Artificial Intelligence CS 165A Thursday, November 8, 2007 Inference in FOL (Ch 9) Brief midterm review Today 1 1 1

What we’ve been talking about Complete and sound inference procedures New inference rules: Universal Instantiation (gets rid of ) Existential Instantiation (gets rid of ) Existential Introduction (adds ) Generalized Modus Ponens Generalized (First-Order) Resolution Complete but semidecidable Unification Finds the substitution(s) necessary to make two sentences match Conjunctive normal form (CNF)

FOL example Domain elements: {Bob, Alice, Carol, Ted} x Sleepy(x) Sleepy(Bob)  Sleepy(Alice)  Sleepy(Carol)  Sleepy(Ted) x Hungry(x) Hungry(Bob)  Hungry(Alice)  Hungry(Carol)  Hungry(Ted) Universal Instantiation: x Sleepy(x) Sleepy(Alice) Existential Instantiation: x Hungry(x) Hungry(k) where k is a constant (not a variable!) Existential Introduction: AtHome(Ted) x AtHome(x)

Two versions of Generalized Resolution Disjunctions For literals pi and qi , where UNIFY(pj , qk) =  Implications For atomic sentences pi, qi, ri, si , where UNIFY(pj , qk) = 

Thursday quiz Note: Everyone gets a 100% for last Thursday! If the domain is natural numbers (0, 1, 2, ...), what is a general unifier (if one exists) for these two atomic sentences: GreaterThan (x, 7) GreaterThan (Squared (y), y) E.g.,  = { a/13, b/4 } What new (to us) dimension does situational calculus allow us to represent and reason about?

Conjunctive normal form (CNF) First we must convert all sentences to Conjunctive Normal Form (CNF) A CNF sentence is a disjunctions of literals Literals: Possibly negated propositions, variables, constants, or predicates So each sentence in the KB is a disjunction of literals, e.g.: P(x)  Q(y) Dog(x)  Cat(y) Big(x)  Slow(x)  Young(x) The KB itself is a conjunction of disjunctions of literals, e.g.: P(x)Q(y)  Dog(x)Cat(y)  Big(x)Slow(x)Young(x)

Another “canonical” form: INF Implicative Normal Form (INF) Each sentence in the KB is an implication with a conjunction of atoms on the left and a disjunction of atoms on the right, e.g.: P(x)  Q(x) P(x)  Q(x)  R(x)  S(x) INF and CNF are logically equivalent

Conversion to Conjunctive Normal Form (CNF) Replace (P  Q) with (P  Q) and (Q  P) Eliminate implications: Replace (P  Q) with (P  Q) Move  inwards: , , , (P  Q), (P  Q) Standardize variables apart: x P(x)  x Q(x) becomes x1 P(x1)  x2 Q(x2) [Give all variables different names] Move quantifiers left in order: x P(x)  y Q(y) becomes x y P(x)  Q(y) Eliminate  by “Skolemization” (coming) Drop universal quantifiers Distribute  over , e.g.: (P  Q)  R becomes (P  R)  (Q  R) [What about (P  Q)  R ?] Flatten nesting: (P  Q)  R becomes P  Q  R

Conversion to Implicative Normal Form (INF) First convert to CNF Convert disjunctions to implications: negative literals to the left, positive literals to the right P  Q  R  S becomes P  Q  R  S P  Q becomes P  Q  False R  S becomes True  R  S Remember that P(x) is logically equivalent to True  P(x) and P(x) is logically equivalent to P(x)  False

Skolemization To “Skolemize” is to remove existential quantifiers by elimination Existential elimination when x is on the outside x Sleepy(x) …becomes… Sleepy(RipVanWinkle) [If what?] More complicated when inside a universal quantifier x Person(x)  y Heart(y)  Has(x, y) “Everyone has a heart” x Person(x)  Heart(H1)  Has(x, H1) “Everyone has the heart H1” Rather, introduce a “Skolem function” H(x) x Person(x)  Heart(H(x))  Has(x, H(x)) where H() does not appear elsewhere in the KB Arguments of the Skolem function: all enclosing universally quantified variables

Skolemization examples By Existential Elimination, y MajorsIn(y, Sociology) MajorsIn(Somedude, Sociology) But it’s not true that x y MajorsIn(y, x) x MajorsIn(Somedude, x) Rather, x y MajorsIn(y, x) x MajorsIn(P(x), x) where P(x) refers to a different constant for every x x y Student(x)  TakesCourses(x)  KnowsAbout(x, y) x Student(x)  TakesCourses(x)  KnowsAbout(x, S(x))

Simple examples How to put these into CNF and INF? CNF INF P(x)  Q(x) P(x)  Q(x) P(x) True  P(x) P(x)  Q(x) True  P(x)  Q(x) P(x)  Q(y) True  P(x)  Q(y) P(x)  Q(y) Q(y)  P(x) P(x), Q(x) True  P(x), True  Q(x)  P(x)  Q(x) P(x) P(x)  Q(x) P(x)  Q(y) P(x)   Q(y) P(x)  Q(x) In practice, we can leave out the “True ” in these sentences

Moving  inwards x S(x)  (is equivalent to…) x S(x) x S(x)  x Hungry(x)   x Hungry(x) x S(x)  x S(x)  x Hungry(x)  x Hungry(x) (P  Q)  (P  Q) (P  Q)  (P  Q) [In CNF that’s two sentences!]

Generalized (first-order) resolution Disjunctions For literals pi and qi , where UNIFY(pj , qk) =  Examples: Note: p and q do not unify Rather, p and q unify pj: Engineer(x) qj: Engineer(Bill)  = { x/Bill } pj: Loves(x, Broccoli) qj: Loves(Joe, y)  = { x/Joe, y/Broccoli }

Generalized resolution example KB: Rich(x)  Famous(x) Rich(x)  Content(x) Famous(x)  Happy(x) Content(x)  Happy(x) Rich(Bob) Is Bob happy? ASK(KB, Happy(Bob)) GMP can’t answer this GR can First put KB in CNF KB in CNF: Rich(x)  Famous(x) Rich(x)  Content(x) Famous(x)  Happy(x) Content(x)  Happy(x) Rich(Bob) and the negated query: Happy(Bob) What’s this method called? Proof by contradiction Refutation

Example (cont.) QED Rich(x)  Famous(x) Rich(x)  Content(x) Famous(x)  Happy(x) Content(x)  Happy(x) Rich(Bob) Happy(Bob) Example (cont.) Rich(x)  Content(x) Content(x)  Happy(x) Rich(x)  Happy(x) { } Rich(Bob) Happy(Bob) {x/Bob} Happy(Bob) False { } QED

Forward and Backward Chaining A reasoning program needs A language for representing knowledge – First-Order Logic Rules – just one, Generalized Resolution Control mechanism– ??? Two general approaches to control Start with the KB and generate new conclusions (which can enable more inferences to be made) E.g., when a new fact is added to the KB Given what we want to prove, find implication sentences that would allow us to conclude it, and attempt to establish their premises E.g., when a goal is to be proved

Forward and Backward Chaining S G Forward chaining  S G Backward chaining 

Forward and Backward Chaining Forward chaining Data driven or data directed New version of TELL(KB, p) Add the sentence p, then apply inference rules to the updated KB until no more rules apply (“chaining” – “chain reaction”) Backward chaining Goal oriented ASK(KB, q) If SUBST(, q) is in KB, return q' = SUBST(, q) Else, find implication sentences p  q then set p as a subgoals Keep doing this, working “backwards” If p is not in KB, look for r  p, then set r as a subgoal Etc….. Backward chaining is the basis for logic programming (e.g., Prolog)

Midterm Review Tuesday during class Closed book exam Please be on time! Closed book exam You may bring one sheet of paper with notes (8.5” x 11”, both sides) Test paper will be supplied (don’t need to bring your own) Calculator not necessary What are you responsible for? Possibly anything that has been covered in the reading (textbook through Ch. 8 and articles), lectures, discussion sessions, and homework assignments But you should probably focus on the lectures….

You will be given… Inference rules for Propositional logic FOL Modus ponens, and-elimination, and-introduction, or-introduction, double-negation elimination, unit resolution, resolution FOL Universal elimination, existential elimination, existential introduction Generalized modus ponens Generalized (first-order) resolution Procedure to convert logic sentences to CNF These are posted on the course web

Examples of things you might be asked Which search algorithm(s) would be most appropriate in this particular situation? What are the main difference(s) between these two search algorithms What does it mean for a search alg. to be optimal? To be complete? Are these heuristics admissible or not? Do {iterative deepening search, A*, minimax, expectimax, …} on this problem. Show the truth table for this propositional logic sentence. Are these logic sentences satisfiable, unsatisfiable, or valid? What does it mean for an inference procedure to be optimal? To be complete? Does the KB entail this sentence S? How are these terms unified? Apply certain inference rules to this KB.