Introduction to Artificial Intelligence 2 nd semester 2016/2017. Chapter 8: First-Order Logic (FOL)

Transcription

1 Introduction to Artificial Intelligence 2 nd semester 2016/2017 Chapter 8: First-Order Logic (FOL) Mohamed B. Abubaker Palestine Technical College Deir El-Balah 1

2 Introduction Propositional logic is used as a representation language to illustrate the basic concepts of logic and knowledge-based agents. is a declarative language because its semantics is based on a truth relation between sentences and possible worlds. is a compositional language The meaning of a sentence is a function of the meaning of its parts The meaning of S 1,4 S 1,2 is related to the meanings of S 1,4 and S 1,2 lacks the expressive power to concisely describe an environment with many objects Cannot say: Squares adjacent to pits are breezy. write a separate rule about breezes and pits for each square, such as 2

3 First-Order Logic (FOL) formal and natural languages We can adopt the foundation of propositional logic and build a more expressive logic on that foundation, borrowing representational ideas from natural language. FOL assumes the world contains: Objects: people, houses, numbers, colors, squares, pits, wumpuses, baseball games, Relations: these can be unary relations or properties such as red, round, prime, or more general n-ary relations such as brother of, bigger than, part of, comes between, Functions: (relations in which there is only one value for a given input) father of, best friend, one more than, plus, 3

4 First-Order Logic (FOL) formal and natural languages The language of first-order logic is built around objects and relations Examples One plus two equals three Objects: one, two, three, one plus two; Relation: equals; Function: plus. ( One plus two is a name for the object that is obtained by applying the function plus to the objects one and two. Three is another name for this object.) Squares neighboring the wumpus are smelly. Objects: wumpus, squares; Property: smelly; Relation: neighboring. 4

5 Formal Languages 5

6 Relationships Models for first-order logic 6

7 First-order Logic (FOL) Syntax 7

8 First-order Logic (FOL) More on Syntax Three kinds of Symbols Constants: objects Predicate: relations Function: functions (can return values other than truth values) Predicates and Functions have arity. Symbols have an interpretation Terms: LeftLeg(Ali), instead of using a constant symbol. Atomic Sentences states facts: Brother(Omar,Ali) P(x,y) is read as x is a P of y Complex Sentences: Brother(Omar,Ali) Brother(Ali,Omar) / King(Omar) King(Ali) 8

9 First-order Logic (FOL) More on Syntax Universal Quantifiers: All kings are persons x King(x) Person(x) is usually pronounced For all.... For all x, if x is a king, then x is a person Typically, is the main connective with A common mistake is to use conjunction instead of implication 9

10 First-order Logic (FOL) More on Syntax Existential Quantifiers: Make a statement about some object in the universe without naming it King Ali has a crown on his head x Crown(x) OnHead(x,Ali) x is pronounced There exists an x such that... or For some x.... x P is true in a given model if P is true in at least one extended interpretation that assigns x to a domain element. Typically, ^ is the main connective with! x means There exists a unique x There exists one and only one x There exists exactly one x Sometimes! is written as 1 10

11 First-order Logic (FOL) Properties of quantifiers x y is the same as y x x y is the same as y x x y is not the same as y x x y Teaches(x,y) For everyone ( all x ) there is someone ( exists y ) whom they teach. i.e, Everybody teaches somebody There might be a different y for each x (y is inside the scope of x) y x Teaches(x,y) There is someone ( exists y ) whom everyone teaches ( all x ). i.e There is someone who is taught by everyone Every x teaches the same y (x is inside the scope of y) 11

12 First-order Logic (FOL) Connections between and Asserting that all x have property P is the same as asserting that does not exist any x that does not have the property P x Likes(x, AI class) is equivalent to x Likes(x, AI class) Everyone likes AI class means that there is no one who does not like AI class Asserting that there exists an x with property P is the same as asserting that not all x do not have the property P x Likes(x, IceCream) is equivalent to x Likes(x, IceCream) The De Morgan rules 12

13 First-order Logic (FOL) Equality term 1 = term 2 is true under a given interpretation if and only if term 1 and term 2 refer to the same object. Father (Ali)=Ahmed To say that Omar has at least two brothers, we would write x,y Brother(x,Omar) Brother(y,Omar) (x=y) x,y Brother(x,Omar) Brother(y,Omar) it is true in the model where Omar has only one brother. 13

14 First-order Logic (FOL) More facts Omar has only two brothers, Ali and Anas Brother(Ali,Omar) Brother(Anas,Omar) Ali = Anas x Brother(x,Omar) (x=ali x=anas) No region in South America borders any region in Europe. x,y In(x, South America) ^ In(y, Europe) border(x, y) No two adjacent countries have the same map color. x,y Country(x)^Country(y)^Adjacent(x,y) (Color(x) = Color(y)) ^ (x=y) 14

15 Using First-order Logic (FOL) Assertions and queries in first-order logic 15

16 Using First-order Logic (FOL) Kinship Domain The son of my father is my brother; One s grandmother is the mother of one s parent; etc.. Domain: People Unary predicates: Male, Female Relations: Parent, Sibling, Brother, Sister, Child, Daughter, Son, Spouse, Wife, Husband, Grandparent, Grandchild, Cousin, Aunt, and Uncle Functions: Mother, Father 16

17 Using First-order Logic (FOL) Kinship Domain one s mother is one s female parent: m,c Mother(c)=m Female(m) Parent(m,c) Male and female are disjoint categories: x Male(x) Female(x) Parent and child are inverse relations: p,c Parent(p,c) Child(c,p) A grandparent is a parent of one s parent: g,c Grandparent(g,c) p Parent(g,p) Parent(p,c) 17

18 Using First-order Logic (FOL) Wumpus World The wumpus agent receives a percept vector with five elements The corresponding first-order sentence stored in the knowledge base must include both the percept and the time at which it occurred. Percept([Stench,Breeze,Glitter,None,None],5) The actions in the wumpus world can be represented by logical terms: Turn(Right), Turn(Left), Forward, Shoot, Grab, Climb To determine which is best, the agent program executes the query ASKVARS( a BestAction(a,5)) 18

19 Using First-order Logic (FOL) Wumpus World The raw percept data implies certain facts about the current state t,s,b,m,c Percept([s,b,Glitter,m,c ],t) Glitter(t) Simple reflex actions: t Glitter(t) BestAction(Grab,t) Instead of encoding: Adjacent(Square 1,1, Square 2,1 ) Adjacency of any two squares can be defined as: x,y,a,b Adjacent([x,y],[a,b]) (x = a (y = b 1 y = b + 1)) (y = b (x = a 1 x = a + 1)). The agent s location changes over time: At(Agent,s,t) to mean that the agent is at square s at time t 19

20 Using First-order Logic (FOL) Wumpus World We can then say that objects can only be at one location at a time: x,s1,s2,t At(x,s1,t) At(x,s2,t) s1 = s2 if the agent is at a square and perceives a breeze, then that square is breezy: s,t At(Agent,s,t) Breeze(t) Breezy(s) first-order logic just needs one axiom: s Breezy(s) r Adjacent(r,s) Pit(r) 20

21 First-order Logic (FOL) Knowledge Engineering 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 21

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