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First-Order Logic Outline Pros and Cons of Propositional Logic First Outline • Why FOL? • Syntax and semantics of FOL First‐Order Logic • Using FOL • Wumpus world in FOL Chapter 8 • Knowledge engineering in FOL (not 8.1) Pros and cons of propositional logic First‐order logic Propositional logic is declarative • Whereas propositional logic assumes the Propositional logic allows partial/disjunctive/negated information world contains facts, first‐order logic (like – (unlike most data structures and databases) natural language) assumes the world contains Propositional logic is compositional – meaning of B1,1 P1,2 is derived from meaning of B1,1 and of P1,2 – Objects: people, houses, numbers, colors, baseball Meaning in propositional logic is context‐independent – (unlike natural language, where meaning depends on context) games, wars, … – Propositional logic has very limited expressive power Relations: red, round, prime, brother of, bigger – (unlike natural language) than, part of, comes between, … – E.g., cannot say "pits cause breezes in adjacent squares“ • except by writing one sentence for each square – Functions: father of, best friend, one more than, plus, … Syntax of FOL: Basic elements Atomic sentences • Constants KingJohn, 2, Udel,... Atomic sentence = predicate (term1,...,termn) • Predicates Brother, >,... or term1 = term2 • Functions Sqrt, LeftLegOf,... Term = constant • Variables x, y, a, b,... or variable • Connectives , , , , or function (term1,...,termn) • Equality = • E.g., Brother(KingJohn,RichardTheLionheart) , • Quantifiers , (Length(LeftLegOf(Richard)), Length(LeftLegOf(KingJohn))) Married(mother(mother(mary)), father(mother(mary))) Representing Some Sentences Complex sentences • John likes Mary. Complex sentences are made from atomic • Bill hit John. sentences using connectives • Florida is a state. S, S1 S2, S1 S2, S1 S2, S1 S2, • The product of a and b is c. • 2+3=7. E.g. Sibling(KingJohn,Richard) • Tweety is yellow. Sibling(Richard,KingJohn) >(1,2) ≤ (1,2) >(1,2) >(1,2) Truth in first‐order logic Models for FOL: Example • Sentences are true with respect to a model and an interpretation • Model contains objects (domain elements) and relations among them • Interpretation specifies referents for constant symbols → objects predicate symbols → relations function symbols → functional relations • An atomic sentence predicate(term1,...,termn) is true iff the objects referred to by term1,...,termn are in the relation referred to by predicate Universal quantification A common mistake to avoid • <variables> <sentence> Typically, is the main connective with Everyone at UDel is smart: x At(x,UDel) Smart(x) • Common mistake: using as the main connective • x P is true in a model m iff P is true with x being each with : possible object in the model x At(x,UDel) Smart(x) • Roughly speaking, equivalent to the conjunction of means “Everyone is at Udel and everyone is smart” instantiations of x At(KingJohn,UDel) Smart(KingJohn) At(Richard,UDel) Smart(Richard) At(NUS,UDel) Smart(NUS) ... Existential quantification Another common mistake to avoid • <variables> <sentence> • Typically, is the main connective with Someone at Udel is smart: x At(x,UDel) Smart(x) • Common mistake: using as the main connective • xPis true in a model m iff P is true with x being some with : possible objec t in the modldel x At(x,UDel) Smart(x) • Roughly speaking, equivalent to the disjunction of instantiations of P is true if there is anyone who is not at UDel! At(KingJohn,UDel) Smart(KingJohn) At(Richard,UDel) Smart(Richard) At(NUS,UDel) Smart(NUS) ... Properties of quantifiers Equality • x yis the same as y x • x yis the same as y x • term1 = term2 is true under a given interpretation if and only if term and term refer to the same object • x yis not the same as y x 1 2 • x yLoves(x,y) – “There is a person who loves everyone in the world” • E.g., definition of Sibling in terms of Parent: • y x Loves(x,y) – “Everyone in the world is loved by at least one person” x,y Sibling(x,y) [(x = y) m,f (m = f) Parent(m,x) • Quantifier duality: each can be expressed using the other Parent(f,x) Parent(m,y) Parent(f,y)] • x Likes(x,IceCream) x Likes(x,IceCream) • x Likes(x,Broccoli) x Likes(x,Broccoli) Using FOL Using FOL The kinship domain: The set domain: • Brothers are siblings • s Set(s) (s = {} ) (x,s2 Set(s2) s = {x|s2}) x,y Brother(x,y) Sibling(x,y) • x,s {x|s} = {} • x,s x s s = {x|s} • One's mother is one's female parent • x,s x s [ y,s2} (s = {y|s2} (x = y x s2))] m,c Mother(c) = m (Female(m) Parent(m,c)) • s1,s2 s1 s2 (x x s1 x s2) • s1,s2 (s1 = s2) (s1 s2 s2 s1) • “Sibling” is symmetric • x,s1,s2 x (s1 s2) (x s1 x s2) x,y Sibling(x,y) Sibling(y,x) • x,s1,s2 x (s1 s2) (x s1 x s2) Knowledge engineering in FOL Summary 1. Identify the task • First‐order logic: 2. Assemble the relevant knowledge – objects and relations are semantic primitives 3. Decide on a vocabulary of predicates, functions, – syntax: constants, functions, predicates, equality, and constants quantifiers 4. Encode general knowledge about the domain 5. Encode a description of the specific problem instance • Increased expressive power: sufficient to 6. Pose queries to the inference procedure and get define wumpus world answers 7. Debug the knowledge base.
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