Computational Logic and Cognitive Science: An Overview


 Lynette Skinner
 2 years ago
 Views:
Transcription
1 Computational Logic and Cognitive Science: An Overview Session 1: Logical Foundations Technical University of Dresden 25th of August, 2008 University of Osnabrück
2 Who we are Helmar Gust Interests: Analogical Reasoning, Logic Programming, ELearning Systems, NeuroSymbolic Integration KaiUwe Kühnberger Interests: Analogical Reasoning, Ontologies, NeuroSymbolic Integration Where we work: University of Osnabrück Institute of Cognitive Science Working Group: Artificial Intelligence
3 Cognitive Science in Osnabrück Institute of Cognitive Science International Study Programs Bachelor Program Master Program Joined degree with Trento/Rovereto PhD Program Doctorate Program Cognitive Science Graduate School Adaptivity in Hybrid Cognitive Systems Web:
4 Who are You? Prerequisites? Logic? Propositional logic, FOL, models? Calculi, theorem proving? Nonclassical logics: manyvalued logic, nonmonotonicity, modal logic? Topics in Cognitive Science? Rationality (bounded, unbounded, heuristics), human reasoning? Cognitive models / architectures (symbolic, neural, hybrid)? Creativity?
5 Overview of the Course First Session (Monday) Foundations: Forms of reasoning, propositional and FOL, properties of logical systems, Boolean algebras, normal forms Second Session (Tuesday) Cognitive findings: Wasonselection task, theories of mind, creativity, causality, types of reasoning, analogies Third Session (Thursday morning) Nonclassical types of reasoning: manyvalued logics, fuzzy logics, modal logics, probabilistic reasoning Fourth Session (Thursday afternoon) Nonmonotonicity Fifth Session (Friday) Analogies, neurosymbolic approaches Wrapup
6 Forms of Reasoning: Deduction, Abduction, Induction Theorem Proving, Sherlock Holmes, and All Swans are White...
7 Basic Types of Inferences: Deduction Deduction: Derive a conclusion from given axioms ( knowledge ) and facts ( observations ). Example: All humans are mortal. Socrates is a human. Therefore, it follows that Socrates is mortal. (axiom) (fact/ premise) (conclusion) The conclusion can be derived by applying the modus ponens inference rule (Aristotelian logic). Theorem proving is based on deductive reasoning techniques.
8 Basic Types of Inferences: Induction Induction: Derive a general rule (axiom) from background knowledge and observations. Example: Socrates is a human Socrates is mortal (background knowledge) (observation/ example) Therefore, I hypothesize that all humans are mortal (generalization) Remarks: Induction means to infer generalized knowledge from example observations: Induction is the inference mechanism for (machine) learning.
9 Basic Types of Inferences: Abduction Abduction: From a known axiom (theory) and some observation, derive a premise. Example: All humans are mortal Socrates is mortal (theory) (observation) Therefore, Socrates must have been a human (diagnosis) Remarks: Abduction is typical for diagnostic and expert systems. If one has the flue, one has moderate fewer. Patient X has moderate fewer. Therefore, he has the flue. Strong relation to causation
10 Deduction Deductive inferences are also called theorem proving or logical inference. Deduction is truth preserving: If the premises (axioms and facts) are true, then the conclusion (theorem) is true. To perform deductive inferences on a machine, a calculus is needed: A calculus is a set of syntactical rewriting rules defined for some (formal) language. These rules must be sound and should be complete. We will focus on firstorder logic (FOL). Syntax of FOL. Semantics of FOL.
11 Propositional Logic and FirstOrder Logic Some rather Abstract Stuff
12 Propositional Logic Formulas: Given is a countable set of atomic propositions AtProp = {p,q,r,...}. The set of wellformed formulas Form of propositional logic is the smallest class such that it holds: p AtProp: p Form ϕ, ψ Form: ϕ ψ Form ϕ, ψ Form: ϕ ψ Form ϕ Form: ϕ Form Semantics: A formula ϕ is valid if ϕ is true for all possible assignments of the atomic propositions occurring in ϕ A formula ϕ is satisfiable if ϕ is true for some assignment of the atomic propositions occurring in ϕ Models of propositional logic are specified by Boolean algebras (A model is a distribution of truthvalues over AtProp making ϕ true)
13 Propositional Logic Hilbertstyle calculus Axioms: p (q p) [p (q r)] [(p q) (p r)] ( p q) (q p) p q p and (p q) q (r p) ((r q) (r p q)) p (p q) and q (p q) (p r) ((q r) (p q r)) Rules: Modus Ponens: If expressions ϕ and ϕ ψ are provable then ψ is also provable. Remark: There are other possible axiomatizations of propositional logic.
14 Propositional Logic Other calculi: Gentzentype calculus Tableauxcalculus Propositional logic is relatively weak: no temporal or modal statements, no rules can be expressed Therefore a stronger system is needed
15 FirstOrder Logic Syntactically wellformed firstorder formulas for a signature Σ = {c 1,...,c n,f 1,...,f m,r 1,...,R l } are inductively defined. The set of Terms is the smallest class such that: A variable x Var is a term, a constant c i {c 1,...,c n } is a term. Var is a countable set of variables. If f i is a function symbol of arity r and t 1,...,t r are terms, then f i (t 1,...,t r ) is a term. The set of Formulas is the smallest class such that: If R j is a predicate symbol of arity r and t 1,...,t r are terms, then R j (t 1,...,t r ) is a formula (atomic formula or literal). For all formulas ϕ and ψ: ϕ ψ, ϕ ψ, ϕ, ϕ ψ, ϕ ψ are formulas. If x Var and ϕ is a formula, then xϕ and xϕ are formulas. Notice that term and formula are rather different concepts. Terms are used to define formulas and not vice versa.
16 Firstorder Logic Semantics (meaning) of FOL formulas. Expressions of FOL are interpreted using an interpretation function I: Σ A(U) I(c i ) U I(f i ) : U arity(fi) U I(R i ) : U arity(ri) {true, false} U is the called the universe or the domain A pair M = <U,I> is called a structure.
17 Firstorder Logic Semantics (meaning) of FOL formulas. Recursive definition for interpreting terms and evaluating truth values of formulas: For c {c 1,...,c n }: [[c i ]] = I(c i ) [[f i (t 1,...,t r )]] = I(f I )([[t 1 ]],...,[[t r ]]) [[R(t 1,...,t r )]] = true iff <[[t 1 ]],...,[[t r ]]> I(R) [[ϕ ψ]] = true iff [[ϕ]] = true and [[ψ]] = true [[ϕ ψ]] = true iff [[ϕ]] = true or [[ψ]] = true [[ ϕ]] = true iff [[ϕ]] = false [[ x ϕ(x)]] = true iff for all d U: [[ϕ(x)]] x=d = true [[ x ϕ(x)]] = true iff there exists d U: [[ϕ(x)]] x=d = true
18 Firstorder Logic Semantics Model If the interpretation of a formula ϕ with respect to a structure M = <U,I> results in the truth value true, M is called a model for ϕ (formal: M ϕ) Validity If every structure M = <U,I> is a model for ϕ we call ϕ valid ( ϕ) Satisfiability If there exists a model M = <U,I> for ϕ we call ϕ satisfiable Example: x y (R(x) R(y) R(x) R(y)) [valid] If x and y are rich then either x is rich or y is rich If x and y are even then either x is even or y is even
19 Firstorder Logic Semantics An example: x (N(x) P(x,c)) [satisfiable] There is a natural number that is smaller than 17. There exists someone who is a student and likes logic. Notice that there are models which make the statement false Logical consequence A formula ϕ is a logical consequence (or a logical entailment) of A = {A 1,...,A n }, if each model for A is also a model for ϕ. We write A ϕ Notice: A ϕ can mean that A is a model for ϕ or that ϕ is a logical consequence of A Therefore people usually use different alphabets or fonts to make this difference visible
20 Theories The theory Th(A) of a set of formulas A: Th(A) := {ϕ A ϕ} Theories are closed under semantic entailment The operator: Th : A Th(A) is a so called closure operator: X Th(X) extensive / inductive X Y Th(X) Th(Y) monotone Th(Th(X)) = Th(X) idempotent
21 Firstorder Logic Semantic equivalences Two formulas ϕ and ψ are semantically equivalent (we write ϕ ψ) if for all interpretations of ϕ and ψ it holds: M is a model for ϕ iff M is a model for ψ. A few examples: ϕ ϕ ϕ ϕ ψ ψ ϕ ϕ (ψ χ) (ϕ ψ) (ϕ χ) The following statements are equivalent (based on the deduction theorem): G is a logical consequence of {A 1,...,A n } A 1... A n G is valid Every structure is a model for this expression. A 1... A n G is not satisfiable. There is no structure making this expression true This can be used in the resolution calculus: If an expression A 1... A n G is not satisfiable, then false can be derived syntactically.
22 Repetition: Semantic Equivalences Here is a list of semantic equivalences (ϕ ψ) (ψ ϕ), (ϕ ψ) (ψ ϕ) (commutativity) (ϕ ψ) χ ϕ (ψ χ), (ϕ ψ) χ ϕ (ψ χ) (associativity) (ϕ (ϕ ψ)) ϕ, (ϕ (ϕ ψ)) ϕ (absorption) (ϕ (ψ χ)) (ϕ ψ) (ϕ χ) (distributivity) (ϕ (ψ χ)) (ϕ ψ) (ϕ χ) (distributivity) ϕ ϕ (double negation) (ϕ ψ) ( ϕ ψ), (ϕ ψ) ( ϕ ψ) (demorgan) ( ϕ), ( ϕ) ϕ ( ϕ) ϕ, ( ϕ) Here are some more semantic equivalences (ϕ ϕ) ϕ, (ϕ ϕ) ϕ (idempotency) ϕ ϕ (tautology) ϕ ϕ (contradiction) xϕ x ϕ, xϕ x ϕ (quantifiers) ( x ϕ ψ) x (ϕ ψ), ( x ϕ ψ) x (ϕ ψ) x(ϕ ψ) ( xϕ xψ) Etc.
23 Properties of Logical Systems Soundness A calculus is sound, if only such conclusions can be derived which also hold in the model In other words: Everything that can be derived is semantically true Completeness A calculus is complete, if all conclusions can be derived which hold in the models In other words: Everything that is semantically true can syntactically be derived Decidability A calculus is decidable if there is an algorithm that calculates effectively for every formula whether such a formula is a theorem or not Usually people are interested in completeness results and decidability results We say a logic is sound/complete/decidable if there exists a calculus with these properties
24 Some Properties of Classical Logic Propositional Logic: Sound and Complete, i.e. everything that can be proven is valid and everything that is valid can be proven Decidable, i.e. there is an algorithm that decides for every input whether this input is a theorem or not Firstorder logic: Complete (Gödel 1930) Undecidable, i.e. no algorithm exists that decides for every input whether this input is a theorem or not (Church 1936) More precisely FOL is semidecidable Models The classical model for FOL are Boolean algebras
25 Boolean Algebras P [[P]] U if arity is 1 (or [[P]] U... U if arity > 1) x 1,...,x n : P(x 1,...,x n ) Q(x 1,...,x n ) [[P]] [[Q]] We can draw Venn diagrams: P Q Regions (e.g. arbitrary subsets) of the ndimensional real space can be interpreted as a Boolean algebra
26 Boolean Algebras The power set (U) has the following properties: It is a partially ordered set with order A B is the largest set X with X A and X B A B is the smallest set X with A X and B X comp(a) is the largest set X with A X = U is the largest set in (U), such that X U for all X (U) is the smallest set in (U), such that X for all X (U)
27 Boolean Algebras The concept of a lattice Definition: A partial order D = <D, > is called a lattice if for each two elements x,y D it holds: sup(x,y) exists and inf(x,y) exists sup(x,y) is the least upper bound of elements x and y inf(x,y) is the greatest lower bound of x and y The concept of a Boolean Algebra Definition: A Boolean algebra is a tuple M = <D,,,,> (or alternatively <D,,,,,>) such that <D, > = <D,, > is a distributive lattice is the top and the bottom element is a complement operation
28 Lindenbaum Algebras The Linbebaum algebra for propositional logic with atomic propositions p and q
29 Normal Forms If there are a lot of different representations of the same statement Are there simple ones? Are there normal forms? Different normal forms for FOL Negation normal form Only negations of atomic formulas Prenex normal form No embedded Quantifiers Conjunctive normal form Only conjunctions of disjunctions Disjunctive normal form Only disjunctions of conjunctions Gentzen normal form Only implications where the condition is an atomic conjunction and the conclusion is an atomic disjunction
30 Normal Forms If there are a lot of different representations of the same statement Are there simple ones? Are there normal forms? Different normal forms for FOL (x:(p(x) y:q(x,y))) Negation normal form x:(p(x) y: q(x,y)) Only negations of atomic formulas Prenex normal form xy:(p(x) : q(x,y)) No embedded Quantifiers Conjunctive normal form p(c x ) q(c x,y) Only conjunctions of disjunctions Disjunctive normal form Only disjunctions of conjunctions Gentzen normal form q(c x,y) p(c x ) Only implications where the condition is an atomic conjunction and the conclusion is an atomic disjunction
31 Clause Form Conjunctive normal form. We know: Every formula of propositional logic can be rewritten as a conjunction of disjunctions of atomic propositions. Similarly every formula of predicate logic can be rewritten as a conjunction of disjunctions of literals (modulo the quantifiers). A formula is in clause form if it is rewritten as a set of disjunctions of (possibly negative) literals. Example: {{p(c x ) },{ q(c x,y)}} Theorem: Every FOL formula F can be transformed into clause form F such that F is satisfiable iff F is satisfiable
32 What is the meaning of these Axioms? x: C(x,x) x,y: C(x,y) C(y,x) x,y: P(x,y) z: (C(z,x) C(z,y)) x,y: O(x,y) z: (P(z,x) P(z,y)) x,y: DC(x,y) C(x,y) x,y: EC(x,y) C(x,y) O(x,y) x,y: PO(x,y) O(x,y) P(x,y) P(y,x) x,y: EQ(x,y) P(x,y) P(y,x) x,y: PP(x,y) P(x,y) P(y,x) x,y: TPP(x,y) PP(x,y) z(ec(z,x) EC(z,y)) x,y: TPPI(x,y) PP(y,x) z(ec(z,y) EC(z,x)) x,y: NTPP(x,y) PP(x,y) z(ec(z,x) EC(z,y)) x,y: NTPPI(x,y) PP(y,x) z(ec(z,y) EC(z,x))
33 Is This a Theorem? x,y,z: NTPP(x,y) NTPP(y,z) NTPP(x,z) Easy to see if we look at models!
34 Relations of Regions of the RCC8 (a canonical model: ndimensional closed discs)
35 Thank you very much!!
CHAPTER 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 13 of 41. More Propositional and Predicate Logic
Lecture 13 of 41 More Propositional and Predicate Logic Monday, 20 September 2004 William H. Hsu, KSU http://www.kddresearch.org http://www.cis.ksu.edu/~bhsu Reading: Sections 8.18.3, Russell and Norvig
More informationFirstOrder Logics and Truth Degrees
FirstOrder Logics and Truth Degrees George Metcalfe Mathematics Institute University of Bern LATD 2014, Vienna Summer of Logic, 1519 July 2014 George Metcalfe (University of Bern) FirstOrder Logics
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 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 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 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 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 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 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 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 informationA Propositional Dynamic Logic for CCS Programs
A Propositional Dynamic Logic for CCS Programs Mario R. F. Benevides and L. Menasché Schechter {mario,luis}@cos.ufrj.br Abstract This work presents a Propositional Dynamic Logic in which the programs are
More informationCorrespondence analysis for strong threevalued logic
Correspondence analysis for strong threevalued logic A. Tamminga abstract. I apply Kooi and Tamminga s (2012) idea of correspondence analysis for manyvalued logics to strong threevalued logic (K 3 ).
More informationMathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson
Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement
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 informationEQUATIONAL LOGIC AND ABSTRACT ALGEBRA * ABSTRACT
EQUATIONAL LOGIC AND ABSTRACT ALGEBRA * Taje I. Ramsamujh Florida International University Mathematics Department ABSTRACT Equational logic is a formalization of the deductive methods encountered in studying
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 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 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 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 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 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 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 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 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 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 informationManyvalued Intuitionistic Implication and Inference Closure in a Bilatticebased Logic
1 Manyvalued Intuitionistic Implication and Inference Closure in a Bilatticebased Logic Zoran Majkić Dept. of Computer Science,UMIACS, University of Maryland, College Park, MD 20742 zoran@cs.umd.edu
More informationResolution. Informatics 1 School of Informatics, University of Edinburgh
Resolution In this lecture you will see how to convert the natural proof system of previous lectures into one with fewer operators and only one proof rule. You will see how this proof system can be used
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 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 informationValidity Checking. Propositional and FirstOrder Logic (part I: semantic methods)
Validity Checking Propositional and FirstOrder Logic (part I: semantic methods) Slides based on the book: Rigorous Software Development: an introduction to program verification, by José Bacelar Almeida,
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 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 informationSJÄLVSTÄNDIGA ARBETEN I MATEMATIK
SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Automated Theorem Proving av Tom Everitt 2010  No 8 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 106 91 STOCKHOLM
More informationPredicate logic. Logic in computer science. Logic in Computer Science (lecture) PART II. first order logic
PART II. Predicate logic first order logic Logic in computer science Seminar: INGK401K5; INHK401; INJK401K4 University of Debrecen, Faculty of Informatics kadek.tamas@inf.unideb.hu 1 / 19 Alphabets Logical
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 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 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 informationCertain Answers as Objects and Knowledge
Certain Answers as Objects and Knowledge Leonid Libkin School of Informatics, University of Edinburgh Abstract The standard way of answering queries over incomplete databases is to compute certain answers,
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 informationUniversity of Ostrava. Reasoning in Description Logic with Semantic Tableau Binary Trees
University of Ostrava Institute for Research and Applications of Fuzzy Modeling Reasoning in Description Logic with Semantic Tableau Binary Trees Alena Lukasová Research report No. 63 2005 Submitted/to
More informationOne More Decidable Class of Finitely Ground Programs
One More Decidable Class of Finitely Ground Programs Yuliya Lierler and Vladimir Lifschitz Department of Computer Sciences, University of Texas at Austin {yuliya,vl}@cs.utexas.edu Abstract. When a logic
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 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 informationFirstOrder Stable Model Semantics and FirstOrder Loop Formulas
Journal of Artificial Intelligence Research 42 (2011) 125180 Submitted 03/11; published 10/11 FirstOrder Stable Model Semantics and FirstOrder Loop Formulas Joohyung Lee Yunsong Meng School of Computing,
More informationPropositional Logic. A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both.
irst Order Logic Propositional Logic A proposition is a declarative sentence (a sentence that declares a fact) that is either true or false, but not both. Are the following sentences propositions? oronto
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 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 informationComputational Methods for Database Repair by Signed Formulae
Computational Methods for Database Repair by Signed Formulae Ofer Arieli (oarieli@mta.ac.il) Department of Computer Science, The Academic College of TelAviv, 4 Antokolski street, TelAviv 61161, Israel.
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 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. 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 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 informationIntroduction to Logic in Computer Science: Autumn 2006
Introduction to Logic in Computer Science: Autumn 2006 Ulle Endriss Institute for Logic, Language and Computation University of Amsterdam Ulle Endriss 1 Plan for Today Now that we have a basic understanding
More informationClassical BI. (A Logic for Reasoning about Dualising Resources) James Brotherston Cristiano Calcagno
Classical BI (A Logic for Reasoning about Dualising Resources) James Brotherston Cristiano Calcagno Dept. of Computing, Imperial College London, UK {jbrother,ccris}@doc.ic.ac.uk Abstract We show how to
More informationArtificial Intelligence
Artificial Intelligence ICS461 Fall 2010 1 Lecture #12B More Representations Outline Logics Rules Frames Nancy E. Reed nreed@hawaii.edu 2 Representation Agents deal with knowledge (data) Facts (believe
More informationUpdating Action Domain Descriptions
Updating Action Domain Descriptions Thomas Eiter, Esra Erdem, Michael Fink, and Ján Senko Institute of Information Systems, Vienna University of Technology, Vienna, Austria Email: (eiter esra michael jan)@kr.tuwien.ac.at
More informationA set is a Many that allows itself to be thought of as a One. (Georg Cantor)
Chapter 4 Set Theory A set is a Many that allows itself to be thought of as a One. (Georg Cantor) In the previous chapters, we have often encountered sets, for example, prime numbers form a set, domains
More informationominimality and Uniformity in n 1 Graphs
ominimality and Uniformity in n 1 Graphs Reid Dale July 10, 2013 Contents 1 Introduction 2 2 Languages and Structures 2 3 Definability and Tame Geometry 4 4 Applications to n 1 Graphs 6 5 Further Directions
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 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 informationINTRODUCTORY SET THEORY
M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H1088 Budapest, Múzeum krt. 68. CONTENTS 1. SETS Set, equal sets, subset,
More informationReview for Final Exam
Review for Final Exam Note: Warning, this is probably not exhaustive and probably does contain typos (which I d like to hear about), but represents a review of most of the material covered in Chapters
More informationPropositional Logic. Definition: A proposition or statement is a sentence which is either true or false.
Propositional Logic Definition: A proposition or statement is a sentence which is either true or false. Definition:If a proposition is true, then we say its truth value is true, and if a proposition is
More informationRepair Checking in Inconsistent Databases: Algorithms and Complexity
Repair Checking in Inconsistent Databases: Algorithms and Complexity Foto Afrati 1 Phokion G. Kolaitis 2 1 National Technical University of Athens 2 UC Santa Cruz and IBM Almaden Research Center Oxford,
More informationQuery Answering in Inconsistent Databases
Query Answering in Inconsistent Databases Leopoldo Bertossi 1 and Jan Chomicki 2 1 School of Computer Science, Carleton University, Ottawa, Canada, bertossi@scs.carleton.ca 2 Dept. of Computer Science
More informationRigorous. Development. Software. Program Verification. & Springer. An Introduction to. Jorge Sousa Pinto. Jose Bacelar Almeida Maria Joao Frade
Jose Bacelar Almeida Maria Joao Frade Jorge Sousa Pinto Simao Melo de Sousa Rigorous Software Development An Introduction to Program Verification & Springer Contents 1 Introduction 1 1.1 A Formal Approach
More informationThe Language of Mathematics
CHPTER 2 The Language of Mathematics 2.1. Set Theory 2.1.1. Sets. set is a collection of objects, called elements of the set. set can be represented by listing its elements between braces: = {1, 2, 3,
More informationWHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly
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 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 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 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 informationAnnouncements. CompSci 230 Discrete Math for Computer Science Sets. Introduction to Sets. Sets
CompSci 230 Discrete Math for Computer Science Sets September 12, 2013 Prof. Rodger Slides modified from Rosen 1 nnouncements Read for next time Chap. 2.32.6 Homework 2 due Tuesday Recitation 3 on Friday
More informationThe Mathematics of GIS. Wolfgang Kainz
The Mathematics of GIS Wolfgang Kainz Wolfgang Kainz Department of Geography and Regional Research University of Vienna Universitätsstraße 7, A00 Vienna, Austria EMail: wolfgang.kainz@univie.ac.at Version.
More informationMATHEMATICAL LOGIC FOR COMPUTER SCIENCE
MATHEMATICAL LOGIC FOR COMPUTER SCIENCE Second Edition WORLD SCIENTIFIC SERIES IN COMPUTER SCIENCE 25: Computer Epistemology A Treatise on the Feasibility of the Unfeasible or Old Ideas Brewed New (T Vamos)
More informationA Beginner s Guide to Modern Set Theory
A Beginner s Guide to Modern Set Theory Martin Dowd Product of Hyperon Software PO Box 4161 Costa Mesa, CA 92628 www.hyperonsoft.com Copyright c 2010 by Martin Dowd 1. Introduction..... 1 2. Formal logic......
More informationRemoving Partial Inconsistency in Valuation Based Systems*
Removing Partial Inconsistency in Valuation Based Systems* Luis M. de Campos and Serafín Moral Departamento de Ciencias de la Computación e I.A., Universidad de Granada, 18071 Granada, Spain This paper
More informationfacultad de informática universidad politécnica de madrid
facultad de informática universidad politécnica de madrid On the Confluence of CHR Analytical Semantics Rémy Haemmerlé Universidad olitécnica de Madrid & IMDEA Software Institute, Spain TR Number CLI2/2014.0
More informationHigh Integrity Software Conference, Albuquerque, New Mexico, October 1997.
MetaAmphion: Scaling up HighAssurance Deductive Program Synthesis Steve Roach Recom Technologies NASA Ames Research Center Code IC, MS 2692 Moffett Field, CA 94035 sroach@ptolemy.arc.nasa.gov Jeff Van
More information4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.
Section 4. Set Theory 4.1. Definitions A set may be viewed as any well defined collection of objects, called elements or members of the set. Sets are usually denoted with upper case letters, A, B, X, Y,
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 informationFirstOrder Theories
FirstOrder Theories Ruzica Piskac Max Planck Institute for Software Systems, Germany piskac@mpisws.org Seminar on Decision Procedures 2012 Ruzica Piskac FirstOrder Theories 1 / 39 Acknowledgments Theories
More informationScalable Automated Symbolic Analysis of Administrative RoleBased Access Control Policies by SMT solving
Scalable Automated Symbolic Analysis of Administrative RoleBased Access Control Policies by SMT solving Alessandro Armando 1,2 and Silvio Ranise 2, 1 DIST, Università degli Studi di Genova, Italia 2 Security
More information3(vi) B. Answer: False. 3(vii) B. Answer: True
Mathematics 0N1 Solutions 1 1. Write the following sets in list form. 1(i) The set of letters in the word banana. {a, b, n}. 1(ii) {x : x 2 + 3x 10 = 0}. 3(iv) C A. True 3(v) B = {e, e, f, c}. True 3(vi)
More informationEquivalence in Answer Set Programming
Equivalence in Answer Set Programming Mauricio Osorio, Juan Antonio Navarro, José Arrazola Universidad de las Américas, CENTIA Sta. Catarina Mártir, Cholula, Puebla 72820 México {josorio, ma108907, arrazola}@mail.udlap.mx
More informationSoftware Verification and Testing. Lecture Notes: Z I
Software Verification and Testing Lecture Notes: Z I Motivation so far: we have seen that properties of software systems can be specified using firstorder logic, set theory and the relational calculus
More informationThe Modal Logic Programming System MProlog
The Modal Logic Programming System MProlog Linh Anh Nguyen Institute of Informatics, University of Warsaw ul. Banacha 2, 02097 Warsaw, Poland nguyen@mimuw.edu.pl Abstract. We present the design of our
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 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 informationParametric Domaintheoretic models of Linear Abadi & Plotkin Logic
Parametric Domaintheoretic models of Linear Abadi & Plotkin Logic Lars Birkedal Rasmus Ejlers Møgelberg Rasmus Lerchedahl Petersen IT University Technical Report Series TR007 ISSN 600 600 February 00
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More informationAlgorithmic Software Verification
Algorithmic Software Verification (LTL Model Checking) Azadeh Farzan What is Verification Anyway? Proving (in a formal way) that program satisfies a specification written in a logical language. Formal
More informationSyntax and Semantics for Business Rules
Syntax and Semantics for Business Rules Xiaofan Liu 1 2, Natasha Alechina 1, and Brian Logan 1 1 School of Computer Science, University of Nottingham, Nottingham, NG8 1BB, UK 2 School of Computer and Communication,
More informationAN INTUITIONISTIC EPISTEMIC LOGIC FOR ASYNCHRONOUS COMMUNICATION. Yoichi Hirai. A Master Thesis
AN INTUITIONISTIC EPISTEMIC LOGIC FOR ASYNCHRONOUS COMMUNICATION by Yoichi Hirai A Master Thesis Submitted to the Graduate School of the University of Tokyo on February 10, 2010 in Partial Fulfillment
More informationA Note on Context Logic
A Note on Context Logic Philippa Gardner Imperial College London This note describes joint work with Cristiano Calcagno and Uri Zarfaty. It introduces the general theory of Context Logic, and has been
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 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 informationµz An Efficient Engine for Fixed points with Constraints
µz An Efficient Engine for Fixed points with Constraints Kryštof Hoder, Nikolaj Bjørner, and Leonardo de Moura Manchester University and Microsoft Research Abstract. The µz tool is a scalable, efficient
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 information