Inference in firstorder logic


 Sydney Lamb
 1 years ago
 Views:
Transcription
1 CS 2710 Foundations of AI Lecture 15 Inference in firstorder logic Milos Hauskrecht 5329 Sennott Square Logical inference in FOL Logical inference problem: Given a knowledge base KB (a set of sentences) and a sentence, does the KB semantically entail? KB? Logical inference problem in the firstorder logic is undecidable!!! No procedure that can decide the entailment for all possible input sentences in a finite number of steps. Methods: Inference rule approach Resolution refutation 1
2 Inference rules for quantifiers Universal elimination x ( a  is a constant symbol ( a) substitutes a variable with a constant symbol x Likes( IceCream) Likes( Ben, IceCream) Existential elimination. x ( ( a) Substitutes a variable with a constant symbol that does not appear elsewhere in the KB x Kill ( Victim) Kill ( Murderer, Victim) Inference rules for quantifiers Universal instantiation (introduction) x is not free in x Introduces a universal variable which does not affect or its assumptions Sister ( Am Jane) x Sister( Am Jane) Existential instantiation (introduction) ( a) a is a ground term in x ( x is not free in Substitutes a ground term in the sentence with a variable and an existential statement Likes( Ben, IceCream) x Likes( IceCream) 2
3 Unification Problem in inference: Universal elimination gives us many opportunities for substituting variables with ground terms x ( a  is a constant symbol ( a) Solution: avoid making blind substitutions of ground terms Make substitutions that help to advance inferences Use substitutions matching similar sentences in KB Make inferences on the variable level Do not substitute ground terms if not necessary Unification takes two similar sentences and computes the substitution that makes them look the same, if it exists UNIFY( p, q) s.t. SUBST(σ, p) SUBST (, q) Generalized inference rules Use substitutions that let us make inferences!!!! Example: Generalized Modus Ponens If there exists a substitution such that SUBST, A ) SUBST (, A ') ( i i A1 A2 An B, A1 ', A2 ', An ' SUBST (, B) for all i=1,2, n Substitution that satisfies the generalized inference rule can be build via unification process Advantage of the generalized rules: they are focused only substitutions that allow the inferences to proceed are tried 3
4 Resolution inference rule Recall: Resolution inference rule is sound and complete (refutationcomplete) for the propositional logic and CNF A B, A C B C Generalized resolution rule is sound and refutation complete for the firstorder logic and CNF w/o equalities (if unsatisfiable the resolution will find the contradiction) 1 2 k, 1 2 n SUBST (, Example: 1 UNIFY(, ) i 1 i 1 i k j 1 fail P(, Q( John) P( John) j 1 j 1 ) n Inference with the resolution rule Matches the KB in CNF Proof by refutation: Prove that KB, is unsatisfiable resolution is refutationcomplete Main procedure (steps): 1. Convert KB, to CNF with ground terms and universal variables only 2. Apply repeatedly the resolution rule while keeping track and consistency of substitutions 3. Stop when empty set (contradiction) is derived or no more new resolvents (conclusions) follow 4
5 Conversion to CNF 1. Eliminate implications, equivalences ( p q) ( p q) 2. Move negations inside (DeMorgan s Laws, double negation) ( p q) p q x p x p ( p q) p q x p x p p p 3. Standardize variables (rename duplicate variables) ( x P( ) ( x Q( ) ( x P( ) ( y Q( ) 4. Move all quantifiers left (no invalid capture possible ) ( x P( ) ( y Q( ) x y P( Conversion to CNF 5. Skolemization (removal of existential quantifiers through elimination) If no universal quantifier occurs before the existential quantifier, replace the variable with a new constant symbol also called Skolem constant y P( A) P( A) B) If a universal quantifier precedes the existential quantifier replace the variable with a function of the universal variable x y P( x P( F( ) F(  a special function  called Skolem function 5
6 Conversion to CNF 6. Drop universal quantifiers (all variables are universally quantified) x P( F( ) P( F( ) 7. Convert to CNF using the distributive laws p ( q r) ( p q) ( p r) The result is a CNF with variables, constants, functions Resolution example KB P(,, P(, R(, α S(A) 6
7 Resolution example KB P(,, P(, R(, α S(A) { y / w} P( Resolution example KB P(,, P(, R(, α S(A) { y / w} P( { x / w} S( 7
8 Resolution example KB P(,, P(, R(, α S(A) { y / w} P( { x / w} S( { z / w} S( Resolution example KB P(,, P(, R(, α S(A) { y / w} P( { x / w} S( { z / w} S( { w/ A} Empty resolution 8
9 Resolution example KB P(,, P(, R(, α S(A) { y / w} P( { x / w} S( { z / w} KB S( { w/ A} Empty resolution Contradiction Dealing with equality Resolution works for the firstorder logic without equalities To incorporate equalities we need an additional inference rule Demodulation rule UNIFY z i, t ) ( k, t1 t2 SUB( SUBST (, t ), SUBST (, t ), ) 1 fail where Example: P( f ( a)), f ( x P( a) Paramodulation rule: more powerful Resolution+paramodulation give a refutationcomplete proof theory for FOL 2 z i 1 occurs in i 2 k 9
10 Sentences in Horn normal form Horn normal form (HNF) in the propositional logic a special type of a clause with at most one positive literal ( A B) ( A C D) Typically written as: ( B A) (( A C) D) A clause with one literal, e.g. A, is also called a fact A clause representing an implication (with a conjunction of positive literals in antecedent and one positive literal in consequent), is also called a rule Resolution rule and modus ponens: Both are complete inference rule for unit inferences for KBs in the Horn normal form. Recall: Not all KBs are convertible to HNF!!! Horn normal form in FOL Firstorder logic (FOL) adds variables and quantifiers, works with terms, predicates HNF in FOL: primitive sentences (propositions) are formed by predicates Generalized modus ponens rule: a substituti on s.t. i SUBST (, ') SUBST (, ) 1', 2', n', 1 2 n SUBST (, ) Generalized resolution and generalized modus ponens: is complete for unit inferences for the KBs in HN; Not all firstorder logic sentences can be expressed in HNF i i 10
11 Forward and backward chaining Two inference procedures based on modus ponens for Horn KBs: Forward chaining Idea: Whenever the premises of a rule are satisfied, infer the conclusion. Continue with rules that became satisfied. Typical usage: If we want to infer all sentences entailed by the existing KB. Backward chaining (goal reduction) Idea: To prove the fact that appears in the conclusion of a rule prove the premises of the rule. Continue recursively. Typical usage: If we want to prove that the target (goal) sentence is entailed by the existing KB. Both procedures are complete for KBs in Horn form!!! Forward chaining example Forward chaining Idea: Whenever the premises of a rule are satisfied, infer the conclusion. Continue with rules that became satisfied Assume the KB with the following rules: KB: : Steamboat ( Sailboat ( Faster ( R2: R3: F1: F2: F3: Sailboat ( RowBoat ( Faster ( Faster ( Faster ( Faster ( Steamboat (Titanic ) Sailboat (Mistral ) RowBoat (PondArrow ) Theorem: Faster ( Titanic, PondArrow )? 11
12 Forward chaining example KB: : R2: R3: Steamboat( Sailboat ( Sailboat ( RowBoat ( Faster( F1: F3: RowBoat(PondArrow )? Forward chaining example KB: : R2: R3: Steamboat( Sailboat ( Sailboat ( RowBoat ( Faster( F1: F3: RowBoat(PondArrow ) Rule is satisfied: F4: Mistral ) 12
13 Forward chaining example KB: : R2: R3: Steamboat( Sailboat ( Sailboat ( RowBoat ( Faster( F1: F3: RowBoat(PondArrow ) Rule is satisfied: F4: Mistral ) Rule R2 is satisfied: F5: Faster( Mistral, PondArro Forward chaining example KB: : R2: R3: Steamboat( Sailboat ( Sailboat ( RowBoat ( Faster( F1: F3: RowBoat(PondArrow ) Rule is satisfied: F4: Mistral ) Rule R2 is satisfied: F5: Faster( Mistral, PondArro Rule R3 is satisfied: F6: PondArro 13
14 Backward chaining example Backward chaining (goal reduction) Idea: To prove the fact that appears in the conclusion of a rule prove the antecedents (if part) of the rule & repeat recursively. KB: : R2: R3: F1: F2: F3: Steamboat( Sailboat ( Sailboat ( RowBoat ( Faster( Sailboat(Mistral) RowBoat(PondArro Theorem: PondArro Backward chaining example PondArro F1: F3: RowBoat(PondArro Sailboat(PondArro Steamboat( Sailboat ( PondArro { x / Titanic, y / PondArrow } 14
15 Backward chaining example PondArro R2 Sailboat(Titanic) F1: F3: RowBoat(PondArro Sailboat(PondArro RowBoat(PondArro Sailboat ( RowBoat ( PondArro { y / Titanic, z / PondArrow } Backward chaining example PondArro R2 R3 F1: F3: RowBoat(PondArro Sailboat(Titanic) Faster( PondArrow ) Sailboat(PondArro RowBoat(PondArro Faster( PondArro { x / Titanic, z / PondArrow} 15
16 Backward chaining example PondArro R2 R3 F1: F3: RowBoat(PondArro Sailboat(Titanic) Faster( PondArrow ) Sailboat(PondArro RowBoat(PondArro Sailboat(Mistral) Steamboat( Sailboat( Mistral ) { x / Titanic, y / Mistral } Backward chaining example PondArro R2 R3 y must be bound to the same term Sailboat(Titanic) Faster( Mistral, PondArro Mistral ) Sailboat(PondArro RowBoat(PondArro Sailboat(Mistral) 16
17 Backward chaining example PondArro R2 R3 F1: F3: RowBoat(PondArro Sailboat(Titanic) Faster( Mistral, PondArro Mistral ) Sailboat(PondArro RowBoat(PondArro R2 Sailboat(Mistral) Sailboat(Mistral) RowBoat(PondArro Backward chaining PondArro R2 R3 Sailboat(Titanic) Faster( Mistral, PondArro Mistral ) Sailboat(PondArro RowBoat(PondArro R2 Sailboat(Mistral) Sailboat(Mistral) RowBoat(PondArro 17
Inf2D 13: ResolutionBased 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
More informationPropositional 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
More informationKnowledge Representation. Propositional logic
CS 2710 Foundations of AI Lecture 10 Knowledge Representation. Propositional logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Knowledgebased agent Knowledge base Inference engine Knowledge
More informationCS 4700: Artificial Intelligence
CS 4700: Foundations of Artificial Intelligence Fall 2017 Instructor: Prof. Haym Hirsh Lecture 14 Today Knowledge Representation and Reasoning (R&N Ch 79) Prelim, Statler Auditorium Tuesday, March 21
More informationIntroduction 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:
More informationFirstOrder Theorem Proving and Vampire. Laura Kovács (Chalmers University of Technology) Andrei Voronkov (The University of Manchester)
FirstOrder Theorem Proving and Vampire Laura Kovács (Chalmers University of Technology) Andrei Voronkov (The University of Manchester) Outline Introduction FirstOrder Logic and TPTP Inference Systems
More informationCS2742 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
More informationIntroduction to Artificial Intelligence. Logical Agents
Introduction to Artificial Intelligence Logical Agents (Logic, Deduction, Knowledge Representation) Bernhard Beckert UNIVERSITÄT KOBLENZLANDAU Winter Term 2004/2005 B. Beckert: KI für IM p.1 Outline Knowledgebased
More informationFirstOrder Theorem Proving and Vampire
FirstOrder Theorem Proving and Vampire Laura Kovács 1,2 and Martin Suda 2 1 TU Wien 2 Chalmers Outline Introduction FirstOrder Logic and TPTP Inference Systems Saturation Algorithms Redundancy Elimination
More informationA Unit Resolution Approach to Knowledge Compilation. 2. Preliminaries
A Unit Resolution Approach to Knowledge Compilation Arindama Singh and Manoj K Raut Department of Mathematics Indian Institute of Technology Chennai600036, India Abstract : Knowledge compilation deals
More informationCITS2211 Discrete Structures Proofs
CITS2211 Discrete Structures Proofs Unit coordinator: Rachel CardellOliver August 13, 2017 Highlights 1 Arguments vs Proofs. 2 Proof strategies 3 Famous proofs Reading Chapter 1: What is a proof? Mathematics
More informationAdvanced Topics in LP and FP
Lecture 3: Logic Programming in Prolog Outline 1 2 3 The control of execution Logic Programming Declarative programming style: Emphasis on What is the solution? instead of How to find the solution? Symbolic
More informationKnowledge and reasoning 2
Knowledge and reasoning 2 DDC65 Artificial intelligence and Lisp Peter Dalenius petda@ida.liu.se Department of Computer and Information Science Linköping University Knowledgebased agents Inside our agent
More informationArgument. whenever all the assumptions are true, then the conclusion is true. If today is Wednesday, then yesterday is Tuesday. Today is Wednesday.
Logic and Proof Argument An argument is a sequence of statements. All statements but the first one are called assumptions or hypothesis. The final statement is called the conclusion. An argument is valid
More informationDiscrete Structures for Computer Science
Discrete Structures for Computer Science William Garrison bill@cs.pitt.edu 6311 Sennott Square Lecture #4: Predicates and Quantifiers Based on materials developed by Dr. Adam Lee Topics n Predicates n
More informationComputational Logic Automated Deduction Fundamentals
Computational Logic Automated Deduction Fundamentals 1 Elements of FirstOrder Predicate Logic First Order Language: An alphabet consists of the following classes of symbols: 1. variables denoted by X,
More informationPropositional Logic. Logical Expressions. Logic Minimization. CNF and DNF. Algebraic Laws for Logical Expressions CSC 173
Propositional Logic CSC 17 Propositional logic mathematical model (or algebra) for reasoning about the truth of logical expressions (propositions) Logical expressions propositional variables or logical
More information3 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
More informationInf2D 06: Logical Agents: Knowledge Bases and the Wumpus World
Inf2D 06: Logical Agents: Knowledge Bases and the Wumpus World School of Informatics, University of Edinburgh 26/01/18 Slide Credits: Jacques Fleuriot, Michael Rovatsos, Michael Herrmann Outline Knowledgebased
More informationPropositional and Predicate Logic
Propositional and Predicate Logic CS 53605: Science of Programming This is for Section 5 Only: See Prof. Ren for Sections 1 4 A. Why Reviewing/overviewing logic is necessary because we ll be using it
More informationLING 501, Fall 2004: Quantification
LING 501, Fall 2004: Quantification The universal quantifier Ax is conjunctive and the existential quantifier Ex is disjunctive Suppose the domain of quantification (DQ) is {a, b}. Then: (1) Ax Px Pa &
More informationPropositional 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
More informationOutline Firstorder logic. Firstorder Logic. Introduction. Recall: 4Queens problem. Firstorder Logic. Firstorder Logic Syntax.
Firstorder Logic CS 486/686 Sept 25, 2008 University of Waterloo Outline Firstorder logic Syntax and semantics Inference Propositionalization with ground inference Lifted resolution 1 2 Introduction
More informationarxiv: v1 [cs.lo] 1 Sep 2017
A DECISION PROCEDURE FOR HERBRAND FORMULAE WITHOUT SKOLEMIZATION arxiv:1709.00191v1 [cs.lo] 1 Sep 2017 TIMM LAMPERT Humboldt University Berlin, Unter den Linden 6, D10099 Berlin email address: lampertt@staff.huberlin.de
More informationSemantics and Pragmatics of NLP
Semantics and Pragmatics of NLP Alex Ewan School of Informatics University of Edinburgh 28 January 2008 1 2 3 Taking Stock We have: Introduced syntax and semantics for FOL plus lambdas. Represented FOL
More informationCOMP4418: Knowledge Representation and Reasoning FirstOrder Logic
COMP4418: Knowledge Representation and Reasoning FirstOrder Logic Maurice Pagnucco School of Computer Science and Engineering University of New South Wales NSW 2052, AUSTRALIA morri@cse.unsw.edu.au COMP4418
More informationUnit 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
More informationResolution and Refutation
Resolution and Refutation York University Department of Computer Science and Engineering York University CSE 3401 V. Movahedi 04_Resolution 1 Propositional Logic Resolution Refutation Predicate Logic
More informationCMPSCI 601: Tarski s Truth Definition Lecture 15. where
@ CMPSCI 601: Tarski s Truth Definition Lecture 15! "$#&%(') *+,!".#/%0'!12 43 5 6 7 8:9 4; 9 9 < = 9 = or 5 6?>A@B!9 2 D for all C @B 9 CFE where ) CGE @BHI LJKK MKK )HG if H ; C if H @ 1 > > > Fitch
More informationCS206 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
More informationFirst 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
More information1. Propositions: Contrapositives and Converses
Preliminaries 1 1. Propositions: Contrapositives and Converses Given two propositions P and Q, the statement If P, then Q is interpreted as the statement that if the proposition P is true, then the statement
More informationAnalyzing Arguments with Truth Tables
Analyzing Arguments with Truth Tables MATH 100 Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Fall 2014 Introduction Euler diagrams are useful for checking the validity of simple
More informationNormal Forms Note: all ppts about normal forms are skipped.
Normal Forms Note: all ppts about normal forms are skipped. Well formed formula (wff) also called formula, is a string consists of propositional variables, connectives, and parenthesis used in the proper
More informationInference in firstorder logic
Inference in firstorder logic Chapter 9 Chapter 9 1 Outline Reducing firstorder inference to propositional inference Unification Generalized Modus Ponens Forward and backward chaining Logic programming
More informationArtificial Intelligence
Artificial Intelligence FirstOrder Logic Marc Toussaint University of Stuttgart Winter 2015/16 (slides based on Stuart Russell s AI course) Firstorder logic (FOL) is exactly what is sometimes been thought
More informationWUCT121. Discrete Mathematics. Logic. Tutorial Exercises
WUCT11 Discrete Mathematics Logic Tutorial Exercises 1 Logic Predicate Logic 3 Proofs 4 Set Theory 5 Relations and Functions WUCT11 Logic Tutorial Exercises 1 Section 1: Logic Question1 For each of the
More informationDiscrete Mathematical Structures: Theory and Applications
Chapter 1: Foundations: Sets, Logic, and Algorithms Discrete Mathematical Structures: Theory and Applications Learning Objectives Learn about sets Explore various operations on sets Become familiar with
More informationGlossary of Logical Terms
Math 304 Spring 2007 Glossary of Logical Terms The following glossary briefly describes some of the major technical logical terms used in this course. The glossary should be read through at the beginning
More informationTutorial 2 Logic of Quantified Statements
Tutorial 2 Logic of Quantified Statements 1. Let V be the set of all visitors to Universal Studios Singapore on a certain day, T (v) be v took the Transformers ride, G(v) be v took the Battlestar Galactica
More information4 Predicate / First Order Logic
4 Predicate / First Order Logic 4.1 Syntax 4.2 Substitutions 4.3 Semantics 4.4 Equivalence and Normal Forms 4.5 Unification 4.6 Proof Procedures 4.7 Implementation of Proof Procedures 4.8 Properties First
More informationCSE20: Discrete Mathematics for Computer Science. Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication
CSE20: Discrete Mathematics for Computer Science Lecture Unit 2: Boolan Functions, Logic Circuits, and Implication Disjunctive normal form Example: Let f (x, y, z) =xy z. Write this function in DNF. Minterm
More informationOn Rules with Existential Variables: Walking the Decidability Line
On Rules with Existential Variables: Walking the Decidability Line JeanFrançois Baget a, Michel Leclère b, MarieLaure Mugnier,b, Eric Salvat c a INRIA Sophia Antipolis, France b University of Montpellier
More informationDefault Logic Autoepistemic Logic
Default Logic Autoepistemic Logic Nonclassical logics and application seminar, winter 2008 Mintz Yuval Introduction and Motivation Birds Fly As before, we are troubled with formalization of Nonabsolute
More informationChapter 3. SLDResolution. 3.1 Informal Introduction
Chapter 3 SLDResolution This chapter introduces the inference mechanism which is the basis of most logic programming systems. The idea is a special case of the inference rule called the resolution principle
More informationConcurrent Inference through Dual Transformation
Concurrent Inference through Dual Transformation GUILHERME BITTENCOURT, Departamento de Automação e Sistemas, Universidade Federal de Santa Catarina, 88040900 Florianópolis, SC, Brazil. Email: gb@lcmi.ufsc.br
More informationThe Arithmetical Hierarchy
Chapter 11 The Arithmetical Hierarchy Think of K as posing the problem of induction for computational devices, for it is impossible to tell for sure whether a given computation will never halt. Thus, K
More informationCS Lecture 19: Logic To Truth through Proof. Prof. Clarkson Fall Today s music: Theme from Sherlock
CS 3110 Lecture 19: Logic To Truth through Proof Prof. Clarkson Fall 2014 Today s music: Theme from Sherlock Review Current topic: How to reason about correctness of code Last week: informal arguments
More informationTruthmaker Maximalism defended again. Eduardo Barrio and Gonzalo RodriguezPereyra
1 Truthmaker Maximalism defended again 1 Eduardo Barrio and Gonzalo RodriguezPereyra 1. Truthmaker Maximalism is the thesis that every truth has a truthmaker. Milne (2005) attempts to refute it using
More informationCS Introduction to Complexity Theory. Lecture #11: Dec 8th, 2015
CS 2401  Introduction to Complexity Theory Lecture #11: Dec 8th, 2015 Lecturer: Toniann Pitassi Scribe Notes by: Xu Zhao 1 Communication Complexity Applications Communication Complexity (CC) has many
More informationLecture 2: FirstOrder Logic  Syntax, Semantics, Resolution
Lecture 2: FirstOrder Logic  Syntax, Semantics, Resolution Ruzica Piskac Ecole Polytechnique Fédérale de Lausanne, Switzerland ruzica.piskac@epfl.ch Seminar on Automated Reasoning 2010 Acknowledgments
More informationThe semantics of ALN knowledge bases is the standard modelbased semantics of rstorder logic. An interpretation I contains a nonempty domain O I. It
Backward Reasoning in Aboxes for Query Answering MarieChristine Rousset L.R.I. Computer Science Laboratory C.N.R.S & University of ParisSud Building 490, 91405, Orsay Cedex, France mcr@lri.fr Abstract
More informationCHAPTER 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
More information6. Conditional derivations
6. Conditional derivations 6.1 An argument from Hobbes In his great work, Leviathan, the philosopher Thomas Hobbes (15881679) gives an important argument for government. Hobbes begins by claiming that
More informationProof Theoretical Reasoning Lecture 3 Modal Logic S5 and Hypersequents
Proof Theoretical Reasoning Lecture 3 Modal Logic S5 and Hypersequents Revantha Ramanayake and Björn Lellmann TU Wien TRS Reasoning School 2015 Natal, Brasil Outline Modal Logic S5 Sequents for S5 Hypersequents
More informationChapter 2. Mathematical Reasoning. 2.1 Mathematical Models
Contents Mathematical Reasoning 3.1 Mathematical Models........................... 3. Mathematical Proof............................ 4..1 Structure of Proofs........................ 4.. Direct Method..........................
More informationArtificial Intelligence Study Manual for the Comprehensive Oral Examination
Artificial Intelligence Study Manual for the Comprehensive Oral Examination Hendrik Fischer Formerly of: Artificial Intelligence Center The University of Georgia Athens, Georgia 30602 henne@uga.edu Brian
More informationCountable and uncountable sets. Matrices.
CS 441 Discrete Mathematics for CS Lecture 11 Countable and uncountable sets. Matrices. Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Arithmetic series Definition: The sum of the terms of the
More informationNotes for the Proof Theory Course
Notes for the Proof Theory Course Master 1 Informatique, Univ. Paris 13 Damiano Mazza Contents 1 Propositional Classical Logic 5 1.1 Formulas and truth semantics.................... 5 1.2 Atomic negation...........................
More informationEtype interpretation without Etype pronoun: How Peirce s Graphs. capture the uniqueness implication of donkey sentences
Etype interpretation without Etype pronoun: How Peirce s Graphs capture the uniqueness implication of donkey sentences Author: He Chuansheng (PhD student of linguistics) The Hong Kong Polytechnic University
More informationDefinition 2. Conjunction of p and q
Proposition Propositional Logic CPSC 2070 Discrete Structures Rosen (6 th Ed.) 1.1, 1.2 A proposition is a statement that is either true or false, but not both. Clemson will defeat Georgia in football
More informationMathematical Induction
Mathematical Induction MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Mathematical Induction Fall 2014 1 / 21 Outline 1 Mathematical Induction 2 Strong Mathematical
More informationLecture Notes on The CurryHoward Isomorphism
Lecture Notes on The CurryHoward Isomorphism 15312: Foundations of Programming Languages Frank Pfenning Lecture 27 ecember 4, 2003 In this lecture we explore an interesting connection between logic and
More informationCS6840: Advanced Complexity Theory Mar 29, Lecturer: Jayalal Sarma M.N. Scribe: Dinesh K.
CS684: Advanced Complexity Theory Mar 29, 22 Lecture 46 : Size lower bounds for AC circuits computing Parity Lecturer: Jayalal Sarma M.N. Scribe: Dinesh K. Theme: Circuit Complexity Lecture Plan: Proof
More informationPeano Arithmetic. CSC 438F/2404F Notes (S. Cook) Fall, Goals Now
CSC 438F/2404F Notes (S. Cook) Fall, 2008 Peano Arithmetic Goals Now 1) We will introduce a standard set of axioms for the language L A. The theory generated by these axioms is denoted PA and called Peano
More informationON EXISTENCE IN SET THEORY
ON EXISTENCE IN SET THEORY RODRIGO A. FREIRE Abstract. The aim of the present paper is to provide a robust classification of valid sentences in set theory by means of existence and related notions and,
More informationSection 1.2 Propositional Equivalences. A tautology is a proposition which is always true. A contradiction is a proposition which is always false.
Section 1.2 Propositional Equivalences A tautology is a proposition which is always true. Classic Example: P P A contradiction is a proposition which is always false. Classic Example: P P A contingency
More informationPrinciples of AI Planning
Principles of AI Planning 4. Normal forms AlbertLudwigsUniversität Freiburg Bernhard Nebel and Robert Mattmüller October 31st, 2012 Similarly to s in propositional logic (DNF, CNF, NNF,... ) we can define
More informationMAXPLANCKINSTITUT INFORMATIK
MAXPLANCKINSTITUT FÜR INFORMATIK On evaluating decision procedures for modal logic Ullrich Hustadt and Renate A. Schmidt MPI I 97 2 003 April 1997 INFORMATIK Im Stadtwald D 66123 Saarbrücken Germany
More informationArgumentative Characterisations of Nonmonotonic Inference in Preferred Subtheories: Stable Equals Preferred
Argumentative Characterisations of Nonmonotonic Inference in Preferred Subtheories: Stable Equals Preferred Sanjay Modgil November 17, 2017 Abstract A number of argumentation formalisms provide dialectical
More informationPredicate Logic. CSE 191, Class Note 02: Predicate Logic Computer Sci & Eng Dept SUNY Buffalo
Predicate Logic CSE 191, Class Note 02: Predicate Logic Computer Sci & Eng Dept SUNY Buffalo c Xin He (University at Buffalo) CSE 191 Discrete Structures 1 / 22 Outline 1 From Proposition to Predicate
More informationComputational Logic for Computer Science
Motivation: formalization  proofs & deduction Computational proofs  logic & deduction Formal proofs Pr Computational Logic for Computer Science Mauricio AyalaRinco n & Fla vio L.C. de Moura Grupo de
More informationNotes on Modal Logic
Notes on Modal Logic Notes for PHIL370 Eric Pacuit October 22, 2012 These short notes are intended to introduce some of the basic concepts of Modal Logic. The primary goal is to provide students in Philosophy
More informationFuzzy Expert Systems Lecture 6 (Fuzzy Logic )
Fuzzy Expert Systems Lecture 6 (Fuzzy Logic ) Unlike Classical Logic, Fuzzy Logic is concerned, in the main, with modes of reasoning which are approximate rather than exact L. A. Zadeh Lecture 6 صفحه Summary
More informationVinter: A VampireBased Tool for Interpolation
Vinter: A VampireBased Tool for Interpolation Kryštof Hoder 1, Andreas Holzer 2, Laura Kovács 2, and Andrei Voronkov 1 1 University of Manchester 2 TU Vienna Abstract. This paper describes the Vinter
More informationAxiomatic and TableauBased Reasoning for Kt(H,R)
Axiomatic and TableauBased Reasoning for Kt(H,R) Renate A. Schmidt 1 School of Computer Science University of Manchester, Manchester, UK John G. Stell School of Computing University of Leeds, Leeds, UK
More informationDe Morgan s a Laws. De Morgan s laws say that. (φ 1 φ 2 ) = φ 1 φ 2, (φ 1 φ 2 ) = φ 1 φ 2.
De Morgan s a Laws De Morgan s laws say that (φ 1 φ 2 ) = φ 1 φ 2, (φ 1 φ 2 ) = φ 1 φ 2. Here is a proof for the first law: φ 1 φ 2 (φ 1 φ 2 ) φ 1 φ 2 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0 0 a Augustus DeMorgan
More informationLECTURE 1. Logic and Proofs
LECTURE 1 Logic and Proofs The primary purpose of this course is to introduce you, most of whom are mathematics majors, to the most fundamental skills of a mathematician; the ability to read, write, and
More informationLogic and Proofs 1. 1 Overview. 2 Sentential Connectives. John Nachbar Washington University December 26, 2014
John Nachbar Washington University December 26, 2014 Logic and Proofs 1 1 Overview. These notes provide an informal introduction to some basic concepts in logic. For a careful exposition, see, for example,
More informationIncomplete version for students of easllc2012 only. 94 FirstOrder Logic. Incomplete version for students of easllc2012 only. 6.5 The Semantic Game 93
65 The Semantic Game 93 In particular, for every countable X M there is a countable submodel N of M such that X N and N = T Proof Let T = {' 0, ' 1,} By Proposition 622 player II has a winning strategy
More informationFinite Model Theory: FirstOrder Logic on the Class of Finite Models
1 Finite Model Theory: FirstOrder Logic on the Class of Finite Models Anuj Dawar University of Cambridge Modnet Tutorial, La Roche, 21 April 2008 2 Finite Model Theory In the 1980s, the term finite model
More informationA ModalLayered Resolution Calculus for K
A ModalLayered Resolution Calculus for K Cláudia Nalon 1, Ullrich Hustadt 2, and Clare Dixon 2 1 Departament of Computer Science, University of Brasília C.P. 4466 CEP:70.910090 Brasília DF Brazil nalon@unb.br
More informationTR : Tableaux for the Logic of Proofs
City University of New York (CUNY) CUNY Academic Works Computer Science Technical Reports Graduate Center 2004 TR2004001: Tableaux for the Logic of Proofs Bryan Renne Follow this and additional works
More informationConjunctive Normal Form and SAT
Notes on SatisfiabilityBased Problem Solving Conjunctive Normal Form and SAT David Mitchell mitchell@cs.sfu.ca September 19, 2013 This is a preliminary draft of these notes. Please do not distribute without
More informationA Unit ResolutionBased Approach to Tractable and Paraconsistent Reasoning
A Unit ResolutionBased Approach to Tractable and Paraconsistent Reasoning ylvie CosteMarquis and Pierre Marquis 1 Abstract. A family of unit resolutionbased paraconsistent inference relations Σ for
More informationLogic and Truth Tables
Logic and Truth Tables What is a Truth Table? A truth table is a tool that helps you analyze statements or arguments in order to verify whether or not they are logical, or true. There are five basic operations
More informationFROM AXIOMS TO STRUCTURAL RULES, THEN ADD QUANTIFIERS.
FROM AXIOMS TO STRUCTURAL RULES, THEN ADD QUANTIFIERS. REVANTHA RAMANAYAKE We survey recent developments in the program of generating proof calculi for large classes of axiomatic extensions of a nonclassical
More informationMATHS 315 Mathematical Logic
MATHS 315 Mathematical Logic Second Semester, 2007 Contents 1 Informal Statement Logic 1 1.1 Statements and truth tables............................... 1 1.2 Tautologies, logical equivalence and logical
More informationPartiality and Adjointness in Modal Logic
Preprint of July 2014. Published in Advances in Modal Logic, Vol. 10, 2014. Partiality and Adjointness in Modal Logic Wesley H. Holliday Department of Philosophy & Group in Logic and the Methodology of
More informationCambridge University Press Dependence Logic. A New Approach to Independence Friendly Logic
Jouko Väänänen Dependence Logic A New Approach to Independence Friendly Logic Chapter 1 Dependence Logic Dependence logic introduces the concept of dependence into first order logic by adding a new kind
More informationFundamentals of Software Engineering
Fundamentals of Software Engineering FirstOrder Logic Ina Schaefer Institute for Software Systems Engineering TU Braunschweig, Germany Slides by Wolfgang Ahrendt, Richard Bubel, Reiner Hähnle (Chalmers
More informationDetecting Backdoor Sets with Respect to Horn and Binary Clauses
Detecting Backdoor Sets with Respect to Horn and Binary Clauses Naomi Nishimura 1,, Prabhakar Ragde 1,, and Stefan Szeider 2, 1 School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L
More informationUNIT 2: A DISCUSSION ON FUNDAMENTALS OF LOGIC
UNIT 2: A DISCUSSION ON FUNDAMENTALS OF LOGIC Introduction: Logic is being used as a tool in a number of situations for a variety of reasons. Logic is found to be extremely useful in decision taking problems.
More informationCOMP6463: λcalculus
COMP6463: λcalculus 1. Basics Michael Norrish Michael.Norrish@nicta.com.au Canberra Research Lab., NICTA Semester 2, 2015 Outline Introduction Lambda Calculus Terms Alpha Equivalence Substitution Dynamics
More informationInterpolantbased Transition Relation Approximation
Interpolantbased Transition Relation Approximation Ranjit Jhala and K. L. McMillan 1 University of California, San Diego 2 Cadence Berkeley Labs Abstract. In predicate abstraction, exact image computation
More informationIntroduction to Kleene Algebra Lecture 15 CS786 Spring 2004 March 15 & 29, 2004
Introduction to Kleene Algebra Lecture 15 CS786 Spring 2004 March 15 & 29, 2004 Completeness of KAT In this lecture we show that the equational theories of the Kleene algebras with tests and the starcontinuous
More informationHybrid Rules with WellFounded Semantics
Hybrid Rules with WellFounded Semantics W lodzimierz Drabent Jan Ma luszyński arxiv:0906.3815v1 [cs.lo] 20 Jun 2009 March 13, 2009 Submitted for publication Abstract A general framework is proposed for
More informationModal and temporal logic
Modal and temporal logic N. Bezhanishvili I. Hodkinson C. Kupke Imperial College London 1 / 83 Overview Part II 1 Soundness and completeness. Canonical models. 3 lectures. 2 Finite model property. Filtrations.
More informationReprinted from the Bulletin of Informatics and Cybernetics Research Association of Statistical Sciences,Vol.41
LEARNABILITY OF XML DOCUMENT TRANSFORMATION RULES USING TYPED EFSS by Noriko Sugimoto Reprinted from the Bulletin of Informatics and Cybernetics Research Association of Statistical Sciences,Vol.41 FUKUOKA,
More informationPEANO AXIOMS FOR THE NATURAL NUMBERS AND PROOFS BY INDUCTION. The Peano axioms
PEANO AXIOMS FOR THE NATURAL NUMBERS AND PROOFS BY INDUCTION The Peano axioms The following are the axioms for the natural numbers N. You might think of N as the set of integers {0, 1, 2,...}, but it turns
More information