For Monday

1. For Monday • Read “lectures” 6,9-12 of Learn Prolog Now: http://www.coli.uni-saarland.de/~kris/learn-prolog-now/ • Prolog Handout 2

2. Exam 1 • Next Monday • Covers through chapter 9 • Topics include: • History • Agents/Environments • Search • Logic/Inference • No Prolog • Take home handed out Wednesday, due at the exam

3. Resolution • Propositional version. {a Ú b, ¬b Ú c} |- a Ú c OR {¬aÞ b, b Þ c} |- ¬a Þ c Reasoning by cases OR transitivity of implication • First­order form • For two literals pj and qk in two clauses • p1Ú ... pj ... Ú pm • q1Ú ... qk ... Ú qn such that q=UNIFY(pj , ¬qk), derive SUBST(q, p1Ú...pj­1Úpj+1...ÚpmÚq1Ú...qk­1 qk+1...Úqn)

4. Implication form • Can also be viewed in implicational form where all negated literals are in a conjunctive antecedent and all positive literals in a disjunctive conclusion. ¬p1Ú...Ú¬pmÚq1Ú...ÚqnÛ p1Ù... Ù pmÞ q1Ú ...Ú qn

5. Conjunctive Normal Form (CNF) • For resolution to apply, all sentences must be in conjunctive normal form, a conjunction of disjunctions of literals (a1Ú ...Ú am) Ù (b1Ú ... Ú bn) Ù ..... Ù (x1Ú ... Ú xv) • Representable by a set of clauses (disjunctions of literals) • Also representable as a set of implications (INF).

6. Example Initial CNF INF P(x) Þ Q(x) ¬P(x) Ú Q(x) P(x) Þ Q(x) ¬P(x) Þ R(x) P(x) Ú R(x) True Þ P(x) Ú R(x) Q(x) Þ S(x) ¬Q(x) Ú S(x) Q(x) Þ S(x) R(x) Þ S(x) ¬R(x) Ú S(x) R(x) Þ S(x)

7. Resolution Proofs • INF (CNF) is more expressive than Horn clauses. • Resolution is simply a generalization of modus ponens. • As with modus ponens, chains of resolution steps can be used to construct proofs. • Factoring removes redundant literals from clauses • S(A) Ú S(A) -> S(A)

8. Sample Proof P(w)  Q(w) Q(y)  S(y) {y/w} P(w)  S(w) True  P(x)  R(x) {w/x} True  S(x)  R(x) R(z)  S(z) {x/A, z/A} True  S(A)

9. Refutation Proofs • Unfortunately, resolution proofs in this form are still incomplete. • For example, it cannot prove any tautology (e.g. PÚ¬P) from the empty KB since there are no clauses to resolve. • Therefore, use proof by contradiction (refutation, reductio ad absurdum). Assume the negation of the theorem P and try to derive a contradiction (False, the empty clause). • (KB Ù ¬P Þ False) Û KB Þ P

10. Sample Proof P(w)  Q(w) Q(y)  S(y) {y/w} P(w)  S(w) True  P(x)  R(x) {w/x} True  S(x)  R(x) R(z)  S(z) {z/x} S(A)  False True  S(x) {x/A} False

11. Resolution Theorem Proving • Convert sentences in the KB to CNF (clausal form) • Take the negation of the proposed theorem (query), convert it to CNF, and add it to the KB. • Repeatedly apply the resolution rule to derive new clauses. • If the empty clause (False) is eventually derived, stop and conclude that the proposed theorem is true.

12. Conversion to Clausal Form • Eliminate implications and biconditionals by rewriting them. p Þ q -> ¬p Ú q p Û q ­> (¬p Ú q) Ù (p Ú ¬q) • Move ¬ inward to only be a part of literals by using deMorgan's laws and quantifier rules. ¬(p Ú q) -> ¬p Ù ¬q ¬(p Ù q) -> ¬p Ú¬q ¬"x p -> $x ¬p ¬$x p -> "x ¬p ¬¬p -> p

13. Conversion continued • Standardize variables to avoid use of the same variable name by two different quantifiers. "x P(x) Ú$x P(x) -> "x1 P(x1) Ú $x2 P(x2) • Move quantifiers left while maintaining order. Renaming above guarantees this is a truth­preserving transformation. "x1 P(x1) Ú $x2 P(x2) -> "x1$x2 (P(x1) Ú P(x2))

14. Conversion continued • Skolemize: Remove existential quantifiers by replacing each existentially quantified variable with a Skolem constant or Skolem function as appropriate. • If an existential variable is not within the scope of any universally quantified variable, then replace every instance of the variable with the same unique constant that does not appear anywhere else. $x (P(x) Ù Q(x)) -> P(C1) Ù Q(C1) • If it is within the scope of n universally quantified variables, then replace it with a unique n­ary function over these universally quantified variables. "x1$x2(P(x1) Ú P(x2)) -> "x1 (P(x1) Ú P(f1(x1))) "x(Person(x) Þ$y(Heart(y) Ù Has(x,y))) -> "x(Person(x) Þ Heart(HeartOf(x)) Ù Has(x,HeartOf(x))) • Afterwards, all variables can be assumed to be universally quantified, so remove all quantifiers.

15. Conversion continued • Distribute Ù over Ú to convert to conjunctions of clauses (aÙb) Ú c -> (aÚc) Ù (bÚc) (aÙb) Ú (cÙd) -> (aÚc) Ù (bÚc) Ù (aÚd) Ù (bÚd) • Can exponentially expand size of sentence. • Flatten nested conjunctions and disjunctions to get final CNF (a Ú b) Ú c -> (a Ú b Ú c) (a Ù b) Ù c -> (a Ù b Ù c) • Convert clauses to implications if desired for readability (¬a Ú ¬b Ú c Ú d) -> a Ù b Þ c Ú d

16. Sample Clause Conversion "x((Prof(x) Ú Student(x)) Þ($y(Class(y) Ù Has(x,y)) Ù$y(Book(y) Ù Has(x,y)))) "x(¬(Prof(x) Ú Student(x)) Ú($y(Class(y) Ù Has(x,y)) Ù$y(Book(y) Ù Has(x,y)))) "x((¬Prof(x) Ù ¬Student(x)) Ú ($y(Class(y) Ù Has(x,y)) Ù$y(Book(y) Ù Has(x,y)))) "x((¬Prof(x) Ù ¬Student(x)) Ú ($y(Class(y) Ù Has(x,y)) Ù$z(Book(z) Ù Has(x,z)))) "x$y$z((¬Prof(x)Ù¬Student(x))Ú ((Class(y) Ù Has(x,y)) Ù (Book(z) Ù Has(x,z)))) (¬Prof(x)Ù¬Student(x))Ú (Class(f(x)) Ù Has(x,f(x)) Ù Book(g(x)) Ù Has(x,g(x))))

17. Clause Conversion (¬Prof(x)Ù¬Student(x))Ú (Class(f(x)) Ù Has(x,f(x)) Ù Book(g(x)) Ù Has(x,g(x)))) (¬Prof(x) Ú Class(f(x))) Ù (¬Prof(x) Ú Has(x,f(x))) Ù (¬Prof(x) Ú Book(g(x))) Ù (¬Prof(x) Ú Has(x,g(x))) Ù (¬Student(x) Ú Class(f(x))) Ù (¬Student(x) Ú Has(x,f(x))) Ù (¬Student(x) Ú Book(g(x))) Ù (¬Student(x) Ú Has(x,g(x))))

18. Sample Resolution Problem • Jack owns a dog. • Every dog owner is an animal lover. • No animal lover kills an animal. • Either Jack or Curiosity killed Tuna the cat. • Did Curiosity kill the cat?

19. In Logic Form A) $x Dog(x) Ù Owns(Jack,x) B) "x ($y Dog(y) Ù Owns(x,y)) Þ AnimalLover(x)) C) "x AnimalLover(x) Þ ("y Animal(y) Þ ¬Kills(x,y)) D) Kills(Jack,Tuna) Ú Kills(Cursiosity,Tuna) E) Cat(Tuna) F) "x(Cat(x) Þ Animal(x)) Query: Kills(Curiosity,Tuna)

20. In Normal Form A1) Dog(D) A2) Owns(Jack,D) B) Dog(y) Ù Owns(x,y) Þ AnimalLover(x) C) AnimalLover(x) Ù Animal(y) Ù Kills(x,y) Þ False D) Kills(Jack,Tuna) Ú Kills(Curiosity,Tuna) E) Cat(Tuna) F) Cat(x) Þ Animal(x) Query: Kills(Curiosity,Tuna) Þ False

21. Resolution Proof

22. Logic Programming • Also called declarative programming • We write programs that say what is to be the result • We don’t specify how to get the result • Based on logic, specifically first order predicate calculus

23. Prolog • Programming in Logic • Developed in 1970’s • ISO standard published in 1996 • Used for: • Artificial Intelligence: expert systems, natural language processing, machine learning, constraint satisfaction, anything with rules • Logic databases • Prototyping

24. Bibliography • Clocksin and Mellish, Programming in Prolog • Bratko, Prolog Programming for Artificial Intelligence • Sterling and Shapiro, The Art of Prolog • O’Keefe, The Craft of Prolog

25. Working with Prolog • You interact with the Prolog listener. • Normally, you operate in a querying mode which produces backward chaining. • New facts or rules can be entered into the Prolog database either by consulting a file or by switching to consult mode and typing them into the listener.

26. Prolog and Logic • First order logic with different syntax • Horn clauses • Does have extensions for math and some efficiency.

27. The parent Predicate • Definition of parent/2 (uses facts only)%parent(Parent,Child).parent(pam, bob).parent(tom, liz).parent(bob, ann).parent(bob, pat).parent(pat, jim).

28. Constants in Prolog • Two kinds of constants: • Numbers (much like numbers in other languages) • Atoms • Alphanumeric strings which begin with a lowercase letter • Strings of special characters (usually used as operators) • Strings of characters enclosed in single quotes

29. Variables in Prolog • Prolog variables begin with capital letters. • We make queries by using variables: ?- parent(bob,X). X = ann • Prolog variables are logic variables, not containers to store values in. • Variables become bound to their values. • The answers from Prolog queries reflect the bindings.

30. Query Resolution • When given a query, Prolog tries to find a fact or rule which matches the query, binding variables appropriately. • It starts with the first fact or rule listed for a given predicate and goes through the list in order. • If no match is found, Prolog returns no.

31. Backtracking • We can get multiple answers to a single Prolog query if multiple items match: ?- parent(X,Y). • We do this by typing a semi-colon after the answer. • This causes Prolog to backtrack, unbinding variables and looking for the next match. • Backtracking also occurs when Prolog attempts to satisfy rules.

32. Rules in Prolog • Example Prolog Rule: offspring(Child, Parent) :- parent(Parent, Child). • You can read “:-” as “if” • Variables with the same name must be bound to the same thing.

33. Rules in Prolog • Suppose we have a set of facts for male/1 and female/1 (such as female(ann).). • We can then define a rule for mother/2 as follows:mother(Mother, Child) :- parent(Mother, Child), female(Mother). • The comma is the Prolog symbol for and. • The semi-colon is the Prolog symbol for or.

34. Recursive Predicates • Consider the notion of an ancestor. • We can define a predicate, ancestor/2, using parent/2 if we make ancestor/2 recursive.

35. Lists in Prolog • The empty list is represented as []. • The first item is called the head of the list. • The rest of the list is called the tail.

36. List Notation • We write a list as: [a, b, c, d] • We can indicate the tail of a list using a vertical bar: L = [a, b, c,d], L = [Head | Tail], L = [ H1, H2 | T ].Head = a, Tail = [b, c, d], H1 = a, H2 = b, T = [c, d]

37. Some List Predicates • member/2 • append/3

38. Try It • reverse(List,ReversedList) • evenlength(List) • oddlength(List)

39. The Anonymous Variable • Some variables only appear once in a rule • Have no relationship with anything else • Can use _ for each such variable

40. Arithmetic in Prolog • Basic arithmetic operators are provided for by built-in procedures: +, -, *, /, mod, // • Note carefully: ?- X = 1 + 2. X = 1 + 2 ?- X is 1 + 2. X = 3

41. Arithmetic Comparison • Comparison operators: > < >= =< (note the order: NOT <=) =:= (equal values) =\= (not equal values)

42. Arithmetic Examples • Retrieving people born 1950-1960: ?- born(Name, Year), Year >= 1950, Year =< 1960. • Difference between = and =:= ?- 1 + 2 =:= 2 + 1. yes ?- 1 + 2 = 2 + 1. no ?- 1 + A = B + 2. A = 2 B = 1

43. Length of a List • Definition of length/2length([], 0).length([_ | Tail], N) :- length(Tail, N1), N is 1 + N1. • Note: all loops must be implemented via recursion

44. Counting Loops • Definition of sum/3sum(Begin, End, Sum) :- sum(Begin, End, Begin, Sum).sum(X, X, Y, Y).sum(Begin, End, Sum1, Sum) :- Begin < End, Next is Begin + 1, Sum2 is Sum1 + Next, sum(Next, End, Sum2, Sum).

45. The Cut (!) • A way to prevent backtracking. • Used to simplify and to improve efficiency.

46. Negation • Can’t say something is NOT true • Use a closed world assumption • Not simply means “I can’t prove that it is true”

47. Dynamic Predicates • A way to write self-modifying code, in essence. • Typically just storing data using Prolog’s built-in predicate database. • Dynamic predicates must be declared as such.

48. Using Dynamic Predicates • assert and variants • retract • Fails if there is no clause to retract • retractall • Doesn’t fail if no clauses