Vibhav Gogate University of Texas, Dallas Artificial Intelligence: CS 4365 First-order Logic

Transcription

1 Artificial Intelligence: CS 4365 First-order Logic Vibhav Gogate University of Texas, Dallas First-order Logic

2 Course Outline: Where are we? Introduction and Agents Search, Heuristics and CSPs Adversarial Search Logical Knowledge Representation and Inference Uncertainty and Bayesian Networks Machine Learning Advanced Topics

3 Knowledge Representation and Reasoning System: Recap Declarative Representation Model = Propositional logic, Probabilistic Propositional logic, First-order logic, Probabilistic first-order logic General-purpose, universal inference engine

4 Why First order logic? Much more expressive and compact that propositional logic. Yet, unlike a natural language, it is formal and precise! English: If you are sitting near a person with Flu, then you will get the Flu. First-order Logic: x, y Flu(x) Neighbor(x, y) Flu(y) Propositional Logic: Flu(student) for each student in class Neighbor(student1,student2) for each pair of students in class One Constraint for each pair of students Size(FOL)= 1 formula Size(PL) = n 2 formulas Imagine an online class with 1 million students!

5 Propositional Logic vs First-Order Logic Propositional Logic First-order logic Ontology Facts (P,Q) Objects, Properties Relations Syntax Atomic Sentences Variables & Quantification Connectives Function symbols Semantics Truth Tables Interpretations Much more complicated Inference DPLL, WalkSAT Unification, Prolog Resolution First-order resolution

6 FOL: Syntax Constants: a, b, dog33. Defines the objects in the domain. Variables: X,Y Refer to an object without naming it. Predicate Symbols: Flu, Neighbor Relations between objects Functions: dad-of Mapping from objects to objects. Terms: dad-of(dog33), x, dog33 Logical expression that refers to an object Atomic Sentence: In(dad-of(dog33,food6) Can be true or false

7 FOL: Syntax Logical Connectives: and, or, not, implies Flu(x) Neighbor(x, y) Flu(y) Quantifiers: Forall ( ) and There exists ( ) Example Dumbo is grey grey(dumbo) Elephants are grey x elephant(x) grey(x) There is a grey elephant x elephant(x) grey(x)

8 Quantifier/Connective Interaction E(x) == x is an elephant G(x) == x has the color grey x E(x) G(x) Every object in the domain is an elephant and is grey. x E(x) G(x) All elephants are grey. x E(x) G(x) There exists an elephant which is grey. x E(x) G(x) Meaningless. is True for any object that is not an elephant. Typically: is the main connective with and with. Common mistake: using as the main connective with and with

9 Nested Quantifiers: Order Matters!! x y P(x, y) y x P(x, y) Examples: Every dog has a tail d t has(d, t) What does t d has(d, t) mean? Every dog shares a tail There is someone who is loved by everyone y x loves(x, y) What does x y loves(x, y) mean? Everybody loves somebody.

10 Connections between Quantifiers Quantifier duality: each can be expressed using the other x Likes(x, IceCream) x Likes(x, IceCream) x Likes(x, Broccoli) x Likes(x, Broccoli)

11 Fun with Sentences Brothers are siblings x, y Brother(x, y) Sibling(x, y). Sibling is reflexive x, y Sibling(x, y) Sibling(y, x) One s mother is one s female parent x, y Mother(x, y) (Female(x) Parent(x, y)) A first cousin is a child of a parent s sibling x, y FirstCousin(x, y) p, ps Parent(p, x) Sibling(ps, p) Parent(ps, y)

12 Wumpus World in Propositional Logic Let P i,j be true if there is a pit in [i,j] Let B i,j be true if there is a breeze in [i,j] Knowledge Base: P 1,1 B 1,1 and so on Pits cause breeze in adjacent squares B 1,1 (P 1,2 P 2,1 ) B 2,1 (P 1,1 P 2,2 P 3,1 ) and so on 64 distinct proposition symbols, 155 sentences

13 Wumpus World in First-order Logic KB: Let P(i,j) be true if there is a pit in [i,j] Let B(i,j) be true if there is a breeze in [i,j] P(1, 1), B(1, 1) and so on Pits cause breezes in adjacent squares i, j B(i, j) P(i, add(j, 1)) P(i, add(j, 1)) P(add(i, 1), j) P(add(i, 1), j)

14 Equality Equality symbol term 1 = term 2 is used to signify that two terms refer to the same object E.g., Father(John) = Henry says that the object referred to by Father(John) and the object referred to by Henry are the same. E.g., definition of (full) Sibling in terms of Parent: x, y Sibling(x, y) [ (x = y) m, f (m = f ) Parent(m, x) Parent(f, x) Parent(m, y) Parent(f, y)]

15 Truth in first-order logic: Semantics Sentences are true with respect to a model and an interpretation Model contains objects and relations among them Interpretation specifies referents for constant symbols objects predicate symbols relations function symbols functional relations An atomic sentence predicate(term 1,..., term n ) is true iff the objects referred to by term 1,..., term n are in the relation referred to by predicate

16 FOL: Semantics Interpretations=Mappings syntactic tokens model elements

17 FOL: Semantics Interpretations=Mappings syntactic tokens model elements

18 An alternative semantics How to capture the fact that Richard has two brothers John and Geoffrey? Brother(John, Richard) Brother(Geoffrey, Richard) Not quite! Brother(John, Richard) Brother(Geoffrey, Richard) John Geoffrey x Brother(x, Richard) (x = John x = Geoffrey) As a result, humans may make mistakes in translating their knowledge into first-order logic, resulting in unintuitive behaviors from logical reasoning systems that use the knowledge. Solution: Restrictions of first-order logic

19 Database Semantics: Restrictions Unique-names assumption: Every constant symbol refers to a distinct object Closed-world assumption: Atomic sentences not known to be true are in fact false Domain-closure: Each model contains no more domain elements than those named by the constant symbols. Now, Brother(John, Richard) Brother(Geoffrey, Richard) indeed states that Richard s two brothers are John and Geoffrey. Pros and Cons: Useful when we are certain about the identity of all the objects described in the KB and when we have all facts in hand. Otherwise can be quite awkward

20 Knowledge Engineering Process of constructing knowledge for the real-world domain of interest. Fun (Read section 8.4 of Russell and Norvig) Project 3 will be a knowledge engineering project.