example and think about KE process for your project domain
19 Ontology Ontology: What Objects Exist and Are Symbolically Representable? Issue: Grouping Objects and Describing Families Grouping objects and describing families Example: sets of sets Russell s paradox: (Four) responses: types, formalism, intuitionism, ZermeloFraenkel set theory Sidebar: natural kinds (p. 232) Issue: Reasoning About Time Modal logics (CIS 301) Interval logics (Section 8.4 R&N p ) Example Domains Grocery shopping (Section 8.5 R&N); similar example in Winston 3e Data models for knowledge discovery in databases (KDD) Data dictionaries See grocery example, especially p
20 Unification: Definitions and Idea Sketch
21 Generalized Modus Ponens
22 Soundness of GMP
23 Summary Points Applications of Knowledge Bases (KBs) and Inference Systems Industrial Strength KBs Building KBs Knowledge Engineering (KE) and protocol analysis Inductive Logic Programming (ILP) and other machine learning techniques Components Ontologies Fact and rule bases Using KBs Systems of Sequent Rules: GMP/AI/UE, Resolution Methodology of Inference Inference as search Forward and backward chaining Fanin, fanout
24 Terminology Logical Languages: WFFs, Quantification Properties of Knowledge Bases (KBs) Satisfiability and validity Entailment and provability Properties of Proof Systems: Soundness and Completeness Knowledge Bases in Practice Knowledge Engineering Ontologies Sequent Rules (Generalized) Modus Ponens AndIntroduction UniversalElimination Methodology of Inference Forward and backward chaining Fanin, fanout (wax on, wax off )
Computational Logic and Cognitive Science: An Overview
Computational Logic and Cognitive Science: An Overview Session 1: Logical Foundations Technical University of Dresden 25th of August, 2008 University of Osnabrück Who we are Helmar Gust Interests: Analogical
More informationCourse Outline Department of Computing Science Faculty of Science. COMP 37103 Applied Artificial Intelligence (3,1,0) Fall 2015
Course Outline Department of Computing Science Faculty of Science COMP 710  Applied Artificial Intelligence (,1,0) Fall 2015 Instructor: Office: Phone/Voice Mail: EMail: Course Description : Students
More informationLogic in general. Inference rules and theorem proving
Logical Agents Knowledgebased agents Logic in general Propositional logic Inference rules and theorem proving First order logic Knowledgebased agents Inference engine Knowledge base Domainindependent
More informationLecture 18 of 42. Lecture 18 of 42
Knowledge Representation Concluded: KE, CIKM, & Representing Events over Time Discussion: Structure Elicitation, Event Calculus William H. Hsu Department of Computing and Information Sciences, KSU KSOL
More information! " # The Logic of Descriptions. Logics for Data and Knowledge Representation. Terminology. Overview. Three Basic Features. Some History on DLs
,!0((,.+#$),%$(&.& *,2($)%&2.'3&%!&, Logics for Data and Knowledge Representation Alessandro Agostini agostini@dit.unitn.it University of Trento Fausto Giunchiglia fausto@dit.unitn.it The Logic of Descriptions!$%&'()*$#)
More information2. The Language of Firstorder Logic
2. The Language of Firstorder Logic KR & R Brachman & Levesque 2005 17 Declarative language Before building system before there can be learning, reasoning, planning, explanation... need to be able to
More informationCHAPTER 7 GENERAL PROOF SYSTEMS
CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes
More informationLecture 8: Resolution theoremproving
Comp24412 Symbolic AI Lecture 8: Resolution theoremproving Ian PrattHartmann Room KB2.38: email: ipratt@cs.man.ac.uk 2014 15 In the previous Lecture, we met SATCHMO, a firstorder theoremprover implemented
More informationSummary Last Lecture. Automated Reasoning. Outline of the Lecture. Definition sequent calculus. Theorem (Normalisation and Strong Normalisation)
Summary Summary Last Lecture sequent calculus Automated Reasoning Georg Moser Institute of Computer Science @ UIBK Winter 013 (Normalisation and Strong Normalisation) let Π be a proof in minimal logic
More informationBeyond Propositional Logic Lukasiewicz s System
Beyond Propositional Logic Lukasiewicz s System Consider the following set of truth tables: 1 0 0 1 # # 1 0 # 1 1 0 # 0 0 0 0 # # 0 # 1 0 # 1 1 1 1 0 1 0 # # 1 # # 1 0 # 1 1 0 # 0 1 1 1 # 1 # 1 Brandon
More informationPredicate logic Proofs Artificial intelligence. Predicate logic. SET07106 Mathematics for Software Engineering
Predicate logic SET07106 Mathematics for Software Engineering School of Computing Edinburgh Napier University Module Leader: Uta Priss 2010 Copyright Edinburgh Napier University Predicate logic Slide 1/24
More informationConsistency, completeness of undecidable preposition of Principia Mathematica. Tanmay Jaipurkar
Consistency, completeness of undecidable preposition of Principia Mathematica Tanmay Jaipurkar October 21, 2013 Abstract The fallowing paper discusses the inconsistency and undecidable preposition of Principia
More informationOverview of the TACITUS Project
Overview of the TACITUS Project Jerry R. Hobbs Artificial Intelligence Center SRI International 1 Aims of the Project The specific aim of the TACITUS project is to develop interpretation processes for
More informationAn Introduction to AI Planning
An Introduction to AI Planning Ute Schmid Applied CS/Cognitive Systems Bamberg University Goals, Methods, Topics of AI Basic concepts of AI Planning Deductive Planning vs. Statebased (Strips) Planning
More informationWOLLONGONG COLLEGE AUSTRALIA. Diploma in Information Technology
First Name: Family Name: Student Number: Class/Tutorial: WOLLONGONG COLLEGE AUSTRALIA A College of the University of Wollongong Diploma in Information Technology Final Examination Spring Session 2008 WUCT121
More informationSoftware Modeling and Verification
Software Modeling and Verification Alessandro Aldini DiSBeF  Sezione STI University of Urbino Carlo Bo Italy 34 February 2015 Algorithmic verification Correctness problem Is the software/hardware system
More informationSchedule. Logic (master program) Literature & Online Material. gic. Time and Place. Literature. Exercises & Exam. Online Material
OLC mputational gic Schedule Time and Place Thursday, 8:15 9:45, HS E Logic (master program) Georg Moser Institute of Computer Science @ UIBK week 1 October 2 week 8 November 20 week 2 October 9 week 9
More informationRemarks on NonFregean Logic
STUDIES IN LOGIC, GRAMMAR AND RHETORIC 10 (23) 2007 Remarks on NonFregean Logic Mieczys law Omy la Institute of Philosophy University of Warsaw Poland m.omyla@uw.edu.pl 1 Introduction In 1966 famous Polish
More informationAutomated Theorem Proving  summary of lecture 1
Automated Theorem Proving  summary of lecture 1 1 Introduction Automated Theorem Proving (ATP) deals with the development of computer programs that show that some statement is a logical consequence of
More informationCHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
More informationCS510 Software Engineering
CS510 Software Engineering Propositional Logic Asst. Prof. Mathias Payer Department of Computer Science Purdue University TA: Scott A. Carr Slides inspired by Xiangyu Zhang http://nebelwelt.net/teaching/15cs510se
More informationFoundational Proof Certificates
An application of proof theory to computer science INRIASaclay & LIX, École Polytechnique CUSO Winter School, Proof and Computation 30 January 2013 Can we standardize, communicate, and trust formal proofs?
More informationTheory of Automated Reasoning An Introduction. AnttiJuhani Kaijanaho
Theory of Automated Reasoning An Introduction AnttiJuhani Kaijanaho Intended as compulsory reading for the Spring 2004 course on Automated Reasononing at Department of Mathematical Information Technology,
More informationThe Foundations: Logic and Proofs. Chapter 1, Part III: Proofs
The Foundations: Logic and Proofs Chapter 1, Part III: Proofs Rules of Inference Section 1.6 Section Summary Valid Arguments Inference Rules for Propositional Logic Using Rules of Inference to Build Arguments
More informationTrust but Verify: Authorization for Web Services. The University of Vermont
Trust but Verify: Authorization for Web Services Christian Skalka X. Sean Wang The University of Vermont Trust but Verify (TbV) Reliable, practical authorization for web service invocation. Securing complex
More informationINDUCTIVE & DEDUCTIVE RESEARCH APPROACH
INDUCTIVE & DEDUCTIVE RESEARCH APPROACH Meritorious Prof. Dr. S. M. Aqil Burney Director UBIT Chairman Department of Computer Science University of Karachi burney@computer.org www.drburney.net Designed
More informationdef: An axiom is a statement that is assumed to be true, or in the case of a mathematical system, is used to specify the system.
Section 1.5 Methods of Proof 1.5.1 1.5 METHODS OF PROOF Some forms of argument ( valid ) never lead from correct statements to an incorrect. Some other forms of argument ( fallacies ) can lead from true
More informationBindings, mobility of bindings, and the quantifier
ICMS, 26 May 2007 1/17 Bindings, mobility of bindings, and the quantifier Dale Miller, INRIASaclay and LIX, École Polytechnique This talk is based on papers with Tiu in LICS2003 & ACM ToCL, and experience
More informationCSE 459/598: Logic for Computer Scientists (Spring 2012)
CSE 459/598: Logic for Computer Scientists (Spring 2012) Time and Place: T Th 10:3011:45 a.m., M109 Instructor: Joohyung Lee (joolee@asu.edu) Instructor s Office Hours: T Th 4:305:30 p.m. and by appointment
More informationFormal Logic, Algorithms, and Incompleteness! Robert Stengel! Robotics and Intelligent Systems MAE 345, Princeton University, 2015
Formal Logic, Algorithms, and Incompleteness! Robert Stengel! Robotics and Intelligent Systems MAE 345, Princeton University, 2015 Learning Objectives!! Principles of axiomatic systems and formal logic!!
More informationArtificial Intelligence. Knowledge Representation
RC Chakraborty 03/03 to 08/3, 2007, Lecture 15 to 22 (8 hrs) Slides 1 to 79 myreaders, http://myreaders.wordpress.com/, rcchak@gmail.com (Revised Feb. 02, 2008) Artificial Intelligence Knowledge Representation
More informationCS Master Level Courses and Areas COURSE DESCRIPTIONS. CSCI 521 RealTime Systems. CSCI 522 High Performance Computing
CS Master Level Courses and Areas The graduate courses offered may change over time, in response to new developments in computer science and the interests of faculty and students; the list of graduate
More informationFormalization of the CRM: Initial Thoughts
Formalization of the CRM: Initial Thoughts Carlo Meghini Istituto di Scienza e Tecnologie della Informazione Consiglio Nazionale delle Ricerche Pisa CRM SIG Meeting Iraklio, October 1st, 2014 Outline Overture:
More informationSound and Complete Inference Rules in FOL
Sound and Complete Inference Rules in FOL An inference rule i is called sound if KB = α whenever KB i α An inference rule i is called complete if KB i α whenever KB = α Generalised ModusPonens (equivalently,
More informationFormal Methods in Security Protocols Analysis
Formal Methods in Security Protocols Analysis Li Zhiwei Aidong Lu Weichao Wang Department of Computer Science Department of Software and Information Systems University of North Carolina at Charlotte Big
More informationIntroduction to Knowledge Fusion and Representation
Introduction to Knowledge Fusion and Representation Introduction 1. A.I. 2. Knowledge Representation 3. Reasoning 4. Logic 5. Information Integration 6. Semantic Web Knowledge Fusion Fall 2004 1 What is
More informationML for the Working Programmer
ML for the Working Programmer 2nd edition Lawrence C. Paulson University of Cambridge CAMBRIDGE UNIVERSITY PRESS CONTENTS Preface to the Second Edition Preface xiii xv 1 Standard ML 1 Functional Programming
More informationPredicate Logic. Example: All men are mortal. Socrates is a man. Socrates is mortal.
Predicate Logic Example: All men are mortal. Socrates is a man. Socrates is mortal. Note: We need logic laws that work for statements involving quantities like some and all. In English, the predicate is
More informationPlanning. Planning. Example Domain: Wumpus World. Wumpus World (WW)
Planning I lecture (Yoonsuck Choe): Material from Russel and Norvig (nd ed.) 7., 7.7: Wumpus world (an example domain) 0.3: Situation calculus : Planning Planning The task of coming up with a sequence
More informationFirstorder logic. Chapter 8. Chapter 8 1
Firstorder logic Chapter 8 Chapter 8 1 Outline Why FOL? Syntax and semantics of FOL Fun with sentences Wumpus world in FOL Chapter 8 2 Pros and cons of propositional logic Propositional logic is declarative:
More informationHandout #1: Mathematical Reasoning
Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or
More informationThis asserts two sets are equal iff they have the same elements, that is, a set is determined by its elements.
3. Axioms of Set theory Before presenting the axioms of set theory, we first make a few basic comments about the relevant first order logic. We will give a somewhat more detailed discussion later, but
More informationOntologies for Enterprise Integration
Ontologies for Enterprise Integration Mark S. Fox and Michael Gruninger Department of Industrial Engineering,University of Toronto, 4 Taddle Creek Road, Toronto, Ontario M5S 1A4 tel:14169786823 fax:14169711373
More informationCS 441 Discrete Mathematics for CS Lecture 5. Predicate logic. CS 441 Discrete mathematics for CS. Negation of quantifiers
CS 441 Discrete Mathematics for CS Lecture 5 Predicate logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Negation of quantifiers English statement: Nothing is perfect. Translation: x Perfect(x)
More informationDEDUCTIVE & INDUCTIVE REASONING
DEDUCTIVE & INDUCTIVE REASONING Expectations 1. Take notes on inductive and deductive reasoning. 2. This is an information based presentation  I simply want you to be able to apply this information to
More informationExample: Backward Chaining. Inference Strategy: Backward Chaining. FirstOrder Logic. Knowledge Engineering. Example: Proof
Inference Strategy: Backward Chaining Idea: Check whether a particular fact q is true. Backward Chaining: Given a fact q to be proven, 1. See if q is already in the KB. If so, return TRUE. 2. Find all
More informationPredicate Logic Review
Predicate Logic Review UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 Grammar A term is an individual constant or a variable. An individual constant is a lowercase letter from the beginning
More informationLecture 2 of 41. Agents and Problem Solving
Lecture 2 of 41 Agents and Problem Solving Monday, 23 August 2004 William H. Hsu, KSU http://www.kddresearch.org http://www.cis.ksu.edu/~bhsu Reading for Next Class: Chapter 3, Appendix A, Russell and
More informationRelational Methodology for Data Mining and Knowledge Discovery
Relational Methodology for Data Mining and Knowledge Discovery Vityaev E.E.* 1, Kovalerchuk B.Y. 2 1 Sobolev Institute of Mathematics SB RAS, Novosibirsk State University, Novosibirsk, 630090, Russia.
More informationCertamen 1 de Representación del Conocimiento
Certamen 1 de Representación del Conocimiento Segundo Semestre 2012 Question: 1 2 3 4 5 6 7 8 9 Total Points: 2 2 1 1 / 2 1 / 2 3 1 1 / 2 1 1 / 2 12 Here we show one way to solve each question, but there
More informationOptimizing Description Logic Subsumption
Topics in Knowledge Representation and Reasoning Optimizing Description Logic Subsumption Maryam FazelZarandi Company Department of Computer Science University of Toronto Outline Introduction Optimization
More informationRigorous Software Development CSCIGA 3033009
Rigorous Software Development CSCIGA 3033009 Instructor: Thomas Wies Spring 2013 Lecture 11 Semantics of Programming Languages Denotational Semantics Meaning of a program is defined as the mathematical
More informationLecture Notes in Discrete Mathematics. Marcel B. Finan Arkansas Tech University c All Rights Reserved
Lecture Notes in Discrete Mathematics Marcel B. Finan Arkansas Tech University c All Rights Reserved 2 Preface This book is designed for a one semester course in discrete mathematics for sophomore or junior
More informationFoundations of Logic and Mathematics
Yves Nievergelt Foundations of Logic and Mathematics Applications to Computer Science and Cryptography Birkhäuser Boston Basel Berlin Contents Preface Outline xiii xv A Theory 1 0 Boolean Algebraic Logic
More informationIAI : Expert Systems
IAI : Expert Systems John A. Bullinaria, 2005 1. What is an Expert System? 2. The Architecture of Expert Systems 3. Knowledge Acquisition 4. Representing the Knowledge 5. The Inference Engine 6. The ReteAlgorithm
More informationFixedPoint Logics and Computation
1 FixedPoint Logics and Computation Symposium on the Unusual Effectiveness of Logic in Computer Science University of Cambridge 2 Mathematical Logic Mathematical logic seeks to formalise the process of
More informationThings That Might Not Have Been Michael Nelson University of California at Riverside mnelson@ucr.edu
Things That Might Not Have Been Michael Nelson University of California at Riverside mnelson@ucr.edu Quantified Modal Logic (QML), to echo Arthur Prior, is haunted by the myth of necessary existence. Features
More informationWOLLONGONG COLLEGE AUSTRALIA. Diploma in Information Technology
First Name: Family Name: Student Number: Class/Tutorial: WOLLONGONG COLLEGE AUSTRALIA A College of the University of Wollongong Diploma in Information Technology MidSession Test Summer Session 00800
More informationON FUNCTIONAL SYMBOLFREE LOGIC PROGRAMS
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical and Mathematical Sciences 2012 1 p. 43 48 ON FUNCTIONAL SYMBOLFREE LOGIC PROGRAMS I nf or m at i cs L. A. HAYKAZYAN * Chair of Programming and Information
More informationPredicate Logic. M.A.Galán, TDBA64, VT03
Predicate Logic 1 Introduction There are certain arguments that seem to be perfectly logical, yet they cannot be specified by using propositional logic. All cats have tails. Tom is a cat. From these two
More informationTopic 2: Structure of KnowledgeBased Systems
Engineering (Ingeniería del Conocimiento) Escuela Politécnica Superior, UAM Course 20072008 Topic 2: Structure of Based Systems Contents 2.1 Components according to the Final User 2.2 Components according
More informationAnswers G53KRR 200910
s G53KRR 200910 1. (a) Express the following sentences in firstorder logic, using unary predicates Fragile, Break, Fall, TennisBall. S1 Fragile things break if they fall S2 Tennis balls are not fragile
More informationSubjects, Predicates and The Universal Quantifier
Introduction to Logic Week Sixteen: 14 Jan., 2008 Subjects, Predicates and The Universal Quantifier 000. Office Hours this term: Tuesdays and Wednesdays 12, or by appointment; 5B120. 00. Results from
More information060010706 Artificial Intelligence 2014
Module1 Introduction Short Answer Questions: 1. Define the term Artificial Intelligence (AI). 2. List the two general approaches used by AI researchers. 3. State the basic objective of bottomup approach
More informationRelations: their uses in programming and computational specifications
PEPS Relations, 15 December 2008 1/27 Relations: their uses in programming and computational specifications Dale Miller INRIA  Saclay & LIX, Ecole Polytechnique 1. Logic and computation Outline 2. Comparing
More information6.080/6.089 GITCS Feb 12, 2008. Lecture 3
6.8/6.89 GITCS Feb 2, 28 Lecturer: Scott Aaronson Lecture 3 Scribe: Adam Rogal Administrivia. Scribe notes The purpose of scribe notes is to transcribe our lectures. Although I have formal notes of my
More informationAikaterini Marazopoulou
Imperial College London Department of Computing Tableau Compiled Labelled Deductive Systems with an application to Description Logics by Aikaterini Marazopoulou Submitted in partial fulfilment of the requirements
More informationDegrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets.
Degrees of Truth: the formal logic of classical and quantum probabilities as well as fuzzy sets. Logic is the study of reasoning. A language of propositions is fundamental to this study as well as true
More informationSEARCHING AND KNOWLEDGE REPRESENTATION. Angel Garrido
Acta Universitatis Apulensis ISSN: 15825329 No. 30/2012 pp. 147152 SEARCHING AND KNOWLEDGE REPRESENTATION Angel Garrido ABSTRACT. The procedures of searching of solutions of problems, in Artificial Intelligence
More information(LMCS, p. 317) V.1. First Order Logic. This is the most powerful, most expressive logic that we will examine.
(LMCS, p. 317) V.1 First Order Logic This is the most powerful, most expressive logic that we will examine. Our version of firstorder logic will use the following symbols: variables connectives (,,,,
More informationIntroduction to Automata Theory. Reading: Chapter 1
Introduction to Automata Theory Reading: Chapter 1 1 What is Automata Theory? Study of abstract computing devices, or machines Automaton = an abstract computing device Note: A device need not even be a
More information4 Domain Relational Calculus
4 Domain Relational Calculus We now present two relational calculi that we will compare to RA. First, what is the difference between an algebra and a calculus? The usual story is that the algebra RA is
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationOntologies and the Web Ontology Language OWL
Chapter 7 Ontologies and the Web Ontology Language OWL vocabularies can be defined by RDFS not so much stronger than the ER Model or UML (even weaker: no cardinalities) not only a conceptual model, but
More informationGeneralized Modus Ponens
Generalized Modus Ponens This rule allows us to derive an implication... True p 1 and... p i and... p n p 1... p i1 and p i+1... p n implies p i implies q implies q allows: a 1 and... a i and... a n implies
More informationPREDICATE LOGIC. 1 Basic Concepts. Jorma K. Mattila LUT, Department of Mathematics and Physics
PREDICATE LOGIC Jorma K. Mattila LUT, Department of Mathematics and Physics 1 Basic Concepts In predicate logic the formalism of propositional logic is extended and is made it more finely build than propositional
More informationLikewise, we have contradictions: formulas that can only be false, e.g. (p p).
CHAPTER 4. STATEMENT LOGIC 59 The rightmost column of this truth table contains instances of T and instances of F. Notice that there are no degrees of contingency. If both values are possible, the formula
More informationPredicate Logic. For example, consider the following argument:
Predicate Logic The analysis of compound statements covers key aspects of human reasoning but does not capture many important, and common, instances of reasoning that are also logically valid. For example,
More informationSatisfiability and Validity
6.825 Techniques in Artificial Intelligence Satisfiability and Validity Lecture 4 1 Last time we talked about propositional logic. There's no better way to empty out a room than to talk about logic. So
More informationDiscrete Mathematics, Chapter : Predicate Logic
Discrete Mathematics, Chapter 1.41.5: Predicate Logic Richard Mayr University of Edinburgh, UK Richard Mayr (University of Edinburgh, UK) Discrete Mathematics. Chapter 1.41.5 1 / 23 Outline 1 Predicates
More informationTechnical Report. Machine learning and automated theorem proving. James P. Bridge. Number 792. November Computer Laboratory
Technical Report UCAMCLTR792 ISSN 14762986 Number 792 Computer Laboratory Machine learning and automated theorem proving James P. Bridge November 2010 15 JJ Thomson Avenue Cambridge CB3 0FD United
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
More informationFoundations of Artificial Intelligence. Knowledge Representation and Reasoning
Foundations of Artificial Intelligence Knowledge Representation and Reasoning Knowledge Based Systems The field of knowledge engineering can be defined as the process of assessing problems, acquiring knowledge
More informationStatic Program Transformations for Efficient Software Model Checking
Static Program Transformations for Efficient Software Model Checking Shobha Vasudevan Jacob Abraham The University of Texas at Austin Dependable Systems Large and complex systems Software faults are major
More informationA Nonmonotonic Typed Multilevel Logic for Multilevel Secure Data / Knowledge Base Management Systems
A Nonmonotonic Typed Multilevel Logic for Multilevel Secure Data / Knowledge Base Management Systems Bhavani Thuraisingham The MITRE Corporation, Bedford, MA 01730 ABSTRACT This paper describes a logic
More informationCourse Syllabus For Operations Management. Management Information Systems
For Operations Management and Management Information Systems Department School Year First Year First Year First Year Second year Second year Second year Third year Third year Third year Third year Third
More informationHILBERT S PROGRAM THEN AND NOW
HILBERT S PROGRAM THEN AND NOW Richard Zach 1 INTRODUCTION Hilbert s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated
More informationMAPI Programa Doutoral em Informática. Rigorous Software Development
MAPI Programa Doutoral em Informática Rigorous Software Development Unidade Curricular em Teoria e Fundamentos Theory and Foundations (UCTF) DIUM, DCCFCUP May, 2012 Abstract This text presents a UCTF
More informationA Problem Course in Mathematical Logic Version 1.6. Stefan Bilaniuk
A Problem Course in Mathematical Logic Version 1.6 Stefan Bilaniuk Department of Mathematics Trent University Peterborough, Ontario Canada K9J 7B8 Email address: sbilaniuk@trentu.ca 1991 Mathematics Subject
More informationIntroduction to Logic: Argumentation and Interpretation. Vysoká škola mezinárodních a veřejných vztahů PhDr. Peter Jan Kosmály, Ph.D. 9. 3.
Introduction to Logic: Argumentation and Interpretation Vysoká škola mezinárodních a veřejných vztahů PhDr. Peter Jan Kosmály, Ph.D. 9. 3. 2016 tests. Introduction to Logic: Argumentation and Interpretation
More information196 Chapter 7. Logical Agents
7 LOGICAL AGENTS In which we design agents that can form representations of the world, use a process of inference to derive new representations about the world, and use these new representations to deduce
More informationGödel s correspondence on proof theory and constructive mathematics
Gödel s correspondence on proof theory and constructive mathematics W. W. Tait The volumes of Gödel s collected papers under review consist almost entirely of a rich selection of his philosophical/scientific
More informationNeighborhood Data and Database Security
Neighborhood Data and Database Security Kioumars Yazdanian, FrkdCric Cuppens email: yaz@ tlscs.cert.fr  cuppens@ tlscs.cert.fr CERT / ONERA, Dept. of Computer Science 2 avenue E. Belin, B.P. 4025,31055
More informationCopyright 2012 MECS I.J.Information Technology and Computer Science, 2012, 1, 5063
I.J. Information Technology and Computer Science, 2012, 1, 5063 Published Online February 2012 in MECS (http://www.mecspress.org/) DOI: 10.5815/ijitcs.2012.01.07 Using Logic Programming to Represent
More informationOKBC: A Programmatic Foundation for Knowledge Base Interoperability 1 2
Appears in the Proceedings of AAAI98, July 2630, Madison, WI. 1 OKBC: A Programmatic Foundation for Knowledge Base Interoperability 1 2 Vinay K. Chaudhri SRI International 333 Ravenswood Avenue Menlo
More informationWe would like to state the following system of natural deduction rules preserving falsity:
A Natural Deduction System Preserving Falsity 1 Wagner de Campos Sanz Dept. of Philosophy/UFG/Brazil sanz@fchf.ufg.br Abstract This paper presents a natural deduction system preserving falsity. This new
More informationCOURSE DESCRIPTION FOR THE COMPUTER INFORMATION SYSTEMS CURRICULUM
COURSE DESCRIPTION FOR THE COMPUTER INFORMATION SYSTEMS CURRICULUM Course Code 2505100 Computing Fundamentals Pass/ Fail Prerequisite None This course includes an introduction to the use of the computer
More informationProblems on Discrete Mathematics 1
Problems on Discrete Mathematics 1 ChungChih Li 2 Kishan Mehrotra 3 Syracuse University, New York L A TEX at January 11, 2007 (Part I) 1 No part of this book can be reproduced without permission from
More informationSemantic Groundedness
Semantic Groundedness Hannes Leitgeb LMU Munich August 2011 Hannes Leitgeb (LMU Munich) Semantic Groundedness August 2011 1 / 20 Luca told you about groundedness in set theory. Now we turn to groundedness
More informationLecture 5: Introduction to Knowledge Representation
Lecture 5: Introduction to Knowledge Representation Dr. Roman V Belavkin BIS4410 Contents 1 Knowledge Engineering Knowledge Engineering Definition 1 (Knowledge Engineering). The process of designing knowledgebased
More information