Knowledge Representation and Reasoning

University "Politehnica" of BucharestDepartment of Computer Science

Fall 2012

Adina Magda Floreahttp://turing.cs.pub.ro/krr_11

curs.cs.pub.ro

Master of Science in Artificial Intelligence, 2012-2014

2

Lecture 1

Lecture outline Course goals Grading Textbooks and readings Syllabus Why KR? KR&R Challenges What is KR&R? Formal logic: why and how Links for the young researcher

3

Course goals

Provide an overview of existing representational frameworks developed within AI, their key concepts and inference methods.

Acquiring skills in representing knowledge

Understanding the principles behind different knowledge representation techniques

Being able to read and understand research literature in the area of KR&R

Being able to complete a project in this research area

4

Grading

Course grades Mid-term exam               20% Final exam                     30% Projects  30%Laboratory                    20%

Requirements: min 7 lab attendances, min 50% of term activity (mid-term ex, projects, lab)

Academic Honesty Policy It will be considered an honor code violation to give or use someone else's code or written answers, either for the assignments or exam tests. If such a case occurs, we will take action accordingly.

5

Textbooks and Readings

Textbooks• Artificial Intelligence: A Modern Approach (2010) by

Stuart Russell and Peter Norvig

• Knowledge Representation and Reasoning by Ronald Brachman and Hector Levesque, Morgan Kaufman, 2004

• Computational Intelligence: a Logical Approach by David Poole, Alain Mackworth, and Randy Goebel, Oxford University Press, 1998

Readings• Reading materials will be assigned to you.• You are expected to do the readings before the class

6

Syllabus

1. General knowledge representation issuesReadings: http://plato.stanford.edu/entries/logic-ai/

2. Logical agents – Logical knowledge representation and reasoning

• First order predicate logic revisited, ATPReadings:AIMA Chapter 7 http://aima.cs.berkeley.edu/newchap07.pdf

• Nonmonotonic logics and reasoning

Readings:Non-monotonic Logic, Stanford Encyclopedia of Philosophy

http://plato.stanford.edu/entries/logic-nonmonotonic/Nonmonotonic Reasoning, G. Brewka, I. Niemela, M. Truszczynskihttp://www.informatik.uni-leipzig.de/~brewka/papers/NMchapter.pdfNonmonotonic Reasoning With WebBased Social Networkshttp://www.mindswap.org/~katz/papers/socialnet-defaults.pdf

7

Syllabus

• Modal logic, logics of knowledge and beliefsReadings: Modal logic on Wikipedia

http://en.wikipedia.org/wiki/Modal_logic+ to be announced

• Semantic networks and description logics, reasoning services

Readings: to be announced

• Knowledge representation for the Semantic Web

Readings:Ontology knowledge representation - from description

logic to OWL Description Logics as Ontology Languages for the Semantic Web

http://lat.inf.tu-dresden.de/research/papers/2005/BaSaJS60.pdf

8

Syllabus

Midterm exam (written examination) – 1h

3. Rule based agents

• Rete: Efficient unification Readings:

The RETE algorithm

http://www.cis.temple.edu/~ingargio/cis587/readings/rete.html

• The Soar model, universal subgoaling and chunking –Readings:

A gentle introduction to Soar, an architecture for human cognition

http://ai.eecs.umich.edu/soar/sitemaker/docs/misc/GentleIntroduction-2006.pdf

• Modern rule based systemsReadings: to be announced

9

Syllabus

4. Probabilistic agents

• Probabilistic knowledge representation and reasoningReadings: to be announced

5. Temporal reasoning• Readings: to be announced

6. Reasoning with actions• Planning• Readings: to be announced

7. Intelligence without representation and reasoning vs. Strong AI• Calls Debate

Final exam

10

Links for the young researcher AI-MAS Links of interest

http://aimas.cs.pub.ro/links

Academic publishinghttp://en.wikipedia.org/wiki/Academic_publishing

Writing a Scientific Paperhttp://www.oup.com/us/samplechapters/0841234620/?view=usa

ISI Web of Knowledgehttp://isiwebofknowledge.com/

Master Journal Listhttp://science.thomsonreuters.com/mjl/

Conference Proceedings Citation Indexhttp://wokinfo.com/products_tools/multidisciplinary/webofscience/cpci/

TED – Ideas worth spreadinghttp://www.ted.com/

11

Lecture 1

Readings for Lecture 1:

http://plato.stanford.edu/entries/logic-ai/

Readings for Lecture 2

AIMA Chapter 7 http://aima.cs.berkeley.edu/newchap07.pdf

12

1. Why KR?

We understand by "knowledge" all kinds of facts about the world.

Knowledge is necessary for intelligent behavior (human beings, robots).

In this course we consider representation of knowledge and how we can use it in making intelligent artifacts.

What is knowledge?

13

2. KR&R Challenges

Challenges of KR&R:

• representation of commonsense knowledge

• the ability of a knowledge-based system to tradeoff computational efficiency for accuracy of inferences

• its ability to represent and manipulate uncertain knowledge and information.

14

3. What is KR?

Randall Davis, Howard Shrobe, Peter Szolovits, MIT

A knowledge representation is most fundamentally a surrogate, a substitute for the thing itself, used to enable an entity to determine consequences by reasoning about the world.

It is a set of ontological commitments, i.e., an answer to the question: In what terms should I think about the world?

15

What is KR?

It is a fragmentary theory of intelligent reasoning, expressed in terms of three components:

• the representation's fundamental conception of intelligent reasoning;

• the set of inferences the representation sanctions;

• the set of inferences it recommends.

16

What is KR?

It is a medium for pragmatically efficient computation, i.e., the computational environment in which reasoning is accomplished. • One contribution to this pragmatic efficiency is

supplied by the guidance a representation provides for organizing information so as to facilitate making the recommended inferences.

It is a medium of human expression, i.e., a language in which we say things about the world.

17

What is KR?

If A represents B, then A stands for B and is usually more easily accessible than B.

Symbolic representations Non-symbolic representations

Symbolic representations – set of propositions or statements that are believed by some agent.

18

4. What is Reasoning?

Reasoning is the use of symbolic representations of some statements in order to derive new ones.

Statements are abstract objects; their representations are concrete objects and can be easily manipulated.

Reasoning should scale well: we need efficient reasoning algorithms

http://plato.stanford.edu/entries/logical-consequence/

19

5. Models of KRR

Symbolic logic and ATP Probabilistic Temporal Rules Structured

20

6. Formal logic

Formal logic is the field of study of entailment relations, formal languages, truth conditions, semantics, and inference.

All propositions/statements are represented as formulae which have a semantics according to the logic in question.

Logical system = Formal language + semantics

Formal logics gives us a framework to discuss different kinds of reasoning.

21

6.1 Logical consequence (entailment)

Proof centered approach to logical consequence: the validity of a reasoning process (argument) amounts to there being a proof of the conclusions from the premises.

22

Logical consequence (entailment)

Model centered approach to logical consequence

Models are abstract mathematical structures that provide possible interpretations for each of the objects in a formal language.

Given a model for a language - define what it is for a sentence in that language to be true (according to that model) or not.

In any model in which the premises are true the conclusion is true too. (Tarski's definition of logical

consequence from 1936.)

23

6.2 Model centered approach

Interpretation of a formula

Model of a formula

Entailment or logical consequence

A formula F is a logical consequence of a set of formulas P1,…Pn iff F is true in all interpretations in which P1,…Pn are true.

P1,… Pn |= L F

T Formula F is a logical consequence of a set of formulas P1,…Pn iff P1,…Pn F is valid.

T Formula F is a logical consequence of a set of formulas P1,…Pn iff P1… Pn ~F is inconsistent.

24

6.3 Proof centered approach

Theorem, deduction

Formal system Inference rule

Premise set

Consequence of

R

R , y = y ,...,y x, x,y i = 1,nn1 n

Ri F F F ,

S =< A, , , >F A

= {y , ... , y1 n } E =0 A

E = E x| y E , y x}1 0 0n

n 1{

E = E x| y E , y x}2 1 1

n

n 1{

E ( i 0)i

25

Proof centered approach

If then is deductible from

|S x

Theorems - the elements of Ei if

Demonstration | R x 

E = ( = )0 A

x Ei

E =0 A x Ei

26

Proof approach important notions

Th() – set of provable theorems in

• Monotonicity

• Idempotence - multiple applications of the operation do not change the result

Th() – a fixed point operator which computes the closure of a set of formulas according to the rules of inference

Th() – the least fixed point of this closure process

27

6.4 Properties of logical systems

Important properties of logical systems:

Consistency - no theorem of the system contradicts another.

Soundness - the system's rules of proof will never allow a false inference from a true premise. If a system is sound and its axioms are true then its theorems are also guaranteed to be true.

Completeness - there are no true sentences in the system that cannot, at least in principle, be proved in the system.

Some logical systems do not have all three properties. Kurt Godel's incompleteness theorems show that no standard formal system of arithmetic can be consistent and complete.

28

Properties of logical systems

A logical system L is complete iff

|= L implies |

(i.e., all valid formulas are provable)

A logical system L is sound iff

| implies |= L

(i.e., no invalid formula is provable)

FOPL

Second order logics

29

7. Logic based representations

2 possible aims• to make the system function according to the logic• to specify and validate the design

Conceptualization of the world / problem Syntax - wffs Semantics - significance, model Model - the domain interpretation for which a formula is true Model - linear or structured M |=S - " is true or satisfied in component S of the structure M"

Model theoryModel theory Generate new wffs that are necessarily true, given that the old wffs

are true - entailment KB |=L Proof theoryProof theory Derive new wffs based on axioms and inference rules

KB |-i

30

PrL, FOPLExtend PrL, PL

Sentential logicof beliefsUses beliefs atoms BA()Index PL with agents

Modal logicModal operators

Logics of knowledgeand beliefModal operators B and K

Dynamic logicModal operatorsfor actions

Temporal logicModal operators for timeLinear timeBranching time

CTL logicBranching timeand action BDI logic

Adds agents, B, D, I

Linear model

Structured models

Situation calculusAdds states, actions

Symbol levelSymbol level

Knowledge levelKnowledge level

Description LogicsSubsumption relationships

31

knowledge propositional first-order

Paul is a man a man(Paul)

Bill is a man b man(Bill)

men are mortal c (x) (man(x) mortal(x))

knowledge first-order second-order

smaller istransitive

( x) (( y) (( z)((<(x,y) <(y,z)

<(x,z)))))

transitive(<)

part-of istransitive

( x) (( y) (( z)((part-of(x,y) part-of(y,z) part-of(x,z)))))

transitive(part-of)

R is transitive iffwhenever R(x,y) andR(y,z) hold, R(x,z)

holds too

not expressible(see however pseudo-

second order)

( R) ((transitive(R) ( x) (( y) (( z)((R(x,y) R(y,z)

R(x,z)))))))

Higher order logic

First order logic

32

8. Automated Reasoning

33

34

A logical puzzle

Someone who lives in Dreadbury Mansion killed Aunt Agatha.Agatha, the butler, and Charles live in Dreadbury Mansion, and are the only people who live therein.A killer always hates his victim, and is never richer than his victim.Charles hates no one that Aunt Agatha hates.Agatha hates everyone except the butler.The butler hates everyone not richer than Aunt Agatha.The butler hates everyone Aunt Agatha hates.No one hates everyone.Agatha is not the butler.

Who killed Aunt Agatha?

35

36

37

38

39

Slides 35-38 are from the slides

First-Order Theorem Proving

Peter Baumgartner

NICTA, Logic and Computation Program, Canberra

Peter.Baumgartner@nicta.com.au