Theorem Proving Semantic Tableaux CIS548 November 15, 2006.

Slides:



Advertisements
Similar presentations
Propositional Equivalences
Advertisements

Exercises for CS1512 Weeks 7 and 8 Propositional Logic 1 (questions)
Exercises for CS1512 Weeks 7 and 8 Propositional Logic 1 (questions + solutions)
Logic & Critical Reasoning
The Logic of Quantified Statements
Chapter 2 Fundamentals of Logic Dept of Information management National Central University Yen-Liang Chen.
For Friday No reading Homework: –Chapter 9, exercise 4 (This is VERY short – do it while you’re running your tests) Make sure you keep variables and constants.
Propositions and Connectives Conditionals and Bi-conditionals Quantifiers.
Formal Logic Proof Methods Direct Proof / Natural Deduction Conditional Proof (Implication Introduction) Reductio ad Absurdum Resolution Refutation.
Computability and Complexity 8-1 Computability and Complexity Andrei Bulatov Logic Reminder.
Logic 3 Tautological Implications and Tautological Equivalences
Propositional Logic. Negation Given a proposition p, negation of p is the ‘not’ of p.
Start with atomic sentences in the KB and apply Modus Ponens, adding new atomic sentences, until “done”.
Adapted from Discrete Math
Predicates and Quantifiers
Discrete Mathematics Goals of a Discrete Mathematics Learn how to think mathematically 1. Mathematical Reasoning Foundation for discussions of methods.
CSCI 115 Chapter 2 Logic. CSCI 115 §2.1 Propositions and Logical Operations.
Mathematical Structures A collection of objects with operations defined on them and the accompanying properties form a mathematical structure or system.
Conjunctive normal form: any formula of the predicate calculus can be transformed into a conjunctive normal form. Def. A formula is said to be in conjunctive.
Chap. 2 Fundamentals of Logic. Proposition Proposition (or statement): an declarative sentence that is either true or false, but not both. e.g. –Margret.
Chapter 5 – Logic CSNB 143 Discrete Mathematical Structures.
Fall 2002CMSC Discrete Structures1 Let’s get started with... Logic !
Theory and Applications
2.3Logical Implication: Rules of Inference From the notion of a valid argument, we begin a formal study of what we shall mean by an argument and when such.
CS344: Introduction to Artificial Intelligence Lecture: Herbrand’s Theorem Proving satisfiability of logic formulae using semantic trees (from Symbolic.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.1, Slide Logic The Study of What’s True or False or Somewhere in Between.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. Section 3.1, Slide Logic The Study of What’s True or False or Somewhere in Between.
Chapter Three Truth Tables 1. Computing Truth-Values We can use truth tables to determine the truth-value of any compound sentence containing one of.
CSNB143 – Discrete Structure LOGIC. Learning Outcomes Student should be able to know what is it means by statement. Students should be able to identify.
Lecture Propositional Equivalences. Compound Propositions Compound propositions are made by combining existing propositions using logical operators.
Fundamentals of Logic 1. What is a valid argument or proof? 2. Study system of logic 3. In proving theorems or solving problems, creativity and insight.
Propositional calculus
Chapter 2 Logic 2.1 Statements 2.2 The Negation of a Statement 2.3 The Disjunction and Conjunction of Statements 2.4 The Implication 2.5 More on Implications.
Chapter 1: The Foundations: Logic and Proofs
C. Varela1 Logic Programming (PLP 11) Predicate Calculus, Horn Clauses, Clocksin-Mellish Procedure Carlos Varela Rennselaer Polytechnic Institute November.
COMP 170 L2 L08: Quantifiers. COMP 170 L2 Outline l Quantifiers: Motivation and Concepts l Quantifiers: Notations and Meaning l Saying things with Quantified.
For Wednesday Read chapter 9, sections 1-3 Homework: –Chapter 7, exercises 8 and 9.
Propositional Logic Predicate Logic
CSNB143 – Discrete Structure Topic 4 – Logic. Learning Outcomes Students should be able to define statement. Students should be able to identify connectives.
LOGIC. Logic in general  Logics are formal languages for representing information such that conclusions can be drawn  Syntax defines the sentences in.
Chapter 2 Fundamentals of Logic 1. What is a valid argument or proof?
1 Section 6.2 Propositional Calculus Propositional calculus is the language of propositions (statements that are true or false). We represent propositions.
3/6/20161 Let’s get started with... Logic !. 3/6/20162 Logic Crucial for mathematical reasoningCrucial for mathematical reasoning Used for designing electronic.
Chapter Ten Relational Predicate Logic. 1. Relational Predicates We now broaden our coverage of predicate logic to include relational predicates. This.
Section 1.4. Propositional Functions Propositional functions become propositions (and have truth values) when their variables are each replaced by a value.
Propositional Logic. Assignment Write any five rules each from two games which you like by using propositional logic notations.
Discrete Math by R.S. Chang, Dept. CSIE, NDHU1 Fundamentals of Logic 1. What is a valid argument or proof? 2. Study system of logic 3. In proving theorems.
CS-7081 Application - 1. CS-7082 Example - 2 CS-7083 Simplifying a Statement – 3.
 Conjunctive Normal Form: A logic form must satisfy one of the following conditions 1) It must be a single variable (A) 2) It must be the negation of.
CENG 424-Logic for CS Introduction Based on the Lecture Notes of Konstantin Korovin, Valentin Goranko, Russel and Norvig, and Michael Genesereth.
Chapter 1 – Part 3 ELEMENTARY LOGIC
2. The Logic of Compound Statements Summary
3. The Logic of Quantified Statements Summary
DISCRETE MATHEMATICS CHAPTER I.
CSNB 143 Discrete Mathematical Structures
2.1 Propositions and Logical Operations
CMSC Discrete Structures
Chapter 8 Logic Topics
Propositional Logic and Methods of Inference
CS 1502 Formal Methods in Computer Science
Chapter 1 Logic and Proof.
Carlos Varela Rensselaer Polytechnic Institute November 10, 2017
Computer Security: Art and Science, 2nd Edition
Discrete Mathematics CMP-200 Propositional Equivalences, Predicates & Quantifiers, Negating Quantified Statements Abdul Hameed
Statements of Symbolic Logic
Predicates and Quantifiers
1.2 Propositional Equivalences
1.3 Propositional Equivalences
Carlos Varela Rennselaer Polytechnic Institute August 30, 2007
Presentation transcript:

Theorem Proving Semantic Tableaux CIS548 November 15, 2006

2006Kutztown University2 Review of Symbolic Logic All sentences fall in one of 3 categories All sentences fall in one of 3 categories  Tautology – true under all truth values of variables  Inconsistency – false under all truth values  Contingency – true under some; false under others Notation Notation  Logical connectives » ^ or & :: and » v :: or » ~ or ¬ :: not »→ :: implication » ↔ :: biconditional  Quantifiers (standard symbols unavailable in ppt) » Ă :: universal quantifier » Ë :: existential quantifier

2006Kutztown University3 Review of Symbolic Logic (cont.) Implication can be replaced Implication can be replaced  P → Q ≡ ~P v Q  Useful for tableaux method Other substitutions that can be made Other substitutions that can be made  p v q ≡ ~(~p ^ ~q) {eliminates v}  p ^ q ≡ ~(~p v ~q) {eliminates ^}  Sheffer stroke – the ultimate (or extreme) » neither-nor ≡ nand {eliminates all connectives} »Ref: » Ref: » Ref:  Not immediately useful for tableaux

2006Kutztown University4 The Tableaux Method Start with negation of sentence Start with negation of sentence  E.g., to prove P v ~P » start with ~ (P v ~P) » let that be the root of a tree Simplify Simplify  Choose any sentence on the branch  Use rules of logic to break it apart  Place pieces onto tree  If atom and its negation appear on a branch, close the branch  If branch has only atoms or negations of atoms » Stop; branch cannot be closed » Sentence is not a tautology

2006Kutztown University5 The Tableaux Method Propositional Logic Rules Action determined by main connective Action determined by main connective  p ^ q :: p & q on the same branch  p v q :: p & q on two separate branches  ~(p ^ q) :: ~p & ~q on two branches  ~(p v q) :: ~p & ~q on the same branch  p → q :: ~p & q on two branches  ~(p → q) :: p & ~q on same branch  p ↔ q :: p → q and q → p on same branch  ~(p ↔ q) :: p ^ ~q and q ^ ~p on two branches

2006Kutztown University6 The Tableaux Method Propositional Logic Example ((p → q) ^ (q → r)) → (p → r) ((p → q) ^ (q → r)) → (p → r)  Place ~((p → q) ^ (q → r)) → (p → r)) at root of tree.  Rule 6 :: ((p → q) ^ (q → r)) & ~(p → r) on branch 1.  Rule 6 :: p & ~r on branch 1.  Rule 1 :: (p → q) & (q → r) on branch 1.  Rule 5 :: ~p on branch 1a; q branch 1b.  Close off branch 1a.  Rule 5 :: ~q on branch 1b1; r branch 1b2.  Close off branch 1b1.  Close off branch 1b2.  All branches are closed  a tautology.

2006Kutztown University7 The Tableaux Method Quantificational Logic Rules To come... To come...