Reading: AIMA Chapter 9 (Inference in FOL)
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2 Review HW#4 First Order Logic (aka The Predicate Calculus) Representing knowledge in FOL Reading: AIMA Chapter 9 (Inference in FOL) Exam#1, Tuesday, October 8 th, SC166, 7:00 pm
3 Must interpret variables, constants, functions, and predicate symbols by associating them with objects, functions and relations in the world. Truth values are relative to an interpretation (model) Express John s father is a diplomat. with diplomat(father-of(john))). Express All birds fly. with ( x) bird(x) => fly(x). 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
4 ( x) p(x) means for all objects x in the domain, p(x) is true that is, it is true in a model m iff p is true with x being each possible object in the model example: All boojums are snarks. expressed by ( x) [boojum(x) snark(x)]. where boojum(x) means x is a boojum snark(x) means x is a snark
5 ( x) p(x) means there exist one or more objects x in the domain s.t. p(x) is true that is, it is true in a model m iff P is true with x being some possible object in the model example: Not all snarks are boojums. expressed by ( x) [snark(x) ~boojum(x)]. where boojum(x) means x is a boojum snark(x) means x is a snark
6 <variables> <sentence> Everyone at Clarkson is smart: x [At(x, Clarkson) Smart(x)]. Roughly speaking, equivalent to the conjunction of instantiations of P [At(KingJohn, Clarkson) Smart(KingJohn)] [At(Richard, Clarkson) Smart(Richard)] [At(Clarkson, Clarkson) Smart(Clarkson)]...
7 Typically, is the main connective with Common mistake: using as the main connective with : x At(x, Clarkson) Smart(x) means Everyone is at Clarkson and also everyone is smart
8 <variables> <sentence> Someone at SLU is smart: ( x) [At(x, SLU) Smart(x)]. Roughly speaking, equivalent to the disjunction of instantiations of P [At(KingJohn, SLU) Smart(KingJohn)] [At(Richard, SLU) Smart(Richard)] [At(SLU, SLU) Smart(SLU)]...
9 Typically, is the main connective with Common mistake: using as the main connective with : x At(x, SLU) Smart(x) is true if there is anyone who is not at SLU!
10 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 Loves(x,y) There is a person who loves everyone in the world (including him/herself) y x Loves(x,y) Everyone in the world is loved by at least one person
11 Quantifier duality: each can be expressed using the other; e.g. consider the predicate likes(x,y) meaning X likes Y. Everyone dislikes parsnips. ( x) [~likes(x,parsnips)]. ~( x) [likes(x,parsnips)]. Everyone likes ice cream. ( x) likes(x,icecream) ~( x) [~likes(x,icecream)]. Someone likes persimmons. ( x) likes(x,persimmons) ~( x) ~likes(x,persimmons)].
12 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 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)]
13 1. Choose the task domain or world you want to represent. the real world Clarkson Univ. microcosm Middle Earth Bizarro World internet virtual world without gravity 2. Associate constants in the language with individuals in the world. john book pizza john23 AIbook
14 3. (a) Relations For each relation you want to represent, associate a predicate symbol in the language. Each n-ary predicate symbol denotes a function from D n to {true,false} e.g. cat(x) pizza(x) likes(x,y) isin(x,cat) isa(pizza, food) father-of(x,y) isa(cat, mammal)
15 3. (b) Functions For each function you want to represent, associate a function symbol in the language. Each n-ary function denotes a function from D n to D s.t. for each tuple (x 1,x 2, x n ) there is a unique y in D s.t. f(x 1, x 2,, x n ) = y dad(x) ssn(x) sqrt(x) signal(t) evaluates to the person who is X s father evalutes to X s social security number evalutes to the square root of x evaluates to the value of the signal at time step t
16 question: What is the difference between a function and a relation? relation true or false father(x,y) means X is the father of Y function returns a value in the domain father(x) returns the unique element in the domain who is X s father Can t use the same symbol to mean both
17 4. You can now write as clauses statements that are true in the intended interpretation; this is axiomatizing the domain, and the clauses are axioms KB set of true sentences 5. You can now ask questions about the intended representation, and interpret the answers using the meaning that you have associated with the symbols. KB i α
18 1. John likes pizza. 2. John likes all kinds of food. 3. Steve only likes easy courses. 4. All science courses are hard. 5. Everybody loves somebody. 6. Abraham is the father of Isaac. 7. John gave the book to Mary. 8. There s a book on the table. 9. There s exactly one book on the table.
19 1. John likes pizza. interpretation: constants john, pizza predicate likes(x,y) means x likes y likes(john, pizza). 2. John likes all kinds of food. predicate food(x) means x is food ( x)[ food(x) => likes(john, x) ].
20 3. Steve only likes easy courses. predicates: course(x) means x is a course, easy(x) means x is easy ( x)[ course(x) easy(x) => likes(steve, x) ] If a course is easy, then Steve likes it (but this doesn t say that he doesn t like hard courses). ( x)[ likes(steve, x) => course(x) easy(x) ] The only things that Steve likes are easy courses. (not pizza, not Mary, not hockey, ) ( x)[ course(x) likes(steve, x) => easy(x) ] If it s a course and Steve likes it, then it must be easy. ( x)[ course(x) ~easy(x) => ~likes(steve, x) ] If it s a hard course, then Steve doesn t like it.
21 3. Steve only likes easy courses. Which sentence best captures the meaning? B or C? Answer: C! Why??? B: ( x)[ course(x) easy(x) => likes(steve, x) ] ( x)[ ~ course(x) ~ easy(x) likes(steve, x) ] If it s a course and it s easy then Steve must like it. C: ( x)[ course(x) likes(steve, x) => easy(x) ] ( x)[ ~ course(x) ~ likes(steve, x) easy(x) ] If it s a course and Steve likes it then it must be easy.
22 4. All science courses are hard. predicate: science(x) means x is in a field of science, hard(x) means x is difficult ( x)[ course(x) science(x) => hard(x) ] ( x)[ course(x) science(x) => ~easy(x) ] 5. Everybody loves somebody. ( x)( y) loves(x,y). says that there is one (possibly more than one) person whom everyone, including him/herself, loves. ( y)( x) loves(x,y). says that for every person, you can find at least one person (possibly him/herself) that he/she loves.
23 3. Steve only likes easy courses. 4. PH432 is a course. 5. PH432 is not easy. 6. likes(steve, PH432). Is it possible for Steve to like PH432? Can sentences 3, 4, 5, and 6 all be true? No. Can sentences B, 4, 5, and 6 all be true? Yes. B: ( x)[ course(x) easy(x) => likes(steve, x) ] ( x)[ ~ course(x) ~ easy(x) likes(steve, x) ] If it s a course and it s easy then Steve must like it.
24 C: ( x)[ course(x) likes(steve, x) => easy(x) ] ( x)[ ~ course(x) ~ likes(steve, x) easy(x) ] 7. BW101 is a course. 8. BW101 is easy. 9. likes(steve, BW101). Is it possible for Steve to like BW101? Can sentences C, 7, 8 and 9 all be true? Sure!
25 What about sentence E? E: ( course(x))[ likes(steve, x) => easy(x) ] Not well-formed: can only quantify a variable, not a predicate ( course(x)) What about sentence F? F: ( x)[ likes(steve, x) => easy(course(x)) ] Not well-formed: course(x) is a predicate. Try the substitution test: ( x)[ likes(steve, x) => easy(true) ] doesn t make sense
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