Resolution Refutation Formal Aspects of Computer Science - Week 9 Resolution Refutation, Relation with Prolog Lee McCluskey, room 2/07

Slides:

Advertisements

Similar presentations

CS4026 Formal Models of Computation Part II The Logic Model Lecture 1 – Programming in Logic.

Advertisements

After the Practical on Clausal Form and Resolution Question 5 Proving the query of Question 4.

Page 1 Nov 09, 2004CAS 701, McMaster Logic Programming and Prolog Presenter: John Xu Date: Nov 09, 2004.

Inference Rules Universal Instantiation Existential Generalization

SLD-resolution Introduction Most general unifiers SLD-resolution

Standard Logical Equivalences

First-Order Logic (FOL) aka. predicate calculus. First-Order Logic (FOL) Syntax User defines these primitives: – Constant symbols (i.e., the "individuals"

Inference in first-order logic Chapter 9. Outline Reducing first-order inference to propositional inference Unification Generalized Modus Ponens Forward.

UIUC CS 497: Section EA Lecture #2 Reasoning in Artificial Intelligence Professor: Eyal Amir Spring Semester 2004.

Automated Reasoning Systems For first order Predicate Logic.

First Order Logic Resolution

Methods of Proof Chapter 7, second half.. Proof methods Proof methods divide into (roughly) two kinds: Application of inference rules: Legitimate (sound)

13 Automated Reasoning 13.0 Introduction to Weak Methods in Theorem Proving 13.1 The General Problem Solver and Difference Tables 13.2 Resolution.

Conflict Resolution  what to do if there is more than 1 matching rule in each inference cycle? match WM with LHS of rules select one rule (conflict resolution)

Artificial Intelligence Chapter 14. Resolution in the Propositional Calculus Artificial Intelligence Chapter 14. Resolution in the Propositional Calculus.

1 Applied Computer Science II Resolution in FOL Luc De Raedt.

Formal Aspects of Computer Science - Week 5 Logic and Reasoning Lee McCluskey, room 2/07

Resolution Refutation Formal Aspects of Computer Science - Week 10 An Automated Theorem Prover Lee McCluskey, room 2/07

The Semantic Web – WEEK 7: Logic, Reasoning and Proof The “Layer Cake” Model – [From Rector & Horrocks Semantic Web cuurse] You are here!

AI - Week 16 Logic and Reasoning in AI: Resolution Refutation Lee McCluskey, room 2/07

1 Automated Reasoning Introduction to Weak Methods in Theorem Proving 13.1The General Problem Solver and Difference Tables 13.2Resolution Theorem.

AI - Week 15 Logic and Reasoning in AI Lee McCluskey, room 2/07

Inference and Resolution for Problem Solving

AI - Week 24 Uncertain Reasoning (quick mention) then REVISION Lee McCluskey, room 2/07

AI – Week 8 AI + 2 Player Games Lee McCluskey, room 3/10

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering A Hole in Goal Trees Notes for: D.W. Loveland and M. Stickel. “A Hole in Goal.

The Semantic Web Week 12 Term 1 Recap Lee McCluskey, room 2/07 Department of Computing And Mathematical Sciences Module Website:

Artificial Intelligence

Formal Aspects of Computer Science – Week 12 RECAP Lee McCluskey, room 2/07

CHA2555 Week2: Knowledge Representation, Logic and Prolog Lee McCluskey First term:

Artificial Intelligence Chapter 14 Resolution in the Propositional Calculus Artificial Intelligence Chapter 14 Resolution in the Propositional Calculus.

Automated Reasoning ARTIFICIAL INTELLIGENCE 6th edition George F Luger

The Semantic Web – WEEK 8: Proof in FOL continued The “Layer Cake” Model – [From Rector & Horrocks Semantic Web cuurse] You are here!

The Semantic Web – WEEK 8: An automatic Proof technique.. The “Layer Cake” Model – [From Rector & Horrocks Semantic Web cuurse] You are here!

School of Computing and Engineering, University of Huddersfield Formal Aspects of Computer Science - CIA 2326 Lee McCluskey, room 2/07

Resolution Refutation Formal Aspects of Computer Science - Week 8 The Law of Resolution and the Resolution Procedure Lee McCluskey, room 2/07

0 1 Todays Topics Resolution – top down and bottom up j-DREW BU procedure Subsumption – change to procedure Infinite Loops RuleML input – Prolog output.

CS1502 Formal Methods in Computer Science Lecture Notes 10 Resolution and Horn Sentences.

Notes for Chapter 12 Logic Programming The AI War Basic Concepts of Logic Programming Prolog Review questions.

Proof Systems KB |- Q iff there is a sequence of wffs D1,..., Dn such that Dn is Q and for each Di in the sequence: a) either Di is in KB or b) Di can.

CSNB234 ARTIFICIAL INTELLIGENCE

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.

1 Chapter 8 Inference and Resolution for Problem Solving.

1CS3754 Class Notes 14B, John Shieh, Figure 2.5: Further steps in the unification of (parents X (father X) (mother bill)) and (parents bill.

Unification Algorithm Input: a finite set Σ of simple expressions Output: a mgu for Σ (if Σ is unifiable) 1. Set k = 0 and  0 = . 2. If Σ  k is a singleton,

CIS2326 Week2: Logic and Prolog Lee McCluskey First term:

The AI War LISP and Prolog Basic Concepts of Logic Programming

Automated Reasoning Early AI explored how to automated several reasoning tasks – these were solved by what we might call weak problem solving methods as.

Kansas State University Department of Computing and Information Sciences CIS 730: Introduction to Artificial Intelligence Lecture 23 Friday, 17 October.

Automated Reasoning Systems For first order Predicate Logic.

Computing & Information Sciences Kansas State University Lecture 15 of 42 CIS 530 / 730 Artificial Intelligence Lecture 15 of 42 William H. Hsu Department.

The Law of Resolution Formal Aspects of Computer Science - Week 7 The Law of Resolution Lee McCluskey, room 2/07

CHA2555 Week2 Practical: Lee McCluskey First term:

1 Knowledge Based Systems (CM0377) Lecture 6 (last modified 20th February 2002)

Inference in First Order Logic. Outline Reducing first order inference to propositional inference Unification Generalized Modus Ponens Forward and backward.

Answer Extraction To use resolution to answer questions, for example a query of the form  X C(X), we must keep track of the substitutions made during.

1 Section 9.2 Logic Programming A logic program is a set of clauses with the restriction that there is exactly one positive literal in each clause. Such.

Proof Methods for Propositional Logic CIS 391 – Intro to Artificial Intelligence.

Copyright 1999Paul F. Reynolds, Jr. Foundations of Logic Programming.

6/11/2016 Linear Resolution and Introduction to First Order Logic Michael Leuschel Softwaretechnik und Programmiersprachen Lecture 5.

Logic Programming Lecture 2: Unification and proof search.

Resolution in the Propositional Calculus

Logical Inference: Through Proof to Truth

CS 621 Artificial Intelligence Lecture 4 – 05/08/05

Biointelligence Lab School of Computer Sci. & Eng.

Biointelligence Lab School of Computer Sci. & Eng.

2004: Topics Covered in COSC 6368

CSNB234 ARTIFICIAL INTELLIGENCE

Logical Inference 4 wrap up

Presentation transcript:

Resolution Refutation Formal Aspects of Computer Science - Week 9 Resolution Refutation, Relation with Prolog Lee McCluskey, room 2/07

Resolution Refutation Recap Last Week we covered: -Resolution -Resolution Refutation This week -Relationship with Prolog -How to get answers from Resolution Refutation (and Prolog) -Another Example

Resolution Refutation Back to Student Example Again ! Put it into Prolog S = student, D = academic, T = teaches Ax ( S(x)=>D(x) ) ; Ax ( (Ey (T(x,y) & D(y) ) => D(x) ) S(Fred) ; T(Jeff,Fred) CLAUSAL FORM: 1.~S(z) V D(z) d(Z) :- s(Z). 2. ~T(x,y) V ~D(y) V D(x) d(X) :- t(X,Y),d(Y). 3. S(Fred) s(fred). 4. T(Jeff,Fred) t(jeff,fred). Query ?d(jeff)

Resolution Refutation How does Prolog Work? d(Z) :- s(Z). d(X) :- t(X,Y),d(Y). s(fred). t(jeff,fred). ? d(jeff) ~d(jeff)d(Z) V ~s(Z). s(jeff)? d(X) V ~t(X,Y) V ~d(Y). ~d(jeff) ~t(jeff,Y) V ~d(Y)t(jeff,fred) ~d(fred)d(Z) V ~s(Z). ~s(fred) dead end Prolog Execution Tree s(fred) contradiction)

Resolution Refutation Example 1. ~S(x) V D(x)3. S(Fred) D(Fred) 2. ~T(x,y) V ~D(y) V D(x) ~T(x,Fred) V D(x)5. ~D(Jeff) ~T(Jeff,Fred)4. T(Jeff,Fred) Subs = Fred / x Subs = Fred / y Subs = Jeff / x..So D(Jeff) follows from our premises

Resolution Refutation Answer Extraction Resolution Refutation (and the PL Prolog) uses the unifications (instantiations via matching) to extract an answer from queries of the form Ex Query-Wff(x) ? (eg member(X, [1,2,3]) ?) The idea is as follows: (1)Attach Query-Wff(x) to the negated query and all its ancestors down the tree to the null clause. (2)Apply all the substitutions to w that were used in the proof tree. This will give an instance of x, say A, such that wff[A/x] follows from W. (3)Of course, if it is possible to reach a contradiction (the null clause) using more than one proof tree then more than one instantiation will be obtained.

Resolution Refutation One More (hard) Example The Martians Problem … Imagine Deep Space 1 travels to Mars and observes many things about the Martians, including the fact that some seem very hostile towards humans. Concrete observations are as follows: (a) All green Martians have antennae. (b) A Martian is friendly to humans if all of its children have antennae. (c) A Martian is green if at least one of its parents is green. On its way back from Mars the robot is hotly pursued by a spacecraft containing green Martians only. Should the robot suspect it is being attacked? Or can the robot reason with its observations to answer the question: `Are all green Martians friendly?'' and hence avert an inter-planetary conflict.

Resolution Refutation Summary Prolog uses a form of resolution within a constrained “search strategy” to work: this is called “SLD” resolution – single literal, depth first”. The refutation procedure is therefore NOT complete. To obtain answers from queries of the form Ex wff(x), we use the answer extraction technique.