Logic III. Introduction to Artificial Intelligence CSE 150
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1 Logic III Introduction to Artificial Intelligence CSE 150
2 Outline Last Lecture Propositional Logic and Sound inference rules Introduction First order logic Symbols, Variables, Functions, Predicates This Lecture FOL Continued Quantification Wumpus in FOL Inference in FOL Unification Generalized Modus Ponens Forward and backward chaining Chapter 7 Reading
3 The Bad News The due date for PA2 is next Thursday The Good News Midterm will be Thursday, May 10th
4 Inference rules An inference rule is sound if the conclusion is true in all cases where the premises are true α β Premise Conclusion 1. Modus Ponens 2. And Introduction 3. And Elimination 4. Double Negation
5 Knowledge Base in Wumpus World S i,j : Stench in cell i,j B i,j : Breeze in cell i,j W i,j : Wumpus in cell i,j Percept Sentences S 1,1 B 1,1 S 2,1 B 2,1 S 1,2 B 1,2 Environment Knowledge (can be written before any sensing) R 1 : S 1,1 W 1,1 W 2,1 W 1,2 R 2 : S 2,1 W 1,1 W 2,1 W 2,2 W 3,1 R 3 : B 1,1 P 1,1 P 2,1 P 1,2 R 4 : B 2,1 P 1,1 P 2,1 P 2,2 P 3,1 R 5 : B 1,2 P 1,1 P 1,2 P 2,2 P 1,3...
6 Inference in Wumpus World Initial KB Percept Sentences S 1,1 B 1,1 S 2,1 B 2,1 S 1,2 B 1,2 Environment Knowledge R 1 : S 1,1 W 1,1 W 2,1 W 1,2 R 2 : S 2,1 W 1,1 W 2,1 W 2,2 W 3,1 R 3 : B 1,1 P 1,1 P 2,1 P 1,2 R 4 : B 2,1 P 1,1 P 2,1 P 2,2 P 3,1 R 5 : B 1,2 P 1,1 P 1,2 P 2,2 P 1,3... Some inferences: Apply Modus Ponens to R 1 Add to KB W 1,1 W 2,1 W 1,2 Apply to this AND-Elimination Add to KB W 1,1 W 2,1 W 1,2
7 Backus-Naur Form of Syntax
8 Formalizing Knowledge The first step in formalizing knowledge is to construct a conceptualization, I.e., a representation of the objects in their world, and their relationships Objects Universe of discourse Functions Functional basis set Relations (Predicates) Relational basis set A conceptualization is a triple < U of D, FBS, RBS > Block World: <{A,B,C,D,E}, {HAT, }, {ON, }> Second step is to express conceptualization using Predicate Calculus A B C D E
9 Sets The set of objects defines a Universe of Discourse. The objects are represented by Constant Symbols. A B C D E e.g. For this blocks world, the universe of discourse is {A,B,C,D,E}
10 Relations Def: A binary relation is a set of ordered pairs Example: Consider set of blocks {A,B,C,D,E} A B C D E The ON relation: ON = {<A,B>, <B,C>, <D,E>} The predicate ON(A,B) means, <A,B> ON ON(A,B) is TRUE, but ON(A,C) and ON(C,D) are FALSE
11 Functions A function is a binary relation such that no two distinct members have the same first element If <X, Y> f X is an argument of f Y is the value of f at X Y is the image of f under X Y = f(x) That is, f(x) is equal to some symbol in Universe of Discourse HAT = {<C,B>, <B, A>, <A, NIL>, <E, D>, <D, NIL> } HAT (C) = B HAT(E) = D A HAT(D) = NIL BC D E
12 Universal quantification <variables> <sentence> For all (every) instances of the variables, the <sentence> is true Everyone at UCSD is smart x At (x, UCSD) Smart(x) x P is equivalent to the conjunction of instantiations of P (I.e., all x s in universe of discourse) At (Jonathan, UCSD) Smart(Jonathan) At (RussellImpagliazzo, UCSD) Smart(RussellImpagliazzo) At (BillClinton, UCSD) Smart(BillClinton) At (GeorgeWBush, UCSD) Smart(GeorgWBush) Universe of discourse may be infinite, compare to propositional logic
13 Universal quantification cont. Typically, is the main connective with A common mistake is to use as the main connective with What does the following mean? x At (x, UCSD) Smart(x) Everyone is at UCSD and everyone is smart
14 Some more examples Birds Fly x Bird (x) Fly(x) predicate predicate
15 Some more examples Birds Fly x Bird (x) Fly(x) predicate predicate But what about penguins x Bird (x) Penguin(x) Fly(x) Fish Swim x Fish (x) Swim(x)
16 Existential quantification <variables> <sentence> There exist instances of the <variables> such that the <sentence> is true Someone at Michigan is smart x At (x, Michigan) Smart(x) x P is equivalent to the disjunction of instantiations of P (I.e., all x s in universe of discourse) At (Rodrigo, Michigan) Smart(Rodrigo) At (Koditschek, Michigan) Smart(Koditschek) At (Gore, Michigan) Smart(Gore) At (Bush, Michigan) Smart(Bush)
17 Existential quantification cont. Typically, is the main connective with A common mistake is to use as the main connective with x At (x, UCSD) Smart(x) is true if there is anyone who is not at UCSD.
18 Another example Some CSE150 student is sleeping x CSE150Student (x) Sleeping(x) Note: There could be more than one sleeping student, but this says that there is at least one. To say One CSE150 student is sleeping x CSE150Student (x) Sleeping(x)^ [ y (CSE150(y) Sleeping(y)) (x=y) ] This is cumbersome, so sometimes a shorthand is used called the uniqueness quantifier! x I.e., there exists a unique x
19 Variables must be quantified In a well-formed formula (wff), all variables must be quantified The following are wffs x At (x, UCSD) Smart(x) x At (x, Michigan) Smart(x) The following are not wffs At (x, UCSD) Smart(x) At (x, Michigan) Smart(x) Occasionally, you ll see expressions with variables that are not quantified. In this case, there is an implicit universal quantifier on those variables.
20 Properties of quantifiers x y is the same as y x (why??) x y is the same as y x (why??) x y is not the same as y x Example Loves predicate: Loves(x,y) is true if x loves y x y Loves(x,y) ``There is a person who loves everyone in the world'' y x Loves(x,y) ``Everyone in the world is loved by at least one person''
21 Quantifier Duality Each quantifier (Universal or Existential) can be expressed using the other x Likes(x, IceCream) Everyone likes Ice Cream x Likes(x, IceCream) [the dual] There does not exist anyone who does not like ice cream Nobody dislikes ice cream x Likes(x, BrusselSprouts) Someone likes Brussel sprouts x Likes(x, BrusselSprouts) Not everyone dislikes Brussel sprouts. [the dual]
22 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 E.g., definition of (full) Sibling in terms of Parent x, y Sibling(x,y) [ (x=y) m,f (m=f) Parent(m,x) Parent(f,x) Parent(m,y) Parent(f,y)]
23 Brothers are siblings Fun with sentences x, y Brother(x,y) Sibling(x,y) ``Sibling'' is reflexive x, y Sibling(x,y) Sibling(y,x) One's mother is one's female parent x, y Mother(x,y) (Female(x) Parent(x,y)) A first cousin is a child of a parent's sibling x, y FirstCousin(x,y) p, ps Parent (p,x) Sibling(ps,p) Parent(ps,y)
24 A tough one Every city has a dogcatcher who has been bitten by every dog in town Consider predicates City(x) Dog(x) Dogcatcher(x,y) LivesIn(x,y) Bit(x,y) x City (x) y Dogcatcher(x,y) ( z (Dog(z) LivesIn(x,z) Bit(y,z))
25 Higher Order Logic In first order logic, we can quantify over objects and then represent predicates and functions on objects. In a higher order logic, we can quantify over predicates and functions. For example: Two objects are equal iff all predicates applied to them are equivalent x,y (x=y) ( p p(x) p(y)) Two functions are equal iff they have the same value for all arguments f, g (f=g) ( x f(x) = g(x) ) These are not allowed in first order logic. While higher order logic is more expressive, we do not know how to reason effectively and undecidable
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