Advanced Topics in LP and FP
|
|
- Benedict Bradley
- 6 years ago
- Views:
Transcription
1 Lecture 3: Logic Programming in Prolog
2 Outline The control of execution
3 Logic Programming Declarative programming style: Emphasis on What is the solution? instead of How to find the solution? Symbolic representation of knowledge (= axioms and derivation rules) Clear separation between data and the inference process, which is built into the execution environment Uniform representation for axioms and inference rules. Modular representation of knowledge The possibility to dinamically modify programs, by adding and removing axioms and rules in the executed program.
4 Logic Programming Execution Based on a restricted part of First-Order Predicate Logic Calculus = satisfiability of goals, using reduction to contradiction Inference rule = resolution with unification Control strategy during proof search backward chaining: from goal to axioms depth traversal of the derivation tree danger of infinite descent along a path which has no solution incomplete search strategy increased efficiency in the usage of memory space
5 Prolog Programs Program = collection of Horn clauses: A 1... A n A true B (rule) (axiom) Lack of explicit negations Closed word assumption = what can not be proved is assumed to be false In contrast the open word assumption considers that we can not say anything about things that can not be proved.
6 Proof steps (1) 1 Initialize the stack of goals with the initial goal (specified by the user) 2 Initialize the answer substitution (used during unification) to be empty (that is, without variable bindings) 3 Extract the goal from the top of the stack of goals and find the first program clause whose conclusion can be unified with the extracted goal 4 Extend the answer substitution with the bindings of the unifier from the previous step, and add the premises of the program clause to the stack of goals. 5 Go to step 3.
7 Proof steps (2) We say that the goal from the top of the stack of goals can not be satisfied if there is no program clause whose head unifies with it. When this happens, the proof process backtracks to the situation when the previous goal was on the top of the stack, and tries to find the next program clause whose head unifies with it. The proof search reports success when the stack of goals gets empty. failure when it can not satisfy the goal from the top of the stack of goals.
8 Example of Prolog program Genealogic example parent(allen,bob). parent(allen,bianca). parent(bob,chris). grandparent(x,y):-parent(x,z), parent(z,y). stands for the following set of Horn clauses: true parent(allen, bob). true parent(allen, bianca). true parent(bob, chris). X. Y. Z.(parent(X, Z) parent(z, Y ) grandparent(x, Y )).
9 Genealogic example: Proof steps Initial goal: Find X, Y such that X is grandparent of Y σ = G = {gp(x, Y )} gp(x 1, Y 1 ) : p(x 1, Z 1 ), p(z 1, Y 1 ) σ = {X X 1, Y Y 1 } G = {p(x 1, Z 1 ), p(z 1, Y 1 )} p(allen, bob) p(allen, bianca) p(bob, chris) σ = {X X 1, Y Y 1, X 1 allen, Z 1 bob} G = {p(bob, Y 1 )} σ = {X X 1, Y Y 1, X 1 allen, Z 1 bob, Y 1 chris} G = {} node(bob, chris) σ = {X X 1, Y Y 1, X 1 allen, Z 1 bianca} G = {p(bianca, Y 1 )} failure σ = {X X 1, Y Y 1, X 1 bob, Z 1 chris} G = {p(chris, Y 1 )} failure success
10 Genealogic example Remarks Order of evaluation = order of trying to prove goals depends on order of clauses in the program order of premises in the bodies of rules. Rule of thumb: place first the premises which are easier to satisfy and are more specific (e.g., those provable directly from axioms)
11 Proofs Control strategies Forward chaining (data-driven): All possible conclusions are proved starting from the initial data The process stops when we reach all goals Backward chaining (goal-driven): Based on using only the rules which can contribute effectively to the satisfaction of a goal: Detect the rules whose conclusion unifies with the goal Attempt to satisfy the premises of this rule, a.s.o.
12 Control strategies The backward chaining algorithm 1. BackwardChaining(rules, G, σ) rules : list of program rules, G : stack of goals, σ : the answer substitution returns the satisfiability of G 2. if G = then 3. return success 4. goal head(g) 5. G tail(g) 6. for-each rule rules do // in program order 7. if unify(goal, conclusion(rule), σ) σ 8. newg premises(rule) G // depth traversal 9. σ σ σ 10. if BackwardChaining(rules, newg, σ )=success 11. return success 12. return failure
13 Example Computing the minimum of two numbers Example Minimum of two numbers: Prolog code 1 min (X, Y, M) :- X =< Y, M is X. 2 min (X, Y, M) :- X > Y, M is Y. 3 4 min2 (X, Y, M) :- X =< Y, M = X. 5 min2 (X, Y, M) :- X > Y, M = Y. 6 7 % definition equivalent with min2 8 min3 (X, Y, X) :- X =< Y. 9 min3 (X, Y, Y) :- X > Y.
14 Computing the minimum of two numbers Usage After consult-ing the text file in which the program was stored:?- min(1+2,3+4,m). M = 3 ; false.?- min(3+4,1+2,m). M = 3.?-min2(1+2,3+4,M). M = 1+2 ; false.?- min2(3+4,1+2,m). M = 1+2.
15 Computing the minimum of two numbers Remarks The conditions X =< Y and X > Y are mutual exclusive How can we eliminate the redundancy? Attempt 1 min4 (X,Y,X) :- X =< Y. min4 (X,Y,Y). Then?- min4(1+2,3+4,m). M = 1+2 ; M = 3+4. the last answer is wrong
16 Computing the minimum of two numbers Improvement The search for a minimum can be interrupted (or cut) after detecting the first satisfiability of a goal. Example min5 (X,Y,X) :- X =< Y,!. min5 (X,Y,Y). Then?- min5(1+2,3+4,m). M = 1+2.
17 The cut operator (!) When encountered the first time it is satisfied. when encountered the second time (during backtracking) failure and blocking of all further attempts to satisfy the goal which was unified with the rule whose body contained the cut operator. Extremely useful to make the execution of logic programs more efficient.
18 The cut operator (!) Example Consider the following Prolog program: 1 girl(mary). 2 girl(ann). 3 4 boy(john). 5 boy(bill). 6 7 pair (X, Y) :- girl (X), boy(y). 8 pair (bella, harry) pair2(x, Y) :- girl (X),!, boy(y). 11 pair2(bella, harry).
19 The cut operator (!) Usage?- pair(x, Y).?- pair2(x, Y). X = mary, X = mary. Y = john ; Y = john ; X = mary, X = mary, Y = bill ; Y = bill. X = ann, Y = john ; X = ann, Y = bill ; X = bella, Y = harry.
20 Negation as failure nott (P) :- P,!, fail. nott (P). Assume P is an atom, for example boy(john) If P is satisfiable: The application of the first rule fails, because of fail in its body. The application of the second Horn clause (which is a fact) is abandoned, because! was encountered before. Outcome: nott(p) is unsatisfiable. If P is unsatisfiable: The application first rule fails before reaching! in its body. The application of the rule succeeds. Outcome: nott(p) is satisfiable.
21 Summary In this lecture we learned The structure of Prolog programs The functionality of a proof The order of evaluation The control algorithm of the proof search Techniques to control the program execution
Logic. Knowledge Representation & Reasoning Mechanisms. Logic. Propositional Logic Predicate Logic (predicate Calculus) Automated Reasoning
Logic Knowledge Representation & Reasoning Mechanisms Logic Logic as KR Propositional Logic Predicate Logic (predicate Calculus) Automated Reasoning Logical inferences Resolution and Theorem-proving Logic
More informationFoundations of Logic Programming
Foundations of Logic Programming Deductive Logic e.g. of use: Gypsy specifications and proofs About deductive logic (Gödel, 1931) Interesting systems (with a finite number of axioms) are necessarily either:
More informationPropositional 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)
More informationArtificial Intelligence Chapter 7: Logical Agents
Artificial Intelligence Chapter 7: Logical Agents Michael Scherger Department of Computer Science Kent State University February 20, 2006 AI: Chapter 7: Logical Agents 1 Contents Knowledge Based Agents
More informationPropositional Resolution
Computational Logic Lecture 4 Propositional Resolution Michael Genesereth Spring 2005 Stanford University Modified by Charles Ling and TA, for CS2209 Use with permission Propositional Resolution Propositional
More informationPropositional Logic: Methods of Proof. Chapter 7, Part II
Propositional Logic: Methods of Proof Chapter 7, Part II Inference in Formal Symbol Systems: Ontology, Representation, ti Inference Formal Symbol Systems Symbols correspond to things/ideas in the world
More informationFirst-Order Theorem Proving and Vampire. Laura Kovács (Chalmers University of Technology) Andrei Voronkov (The University of Manchester)
First-Order Theorem Proving and Vampire Laura Kovács (Chalmers University of Technology) Andrei Voronkov (The University of Manchester) Outline Introduction First-Order Logic and TPTP Inference Systems
More informationBrief Introduction to Prolog
Brief to Prolog Joana Côrte-Real jcr@dcc.fc.up.pt CRACS & INESC TEC Faculty of Sciences University of Porto University of Aizu 5th December 2014 1 / 27 Overview 1 to Prolog Prolog Syntax Tutorial 1 2 Lists
More informationStrong AI vs. Weak AI Automated Reasoning
Strong AI vs. Weak AI Automated Reasoning George F Luger ARTIFICIAL INTELLIGENCE 6th edition Structures and Strategies for Complex Problem Solving Artificial intelligence can be classified into two categories:
More informationProlog and Logic Programming. CS152 Chris Pollett Dec. 3, 2008.
Prolog and Logic Programming CS152 Chris Pollett Dec. 3, 2008. Outline Logic and Logic Programs Horn Clauses Resolution and Unification Prolog Introduction So far this semester we have considered three
More informationKnowledge base (KB) = set of sentences in a formal language Declarative approach to building an agent (or other system):
Logic Knowledge-based agents Inference engine Knowledge base Domain-independent algorithms Domain-specific content Knowledge base (KB) = set of sentences in a formal language Declarative approach to building
More informationLogic (3A) Young W. Lim 12/5/13
Copyright (c) 2013. 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 later version published by the Free Software
More informationInference in first-order logic
CS 2710 Foundations of AI Lecture 15 Inference in first-order logic Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Logical inference in FOL Logical inference problem: Given a knowledge base KB
More informationInference Methods In Propositional Logic
Lecture Notes, Artificial Intelligence ((ENCS434)) University of Birzeit 1 st Semester, 2011 Artificial Intelligence (ENCS434) Inference Methods In Propositional Logic Dr. Mustafa Jarrar University of
More informationChapter 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
More informationCS 771 Artificial Intelligence. Propositional Logic
CS 771 Artificial Intelligence Propositional Logic Why Do We Need Logic? Problem-solving agents were very inflexible hard code every possible state E.g., in the transition of 8-puzzle problem, knowledge
More informationDeductive 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
More informationAdvanced 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
More informationPropositional and Predicate Logic - V
Propositional and Predicate Logic - V Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - V WS 2016/2017 1 / 21 Formal proof systems Hilbert s calculus
More informationInference Methods In Propositional Logic
Lecture Notes, Advanced Artificial Intelligence (SCOM7341) Sina Institute, University of Birzeit 2 nd Semester, 2012 Advanced Artificial Intelligence (SCOM7341) Inference Methods In Propositional Logic
More informationIntelligent Agents. Pınar Yolum Utrecht University
Intelligent Agents Pınar Yolum p.yolum@uu.nl Utrecht University Logical Agents (Based mostly on the course slides from http://aima.cs.berkeley.edu/) Outline Knowledge-based agents Wumpus world Logic in
More informationEE562 ARTIFICIAL INTELLIGENCE FOR ENGINEERS
EE562 ARTIFICIAL INTELLIGENCE FOR ENGINEERS Lecture 10, 5/9/2005 University of Washington, Department of Electrical Engineering Spring 2005 Instructor: Professor Jeff A. Bilmes Logical Agents Chapter 7
More informationPropositional 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
More informationLogical Agents. Chapter 7
Logical Agents Chapter 7 Outline Knowledge-based agents Wumpus world Logic in general - models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability Inference rules and theorem
More informationFirst-Order Theorem Proving and Vampire
First-Order Theorem Proving and Vampire Laura Kovács 1,2 and Martin Suda 2 1 TU Wien 2 Chalmers Outline Introduction First-Order Logic and TPTP Inference Systems Saturation Algorithms Redundancy Elimination
More informationAn Early (1971) Conversation. Logic Programming (PLP 11.3) Horn Clauses Introduction to Prolog: Resolution, Unification.
Logic Programming (PLP 11.3) Horn Clauses Introduction to Prolog: Resolution, Unification Carlos Varela Rennselaer Polytechnic Institute November 9, 2006 C. Varela 1 An Early (1971) Conversation Cats kill
More informationProlog. Resolution (for beginners?) Resolution (cont.) Resolution (cont.) Unification. Resolution (still cont.) Resolution & Unification
Resolution & Unification Prolog Computer Science Resolution (for beginners?) Resolution is a theorem proving method developed by Robinson based on representing logical formulas as clauses Clauses are build
More informationInf2D 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
More informationPropositional 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
More informationLogical agents. Chapter 7. Chapter 7 1
Logical agents Chapter 7 Chapter 7 1 Outline Knowledge-based agents Logic in general models and entailment Propositional (oolean) logic Equivalence, validity, satisfiability Inference rules and theorem
More informationOutline. Logical Agents. Logical Reasoning. Knowledge Representation. Logical reasoning Propositional Logic Wumpus World Inference
Outline Logical Agents ECE57 Applied Artificial Intelligence Spring 007 Lecture #6 Logical reasoning Propositional Logic Wumpus World Inference Russell & Norvig, chapter 7 ECE57 Applied Artificial Intelligence
More informationLecture 10: Even more on predicate logic" Prerequisites for lifted inference: Skolemization and Unification" Inference in predicate logic"
CS440/ECE448: Intro to Artificial Intelligence Lecture 10: Even more on predicate logic Prof. Julia Hockenmaier juliahmr@illinois.edu http://cs.illinois.edu/fa11/cs440 Inference in predicate logic All
More informationLogical Inference 2 rule based reasoning
Logical Inference 2 rule based reasoning Chapter 9 Some material adopted from notes by Andreas Geyer-Schulz,, Chuck Dyer, and Mary Getoor Automated inference for FOL Automated inference for FOL is harder
More information7.5.2 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
More informationPropositional inference, propositional agents
ropositional inference, propositional agents Chapter 7.5 7.7 Chapter 7.5 7.7 1 Outline Inference rules and theorem proving forward chaining backward chaining resolution Efficient model checking algorithms
More informationCS 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
More informationLogical Consequences of Formulae. Definite Clauses
Motivation Logical Consequences of Formulae Recall: F is a logical consequence of P (i.e. P = F ) iff Every model of P is also a model of F. Since there are (in general) infinitely many possible interpretations,
More informationClass Assignment Strategies
Class Assignment Strategies ì Team- A'ack: Team a'ack ì Individualis2c: Search for possible ì Poli2cal: look at others and make decision based on who is winning, who is loosing, and conversa;on ì Emo2on
More informationCSC384: Intro to Artificial Intelligence Knowledge Representation II. Required Readings: 9.1, 9.2, and 9.5 Announcements:
CSC384: Intro to Artificial Intelligence Knowledge Representation II Required Readings: 9.1, 9.2, and 9.5 Announcements: 1 Models Examples. Environment A Language (Syntax) Constants: a,b,c,e Functions:
More informationTop-Down Execution CS240B Notes
Top-Down Execution CS240B Notes Notes based on Section 9.4 of Advanced Database Systems Morgan Kaufmann, 1997 C. Zaniolo, March 2002 p.1/22 Top-Down Execution of Datalog A strict bottom-up execution strategy
More information7. Propositional Logic. Wolfram Burgard and Bernhard Nebel
Foundations of AI 7. Propositional Logic Rational Thinking, Logic, Resolution Wolfram Burgard and Bernhard Nebel Contents Agents that think rationally The wumpus world Propositional logic: syntax and semantics
More informationLecture Notes on Logic Programming
Lecture Notes on Logic Programming 15-317: Constructive Logic Frank Pfenning Lecture 13 October 13, 2009 1 Computation vs Deduction Logic programming is a particular way to approach programming Other paradigms
More informationITEC Prolog Programming
1 ITEC 198 - Prolog Programming Dr. Maung M. Htay Department of Information Technology 1/18/16 Text Prolog Programming for Artificial Intelligence by Ivan Bratko Third Edition, Addison Wesley 2 References
More informationLogical Inference. Artificial Intelligence. Topic 12. Reading: Russell and Norvig, Chapter 7, Section 5
rtificial Intelligence Topic 12 Logical Inference Reading: Russell and Norvig, Chapter 7, Section 5 c Cara MacNish. Includes material c S. Russell & P. Norvig 1995,2003 with permission. CITS4211 Logical
More informationINF5390 Kunstig intelligens. Logical Agents. Roar Fjellheim
INF5390 Kunstig intelligens Logical Agents Roar Fjellheim Outline Knowledge-based agents The Wumpus world Knowledge representation Logical reasoning Propositional logic Wumpus agent Summary AIMA Chapter
More informationCS 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,
More informationArtificial 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 informationLogical Agents (I) Instructor: Tsung-Che Chiang
Logical Agents (I) Instructor: Tsung-Che Chiang tcchiang@ieee.org Department of Computer Science and Information Engineering National Taiwan Normal University Artificial Intelligence, Spring, 2010 編譯有誤
More informationOutline. Logical Agents. Logical Reasoning. Knowledge Representation. Logical reasoning Propositional Logic Wumpus World Inference
Outline Logical Agents ECE57 Applied Artificial Intelligence Spring 008 Lecture #6 Logical reasoning Propositional Logic Wumpus World Inference Russell & Norvig, chapter 7 ECE57 Applied Artificial Intelligence
More informationCS:4420 Artificial Intelligence
CS:4420 Artificial Intelligence Spring 2018 Propositional Logic Cesare Tinelli The University of Iowa Copyright 2004 18, Cesare Tinelli and Stuart Russell a a These notes were originally developed by Stuart
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Wolfram Burgard, Maren Bennewitz, and Marco Ragni Albert-Ludwigs-Universität Freiburg Contents 1 Agents
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Joschka Boedecker and Wolfram Burgard and Bernhard Nebel Albert-Ludwigs-Universität Freiburg May 17, 2016
More informationTitle: Logical Agents AIMA: Chapter 7 (Sections 7.4 and 7.5)
B.Y. Choueiry 1 Instructor s notes #12 Title: Logical Agents AIMA: Chapter 7 (Sections 7.4 and 7.5) Introduction to Artificial Intelligence CSCE 476-876, Fall 2018 URL: www.cse.unl.edu/ choueiry/f18-476-876
More informationResolution (14A) Young W. Lim 8/15/14
Resolution (14A) Young W. Lim Copyright (c) 2013-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationMathematical Logic Prof. Arindama Singh Department of Mathematics Indian Institute of Technology, Madras. Lecture - 15 Propositional Calculus (PC)
Mathematical Logic Prof. Arindama Singh Department of Mathematics Indian Institute of Technology, Madras Lecture - 15 Propositional Calculus (PC) So, now if you look back, you can see that there are three
More informationPropositional Logic: Review
Propositional Logic: Review Propositional logic Logical constants: true, false Propositional symbols: P, Q, S,... (atomic sentences) Wrapping parentheses: ( ) Sentences are combined by connectives:...and...or
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 informationWarm-Up Problem. Is the following true or false? 1/35
Warm-Up Problem Is the following true or false? 1/35 Propositional Logic: Resolution Carmen Bruni Lecture 6 Based on work by J Buss, A Gao, L Kari, A Lubiw, B Bonakdarpour, D Maftuleac, C Roberts, R Trefler,
More informationRevised by Hankui Zhuo, March 21, Logical agents. Chapter 7. Chapter 7 1
Revised by Hankui Zhuo, March, 08 Logical agents Chapter 7 Chapter 7 Outline Wumpus world Logic in general models and entailment Propositional (oolean) logic Equivalence, validity, satisfiability Inference
More informationDeliberative 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/
More informationLogical agents. Chapter 7. Chapter 7 1
Logical agents Chapter 7 Chapter 7 Outline Knowledge-based agents Wumpus world Logic in general models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability Inference rules
More information1 FUNDAMENTALS OF LOGIC NO.10 HERBRAND THEOREM Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ 2 So Far Propositional Logic Logical connectives (,,, ) Truth table Tautology
More informationClosed Book Examination. Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. M.Sc. in Advanced Computer Science
Closed Book Examination COMP60121 Appendix: definition sheet (3 pages) Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE M.Sc. in Advanced Computer Science Automated Reasoning Tuesday 27 th
More informationPropositional 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
More informationPropositional 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
More informationProof Rules for Correctness Triples
Proof Rules for Correctness Triples CS 536: Science of Programming, Fall 2018 A. Why? We can t generally prove that correctness triples are valid using truth tables. We need proof axioms for atomic statements
More informationLogic in AI Chapter 7. Mausam (Based on slides of Dan Weld, Stuart Russell, Subbarao Kambhampati, Dieter Fox, Henry Kautz )
Logic in AI Chapter 7 Mausam (Based on slides of Dan Weld, Stuart Russell, Subbarao Kambhampati, Dieter Fox, Henry Kautz ) 2 Knowledge Representation represent knowledge about the world in a manner that
More informationLogic (3A) Young W. Lim 11/2/13
Copyright (c) 2013. 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 later version published by the Free Software
More informationConvert to clause form:
Convert to clause form: Convert the following statement to clause form: x[b(x) ( y [ Q(x,y) P(y) ] y [ Q(x,y) Q(y,x) ] y [ B(y) E(x,y)] ) ] 1- Eliminate the implication ( ) E1 E2 = E1 E2 x[ B(x) ( y [
More informationLogical Agents. Chapter 7
Logical Agents Chapter 7 Outline Knowledge-based agents Wumpus world Logic in general - models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability Inference rules and theorem
More informationTDT4136 Logic and Reasoning Systems
TDT436 Logic and Reasoning Systems Chapter 7 - Logic gents Lester Solbakken solbakke@idi.ntnu.no Norwegian University of Science and Technology 06.09.0 Lester Solbakken TDT436 Logic and Reasoning Systems
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 8. Satisfiability and Model Construction Davis-Putnam-Logemann-Loveland Procedure, Phase Transitions, GSAT Joschka Boedecker and Wolfram Burgard and Bernhard Nebel
More informationCOMP219: 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
More informationLogic (3A) Young W. Lim 10/31/13
Copyright (c) 2013. 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 later version published by the Free Software
More informationLogic (3A) Young W. Lim 10/29/13
Copyright (c) 2013. 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 later version published by the Free Software
More informationAdam Blank Spring 2017 CSE 311. Foundations of Computing I
Adam Blank Spring 2017 CSE 311 Foundations of Computing I Pre-Lecture Problem Suppose that p, and p (q r) are true. Is q true? Can you prove it with equivalences? CSE 311: Foundations of Computing Lecture
More informationCogSysI Lecture 8: Automated Theorem Proving
CogSysI Lecture 8: Automated Theorem Proving Intelligent Agents WS 2004/2005 Part II: Inference and Learning Automated Theorem Proving CogSysI Lecture 8: Automated Theorem Proving p. 200 Remember......
More informationLogical reasoning - Can we formalise our thought processes?
Logical reasoning - Can we formalise our thought processes? Why study (mathematical) logic? Logic: the science or method of reasoning Logical: able to reason correctly Human reasoning poorly understood,
More informationBachelor/Master Exam Version V3B
Prof aadr Jürgen Giesl Carsten Fuhs, Carsten Otto, Thomas Ströder Bachelor/Master Exam Version V3B First Name: Last Name: Immatriculation Number: Course of Studies (please mark exactly one): Informatik
More informationDefinite Logic Programs: Derivation and Proof Trees. CSE 505 Computing with Logic Stony Brook University
Definite Logic Programs: Derivation and Proof Trees CSE 505 Computing with Logic Stony Brook University http://www.cs.stonybrook.edu/~cse505 1 Refutation in Predicate Logic parent(pam, bob). parent(tom,
More information4 Derivations in the Propositional Calculus
4 Derivations in the Propositional Calculus 1. Arguments Expressed in the Propositional Calculus We have seen that we can symbolize a wide variety of statement forms using formulas of the propositional
More informationArtificial Intelligence Knowledge Representation I
rtificial Intelligence Knowledge Representation I Lecture 6 Issues in Knowledge Representation 1. How to represent knowledge 2. How to manipulate/process knowledge (2) Can be rephrased as: how to make
More informationCOMP219: Artificial Intelligence. Lecture 20: Propositional Reasoning
COMP219: Artificial Intelligence Lecture 20: Propositional Reasoning 1 Overview Last time Logic for KR in general; Propositional Logic; Natural Deduction Today Entailment, satisfiability and validity Normal
More informationLecture Notes on Logic Programming
Lecture Notes on Logic Programming 15-317: Constructive Logic Frank Pfenning Lecture 13 October 2, 2015 1 Computation vs Deduction The previous lectures explored a connection between logic and computation
More informationLogical Agents. Outline
ogical gents Chapter 6, Ie Chapter 7 Outline Knowledge-based agents Wumpus world ogic in general models and entailment ropositional (oolean) logic Equivalence, validity, satisfiability Inference rules
More informationArtificial Intelligence Chapter 9: Inference in First-Order Logic
Artificial Intelligence Chapter 9: Inference in First-Order Logic Andreas Zell After the Textbook: Artificial Intelligence, A Modern Approach by Stuart Russell and Peter Norvig (3 rd Edition) 9 Inference
More informationProof Methods for Propositional Logic
Proof Methods for Propositional Logic Logical equivalence Two sentences are logically equivalent iff they are true in the same models: α ß iff α β and β α Russell and Norvig Chapter 7 CS440 Fall 2015 1
More informationDiscrete Mathematics
Discrete Mathematics Jeremy Siek Spring 2010 Jeremy Siek Discrete Mathematics 1 / 24 Outline of Lecture 3 1. Proofs and Isabelle 2. Proof Strategy, Forward and Backwards Reasoning 3. Making Mistakes Jeremy
More informationNICTA Advanced Course. Theorem Proving Principles, Techniques, Applications
NICTA Advanced Course Theorem Proving Principles, Techniques, Applications λ 1 CONTENT Intro & motivation, getting started with Isabelle Foundations & Principles Lambda Calculus Higher Order Logic, natural
More informationPropositional 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
More informationUnification and occur check. The programming language Prolog
Resolution and Logic Programming Ground resolution Unification and occur check General Resolution Logic Programming SLD-resolution The programming language Prolog Syntax Arithmetic Lists Slide 1 Motivation
More informationLogical Agents: Propositional Logic. Chapter 7
Logical Agents: Propositional Logic Chapter 7 Outline Topics: Knowledge-based agents Example domain: The Wumpus World Logic in general models and entailment Propositional (Boolean) logic Equivalence, validity,
More informationInference 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
More informationLogic Programming (PLP 11) Predicate Calculus, Horn Clauses, Clocksin-Mellish Procedure
Logic Programming (PLP 11) Predicate Calculus, Horn Clauses, Clocksin-Mellish Procedure Carlos Varela Rennselaer Polytechnic Institute November 7, 2016 C. Varela 1 An Early (1971) Conversation USER: Cats
More informationProof strategies, or, a manual of logical style
Proof strategies, or, a manual of logical style Dr Holmes September 27, 2017 This is yet another version of the manual of logical style I have been working on for many years This semester, instead of posting
More informationComputational Logic Automated Deduction Fundamentals
Computational Logic Automated Deduction Fundamentals 1 Elements of First-Order Predicate Logic First Order Language: An alphabet consists of the following classes of symbols: 1. variables denoted by X,
More informationCSC242: Intro to AI. Lecture 11. Tuesday, February 26, 13
CSC242: Intro to AI Lecture 11 Propositional Inference Propositional Inference Factored Representation Splits a state into variables (factors, attributes, features, things you know ) that can have values
More informationArtificial Intelligence
Artificial Intelligence Propositional Logic Marc Toussaint University of Stuttgart Winter 2015/16 (slides based on Stuart Russell s AI course) Outline Knowledge-based agents Wumpus world Logic in general
More informationPropositional 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
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 31. Propositional Logic: DPLL Algorithm Malte Helmert and Gabriele Röger University of Basel April 24, 2017 Propositional Logic: Overview Chapter overview: propositional
More information