# MAI0203 Lecture 7: Inference and Predicate Calculus

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 MAI0203 Lecture 7: Inference and Predicate Calculus Methods of Artificial Intelligence WS 2002/2003 Part II: Inference and Knowledge Representation II.7 Inference and Predicate Calculus MAI0203 Lecture 7: Inference and Predicate Calculus p.175

2 Reasoning and Inference Reasoning is the process of using facts and inference rules to produce conclusions. A knowledge-based system typically consists of two components: knowledge representation inference. Knowledge representation is domain-dependent and must be acquired and formalized for each new domain of interest (knowledge engineering). Inference mechanisms are independent of a concrete domain but they rely on specific formats of knowledge representation. First order logic as representation formalism and resolution as inference mechanism are the foundation of many knowledge-based systems (e.g. expert systems, question-answering systems). MAI0203 Lecture 7: Inference and Predicate Calculus p.176

3 Basic Types of Inference: Deduction (Charles Peirce) Deduction: Derive a conclusion from given axioms ( knowledge ) and facts ( observations ). All humans are mortal. (axiom) Socrates is a human. (fact/premise) Therefore, it follows that Socrates is mortal. (conclusion) The conclusion can be derived by applying the modus ponens inference rule (Aristotelian/propositional logic). MAI0203 Lecture 7: Inference and Predicate Calculus p.177

4 Basic Types of Inference: Induction Induction: Derive a general rule (axiom) from background knowledge and observations. Socrates is a human. (background knowledge) Socrates is mortal. (observation/example) Therefore, I hypothesize that all humans are mortal. (generalization) Induction means to infer (unsure) generalized knowledge from example observations. Induction is the inference mechanism for learning! (see lesson on Machine Learning) Analogy is a special kind of induction. MAI0203 Lecture 7: Inference and Predicate Calculus p.178

5 Basic Types of Inference: Abduction Abduction: From a known axiom (theory) and some observation, derive a premise. All humans are mortal. (theory) Socrates is mortal. (observation) Therefore, Socrates must have been a human. (diagnosis) Abduction is typical for diagnostic systems/expert systems. (It is also the preferred reasoning method of Sherlock Holmes.) Simple medical diagnosis: If one has the flue, one has moderate fewer. Patient has moderate fewer. Therefore, he has the flue. MAI0203 Lecture 7: Inference and Predicate Calculus p.179

6 Deduction Deductive inference is also called theorem proving or logic inference. Deduction is truth preserving: If the premises (axioms and facts) are true, then the conclusion (also called theorem) is true. (given sound and complete induction rules!) Note: The truth of the premises is typically not the logical truth (premises in general are not tautologies). In deduction we infer special theorems from a more general theory. Deduction is knowledge extraction while induction is construction of (hypothetical) knowledge. MAI0203 Lecture 7: Inference and Predicate Calculus p.180

7 Deduction Calculus To perform deductive inference on a machine, a calculus is needed: A calculus is a set of syntactical rewrite-rules defined for some language. These rules must be sound and complete with respect to the task they should perform (to a model). Soundness and completeness are semantic properties! We will focus on the resolution calculus for first order logic (FOL). Syntax of FOL (language for knowledge representation definition of calculus) Semantics of FOL (meaning of FOL sentences properties of calculus) (Basic knowledge in propositional and first order logic MAI0203 Lecture is presumed.) 7: Inference and Predicate Calculus p.181

8 Syntax of FOL/Terms Inductive definition: Terms: A variable is a term. If is a function symbol with arity and... terms, then... is a term. (including constant symbols as 0-ary function symbols) That are all terms. are MAI0203 Lecture 7: Inference and Predicate Calculus p.182

9 Syntax of FOL/Formulas Inductive definition:. Formulas: if is a predicate symbol with arity and... are terms, then.. is a formula. (atomic formula) For all formulas and,,,, and are formula. (connectives not, and, or, implies, equivalent ) If is a variable and is a formula, then and are formulas. (existential and universal quantifier, exists, for all ) That are all formulas. MAI0203 Lecture 7: Inference and Predicate Calculus p.183

10 Remarks on Syntax of FOL Formula are constructed over terms. Never confuse this categories! Additionally, parentheses can be used to group sub-expressions. Expressions which obey the given inductive definition are called well-formed formulas (wwfs). The closure that are all terms/formulas is necessary to exclude all other kinds of (not well-founded) expressions. We refer to atomic formulas also as atoms. Positive and negated atoms (, ) are called positive/negative literals. MAI0203 Lecture 7: Inference and Predicate Calculus p.184

11 Remarks on Syntax of FOL cont. A variable which is in the scope of a quantor is called bound, otherwise it is is called free. is free and and are bound. A formula without free variables is called sentence. Propositional logic is a special case of FOL: use only unary predicate symbols (then there are no terms, no variable and no quantors) or just forbid variables and quantors (use only grounded formulas). MAI0203 Lecture 7: Inference and Predicate Calculus p.185

12 Semantics of FOL Syntax defines how to form well-formed expressions. Semantics gives meaning to expressions. Expressions are interpreted using an interpretation function over a domain (set of objects). A pair is called a structure (or algebra). Function symbols are interpreted as functions and predicate symbols as predicates/relations over a domain. MAI0203 Lecture 7: Inference and Predicate Calculus p.186

13 Example Interpretation x on(x, A) clear(a) y clear topof(y) Constant symbol A is interpreted as a block in a blocksworld. Function symbol topof(x) is interpreted as a function which returns the block which is lying on top of block or if no block is lying on. Predicate symbol clear(x) is interpreted as unary relation which holds if no other block is lying on block. Predicate symbol on(x,y) is interpreted as binary relation which holds if a block is lying immediatly on top of a block. MAI0203 Lecture 7: Inference and Predicate Calculus p.187

14 Example Illustration Possible interpretations for the formulas: on(a, Table) clear(b) clear(b) clear(c) on(a, Table) clear(b) clear(c) clear(b) clear(c) A C B C C A B A B MAI0203 Lecture 7: Inference and Predicate Calculus p.188

15 Intuition about Semantics Natural language sentence describes situation in the world B Block B lies on block A A C FOL formula is interpreted by a structure Blocksworld = with as set of blocks and as defined above. MAI0203 Lecture 7: Inference and Predicate Calculus p.189

16 Propositional Logic vs. FOL In propositional logic, formula are interpreted directly by truth values. In FOL we first must map the components of a formula stepwise into a structure. Constant symbols are mapped to objects in the domain. For interpreted functions, their result can be determined within the structure (evaluation of grounded terms). The result is again an object in the domain. Formula over terms are mapped to truth values. Now the difference between terms and formula should be clear: terms evaluate to values, formula to truth values! MAI0203 Lecture 7: Inference and Predicate Calculus p.190

17 Propositional Logic vs. FOL cont. Mapping of formula to truth values: To assign a truth value, first, the arguments of a predicate symbol are interpreted, afterwards it is determined whether the corresponding relation holds in the structure. If all atomic formula are associated with truth values, more complex formulas can be interpreted as known over the definition of the junctors. For an existentially quantified formula there needs to be at least one element in the domain for which the formula is true. For universally quantified formulas, the formula must be true for all domain objects. MAI0203 Lecture 7: Inference and Predicate Calculus p.191

18 Semantics of the Connectives A B A A B A B A B A B binds stronger than and which bind stronger than and MAI0203 Lecture 7: Inference and Predicate Calculus p.192

19 Model, Satisfiability, Validity If the interpretation of a formula with respect to a structure results in the truth value true, the structure is called a model. We write. If every structure for write. is a model, we call If there exists at least one model for satisfiable., we call valid and Remark: In general structures can be defined over domains which are arbitrary sets. That is, the sets can contain lots of irrelevant objects. There is a canonic way to construct structures over a so called Herbrand Universe. MAI0203 Lecture 7: Inference and Predicate Calculus p.193

20 Illustration is valid e.g. For all humans it holds that if two persons are rich then at least one of them is rich. e.g. For all natural numbers holds that if both numbers are even then at least one of them is even. is satisfiable e.g. It exists a natural number smaller than 17. e.g. It exists a human being who is a student and who likes logic. MAI0203 Lecture 7: Inference and Predicate Calculus p.194

21 Semantical Entailment A formula is called logical consequence (or entailment) of a set of formula, if each model of is also a model of. Note: We write to denote the model relation and also to denote the entailment relation. The following propositions are equivalent: is a logical consequence of is valid (tautology).. 3. is not satisfiable (a contradiction). This relation between logical consequences and syntactical expressions can be exploited for syntactical proofs. We write if formula can be derived from the set of formulas. MAI0203 Lecture 7: Inference and Predicate Calculus p.195

22 Resolution Calculus The resolution calculus consists of a single rule (and does not possess any axioms). Resolution is defined for clauses (each formula is a disjunction of positive and negative literals). All formulas must hold: conjunction of clauses. Proof by contradiction, exploiting the equivalence given above. If is not satisfiable, then false (the empty clause) can be derived:. MAI0203 Lecture 7: Inference and Predicate Calculus p.196

23 Resolution Calculus cont. Resolution rule in propositional logic: Resolution rule for clauses: ( is a substitution of variables such that of the conjunction) is identical in both parts The general idea is to cut out a literal which appears positive in one disjunction and negative in the other. MAI0203 Lecture 7: Inference and Predicate Calculus p.197

24 Resolution in Propositional Logic Illustration: All humans are mortal. Socrates is a human. Socrates is mortal. To proof Human Human Mortal Mortal Mortal Human : Mortal. Human Human Mortal Human Mortal Mortal Mortal MAI0203 Lecture 7: Inference and Predicate Calculus p.198

25 Resolution in FOL Illustration: All humans are mortal. Socrates is a human. Socrates is mortal. To proof Human(x) Mortal(x) (skip bound universal quantifier) Human(S) Mortal(S) Mortal(S) : Human(x) Human(x) Human(S) Mortal(S) Mortal(x). Human(x) Mortal(x) Mortal(x) Mortal(S) MAI0203 Lecture 7: Inference and Predicate Calculus p.199

26 Resolution in FOL cont. Resolution was introduced by Robinson (1965) as a mechanic way to perform logical proofs. We need to understand: Transformation of FOL formulas in clauses by applying equivalence rules. Substitution and unification. MAI0203 Lecture 7: Inference and Predicate Calculus p.200

27 Semantic Equivalence Two formulas and are called equivalent, if for each interpretation of und holds that is valid iff is valid. We write. Theorem: Let be. Let be a formula where appears as a sub-formula. Let be a formula derived from by replacing by. Then it holds. Equivalences can be used to rewrite formula. MAI0203 Lecture 7: Inference and Predicate Calculus p.201

28 Semantic Equivalence cont. (idempotency) (commutativity),,, (associativity) (absorption),, (distributivity) (double negation) (de Morgan) (remove implication), true (tautology) false (contradiction) Remark: This is the tertium non datur principle of classical logic. MAI0203 Lecture 7: Inference and Predicate Calculus p.202

29 Semantic Equivalence cont., tautology; tautology contradiction; contradiction, if, if, if, if it holds:,, is not free in If,,, MAI0203 Lecture 7: Inference and Predicate Calculus p.203

30 Classical Logic Propositional logic and FOL are classical logics. Classical logic is bivalent and monotonic: There are only two truth values true and false. Because of the tertium non datur, derived conclusions cannot be changed by new facts or conclusions (vs. multi-valued and non-monotonic logics). In classical logic, everything follows from a contradiction (ex falso quod libet). A theorem can be proven by contradiction. In contrast, in intuitionistic logic, all proofs must be constructive! MAI0203 Lecture 7: Inference and Predicate Calculus p.204

### Intelligent Agents. First Order Logic. Ute Schmid. Cognitive Systems, Applied Computer Science, Bamberg University. last change: 19.

Intelligent Agents First Order Logic Ute Schmid Cognitive Systems, Applied Computer Science, Bamberg University last change: 19. Mai 2015 U. Schmid (CogSys) Intelligent Agents last change: 19. Mai 2015

### Propositional Logic Language

Propositional Logic Language A logic consists of: an alphabet A, a language L, i.e., a set of formulas, and a binary relation = between a set of formulas and a formula. An alphabet A consists of a finite

### Advanced Topics in LP and FP

Lecture 1: Prolog and Summary of this lecture 1 Introduction to Prolog 2 3 Truth value evaluation 4 Prolog Logic programming language Introduction to Prolog Introduced in the 1970s Program = collection

### Propositional Logic. Jason Filippou UMCP. ason Filippou UMCP) Propositional Logic / 38

Propositional Logic Jason Filippou CMSC250 @ UMCP 05-31-2016 ason Filippou (CMSC250 @ UMCP) Propositional Logic 05-31-2016 1 / 38 Outline 1 Syntax 2 Semantics Truth Tables Simplifying expressions 3 Inference

### Propositional Logics and their Algebraic Equivalents

Propositional Logics and their Algebraic Equivalents Kyle Brooks April 18, 2012 Contents 1 Introduction 1 2 Formal Logic Systems 1 2.1 Consequence Relations......................... 2 3 Propositional Logic

### The Logic of Compound Statements cont.

The Logic of Compound Statements cont. CSE 215, Computer Science 1, Fall 2011 Stony Brook University http://www.cs.stonybrook.edu/~cse215 Refresh from last time: Logical Equivalences Commutativity of :

### 02 Propositional Logic

SE 2F03 Fall 2005 02 Propositional Logic Instructor: W. M. Farmer Revised: 25 September 2005 1 What is Propositional Logic? Propositional logic is the study of the truth or falsehood of propositions or

### COMP219: Artificial Intelligence. Lecture 19: Logic for KR

COMP219: Artificial Intelligence Lecture 19: Logic for KR 1 Overview Last time Expert Systems and Ontologies Today Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof

### Mathematics for linguists

Mathematics for linguists WS 2009/2010 University of Tübingen January 7, 2010 Gerhard Jäger Mathematics for linguists p. 1 Inferences and truth trees Inferences (with a finite set of premises; from now

### Deductive Systems. Lecture - 3

Deductive Systems Lecture - 3 Axiomatic System Axiomatic System (AS) for PL AS is based on the set of only three axioms and one rule of deduction. It is minimal in structure but as powerful as the truth

### An Introduction to Modal Logic III

An Introduction to Modal Logic III Soundness of Normal Modal Logics Marco Cerami Palacký University in Olomouc Department of Computer Science Olomouc, Czech Republic Olomouc, October 24 th 2013 Marco Cerami

### Overview. Knowledge-Based Agents. Introduction. COMP219: Artificial Intelligence. Lecture 19: Logic for KR

COMP219: Artificial Intelligence Lecture 19: Logic for KR Last time Expert Systems and Ontologies oday Logic as a knowledge representation scheme Propositional Logic Syntax Semantics Proof theory Natural

### A. Propositional Logic

CmSc 175 Discrete Mathematics A. Propositional Logic 1. Statements (Propositions ): Statements are sentences that claim certain things. Can be either true or false, but not both. Propositional logic deals

### Propositional Logic: Logical Agents (Part I)

Propositional Logic: Logical Agents (Part I) First Lecture Today (Tue 21 Jun) Read Chapters 1 and 2 Second Lecture Today (Tue 21 Jun) Read Chapter 7.1-7.4 Next Lecture (Thu 23 Jun) Read Chapters 7.5 (optional:

### CSC Discrete Math I, Spring Propositional Logic

CSC 125 - Discrete Math I, Spring 2017 Propositional Logic Propositions A proposition is a declarative sentence that is either true or false Propositional Variables A propositional variable (p, q, r, s,...)

### 7.5.2 Proof by Resolution

137 7.5.2 Proof by Resolution The inference rules covered so far are sound Combined with any complete search algorithm they also constitute a complete inference algorithm However, removing any one inference

### Logic Part I: Classical Logic and Its Semantics

Logic Part I: Classical Logic and Its Semantics Max Schäfer Formosan Summer School on Logic, Language, and Computation 2007 July 2, 2007 1 / 51 Principles of Classical Logic classical logic seeks to model

### The Importance of Being Formal. Martin Henz. February 5, Propositional Logic

The Importance of Being Formal Martin Henz February 5, 2014 Propositional Logic 1 Motivation In traditional logic, terms represent sets, and therefore, propositions are limited to stating facts on sets

### Reasoning. Inference. Knowledge Representation 4/6/2018. User

Reasoning Robotics First-order logic Chapter 8-Russel Representation and Reasoning In order to determine appropriate actions to take, an intelligent system needs to represent information about the world

### 10/5/2012. Logic? What is logic? Propositional Logic. Propositional Logic (Rosen, Chapter ) Logic is a truth-preserving system of inference

Logic? Propositional Logic (Rosen, Chapter 1.1 1.3) TOPICS Propositional Logic Truth Tables Implication Logical Proofs 10/1/12 CS160 Fall Semester 2012 2 What is logic? Logic is a truth-preserving system

### Propositional Logic Not Enough

Section 1.4 Propositional Logic Not Enough If we have: All men are mortal. Socrates is a man. Does it follow that Socrates is mortal? Can t be represented in propositional logic. Need a language that talks

### Logic. Introduction to Artificial Intelligence CS/ECE 348 Lecture 11 September 27, 2001

Logic Introduction to Artificial Intelligence CS/ECE 348 Lecture 11 September 27, 2001 Last Lecture Games Cont. α-β pruning Outline Games with chance, e.g. Backgammon Logical Agents and thewumpus World

### Logic for Computer Science Handout Week 8 DERIVED RULE MODUS TOLLENS DERIVED RULE RAA DERIVED RULE TND

Logic for Computer Science Handout Week 8 DERIVED RULE MODUS TOLLENS We have last week completed the introduction of a calculus, which is sound and complete. That is, we can syntactically establish the

### Logic. Propositional Logic: Syntax

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

### Outline. Formale Methoden der Informatik First-Order Logic for Forgetters. Why PL1? Why PL1? Cont d. Motivation

Outline Formale Methoden der Informatik First-Order Logic for Forgetters Uwe Egly Vienna University of Technology Institute of Information Systems Knowledge-Based Systems Group Motivation Syntax of PL1

### Classical Propositional Logic

The Language of A Henkin-style Proof for Natural Deduction January 16, 2013 The Language of A Henkin-style Proof for Natural Deduction Logic Logic is the science of inference. Given a body of information,

### Propositional Logic: Methods of Proof (Part II)

Propositional Logic: Methods of Proof (Part II) You will be expected to know Basic definitions Inference, derive, sound, complete Conjunctive Normal Form (CNF) Convert a Boolean formula to CNF Do a short

### Propositional Reasoning

Propositional Reasoning CS 440 / ECE 448 Introduction to Artificial Intelligence Instructor: Eyal Amir Grad TAs: Wen Pu, Yonatan Bisk Undergrad TAs: Sam Johnson, Nikhil Johri Spring 2010 Intro to AI (CS

### Propositional Logic: Logical Agents (Part I)

Propositional Logic: Logical Agents (Part I) This lecture topic: Propositional Logic (two lectures) Chapter 7.1-7.4 (this lecture, Part I) Chapter 7.5 (next lecture, Part II) Next lecture topic: First-order

### Knowledge based Agents

Knowledge based Agents Shobhanjana Kalita Dept. of Computer Science & Engineering Tezpur University Slides prepared from Artificial Intelligence A Modern approach by Russell & Norvig Knowledge Based Agents

### CS 380: ARTIFICIAL INTELLIGENCE

CS 380: RTIFICIL INTELLIGENCE PREDICTE LOGICS 11/8/2013 Santiago Ontañón santi@cs.drexel.edu https://www.cs.drexel.edu/~santi/teaching/2013/cs380/intro.html Summary of last day: Logical gents: The can

### First Order Logic: Syntax and Semantics

CS1081 First Order Logic: Syntax and Semantics COMP30412 Sean Bechhofer sean.bechhofer@manchester.ac.uk Problems Propositional logic isn t very expressive As an example, consider p = Scotland won on Saturday

### Introduction to Artificial Intelligence. Logical Agents

Introduction to Artificial Intelligence Logical Agents (Logic, Deduction, Knowledge Representation) Bernhard Beckert UNIVERSITÄT KOBLENZ-LANDAU Winter Term 2004/2005 B. Beckert: KI für IM p.1 Outline Knowledge-based

### CS206 Lecture 03. Propositional Logic Proofs. Plan for Lecture 03. Axioms. Normal Forms

CS206 Lecture 03 Propositional Logic Proofs G. Sivakumar Computer Science Department IIT Bombay siva@iitb.ac.in http://www.cse.iitb.ac.in/ siva Page 1 of 12 Fri, Jan 03, 2003 Plan for Lecture 03 Axioms

### Equivalents of Mingle and Positive Paradox

Eric Schechter Equivalents of Mingle and Positive Paradox Abstract. Relevant logic is a proper subset of classical logic. It does not include among itstheoremsanyof positive paradox A (B A) mingle A (A

### Propositional Logic: Models and Proofs

Propositional Logic: Models and Proofs C. R. Ramakrishnan CSE 505 1 Syntax 2 Model Theory 3 Proof Theory and Resolution Compiled at 11:51 on 2016/11/02 Computing with Logic Propositional Logic CSE 505

### Inference in first-order logic. Production systems.

CS 1571 Introduction to AI Lecture 17 Inference in first-order logic. Production systems. Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Sentences in Horn normal form Horn normal form (HNF) in

### Propositional Calculus: Formula Simplification, Essential Laws, Normal Forms

P Formula Simplification, Essential Laws, Normal Forms Lila Kari University of Waterloo P Formula Simplification, Essential Laws, Normal CS245, Forms Logic and Computation 1 / 26 Propositional calculus

### 2.2: Logical Equivalence: The Laws of Logic

Example (2.7) For primitive statement p and q, construct a truth table for each of the following compound statements. a) p q b) p q Here we see that the corresponding truth tables for two statement p q

### Introduction to Logic in Computer Science: Autumn 2007

Introduction to Logic in Computer Science: Autumn 2007 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Tableaux for First-order Logic The next part of

### Introduction to Metalogic

Philosophy 135 Spring 2008 Tony Martin Introduction to Metalogic 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: Remarks: (i) sentence letters p 0, p 1, p 2,... (ii)

### 3 The Semantics of the Propositional Calculus

3 The Semantics of the Propositional Calculus 1. Interpretations Formulas of the propositional calculus express statement forms. In chapter two, we gave informal descriptions of the meanings of the logical

### Propositional Logic. Logic. Propositional Logic Syntax. Propositional Logic

Propositional Logic Reading: Chapter 7.1, 7.3 7.5 [ased on slides from Jerry Zhu, Louis Oliphant and ndrew Moore] Logic If the rules of the world are presented formally, then a decision maker can use logical

### CS 380: ARTIFICIAL INTELLIGENCE PREDICATE LOGICS. Santiago Ontañón

CS 380: RTIFICIL INTELLIGENCE PREDICTE LOGICS Santiago Ontañón so367@drexeledu Summary of last day: Logical gents: The can reason from the knowledge they have They can make deductions from their perceptions,

### Packet #1: Logic & Proofs. Applied Discrete Mathematics

Packet #1: Logic & Proofs Applied Discrete Mathematics Table of Contents Course Objectives Page 2 Propositional Calculus Information Pages 3-13 Course Objectives At the conclusion of this course, you should

### Propositional Logic. Fall () Propositional Logic Fall / 30

Propositional Logic Fall 2013 () Propositional Logic Fall 2013 1 / 30 1 Introduction Learning Outcomes for this Presentation 2 Definitions Statements Logical connectives Interpretations, contexts,... Logically

### 3 Propositional Logic

3 Propositional Logic 3.1 Syntax 3.2 Semantics 3.3 Equivalence and Normal Forms 3.4 Proof Procedures 3.5 Properties Propositional Logic (25th October 2007) 1 3.1 Syntax Definition 3.0 An alphabet Σ consists

### Computational Logic. Recall of First-Order Logic. Damiano Zanardini

Computational Logic Recall of First-Order Logic Damiano Zanardini UPM European Master in Computational Logic (EMCL) School of Computer Science Technical University of Madrid damiano@fi.upm.es Academic

### CS2742 midterm test 2 study sheet. Boolean circuits: Predicate logic:

x NOT ~x x y AND x /\ y x y OR x \/ y Figure 1: Types of gates in a digital circuit. CS2742 midterm test 2 study sheet Boolean circuits: Boolean circuits is a generalization of Boolean formulas in which

### Propositional Logic Logical Implication (4A) Young W. Lim 4/21/17

Propositional Logic Logical Implication (4A) Young W. Lim Copyright (c) 2016-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation

### Propositional logic. Programming and Modal Logic

Propositional logic Programming and Modal Logic 2006-2007 4 Contents Syntax of propositional logic Semantics of propositional logic Semantic entailment Natural deduction proof system Soundness and completeness

### Logic: The Big Picture

Logic: The Big Picture A typical logic is described in terms of syntax: what are the legitimate formulas semantics: under what circumstances is a formula true proof theory/ axiomatization: rules for proving

### Deliberative Agents Knowledge Representation I. Deliberative Agents

Deliberative Agents Knowledge Representation I Vasant Honavar Bioinformatics and Computational Biology Program Center for Computational Intelligence, Learning, & Discovery honavar@cs.iastate.edu www.cs.iastate.edu/~honavar/

### A Little Deductive Logic

A Little Deductive Logic In propositional or sentential deductive logic, we begin by specifying that we will use capital letters (like A, B, C, D, and so on) to stand in for sentences, and we assume that

### Introduction to Artificial Intelligence Propositional Logic & SAT Solving. UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2010

Introduction to Artificial Intelligence Propositional Logic & SAT Solving UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2010 Today Representation in Propositional Logic Semantics & Deduction

### Computation and Logic Definitions

Computation and Logic Definitions True and False Also called Boolean truth values, True and False represent the two values or states an atom can assume. We can use any two distinct objects to represent

### Introduction to fuzzy logic

Introduction to fuzzy logic Andrea Bonarini Artificial Intelligence and Robotics Lab Department of Electronics and Information Politecnico di Milano E-mail: bonarini@elet.polimi.it URL:http://www.dei.polimi.it/people/bonarini

### First order Logic ( Predicate Logic) and Methods of Proof

First order Logic ( Predicate Logic) and Methods of Proof 1 Outline Introduction Terminology: Propositional functions; arguments; arity; universe of discourse Quantifiers Definition; using, mixing, negating

### Some consequences of compactness in Lukasiewicz Predicate Logic

Some consequences of compactness in Lukasiewicz Predicate Logic Luca Spada Department of Mathematics and Computer Science University of Salerno www.logica.dmi.unisa.it/lucaspada 7 th Panhellenic Logic

### INTRODUCTION TO PREDICATE LOGIC HUTH AND RYAN 2.1, 2.2, 2.4

INTRODUCTION TO PREDICATE LOGIC HUTH AND RYAN 2.1, 2.2, 2.4 Neil D. Jones DIKU 2005 Some slides today new, some based on logic 2004 (Nils Andersen), some based on kernebegreber (NJ 2005) PREDICATE LOGIC:

### Unit I LOGIC AND PROOFS. B. Thilaka Applied Mathematics

Unit I LOGIC AND PROOFS B. Thilaka Applied Mathematics UNIT I LOGIC AND PROOFS Propositional Logic Propositional equivalences Predicates and Quantifiers Nested Quantifiers Rules of inference Introduction

### Example. Logic. Logical Statements. Outline of logic topics. Logical Connectives. Logical Connectives

Logic Logic is study of abstract reasoning, specifically, concerned with whether reasoning is correct. Logic focuses on relationship among statements as opposed to the content of any particular statement.

### Chapter 1: The Logic of Compound Statements. January 7, 2008

Chapter 1: The Logic of Compound Statements January 7, 2008 Outline 1 1.1 Logical Form and Logical Equivalence 2 1.2 Conditional Statements 3 1.3 Valid and Invalid Arguments Central notion of deductive

### Proofs: A General How To II. Rules of Inference. Rules of Inference Modus Ponens. Rules of Inference Addition. Rules of Inference Conjunction

Introduction I Proofs Computer Science & Engineering 235 Discrete Mathematics Christopher M. Bourke cbourke@cse.unl.edu A proof is a proof. What kind of a proof? It s a proof. A proof is a proof. And when

### Logic Background (1A) Young W. Lim 12/14/15

Young W. Lim 12/14/15 Copyright (c) 2014-2015 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

### First Order Logic Semantics (3A) Young W. Lim 9/17/17

First Order Logic (3A) Young W. Lim Copyright (c) 2016-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version

### Propositional Logic Part 1

Propositional Logic Part 1 Yingyu Liang yliang@cs.wisc.edu Computer Sciences Department University of Wisconsin, Madison [Based on slides from Louis Oliphant, Andrew Moore, Jerry Zhu] slide 1 5 is even

### CS100: DISCRETE STRUCTURES. Lecture 5: Logic (Ch1)

CS100: DISCREE SRUCURES Lecture 5: Logic (Ch1) Lecture Overview 2 Statement Logical Connectives Conjunction Disjunction Propositions Conditional Bio-conditional Converse Inverse Contrapositive Laws of

### Chapter 4: Classical Propositional Semantics

Chapter 4: Classical Propositional Semantics Language : L {,,, }. Classical Semantics assumptions: TWO VALUES: there are only two logical values: truth (T) and false (F), and EXTENSIONALITY: the logical

### Propositional Logic Arguments (5A) Young W. Lim 2/23/17

Propositional Logic (5A) Young W. Lim Copyright (c) 2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version

### HANDOUT AND SET THEORY. Ariyadi Wijaya

HANDOUT LOGIC AND SET THEORY Ariyadi Wijaya Mathematics Education Department Faculty of Mathematics and Natural Science Yogyakarta State University 2009 1 Mathematics Education Department Faculty of Mathematics

### Propositional logic II.

Lecture 5 Propositional logic II. Milos Hauskrecht milos@cs.pitt.edu 5329 ennott quare Propositional logic. yntax yntax: ymbols (alphabet) in P: Constants: True, False Propositional symbols Examples: P

### Kecerdasan Buatan M. Ali Fauzi

Kecerdasan Buatan M. Ali Fauzi Artificial Intelligence M. Ali Fauzi Logical Agents M. Ali Fauzi In which we design agents that can form representations of the would, use a process of inference to derive

### Propositional Equivalence

Propositional Equivalence Tautologies and contradictions A compound proposition that is always true, regardless of the truth values of the individual propositions involved, is called a tautology. Example:

### Propositional Logic: Methods of Proof (Part II)

Propositional Logic: Methods of Proof (Part II) This lecture topic: Propositional Logic (two lectures) Chapter 7.1-7.4 (previous lecture, Part I) Chapter 7.5 (this lecture, Part II) (optional: 7.6-7.8)

### Artificial Intelligence

Artificial Intelligence Propositional Logic Marc Toussaint University of Stuttgart Winter 2016/17 (slides based on Stuart Russell s AI course) Motivation: Most students will have learnt about propositional

### More Propositional Logic Algebra: Expressive Completeness and Completeness of Equivalences. Computability and Logic

More Propositional Logic Algebra: Expressive Completeness and Completeness of Equivalences Computability and Logic Equivalences Involving Conditionals Some Important Equivalences Involving Conditionals

### Logic as a Tool Chapter 1: Understanding Propositional Logic 1.1 Propositions and logical connectives. Truth tables and tautologies

Logic as a Tool Chapter 1: Understanding Propositional Logic 1.1 Propositions and logical connectives. Truth tables and tautologies Valentin Stockholm University September 2016 Propositions Proposition:

### Announcements. CS311H: Discrete Mathematics. Propositional Logic II. Inverse of an Implication. Converse of a Implication

Announcements CS311H: Discrete Mathematics Propositional Logic II Instructor: Işıl Dillig First homework assignment out today! Due in one week, i.e., before lecture next Wed 09/13 Remember: Due before

### What is Logic? Introduction to Logic. Simple Statements. Which one is statement?

What is Logic? Introduction to Logic Peter Lo Logic is the study of reasoning It is specifically concerned with whether reasoning is correct Logic is also known as Propositional Calculus CS218 Peter Lo

### CHAPTER 10. Gentzen Style Proof Systems for Classical Logic

CHAPTER 10 Gentzen Style Proof Systems for Classical Logic Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. By humans, not mentioning

### Proseminar on Semantic Theory Fall 2013 Ling 720 Propositional Logic: Syntax and Natural Deduction 1

Propositional Logic: Syntax and Natural Deduction 1 The Plot That Will Unfold I want to provide some key historical and intellectual context to the model theoretic approach to natural language semantics,

### First Order Logic: Syntax and Semantics

irst Order Logic: Syntax and Semantics COMP30412 Sean Bechhofer sean.bechhofer@manchester.ac.uk Logic Recap You should already know the basics of irst Order Logic (OL) It s a prerequisite of this course!

### Propositional logic. Programming and Modal Logic

Propositional logic Programming and Modal Logic 2006-2007 4 Contents Syntax of propositional logic Semantics of propositional logic Semantic entailment Natural deduction proof system Soundness and completeness

### Propositional Logic Resolution (6A) Young W. Lim 12/31/16

Propositional Logic Resolution (6A) Young W. Lim Copyright (c) 2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License,

### Introduction to Intelligent Systems

Logical Agents Objectives Inference and entailment Sound and complete inference algorithms Inference by model checking Inference by proof Resolution Forward and backward chaining Reference Russel/Norvig:

### 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

### Discrete Structures for Computer Science

Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #6: Rules of Inference Based on materials developed by Dr. Adam Lee Today s topics n Rules of inference

### Chapter 7 R&N ICS 271 Fall 2017 Kalev Kask

Set 6: Knowledge Representation: The Propositional Calculus Chapter 7 R&N ICS 271 Fall 2017 Kalev Kask Outline Representing knowledge using logic Agent that reason logically A knowledge based agent Representing

### CHAPTER 11. Introduction to Intuitionistic Logic

CHAPTER 11 Introduction to Intuitionistic Logic Intuitionistic logic has developed as a result of certain philosophical views on the foundation of mathematics, known as intuitionism. Intuitionism was originated

### Boolean Algebra and Proof. Notes. Proving Propositions. Propositional Equivalences. Notes. Notes. Notes. Notes. March 5, 2012

March 5, 2012 Webwork Homework. The handout on Logic is Chapter 4 from Mary Attenborough s book Mathematics for Electrical Engineering and Computing. Proving Propositions We combine basic propositions

### Inf2D 13: Resolution-Based Inference

School of Informatics, University of Edinburgh 13/02/18 Slide Credits: Jacques Fleuriot, Michael Rovatsos, Michael Herrmann Last lecture: Forward and backward chaining Backward chaining: If Goal is known

### Predicate Calculus. Formal Methods in Verification of Computer Systems Jeremy Johnson

Predicate Calculus Formal Methods in Verification of Computer Systems Jeremy Johnson Outline 1. Motivation 1. Variables, quantifiers and predicates 2. Syntax 1. Terms and formulas 2. Quantifiers, scope

### Deduction by Daniel Bonevac. Chapter 3 Truth Trees

Deduction by Daniel Bonevac Chapter 3 Truth Trees Truth trees Truth trees provide an alternate decision procedure for assessing validity, logical equivalence, satisfiability and other logical properties

### Computational Logic. Davide Martinenghi. Spring Free University of Bozen-Bolzano. Computational Logic Davide Martinenghi (1/30)

Computational Logic Davide Martinenghi Free University of Bozen-Bolzano Spring 2010 Computational Logic Davide Martinenghi (1/30) Propositional Logic - sequent calculus To overcome the problems of natural

### Discrete Mathematical Structures. Chapter 1 The Foundation: Logic

Discrete Mathematical Structures Chapter 1 he oundation: Logic 1 Lecture Overview 1.1 Propositional Logic 1.2 Propositional Equivalences 1.3 Quantifiers l l l l l Statement Logical Connectives Conjunction

### 2/13/2012. Logic: Truth Tables. CS160 Rosen Chapter 1. Logic?

Logic: Truth Tables CS160 Rosen Chapter 1 Logic? 1 What is logic? Logic is a truth-preserving system of inference Truth-preserving: If the initial statements are true, the inferred statements will be true