논리와 응용 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)

Knowledge Base A set of representations of facts about the world Knowledge representation language – tell : what has been told to the knowledge base previously – ask : a question and the answer Inference : what follows from what the KB has been Telled Background knowledge : a knowledge base which may initially contained Sentence : individual representation of a fact

Knowledge base The knowledge level :: saying what it knows to KB  “ Golden Gates Bridge links San Francisco and Marin Country The logical level :: the knowledge is encoding into sentences  Links(GGBridge, SF, Marin) The implementation level :: the level that runs on the agent architecture (data structures to represent knowledge or facts)

Knowledge declarative/procedural – love(john, mary). – can_fly(X) :- bird(X), not(can_fly(X)), !. learning : general knowledge about the environment given a series of percepts Commonsense knowledge

Specifying the environment Figure 6.2 A typical wumpus world

Domain specific knowledge – In the squares directly adjacent to a pit, the agent will perceive a breeze Commonsense knowledge – logical reasoning – stench(1,2) & ~setnch(2,1)  ~wumpus(2,2) – wumpus(1,3)  stench(2,1) & stench(2,3) & stench(1,4)

Inference in Wumpus world(I) 4,13,12,11,1 4,23,22,21,2 4,33,32,31,3 4,43,42,41,4 OK A = Agent B = Breeze G = Glitter, Gold OK = Safe square P = Pit S = Stench V = Visited W = Wumpus 4,13,12,11,1 4,23,22,21,2 4,33,32,31,3 4,43,42,41,4 OK A A B V P ? Figure 6.3 The first step taken by the agent in the wumpus world. (a)The initial situation, after percept [None, None, None, None, None]. (b)After one move, with percept [None, Breeze, None, None, None].

Inference in Wumpus world (II) 4,13,12,11,1 4,23,22,21,2 4,33,32,31,3 4,43,42,41,4 OK A = Agent B = Breeze G = Glitter, Gold OK = Safe square P = Pit S = Stench V = Visited W = Wumpus 4,13,12,11,1 4,23,22,21,2 4,33,32,31,3 4,43,42,41,4 OK A VV V Figure 6.4 Two later stages in the progress of the agent. (a)After the third move, with percept [Stench, None, None, None, None]. (b)After the fifth move, with percept [Stench, Breeze, Glitter, None, None]. VV B OK W! B OK V S P ! P ? A W ! S G B

Representation, Reasoning, and Logic Syntax : the possible configurations that constitute sentences Semantics : the facts in the world to which the sentences refer

The logical reasoning Figure 6.5 The connection between sentences and facts is provided by the semantics of the language. The property of one fact following from some other facts is mirrored by the property of one sentence being entailed by some other sentences. Logical inference generates new sentences that are entailed by existing sentences.

Wrong logical reasoning FIRST VILLAGER: We have found a witch. May we burn her? ALL: A witch! Burn her! BEDEVERE: Why do you think she is a witch? SECOND VILLAGER: She turned me into a newt. BEDEVERE: A newt? SECOND VILLAGER (after looking at himself for some time): I got better. ALL: Burn her anyway. BEDEVERE: Quiet! Quiet! There are ways of telling whether she is a witch. BEDEVERE: Tell me … What do you do with witches? ALL: Burn them. BEDEVERE: And what do you burn, apart from witches? FOURTH VILLAGER: … Wood? BEDEVERE: So why do witches burn? SECOND VILLAGER: (pianissimo) Because they’re made of wood? BEDEVERE: Good. ALL: I see. Yes, of course. BEDEVERE: So how can we tell if she is made of wood? FIRST VILLAGER: Make a bridge out of her. BEDEVERE: Ah … but can you not also make bridges out of stone? ALL: Yes, of course … um … er … BEDEVERE: Does wood sink in water? ALL: No, no, it floats. Throw her in the pond. BEDEVERE: Wait. Wait … tell me, what also floats on water? ALL: Bread? No, no no. Apples … gravy … very small rocks … BEDEVERE: No, no no. KING ARTHUR: A duck! (They all turn and look at ARTHUR. BEDEVERE looks up very impressed.) BEDEVERE: Exactly. So … logically … FIRST VILLAGER (beginning to pick up the thread): If she.. Weight the same as a duck … she’s made of wood. BEDEVERE: And therefore? ALL: A witch!

Ontological and epistemological commitments Ontological commitments :: to do with the nature of reality –Propositional logic(true/false), Predicate logic, Temporal logic Epistemological commitments :: to do with the possible states of knowledge an agent can have using various types of logic – degree of belief – fuzzy logic

Commitments LanguageOntological Commitment (What exists in the world) Epistemological Commitment (What an agent believes about facts) Propositional logic First-order logic Temporal logic Probability theory Fuzzy logic facts facts, objects, relations times facts degree of truth true/false/unknown degree of belief 0 … 1 Formal languages and their and ontological and epistemological commitments

Inference Rules for propositional logic  Modus Ponens or Implication-Elimination: (From an implication and the premise of the implication, you can infer the conclusion.)  And-Elimination: (From a conjunction, you can infer any of the conjuncts.)  And-Introduction: (From a list of sentences, you can infer their conjunction.)  Or-Introduction: (From a sentence, you can infer its disjunction with anything else at all.)  Double-Negation Elimination: (From a doubly negated sentence, you can infer a positive sentence.)  Unit Resolution: (From a disjunction, if one of the disjuncts is false, then you can infer the other one is true.) Figure 6.13 Seven inference for propositional logic. The unit resolution rule is a special case of the resolution rule, which in turn is a special case of the full resolution rule for first-order logic discussed in Chapter 9.   => ,  i i  1   2  …   n  1,  2, …,  n  1   2  …   n ii      ,          ,       =>    => ,  =>   Resolution: (This is the most difficult. Because  cannot be both true and false, one of the other disjucts must be true in one of the premises. Or equivalently, implication is transitive.) or equivalently

Complexity of propositional inference NP-complete Monotonicity –If KB 1 ╞  then (KB 1 ∪ KB 2 ) ╞  Horn clause logic – polynomial time complexity – P 1 ∧ P 2 ∧ …. ∧ P n ⇒ Q

Wumpus world Initial state ~S1,1 ~B1,1 ~S2,1 B2,1 S1,2 ~B1,2 Rule R1: ~S1,1 -> ~W1,1 & ~W1,2 & ~W2,1 R2: ~S2,1 -> ~W1,1 & ~W2,1 & ~W2,2 & ~W3,1 R3: ~S1,2 -> ~W1,1 & ~W1,2 & ~W2,2 & ~W1,3 R4: S1,2 -> W1,3 V W1,2 V W2,2 V W1,2

Finding the wumpus Inference process – Modus ponens : ~ S 1,1 and R 1  ~ W1,1 & ~W1,2 & ~W2,1 – And-Elimination ~W1,1 ~W1,2 ~W2,1 – Modus ponens and And-Elimination: ~W2,2 ~W2,1 ~W3,1 – Modus ponens S1,2 and R4  W1,3 V W1,2 V W2,2 V W1,1

Inference process(cont.) – unit resolution ~W1,1 and W1,3 V W1,2 V W2,2 V W1,1  W1,3 V W1,2 V W2,2 – unit resolution ~W2,2 and W1,3 V W1,2 V W2,2  W1,3 V W1,2 – unit resolution ~W1,2 and W1,3 V W1,2  W1,3

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}