First-Order Logic Chap8 1. Pros and Cons of Prop. Logic

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1 Chapter8 First-Order Logic Chap8 1 Pros and Cons of Prop. Logic PL is declarative Knowledge and inference are separate and inference is entirely domain-independent. PL is compositional Meaning of a sentence is a function of the meaning of its parts PL allows partial/ disjunctive/negated information The meaning of PL is context-independent. PL has very limited expressive power e.g. cannot say pits cause breezes in adjacent squares except by writing one sentence for each square Chap8 2 1

2 First-Order Logic PL assumes world contains facts. FOL= PL + predicates, quantifiers, variables, and equality FOL assumes the world contains Objects: things with individual identities e.g. people, numbers, blocks A, B, C, D Properties: distinguishing objects from others e.g. red, round, prime, smelly, Inter-relations among objects - Functions : a special kind of relation in which there is only one value for a given input. e.g. father of, best friend, one more than, - Relations e.g. brother of, bigger than, on(d, A), on(a, Table), Chap8 3 Logics in General Ontological Commitment: What it assums about the nature of reality. Epistemological Commitment: The possible states of knowledge that it allows wrt each fact Chap8 4 2

3 Examples: Conceptualization One plus two equals three - Objects: one, two, three, one plus two - Function: plus - Relation: equals Squares neighboring the wumpus are smelly - Objects: wumpus, square - Property: smelly - Relations: neighboring Evil King John ruled England in Objects: John, England, Properties: evil, king - Relations: ruled All NTU students and their parents are smart Chap8 5 Models for FOL: Example 5 objects: Richard, John, 2 binary relations: brother, on head 3 unary relations: person, king, crown 1 unary function: left leg Chap8 6 3

4 Syntax of FOL Sentence Atomic Complex Complex (Sentence) Sentence Atomic Predicate( Term, K ) Term = Term Term Function(T erm, K ) Connective Quantifier Constant A X Sentence Constant Variable John Variable a x sk Connective Quantifier Variable, K Sentence Predicate Before HasColor Raining L K Functio n Mother LeftLegOf 1 Sentence L Chap8 7 Semantics of FOL 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 Chap8 8 4

5 Sentences and Terms Sentence: represent a fact - Atomic sentence: a predicate symbol followed by a parenthesized list of terms. i.e. Predicate(Term, ) - Complex sentence: constructed via logical connectives - Quantified sentence: expressing properties of entire collections of objects, rather than having enumerate the objects by name. Term: represent an object - Constant (0-ary function constant) - Variable (0-ary function variable) - Function(Term, ) Chap8 9 Sentences and Terms (cont.) Interpretation: - Each constant names exactly one object. - Not all objects have names. - Some objects have multiple names. - A relation is defined as a set of tuples of objects that satisfy it, e.g. the relation of brotherhood {< King John, Richard the Lionheart>, <Richard the Lionheart, King John>} - An n-ary function maps n objects into another object Chap8 10 5

6 FOL Examples Wendy is tall. Durante s nose is big. John loves his dog. tall(wendy) big(nose - of(durante)) loves(john,dog - of(john)) John is the brother of Richard and vice versa. Brother(Richard,John) Brother(John,Richard) Chap8 11 FOL Examples (cont.-1) All kings are persons. x Kings(x) Persons(x) x Kings(x) Persons(x) correct wrong All the following must be true: (correct) Richard is a king Richard is a person. King John is a king King John is a person. Richard s left leg is a king Richard s left leg is a person. John s left leg is a king John s left leg is a person. The crown is a king The crown is a person. (wrong) Richard is a king Richard is a person. King John is a king King John is a person. Richard s left leg is a king Richard s left leg is a person. John s left leg is a king John s left leg is a person. The crown is a king The crown is a person Chap8 12 6

7 FOL Examples (cont.-2) King John has a crown on his head. x Crown(x) OnHead(x,John) x Crown(x) OnHead(x,John) correct wrong At least one of the following must be true: (correct) Richard is a crown Richard is on John s head. King John is a crown King John is on John s head. Richard s left leg is a crown Richard s left leg is on John s head. John s left leg is a crown John s left leg is on John s head. The crown is a crown The crown is on John s head. (wrong) Richard is a crown Richard is on John s head. King John is a crown King John is on John s head. Richard s left leg is a crown Richard s left leg is on John s head John s left leg is a crown John s left leg is on John s head. The crown is a crown The crown is on John s head Chap8 13 Representing Sentences in FOL One plus two equals three. Squares neighboring the wumpus are smelly. Evil King John ruled England in Spot is a cat. All cats are mammals. Spot has a sister who is a cat. A person's brother has that person as a sibling. Everybody loves somebody. There is someone who is loved by everyone. Everyone likes ice cream. There is no one who does not like ice cream. Spot has at least two sisters Chap8 14 7

8 Compound Sentences Negation loves(john,dog - of(john)) Conjunction Disjunction loves(john,dog - of(john)) loves(john,country - of(john)) i [odd( i) even( i)] Implication loves(john,country - of(john)) loves(john,dog - of(john)) The truth or falsity of a compound sentence s can be determined from the truth or falsity of the component sentences of s. atoms + logical connectives Predicate Logic Chap8 15 Quantification Universal quantification - the sentence remains true for all values of the variable. loves(john,everything) - John loves everything. x. loves(john,x) - Everything loves everything. x. y. loves(x,y) xy. loves(x, y) - John loves all fuzzy things. x. fuzzy(x) loves(john,x) - All numbers are either odd or even. i [odd(i) even(i)] Existential quantification - the sentence is true for some value(s) of the variable. - John loves something. x. loves(john,x) Chap8 16 8

9 Quantifiers Quantifiers and can be thought of as the infinitary versions of and respectively. A sentence x. p(x) holds in a model M if and only if p(z) holds for every object Z in the domain of discourse. Similarly, a sentence x. p(x) holds in a model M if and only if there is some object Z for which p is valid. (Ex1) If a person is the parent of another person, then the other person is the child of the person. (Ex2) Everybody loves somebody. x, y Parent(x, y) Child(y, x) x y Loves(x, y) (Ex3) There is someone who is loved by everyone. x y Loves(y, x) Chap8 17 Quantifiers (cont.) de Morgan's Laws For any two sentences p and q, the following two expressions are equivalent. (p q) p q Similarly, the following two are equivalent. (p q) p q Negation involving quantifiers x P x P x P x P x P x P x P x P Chap8 18 9

10 Compositional Semantics Given a model M, FOL allows us to determine whether a sentence φ is true or false relative to an interpretation I and a variable assignment U. The truth/falsity of any atom is defined by M. φ φ is true in M iff is not true in M. a a1 is true iff is true in M and is true in M. 1 a 2 a 1 is true iff at least one of and is true in M. 1 a 2 a 2 a a Chap8 19 A common mistake to avoid Typically, is the main connective with Common mistake: using as the main connective with e.g. Everyone at NUS is smart. x At(x,NUS) Smart(x) --- (O) x At(x,NUS) Smart(x) --- (X) means Everyone is at NUS and everyone is smart Chap

11 A common mistake to avoid Typically, is the main connective with Common mistake: using as the main connective with e.g. Everyone at NUS is smart. x At(x,NUS) Smart(x) --- (O) x At(x,NUS) Smart(x) --- (X) means Everyone is at NUS and everyone is smart Chap8 21 Another common mistake to avoid Typically, is the main connective with Common mistake: using as the main connective with e.g. Someone at NUS is smart. x At(x,NUS) Smart(x) --- (O) x At(x,NUS) Smart(x) --- (X) is true if there is anyone who is not at NUS! Chap

12 Equality Equality: make statements to the effect that two terms refer to the same object. Father(Joh n) = Henry (Ex1) Spot has at least two sisters. x,y Sister(Spot, x) Sister(Spot, y) x,y Sister(Spot, x) Sister(Spot, y) (x = y) (Ex2) There is exactly one King. x King(x) y King(y) x = y! x King(x) wrong correct Chap8 23 Extensions Higher-order logic - quantify over relations and functions e.g. x, y (x = y) ( p p(x) p(y)) Two objects are equal if and only if all properties applied to them are equivalent. e.g. f, g (f = g) ( x f(x) = g(x)) Two functions are equal if and only if they have the same value for all arguments. Lambda calculus - introduce anonymous functions e.g. 2 2 (λλxy x y )(25,24) Uniqueness quantifier:! = = Chap

13 Using First-Order Logic The kinship domain - One s mother is one s female parent. m, c Mather(c) = m Female(m) Parent(m, c) - One s husband is one s male spouse. w, h Husband(h, w) Male(h) Spouse(h,w ) - A sibling is another child of one s parent. x, y Sibling(x, y) x y p Parent(p, x) Parent(p, y) Chap8 25 Using First-Order Logic(cont.-1) The set domain Constant: {} Unary predicate: Set Binary predicate: x s (member) s 1 s 2 (subset) Binary functions: s 1 s 2 (intersection) s 1 s 2 (union) {x s} (adjoin) The only sets are empty set and those made by adjoining something to a set. s Set(s) (s = {}) ( x,s2 Set(s2) s = {x The empty set has no elements adjoined into it. x, s {x s } = {} s2 }) A set is a subset of another if and only if all of the first set s members are members of the second set. s1, s2 s1 s2 ( x x s1 x s2) Chap

14 Interacting with FOL KB Suppose a wumpus-world agent is using an FOL KB And perceives a stench and a breeze, glitter at t = 5 Tell(KB, percept([stench, Breeze, Glitter, None, None], 5)) Ask(KB, a BestAction(a, 5)) Does the KB entail any particular actions at t = 5? Answer: Yes, {a/grab} Ask(KB, S) returns some/all σ (substitution, binding list) such that KB = Sσ Chap8 27 KB for the Wumpus World Percept t, s, g,m, c Percept([s, Breeze, g, m, c], t) Breeze(t) t, s, b,m, c Percept([s, b, Glitter, m, c], t) Glitter(t) Reflex t Glitter(t) BestAction(Grab, t) Environment x, y, a, b Adjacent([x,y], [a,b]) [a,b] {[x+1, y], [x-1, y], [x, y+1] [x, y-1]} Properties of locations s, t At(Agent, s, t) Breeze(t) Breezy(s) Chap

15 KB for the Wumpus World (cont.) Locations are breezy near a pit. - Diagnostic rules (infer cause from effect) s Breezy(s) r Adjacent(r, s) Pit(r) - Causal rules (infer effect from cause) r Pit(r) [ s Adjacent(r, s) Breezy(s)] s [ r Adjacent(r, s) Pit(r) ] Breezy(s) Chap8 29 Knowledge Engineering in FOL 1. Identify the task 2. Assemble the relevant knowledge 3. Decide on a vocabulary of predicates, functions, and constants 4. Encode general knowledge about the domain 5. Encode a description of the specific problem instance 6. Pose queries to the inference procedure and get answers 7. Debug the knowledge base Chap

16 The Electronic Circuits Domain One-bit full adder Chap8 31 The Electronic Circuits Domain (cont.-1) Identify the task Does the circuit actually add properly? (circuit verification) Assemble the relevant knowledge Composed of wires and gates; Types of gates (AND, OR, XOR, NOT) Connections between terminals Irrelevant: size, shape, color, cost of gates Decide on a vocabulary Alternatives: Type(X 1 ) = XOR Type(X 1, XOR) XOR(X 1 ) Chap

17 The Electronic Circuits Domain (cont.-2) Encode general knowledge of the domain - t 1,t 2 Connected(t 1, t 2 ) Signal(t 1 ) = Signal(t 2 ) - t Signal(t) = 1 Signal(t) = t 1,t 2 Connected(t 1, t 2 ) Connected(t 2, t 1 ) - g Type(g) = OR Signal(Out(1,g)) = 1 n Signal(In(n,g)) = 1 - g Type(g) = AND Signal(Out(1,g)) = 0 n Signal(In(n,g)) = 0 - g Type(g) = XOR Signal(Out(1,g)) = 1 Signal(In(1,g)) Signal(In(2,g)) - g Type(g) = NOT Signal(Out(1,g)) Signal(In(1,g)) Chap8 33 The Electronic Circuits Domain (cont.-3) Encode the specific problem instance Type(X 1 ) = XOR Type(A 1 ) = AND Type(O 1 ) = OR Type(X 2 ) = XOR Type(A 2 ) = AND Connected(Out(1,X 1 ),In(1,X 2 )) Connected(In(1,C 1 ),In(1,X 1 )) Connected(Out(1,X 1 ),In(2,A 2 )) Connected(In(1,C 1 ),In(1,A 1 )) Connected(Out(1,A 2 ),In(1,O 1 )) Connected(In(2,C 1 ),In(2,X 1 )) Connected(Out(1,A 1 ),In(2,O 1 )) Connected(In(2,C 1 ),In(2,A 1 )) Connected(Out(1,X 2 ),Out(1,C 1 )) Connected(In(3,C 1 ),In(2,X 2 )) Connected(Out(1,O 1 ),Out(2,C 1 )) Connected(In(3,C 1 ),In(1,A 2 )) Chap

18 The Electronic Circuits Domain (cont.-4) Pose queries to the inference procedure What are the possible sets of values of all the terminals for the adder circuit? i 1,i 2,i 3,o 1,o 2 Signal(In(1,C_1)) = i 1 Signal(In(2,C 1 )) = i 2 Signal(In(3,C 1 )) = i 3 Signal(Out(1,C 1 )) = o 1 Signal(Out(2,C 1 )) = o 2 Debug the knowledge base May have omitted assertions like Chap8 35 Summary First-order logic: - objects and relations are semantic primitives - syntax: constants, functions, predicates, equality, quantifiers Increased expressive power sufficient to define wumpus world Chap