Knowledge Representation

1. Knowledge Representation Praveen Paritosh CogSci 207: Fall 2003: Week 1 Thu, Sep 30, 2004

2. Some Representations

3. Elements of a Representation • Represented world: about what? • Representing world: using what? • Representing rules: how to map? • Process that uses the representation: conventions and systems that use the representations resulting from above. • Analog versus Symbolic

4. Marr’s levels of description • Computational: What is the goal of the computation, why is it appropriate, and what is the logic of the strategy by which it can be carried out? • Algorithmic: How can this computational theory be implemented? In particular, what is the representation for the input and output, and what is the algorithm for the transformation? • Implementation: How can the representation and algorithm be realized physically?

5. Marr’s levels of description – 2 • Computational: a lot of cognitive psychology • Algorithmic: a lot of cognitive science • Implementation: neuroscience

6. A closer look

7. Overview • How knowledge representation works • Basics of logic (connectives, model theory, meaning) • Basics of knowledge representation • Why use logic instead of natural language? • Quantifiers • Organizing large knowledge bases • Ontology • Microtheories • Resource: OpenCyc tutorial materials

8. How Knowledge Representation Works • Intelligence requires knowledge • Computational models of intelligence require models of knowledge • Use formalisms to write down knowledge • Expressive enough to capture human knowledge • Precise enough to be understood by machines • Separate knowledge from computational mechanisms that process it • Important part of cognitive model is what the organism knows

9. How knowledge representations are used in cognitive models • Contents of KB is part of cognitive model • Some models hypothesize multiple knowledge bases. Questions, requests Examples, Statements Answers, analyses Inference Mechanism(s) Learning Mechanism(s) Knowledge Base

10. What’s in the knowledge base? • Facts about the specifics of the world • Northwestern is a private university • The first thing I did at the party was talk to John. • Rules (aka axioms) that describe ways to infer new facts from existing facts • All triangles have three sides • All elephants are grey • Facts and rules are stated in a formal language • Generally some form of logic (aka predicate calculus)

11. Propositional logic • A step towards understanding predicate calculus • Statements are just atomic propositions, with no structure • Propositions can be true or false • Statements can be made into larger statements via logical connectives. • Examples: • C = “It’s cold outside” ; C is a proposition • O = “It’s October” ; O is a proposition • If O then C ;if it’s October then it’s cold outside

12. Symbols for logical connectives • Negation: not, , ~ • Conjunction: and,  • Disjunction: or,  • Implication: implies, , • Biconditional: iff, ------------------------------------------------------------ • Universal quantifier: forall,  • Existential quantifier: exists, 

13. Semantics of connectives • For propositional logic, can define in terms of truth tables

14. Implication and biconditional AB  AB AB  (AB)(BA)

15. Rules of inference • There are many rules that enable new propositions to be derived from existing propositions • Modus Ponens: PQ, P, derive Q • deMorgan’s law: (AB), derive AB • Some properties of inference rules • Soundness: An inference rule is sound if it always produces valid results given valid premises • Completeness: A system of inference rules is complete if it derives everything that logically follows from the axioms.

16. Predicate calculus • Same connectives • Propositions have structure: Predicate/Function + arguments. • R, 2 ; Terms. Terms are not individuals, not propositions • Red(R), (Red R) ; A proposition, written in two ways • (southOf UnicornCafe UniHall) ;a proposition • (+ 2 2) ; Term, since the function + ranges over numbers • Quantifiers enable general axioms to be written • (forall ?x (iff (Triangle ?x) (and (polygon ?x) (numberOfSides ?x 3)))

17. Model Theory • Meaning of a theory = set of models that satisfy it. • Model = set of objects and relationships • If statement is true in KB, then the corresponding relationship(s) hold between the corresponding objects in the modeled world • The objects and relationships in a model can be formal constructs, or pieces of the physical world, or whatever • Meaning of a predicate = set of things in the models for that theory which correspond to it. • E.g., above means “above”, sort of

18. Caution: Meaning pertains to simplest model • There is usually an intended model, i.e., what one is representing. • A sparse set of axioms can be satisfied by dramatically simpler worlds than those intended • Example: Classic blocks world axioms have ordered pairs of integers as a model • (<position on table> <height>)  block • (on A B)  p(A) = p(B) & h(A) = h(B)+1 • (above A B)  p(A) = p(B) & h(A) > h(B) • Moral: Use dense, rich set of axioms

19. Misconceptions about meaning • “Predicates have definitions” • Most don’t. Their meaning is constrained by the sum total of axioms that mention them. • “Logic is too discrete to capture the dynamic fluidity of how our concepts change as we learn” • If you think of the set of axioms that constrain the meaning of a predicate as large, then adding (and removing) elements of that set leads to changes in its models. • Sometimes small changes in the set of axioms can lead to large changes in the set of models. This is the logical version of a discontinuity.

20. Representations as Sculptures • How does one make a statue of an elephant? • Start with a marble block. Carve away everything that does not look like an elephant. • How does one represent a concept? • Start with a vocabulary of predicates and other axioms. Add axioms involving the new predicate until it fits your intended model well. • Knowledge representation is an evolutionary process • It isn’t quick, but incremental additions lead to incremental progress • All representations are by their nature imperfect

21. Introduction to Cyc’s KR system • These materials are based on tutorial materials developed by Cycorp, for training knowledge entry people and ontological engineers • For this class, we have simplified them somewhat. • In examinations, you will only be responsible for the simplified versions

22. NL vs. Logic: Expressiveness NL: Jim’s injury resulted from his falling. Jim’s falling caused his injury. Jim’s injury was a consequence of his falling. Jim’s falling occurred before his injury. NL: Write the rule for every expression? Logic: identify the common concepts, e.g. the relation: x caused y Write rules about the common concepts, e.g. x caused y  x temporally precedes y

23. NL vs. Logic: Ambiguity and Precision • x is at the bank. • river bank? • financial institution? • x is running. • changing location? • operating? • a candidate for office? NL: Ambiguous Logic: Precise x is running-InMotion  x is changing location x is running-DeviceOperating x is operating x is running-AsCandidate  x is a candidate Reasoning: Figuring out what must be true, given what is known. Requires precision of meaning.

24. NL vs. Logic:Calculus of Meaning Logic:Well-understood operators enable reasoning: Logical constants: not, and, or, all, some Not (All men are taller than all women). All men are taller than 12”. Some women are taller than 12”. Not (All A are F than all B). All A are F than x. Some B are F than x.

25. A sampling of some constants: Dog, SnowSkiing, PhysicalAttribute BillClinton,Rover, DisneyLand-TouristAttraction likesAsFriend, bordersOn, objectHasColor, and, not, implies, forAll RedColor, Soil-Sandy Syntax: Terms (aka Constants) Terms denote specific individuals or collections (relations, people, computer programs, types of cars . . . ) Each Terms is a character string prefixed by These denote collections These denote individuals : • Partially Tangible Individuals • Relations • Attribute Values

26. Syntax: Propositions Propositions: a relation applied to some arguments, enclosed in parentheses • Also called formulas, sentences… • Examples: • (isa GeorgeWBush Person) • (likesAsFriend GeorgeWBush AlGore) • (BirthFn JacquelineKennedyOnassis)

27. Syntax: Non-Atomic Terms • New terms can be made by applying functions to other things • In the Cyc system, functions typically end in “Fn” • Examples of functions: • BirthFn, GovernmentFn, BorderBetweenFn • Examples of Non-Atomic Terms: • (GovernmentFn France) • (BorderBetweenFn France Switzerland) • (BirthFn JacquelineKennedyOnassis) Non-atomic Terms can be used in statements like any other term • (residenceOfOrganization (GovernmentFn France) CityOfParisFrance)

28. Why Use NATs? • Uniformity • All kinds of fruits, nuts, etc., are represented in the same, compositional way: (FruitFn PLANT) * • Inferential Efficiency • Forward rules can automatically conclude many useful assertions about NATs as soon as they are created, based on the function and arguments used to create the NAT. • what kind of thing that NAT represents • how to refer to the NAT in English • …

29. Well-formedness: Arity • Arity constraints are represented in CycL with the predicate arity: • (arity performedBy 2) Represents the fact that performedBy takes two arguments, e.g.: (performedBy AssassinationOfPresidentLincoln JohnWilkesBooth) • (arity BirthFn 1) Represents the fact that BirthFn takes one arguments, e.g.: (BirthFn JacquelineKennedyOnassis)

30. Well-Formedness: Argument Type Argument type constraints are represented in CycL with the following 2 predicates: 1 argIsa (argIsa performedBy 1 Action) means that the first argument of performedBy must be an individual Action, such as the assassination of Lincoln in: (performedBy AssassinationOfPresidentLincoln JohnWilkesBooth) 2 argGenl (argGenl penaltyForInfraction 2 Event) means that the second argument of penaltyForInfraction must be a type of Event, such as the collection of illegal equipment use events in: (penaltyForInfraction SportsEvent IllegalEquipmentUse Disqualification)

31. Why constraints are important • They guide reasoning • (performedBy PaintingTheHouse Brick2) • (performedBy MarthaStewart CookingAPie) • They constrain learning

32. Compound propositions • Connectives from propositional logic can be used to make more complex statements • (and (performedBy GettysburgAddress Lincoln) (objectHasColor Rover TanColor)) • (or (objectHasColor Rover TanColor) (objectHasColor Rover BlackColor)) • (implies (mainColorOfObject Rover TanColor) (not (mainColorOfObject Rover RedColor))) • (not (performedBy GettysburgAddress BillClinton))

33. Variables and Quantifiers • General statements can be made by using variables and quantifiers • Variables in logic are like variables in algebra • Sentences involving concepts like “everybody,” “something,” and “nothing” require variables and quantifiers: Everybody loves somebody. Nobody likes spinach. Some people like spinach and some people like broccoli, but no one likes them both.

34. Quantifiers • Adding variables and quantifiers, we can represent more general knowledge. • Two main quantifiers: 1. Universal Quantifer -- forAll Used to represent very general facts, like: All dogs are mammals Everyone loves dogs 2. Existential Quantifier -- thereExists Used to assert that something exists, to state facts like: Someone is bored Some people like dogs

35. Quantifiers • Universal Quantifier (forAll ?THING (isa ?THING Thing)) • Existential Quantifier: (thereExists ?JOE (isa ?JOE Poodle)) • Others defined in CycL: (thereExistsExactly 12 ?ZOS (isa ?ZOS ZodiacSign)) (thereExistsAtLeast 9 ?PLNT (isa ?PLNT Planet)) Everything is a thing. Something is a poodle. There are exactly 12 zodiac signs There are at least 9 planets

36. Implicit Universal Quantification All variables occurring “free” in a formula are understood by Cyc to be implicitly universally quantified. So, to CYC, the following two formulas represent the same fact: (forAll ?X (implies (isa ?X Dog) (isa ?X Animal)) (implies (isa ?X Dog) (isa ?X Animal))

37. Pop Quiz #1 • What does this formula mean? (thereExists ?PLANET (and (isa ?PLANET Planet) (orbits ?PLANET Sun)))

38. Pop Quiz #1 • What does this formula mean? (thereExists ?PLANET (and (isa ?PLANET Planet) (orbits ?PLANET Sun))) “There is at least one planet orbiting the Sun.”

39. Pop Quiz #2 • What does this formula mean? (forAll ?PERSON1(implies (isa ?PERSON1 Person) (thereExists ?PERSON2 (and (isa ?PERSON2 Person) (loves ?PERSON1 ?PERSON2)))

40. Pop Quiz #2 • What does this formula mean? (forAll ?PERSON1(implies (isa ?PERSON1 Person) (thereExists ?PERSON2 (and (isa ?PERSON2 Person) (loves ?PERSON1 ?PERSON2))) “Everybody loves somebody.”

41. Pop Quiz #3 • How about this one? (implies (isa ?PERSON1 Person) (thereExists ?PERSON2 (and (isa ?PERSON2 Person) (loves ?PERSON2 ?PERSON1))))

42. Pop Quiz #3 • How about this one? (implies (isa ?PERSON1 Person) (thereExists ?PERSON2 (and (isa ?PERSON2 Person) (loves ?PERSON2 ?PERSON1)))) “Everyone is loved by someone.”

43. Pop Quiz #4 And this? (implies (isa ?PRSN Person) (loves ?PRSN ?PRSN))

44. Pop Quiz #4 And this? (implies (isa ?PRSN Person) (loves ?PRSN ?PRSN)) “Everyone loves his (or her) self.”