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INTRODUCTION TO PROLOG PRINCIPLES OF PROGRAMMING LANGUAGES Norbert Zeh Winter 2018 Dalhousie University 1/44 A set of facts, predicates and relations that are known to hold, and To run a Prolog program, you pose a query. The program

  1. • If T 1 and T 2 are complex terms, they unify if • If none of the above rules applies to T 1 and T 2 , then T 1 and T 2 do not unify. UNIFICATION (2) Formal definition: • Two identical terms unify. • A variable unifies with any other term. • They have the same functor and arity, • Their corresponding arguments unify, and • The resulting variable instantiations are compatible. 10/44

  2. • If T 1 and T 2 are complex terms, they unify if • If none of the above rules applies to T 1 and T 2 , then T 1 and T 2 do not unify. UNIFICATION (2) Formal definition: • Two identical terms unify. • A variable unifies with any other term. • They have the same functor and arity, • Their corresponding arguments unify, and • The resulting variable instantiations are compatible. 10/44

  3. • If T 1 and T 2 are complex terms, they unify if • If none of the above rules applies to T 1 and T 2 , then T 1 and T 2 do not unify. UNIFICATION (2) Formal definition: • Two identical terms unify. • A variable unifies with any other term. • They have the same functor and arity, • Their corresponding arguments unify, and • The resulting variable instantiations are compatible. 10/44

  4. • If none of the above rules applies to T 1 and T 2 , then T 1 and T 2 do not unify. UNIFICATION (2) Formal definition: • Two identical terms unify. • A variable unifies with any other term. • They have the same functor and arity, • Their corresponding arguments unify, and • The resulting variable instantiations are compatible. 10/44 • If T 1 and T 2 are complex terms, they unify if

  5. UNIFICATION (2) Formal definition: • Two identical terms unify. • A variable unifies with any other term. • They have the same functor and arity, • Their corresponding arguments unify, and • The resulting variable instantiations are compatible. 10/44 • If T 1 and T 2 are complex terms, they unify if • If none of the above rules applies to T 1 and T 2 , then T 1 and T 2 do not unify.

  6. UNIFICATION (3) X = Y, Y = a _9341 = a true X = _9341 Y = _9341 _9341 = a X = Y, X = a, Y = b _9341 = a, _9341 = b a = b X = _9341 Y = _9341 _9341 = a 11/44

  7. UNIFICATION (3) X = Y, Y = a _9341 = a true X = _9341 Y = _9341 _9341 = a X = Y, X = a, Y = b _9341 = a, _9341 = b a = b X = _9341 Y = _9341 _9341 = a 11/44

  8. UNIFICATION (3) X = Y, Y = a _9341 = a true X = _9341 Y = _9341 _9341 = a X = Y, X = a, Y = b _9341 = a, _9341 = b a = b X = _9341 Y = _9341 _9341 = a 11/44

  9. UNIFICATION (3) X = Y, Y = a _9341 = a true X = _9341 Y = _9341 _9341 = a X = Y, X = a, Y = b _9341 = a, _9341 = b a = b X = _9341 Y = _9341 _9341 = a 11/44

  10. UNIFICATION (3) X = Y, Y = a _9341 = a true X = _9341 Y = _9341 _9341 = a X = Y, X = a, Y = b _9341 = a, _9341 = b a = b X = _9341 Y = _9341 _9341 = a 11/44

  11. UNIFICATION (3) X = Y, Y = a _9341 = a true X = _9341 Y = _9341 _9341 = a X = Y, X = a, Y = b _9341 = a, _9341 = b a = b X = _9341 Y = _9341 _9341 = a 11/44

  12. UNIFICATION (3) X = Y, Y = a _9341 = a true X = _9341 Y = _9341 _9341 = a X = Y, X = a, Y = b _9341 = a, _9341 = b a = b X = _9341 Y = _9341 _9341 = a 11/44 †

  13. OCCURS CHECK What about the query X = f(X) ? Logically, this should fail because there is no (finite) instantiation of X that makes the two sides equal. In Prolog, this query succeeds with the answer X = f(X) . In the interest of efficiency, Prolog does not check whether a variable occurs in its own replacement. If you want to test for unification with occurs check, use unify_with_occurs_check/2 , so ?- unify_with_occurs_check(X, f(X)). false. 12/44

  14. OCCURS CHECK What about the query X = f(X) ? Logically, this should fail because there is no (finite) instantiation of X that makes the two sides equal. In Prolog, this query succeeds with the answer X = f(X) . In the interest of efficiency, Prolog does not check whether a variable occurs in its own replacement. If you want to test for unification with occurs check, use unify_with_occurs_check/2 , so ?- unify_with_occurs_check(X, f(X)). false. 12/44

  15. OCCURS CHECK What about the query X = f(X) ? Logically, this should fail because there is no (finite) instantiation of X that makes the two sides equal. In Prolog, this query succeeds with the answer X = f(X) . In the interest of efficiency, Prolog does not check whether a variable occurs in its own replacement. If you want to test for unification with occurs check, use unify_with_occurs_check/2 , so ?- unify_with_occurs_check(X, f(X)). false. 12/44

  16. OCCURS CHECK What about the query X = f(X) ? Logically, this should fail because there is no (finite) instantiation of X that makes the two sides equal. In Prolog, this query succeeds with the answer X = f(X) . In the interest of efficiency, Prolog does not check whether a variable occurs in its own replacement. If you want to test for unification with occurs check, use unify_with_occurs_check/2 , so ?- unify_with_occurs_check(X, f(X)). false. 12/44

  17. OCCURS CHECK What about the query X = f(X) ? Logically, this should fail because there is no (finite) instantiation of X that makes the two sides equal. In Prolog, this query succeeds with the answer X = f(X) . In the interest of efficiency, Prolog does not check whether a variable occurs in its own replacement. If you want to test for unification with occurs check, use unify_with_occurs_check/2 , so ?- unify_with_occurs_check(X, f(X)). false. 12/44

  18. BACKTRACKING f(_5137), g(_5137), h(_5137) _5137 = c _5137 = b _5137 = a Y = _5137 true true h(c) h(b) h(a) g(c), h(c) g(a), h(a) g(b), h(b) k(Y) To find the answers to a query, Prolog applies depth-first search with unification. f(X), g(X), h(X). k(X) :- h(c). h(a). g(c). g(b). g(a). f(c). f(b). f(a). database from top to bottom. When searching for a fact or rule that unifies with a goal, it searches the 13/44

  19. BACKTRACKING f(_5137), g(_5137), h(_5137) _5137 = c _5137 = b _5137 = a Y = _5137 true true h(c) h(b) h(a) g(c), h(c) g(a), h(a) g(b), h(b) k(Y) To find the answers to a query, Prolog applies depth-first search with unification. f(X), g(X), h(X). k(X) :- h(c). h(a). g(c). g(b). g(a). f(c). f(b). f(a). database from top to bottom. When searching for a fact or rule that unifies with a goal, it searches the 13/44

  20. BACKTRACKING f(_5137), g(_5137), h(_5137) _5137 = c _5137 = b _5137 = a Y = _5137 true true h(c) h(b) h(a) g(c), h(c) g(a), h(a) g(b), h(b) k(Y) To find the answers to a query, Prolog applies depth-first search with unification. f(X), g(X), h(X). k(X) :- h(c). h(a). g(c). g(b). g(a). f(c). f(b). f(a). database from top to bottom. When searching for a fact or rule that unifies with a goal, it searches the 13/44

  21. BACKTRACKING f(_5137), g(_5137), h(_5137) _5137 = c _5137 = b _5137 = a Y = _5137 true true h(c) h(b) h(a) g(c), h(c) g(a), h(a) g(b), h(b) k(Y) To find the answers to a query, Prolog applies depth-first search with unification. f(X), g(X), h(X). k(X) :- h(c). h(a). g(c). g(b). g(a). f(c). f(b). f(a). database from top to bottom. When searching for a fact or rule that unifies with a goal, it searches the 13/44

  22. BACKTRACKING f(_5137), g(_5137), h(_5137) _5137 = c _5137 = b _5137 = a Y = _5137 true true h(c) h(b) h(a) g(c), h(c) g(a), h(a) g(b), h(b) k(Y) To find the answers to a query, Prolog applies depth-first search with unification. f(X), g(X), h(X). k(X) :- h(c). h(a). g(c). g(b). g(a). f(c). f(b). f(a). database from top to bottom. When searching for a fact or rule that unifies with a goal, it searches the 13/44

  23. BACKTRACKING f(_5137), g(_5137), h(_5137) _5137 = c _5137 = b _5137 = a Y = _5137 true true h(c) h(b) h(a) g(c), h(c) g(a), h(a) g(b), h(b) k(Y) To find the answers to a query, Prolog applies depth-first search with unification. f(X), g(X), h(X). k(X) :- h(c). h(a). g(c). g(b). g(a). f(c). f(b). f(a). database from top to bottom. When searching for a fact or rule that unifies with a goal, it searches the 13/44

  24. BACKTRACKING f(_5137), g(_5137), h(_5137) _5137 = c _5137 = b _5137 = a Y = _5137 true true h(c) h(b) h(a) g(c), h(c) g(a), h(a) g(b), h(b) k(Y) To find the answers to a query, Prolog applies depth-first search with unification. f(X), g(X), h(X). k(X) :- h(c). h(a). g(c). g(b). g(a). f(c). f(b). f(a). database from top to bottom. When searching for a fact or rule that unifies with a goal, it searches the 13/44

  25. BACKTRACKING f(_5137), g(_5137), h(_5137) _5137 = c _5137 = b _5137 = a Y = _5137 true true h(c) h(b) h(a) g(c), h(c) g(a), h(a) g(b), h(b) k(Y) To find the answers to a query, Prolog applies depth-first search with unification. f(X), g(X), h(X). k(X) :- h(c). h(a). g(c). g(b). g(a). f(c). f(b). f(a). database from top to bottom. When searching for a fact or rule that unifies with a goal, it searches the 13/44

  26. BACKTRACKING f(_5137), g(_5137), h(_5137) _5137 = c _5137 = b _5137 = a Y = _5137 true true h(c) h(b) h(a) g(c), h(c) g(a), h(a) g(b), h(b) k(Y) To find the answers to a query, Prolog applies depth-first search with unification. f(X), g(X), h(X). k(X) :- h(c). h(a). g(c). g(b). g(a). f(c). f(b). f(a). database from top to bottom. When searching for a fact or rule that unifies with a goal, it searches the 13/44

  27. BACKTRACKING To find the answers to a query, Prolog applies depth-first search with unification. _5137 = c _5137 = b _5137 = a Y = _5137 true true h(c) h(b) h(a) g(c), h(c) g(a), h(a) g(b), h(b) f(_5137), g(_5137), h(_5137) k(Y) f(X), g(X), h(X). k(X) :- h(c). h(a). g(c). g(b). g(a). f(c). f(b). f(a). database from top to bottom. When searching for a fact or rule that unifies with a goal, it searches the 13/44 †

  28. BACKTRACKING To find the answers to a query, Prolog applies depth-first search with unification. _5137 = c _5137 = b _5137 = a Y = _5137 true true h(c) h(b) h(a) g(c), h(c) g(a), h(a) g(b), h(b) f(_5137), g(_5137), h(_5137) k(Y) f(X), g(X), h(X). k(X) :- h(c). h(a). g(c). g(b). g(a). f(c). f(b). f(a). database from top to bottom. When searching for a fact or rule that unifies with a goal, it searches the 13/44 †

  29. BACKTRACKING To find the answers to a query, Prolog applies depth-first search with unification. _5137 = c _5137 = b _5137 = a Y = _5137 true true h(c) h(b) h(a) g(c), h(c) g(a), h(a) g(b), h(b) f(_5137), g(_5137), h(_5137) k(Y) f(X), g(X), h(X). k(X) :- h(c). h(a). g(c). g(b). g(a). f(c). f(b). f(a). database from top to bottom. When searching for a fact or rule that unifies with a goal, it searches the 13/44 †

  30. BACKTRACKING To find the answers to a query, Prolog applies depth-first search with unification. _5137 = c _5137 = b _5137 = a Y = _5137 true true h(c) h(b) h(a) g(c), h(c) g(a), h(a) g(b), h(b) f(_5137), g(_5137), h(_5137) k(Y) f(X), g(X), h(X). k(X) :- h(c). h(a). g(c). g(b). g(a). f(c). f(b). f(a). database from top to bottom. When searching for a fact or rule that unifies with a goal, it searches the 13/44 †

  31. LISTS Sequences and collections are represented as lists. Since list elements can themselves be lists, we can use lists to represent complicated data structures such as trees (even though they are often better represented as deeply nested complex terms). • Empty list: [] • Head and tail: [a|[b,c,d]] = [a,b,c,d] [a|[]] = [a] • Multiple heads: [a,b|[c,d]] = [a,b,c,d] 14/44

  32. LISTS Sequences and collections are represented as lists. Since list elements can themselves be lists, we can use lists to represent complicated data structures such as trees (even though they are often better represented as deeply nested complex terms). • Empty list: [] • Head and tail: [a|[b,c,d]] = [a,b,c,d] [a|[]] = [a] • Multiple heads: [a,b|[c,d]] = [a,b,c,d] 14/44

  33. LISTS Sequences and collections are represented as lists. Since list elements can themselves be lists, we can use lists to represent complicated data structures such as trees (even though they are often better represented as deeply nested complex terms). • Empty list: [] • Head and tail: [a|[b,c,d]] = [a,b,c,d] [a|[]] = [a] • Multiple heads: [a,b|[c,d]] = [a,b,c,d] 14/44

  34. CONTROL FLOW CONSTRUCTS The notion of “control flow” is much weaker in Prolog than even in a functional language because we are (mostly) not concerned with the order in which the Prolog interpreter does things. What we do need is a way to build up arbitrarily complex relations … inductively … using recursion. 15/44

  35. CONTROL FLOW CONSTRUCTS The notion of “control flow” is much weaker in Prolog than even in a functional language because we are (mostly) not concerned with the order in which the Prolog interpreter does things. What we do need is a way to build up arbitrarily complex relations … inductively … using recursion. 15/44

  36. CONTROL FLOW CONSTRUCTS The notion of “control flow” is much weaker in Prolog than even in a functional language because we are (mostly) not concerned with the order in which the Prolog interpreter does things. What we do need is a way to build up arbitrarily complex relations … inductively … using recursion. 15/44

  37. CONTROL FLOW CONSTRUCTS The notion of “control flow” is much weaker in Prolog than even in a functional language because we are (mostly) not concerned with the order in which the Prolog interpreter does things. What we do need is a way to build up arbitrarily complex relations … inductively … using recursion. 15/44

  38. RECURSION Summing a list of integers: sum([], 0). sum([X|Xs], Sum) :- sum(Xs, Sum1), Sum is Sum1 + X. Better: sum([], 0). sum([X|Xs], Sum) :- Sum #= Sum1 + X, sum(Xs, Sum1). 16/44

  39. RECURSION Summing a list of integers: sum([], 0). sum([X|Xs], Sum) :- sum(Xs, Sum1), Sum is Sum1 + X. Better: sum([], 0). sum([X|Xs], Sum) :- Sum #= Sum1 + X, sum(Xs, Sum1). 16/44

  40. MAPPING A PREDICATE OVER A LIST OR LISTS odd(X) :- 1 is X mod 2. ?- maplist(odd,[1,3,5]). true. ?- maplist(odd,[1,2,3]). false. ?- maplist(<,[1,3,5],[2,7,8]). true. ?- maplist(<,[1,3,9],[2,7,8]). false. add(X,Y,Sum) :- Sum is X+Y. ?- maplist(add,[1,3,5],[4,8,9],Sums). Sums = [5,11,14]. 17/44

  41. MAPPING A PREDICATE OVER A LIST OR LISTS odd(X) :- 1 is X mod 2. ?- maplist(odd,[1,3,5]). true. ?- maplist(odd,[1,2,3]). false. ?- maplist(<,[1,3,5],[2,7,8]). true. ?- maplist(<,[1,3,9],[2,7,8]). false. add(X,Y,Sum) :- Sum is X+Y. ?- maplist(add,[1,3,5],[4,8,9],Sums). Sums = [5,11,14]. 17/44

  42. MAPPING A PREDICATE OVER A LIST OR LISTS odd(X) :- 1 is X mod 2. ?- maplist(odd,[1,3,5]). true. ?- maplist(odd,[1,2,3]). false. ?- maplist(<,[1,3,5],[2,7,8]). true. ?- maplist(<,[1,3,9],[2,7,8]). false. add(X,Y,Sum) :- Sum is X+Y. ?- maplist(add,[1,3,5],[4,8,9],Sums). Sums = [5,11,14]. 17/44

  43. BUILT-IN PREDICATES Primitives: • true , false • fail (is the same as false ) Unification: • = (arguments unify), \= (arguments do not unify) Arithmetic and numeric comparisons: (Use with caution.) • + , - , * , / , // • < , > , >= , =< , =:= , =\= • 5 \= 2+3 but X is 2+3, 5 = X Lots more 18/44

  44. BUILT-IN PREDICATES Primitives: • true , false • fail (is the same as false ) Unification: • = (arguments unify), \= (arguments do not unify) Arithmetic and numeric comparisons: (Use with caution.) • + , - , * , / , // • < , > , >= , =< , =:= , =\= • 5 \= 2+3 but X is 2+3, 5 = X Lots more 18/44

  45. BUILT-IN PREDICATES Primitives: • true , false • fail (is the same as false ) Unification: • = (arguments unify), \= (arguments do not unify) Arithmetic and numeric comparisons: (Use with caution.) • + , - , * , / , // • < , > , >= , =< , =:= , =\= • 5 \= 2+3 but X is 2+3, 5 = X Lots more 18/44

  46. BUILT-IN PREDICATES Primitives: • true , false • fail (is the same as false ) Unification: • = (arguments unify), \= (arguments do not unify) Arithmetic and numeric comparisons: (Use with caution.) • + , - , * , / , // • < , > , >= , =< , =:= , =\= • 5 \= 2+3 but X is 2+3, 5 = X Lots more 18/44

  47. CONTROL FLOW: GOAL ORDERING (1) Given the facts f(e). g(a). g(b). g(c). g(d). g(e). the following two predicates are logically the same: h1(X) :- f(X), g(X). h2(X) :- g(X), f(X). Which one is more efficient? • h1 instantiates X = e and then succeeds because g(e) holds. • h2 instantiates X = a , X = b , … and fails on all instantiations except X = e . 19/44

  48. CONTROL FLOW: GOAL ORDERING (1) Given the facts f(e). g(a). g(b). g(c). g(d). g(e). the following two predicates are logically the same: h1(X) :- f(X), g(X). h2(X) :- g(X), f(X). Which one is more efficient? • h1 instantiates X = e and then succeeds because g(e) holds. • h2 instantiates X = a , X = b , … and fails on all instantiations except X = e . 19/44

  49. CONTROL FLOW: GOAL ORDERING (1) Given the facts f(e). g(a). g(b). g(c). g(d). g(e). the following two predicates are logically the same: h1(X) :- f(X), g(X). h2(X) :- g(X), f(X). Which one is more efficient? • h1 instantiates X = e and then succeeds because g(e) holds. • h2 instantiates X = a , X = b , … and fails on all instantiations except X = e . 19/44

  50. CONTROL FLOW: GOAL ORDERING (2) :- descend2(Z,Y), child(X,Z). • descend2(anne,bridget) does not terminate. • descend1(anne,bridget) succeeds. What happens? Now ask the queries descend1(anne,bridget) and descend2(anne,bridget) . descend2(X,Y) :- child(X,Y). descend1(X,Y) :- child(X,Y). :- child(X,Z), descend1(Z,Y). Consider the facts descend2(X,Y) descend1(X,Y) and the following logically equivalent definitions of a descendant relationship: child(donna,emily). child(caroline,donna). child(bridget,caroline). child(anne,bridget). 20/44

  51. CONTROL FLOW: GOAL ORDERING (2) :- descend2(Z,Y), child(X,Z). • descend2(anne,bridget) does not terminate. • descend1(anne,bridget) succeeds. What happens? Now ask the queries descend1(anne,bridget) and descend2(anne,bridget) . descend2(X,Y) :- child(X,Y). descend1(X,Y) :- child(X,Y). :- child(X,Z), descend1(Z,Y). Consider the facts descend2(X,Y) descend1(X,Y) and the following logically equivalent definitions of a descendant relationship: child(donna,emily). child(caroline,donna). child(bridget,caroline). child(anne,bridget). 20/44

  52. CONTROL FLOW: GOAL ORDERING (2) :- descend2(Z,Y), child(X,Z). • descend2(anne,bridget) does not terminate. • descend1(anne,bridget) succeeds. What happens? Now ask the queries descend1(anne,bridget) and descend2(anne,bridget) . descend2(X,Y) :- child(X,Y). descend1(X,Y) :- child(X,Y). :- child(X,Z), descend1(Z,Y). Consider the facts descend2(X,Y) descend1(X,Y) and the following logically equivalent definitions of a descendant relationship: child(donna,emily). child(caroline,donna). child(bridget,caroline). child(anne,bridget). 20/44

  53. CONTROL FLOW: CUT ! (read “cut”) is a predicate that always succeeds, but with a side effect: • It commits Prolog to all choices (unification of variables) that were made since the parent goal was unified with the left-hand side of the rule. • This includes the choice to use this particular rule. 21/44

  54. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). p(X) :- f(X). c(1), d(1), e(1) ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) 22/44

  55. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). p(X) :- f(X). c(1), d(1), e(1) ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) 22/44

  56. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). p(X) :- f(X). c(1), d(1), e(1) ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) 22/44

  57. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). p(X) :- f(X). c(1), d(1), e(1) ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) 22/44

  58. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). p(X) :- f(X). c(1), d(1), e(1) ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) 22/44

  59. CUT: FIRST EXAMPLE (1) true true a(_1) f(_3) d(1), e(1) d(2), e(2) e(2) X = _1 c(2), d(2), e(2) X = _2 X = _3 _1 = 1 _2 = 1 _2 = 2 _3 = 3 true c(1), d(1), e(1) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). p(X) :- f(X). b(_2), c(_2), d(_2), e(_2) ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) 22/44

  60. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). p(X) :- f(X). c(1), d(1), e(1) ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) 22/44

  61. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). p(X) :- f(X). c(1), d(1), e(1) ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) 22/44

  62. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). p(X) :- f(X). c(1), d(1), e(1) ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) 22/44

  63. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). p(X) :- f(X). c(1), d(1), e(1) ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) 22/44

  64. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). 22/44 c(1), d(1), e(1) p(X) :- f(X). ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) †

  65. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). 22/44 c(1), d(1), e(1) p(X) :- f(X). ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) †

  66. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). 22/44 c(1), d(1), e(1) p(X) :- f(X). ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) †

  67. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). 22/44 c(1), d(1), e(1) p(X) :- f(X). ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) †

  68. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). 22/44 c(1), d(1), e(1) p(X) :- f(X). ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) †

  69. CUT: FIRST EXAMPLE (1) true true a(_1) f(_3) d(1), e(1) d(2), e(2) e(2) X = _1 a(1). X = _2 X = _3 _1 = 1 _2 = 1 _2 = 2 _3 = 3 true c(2), d(2), e(2) c(1), d(1), e(1) p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). 22/44 b(_2), c(_2), d(_2), e(_2) p(X) :- f(X). ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) †

  70. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). 22/44 c(1), d(1), e(1) p(X) :- f(X). ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) †

  71. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). 22/44 c(1), d(1), e(1) p(X) :- f(X). ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) †

  72. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). 22/44 c(1), d(1), e(1) p(X) :- f(X). ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) †

  73. CUT: FIRST EXAMPLE (1) true _3 = 3 _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 true e(2) d(2), e(2) d(1), e(1) f(_3) a(_1) true c(2), d(2), e(2) a(1). p(X) :- b(X), c(X), d(X), e(X). b(1). b(2). c(1). c(2). d(2). e(2). f(3). p(X) :- a(X). 22/44 c(1), d(1), e(1) p(X) :- f(X). ?- p(X). X = 1 ; X = 2 ; X = 3. p(X) b(_2), c(_2), d(_2), e(_2) †

  74. CUT: FIRST EXAMPLE (2) c(1), !, d(1), e(1) _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 d(1), e(1) !, d(1), e(1) f(_3) a(_1) true c(2), !, d(2), e(2) b(_2), c(_2), !, d(_2), e(_2) a(1). p(X) false. X = 1 ; ?- p(X). p(X) :- f(X). p(X) :- b(X), c(X), !, d(X), e(X). p(X) :- a(X). f(3). e(2). d(2). c(1). c(2). b(1). b(2). 23/44

  75. CUT: FIRST EXAMPLE (2) c(1), !, d(1), e(1) _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 d(1), e(1) !, d(1), e(1) f(_3) a(_1) true c(2), !, d(2), e(2) b(_2), c(_2), !, d(_2), e(_2) a(1). p(X) false. X = 1 ; ?- p(X). p(X) :- f(X). p(X) :- b(X), c(X), !, d(X), e(X). p(X) :- a(X). f(3). e(2). d(2). c(1). c(2). b(1). b(2). 23/44

  76. CUT: FIRST EXAMPLE (2) c(1), !, d(1), e(1) _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 d(1), e(1) !, d(1), e(1) f(_3) a(_1) true c(2), !, d(2), e(2) b(_2), c(_2), !, d(_2), e(_2) a(1). p(X) false. X = 1 ; ?- p(X). p(X) :- f(X). p(X) :- b(X), c(X), !, d(X), e(X). p(X) :- a(X). f(3). e(2). d(2). c(1). c(2). b(1). b(2). 23/44

  77. CUT: FIRST EXAMPLE (2) c(1), !, d(1), e(1) _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 d(1), e(1) !, d(1), e(1) f(_3) a(_1) true c(2), !, d(2), e(2) b(_2), c(_2), !, d(_2), e(_2) a(1). p(X) false. X = 1 ; ?- p(X). p(X) :- f(X). p(X) :- b(X), c(X), !, d(X), e(X). p(X) :- a(X). f(3). e(2). d(2). c(1). c(2). b(1). b(2). 23/44

  78. CUT: FIRST EXAMPLE (2) c(1), !, d(1), e(1) _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 d(1), e(1) !, d(1), e(1) f(_3) a(_1) true c(2), !, d(2), e(2) b(_2), c(_2), !, d(_2), e(_2) a(1). p(X) false. X = 1 ; ?- p(X). p(X) :- f(X). p(X) :- b(X), c(X), !, d(X), e(X). p(X) :- a(X). f(3). e(2). d(2). c(1). c(2). b(1). b(2). 23/44

  79. CUT: FIRST EXAMPLE (2) c(1), !, d(1), e(1) _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 d(1), e(1) !, d(1), e(1) f(_3) a(_1) true c(2), !, d(2), e(2) b(_2), c(_2), !, d(_2), e(_2) a(1). p(X) false. ; X = 1 ?- p(X). p(X) :- f(X). p(X) :- b(X), c(X), !, d(X), e(X). p(X) :- a(X). f(3). e(2). d(2). c(1). c(2). b(1). b(2). 23/44

  80. CUT: FIRST EXAMPLE (2) c(1), !, d(1), e(1) _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 d(1), e(1) !, d(1), e(1) f(_3) a(_1) true c(2), !, d(2), e(2) b(_2), c(_2), !, d(_2), e(_2) a(1). p(X) false. X = 1 ; ?- p(X). p(X) :- f(X). p(X) :- b(X), c(X), !, d(X), e(X). p(X) :- a(X). f(3). e(2). d(2). c(1). c(2). b(1). b(2). 23/44

  81. CUT: FIRST EXAMPLE (2) c(1), !, d(1), e(1) _2 = 2 _2 = 1 _1 = 1 X = _3 X = _2 X = _1 d(1), e(1) !, d(1), e(1) f(_3) a(_1) true c(2), !, d(2), e(2) b(_2), c(_2), !, d(_2), e(_2) a(1). p(X) false. X = 1 ; ?- p(X). p(X) :- f(X). p(X) :- b(X), c(X), !, d(X), e(X). p(X) :- a(X). f(3). e(2). d(2). c(1). c(2). b(1). b(2). 23/44

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