10 주 강의 First-order Logic

Limitation of propositional logic A very limited ontology  to need to the representation power  first-order logic

First-order logic A stronger set of ontological commitments A world in FOL consists of objects, properties, relations, functions Objects  people, houses, number, colors, Bill Clinton Relations  brother of, bigger than, owns, love Properties  red, round, bogus, prime Functions  father of, best friend, third inning of

Examples “ One plus two equals three ” – objects :: one, two, three, one plus two – Relation :: equal – Function :: plus “ Squares neighboring the wumpus are smelly –Objects :: wumpus, square –Property :: smelly –Relation :: neighboring

First order logics Objects 와 relations 시간, 사건, 카테고리 등은 고려하지 않음 영역에 따라 자유로운 표현이 가능함  ‘ king ’ 은 사람의 property 도 될 수 있고, 사람과 국가 를 연결하는 relation 이 될 수도 있다 일차술어논리는 잘 알려져 있고, 잘 연구된 수 학적 모형임

Syntax and Semantics Sentence  AtomicSentence | Sentence Connective Sentence | Auantifier Variable,…Sentence |  Sentence | (Sentence) AtomicSentence  Predicate(Term,…) | Term=Term Term  Function (Term,…) | Constant | Variable Connective   |  |  |  Quantifier   |  Constant  A | X 1 | John | … Variable  a | x | s | … Predicate  Before | HanColor | Raining | … Function  Mother | LeftLegOf | … Figure 7.1 The syntax of first-order logic (with equality) in BNF (Backus-Naur Form).

예 Constant symbols :: A, B, John, Predicate symbols :: Round, Brother Function symbols :: Cosine, FatherOf Terms :: King John, Richard ’ s left leg Atomic sentences :: Brother(Richard,John), Married(FatherOf(Richard), MotherOf(John)) Complex sentences :: Older(John,30)=>~younger(John,30)

Quantifiers World = {a, b, c} Universal quantifier ( ∀ ) ∀ x Cat(x) => Mammal(x)  Cat(a) => Mammal(a) & Cat(a) => Mammal(a) Existential quantifier ( ∃ ) ∃ x Sister(x, Sopt) & Cat(x)

Nested quantifiers ∀ x,y Parent(x,y) => Child(y,x) ∀ x,y Brother(x,y) => Sibling(y,x) ∀ x ∃ y Loves(x,y) ∃ y ∀ x Loves(x,y)

De Morgan ’ s Rule ∀ x ~P  ~ ∃ x P ~P&~Q  ~(P v Q) ~ ∀ x P  ∃ x ~P ~(P&Q)  ~P v ~Q ∀ x P  ~ ∃ x ~P P&Q  ~(~P v ~ Q) ∃ x P  ~ ∀ x ~P P v Q  ~(~P&~Q)

Equality Identity relation Father(John) = Henry ∃ x,y Sister(Spot,x) & Sister(Spot,y) & ~(x=y) ≠ ∃ x,y Sister(Spot,x) & Sister(Spot,y)

Higher-order logic ∀ x,y (x=y)  ( ∀ p p(x)  p(y)) ∀ f,g (f=g)  ( ∀ x f(x)  g(x)) ∀

-expression x,y x 2 – y 2 -expression can be applied to arguments to yield a logical term in the same way that a function can be ( x,y x 2 – y 2 )(25,24) = = 49 x,y Gender(x) ≠Gender(y) & Address(x) = Address(y)

∃ ! (The uniqueness quantifier) ∃ !x King(x) ∃ x King(x) & ∀ y King(y) => x=y world 를 고려하여 보여주면 => object 가 1, 2, 3 개일 때 {a} w0  king={}, w1  king={a}  w1 만 model {a,b} w0  king={}, w1  king={a}, w2  {b}, w3  {a,b}  w1, w2 만 model

Representation of sentences by FOPL One ’ s mother is one ’ s female parent ∀ m,c Mother(c)=m  Female(m) & Parent(m) One ’ s husband is one ’ s male spouse ∀ w,h Husband(h,w)  Male(h) & Spouse(h,w) Male and female are disjoint categories ∀ x Male(x)  ~Female(x) A grandparent is a parent of one ’ s parent ∀ g,c Grandparent(g,c)  ∃ p parent(g,p) & parent(p,g)

Representation of sentences by FOPL A sibling is another child of one ’ s parents ∀ x,y Sibling(x,y)  x≠y & ∃ p Parent(p,x) & Parent(p,y) Symmetric relations ∀ x,y Sibling(x,y)  Sibling(y,x)

The domain of sets (I) The only sets are the empty set and those made by adjoining something to a set : ∀ s Set(s)  (s=EmptySet) v ( ∃ x,s2 Set(s2) & s=Adjoin(x,s2)) The empty set has no elements adjoined into it. ~ ∃ x,s Adjoin(x,s)=EmptySet Adjoining an element already in the set has no effect ∀ x,s Member(x,s)  s=Adjoin(x,s) The only members of a set are the elements that were adjoined into it ∀ x,s Member(x,s)  ∃ y,s2 (s=Adjoin(y,s2) & (x=y v Member(x,s)))

The domain of sets (II) A set is a subset of another if and only if all of the first set ’ s are members of the second set : ∀ s1,s2 Subset(s1,s2)  ( ∀ x Member(x,s1) => member(x,s2)) Two sets are equal if and only if each is a subset of the other: ∀ s1,s2 (s1=s2)  (Subset(s1,s2) & Subset(s2,s1))

The domain of sets (III) An object is a member of the intersection of two sets if and only if it is a member of each of sets : ∀ x,s1,s2 Member(x,Intersection(s1,s2))  Member(x,s1) & Member(x,s2) An object is a member of the union of two sets if and only if it is a member of either set : ∀ x,s1,s2 Member(x,Union(s1,s2))  Member(x,s1) v Member(x,s2)

Asking questions and getting answers Tell(KB, ( ∀ m,c Mother(c)=m  Female(m) & Parent(m,c)) ) …… Tell(KB, (Female(Maxi) & Parent(Maxi,Spot) & Parent(Spot,Boots))) Ask(KB,Grandparent(Maxi,Boots) Ask(KB, ∃ x Child(x, Spot)) Ask(KB, ∃ x Mother(x)=Maxi) Substitution, unification, {x/Boots}