1. An introduction to Logic Programming Chapter 6

2. So far… The computational process was about finding values And now for something completely different. 2

3. Chapter topics 1.Introduction 2.Relational Logic Programming: specify relations among entities 3.Logic Programming: data structures: lists, binary trees, symbolic expressions, natural numbers (church) 4.Prolog: arithmetic, cuts, negation 5.Meta-circular Interpreters for Logic Programming 3

4. How it Came to be Kowalski's Observation (early 70s): An axiom H  B 1 and B 2... and B n represents a procedure H is the procedure’s head and the B i ’s are its body To solve (execute) H, we recursively solve B 1... B n In a quest to automate the process of proving logical statements, Colmerauer and his group embodied this observation into Prolog. 4

5. Actually two Prologs Pure logic programming – No types, no primitives. Full Prolog – Types, arithmetic primitives, data structures and more 5

6. Logic Programming: Introduction Every programming language has: Syntax: set of formulas (facts and rules) Semantics (set of answers to queries) Operational Semantics (how to get an answer): – Unification – Backtracking 6

7. Logic Programming: Introduction Mind switch: – Formula ⇔ procedure declaration – Query ⇔ procedure call – Proving ⇔ computation 7

8. Relational Logic Programming a computational model based on Kowalski's interpretation The Prolog language ('70) - contains RLP + additional programming features 8

9. Relational LP - Syntax atomic formula: predicate_symbol(term 1,..., term n ) predicate symbols start with lowercase terms: – symbols (representing individual constants) – variables (start with uppercase or _ for anonymous) 9

10. Relational LP - Semantics Values: – all atomic values are symbols (typeless) – Very few primitive predicates, such as = (unification), \=, true, false Computation output: – an answer to a query. – An answer to a query is a (possibly partial) substitution (assignment) of query variables. 10

11. Relational LP Syntax - formulas atomic formula: Syntax: predicate_symbol(term 1,...,term n ) Examples: male(moshe) color(red) parent(reuven, moshe) parent(moshe, rina) parent(Parent, Child) ancestor(A,D) address(_City, hertzel, 20) The only difference between predicates and individual constant symbols are their context/location. 11

12. Relational LP Syntax - Procedures A fact is an assertion of an atomic formula. Syntax: H. where H is an atomic formula. Examples: parent(rina, moshe). color(red). ancestor(A,A). Variables in facts are universally quantified. "for all A, it holds that ancestor(A,A)". Procedures are an ordered collection of axioms (facts and rules) sharing the same predicate name and arity. 12

13. % Signature: parent(Parent, Child)/2 % Purpose: Parent is a parent of Child parent(rina, moshe). parent(rina, rachel). parent(rachel, yossi). parent(reuven, moshe). % Signature: female(Person)/1 % Purpose: Person is a female. female(rina). female(rachel). 13 Predicates have arity (no. of parameters). specified in /n in the comment above the procedure. not necessarily unique

14. Relational LP Syntax - Queries A query has the syntax: ?- Q 1, Q 2,..., Q n. where the Q i are atomic formulas. Meaning: Assuming the program axioms, do Q 1 and... and Q n hold? ',' means conjunction. For example, ?- parent(rina, moshe). "Is rina a parent of moshe?” A computation is a proof of a query, returns: true ; false. user requests another answer 14

15. Relational LP Syntax - Queries A query has the syntax: ?- Q 1, Q 2,..., Q n. where the Q i are atomic formulas. Meaning: Assuming the program axioms, do Q 1 and... and Q n hold as well? ',' means conjunction. For example, ?- parent(rina,X). "Does there exist an X which is a child of rina?" X = moshe ; X = rachel. Variables in queries are existentially quantified. 15

16. Relational LP Syntax - Queries "Is there an X which is a child of rina, and is also a parent of some Y?" ?- parent(rina,X),parent(X,Y). X = rachel, Y = yossi. "Find two parents of moshe?": ?- parent(X,moshe),parent(Y,moshe). X = rina, Y = rina ; X = rina, Y = reuven ; X = reuven, Y = rina ; X = reuven, Y = reuven. 16

17. Relational LP Syntax - Queries "Find two different parents of moshe?": ?- parent(X,moshe),parent(Y,moshe),X \= Y. X = rina, Y = reuven ; X = reuven, Y = rina ; false. ?- parent(X,moshe), X \= Y, parent(Y,moshe). false. ?- X=3. X = 3. ?- X\=3. false. ?- 4\=3. true. 17

18. Relational LP - Syntax % Signature: loves(Someone, Somebody)/2 % Purpose: Someone loves Somebody loves(rina,Y). % rina loves everybody. loves(moshe, rachel). loves(moshe, rina). loves(Y,Y). % everybody loves himself Variables in axioms are universally quantified. "for all Y loves(rina,Y)" can be renamed "for all X loves(rina,X)" Using a variable in a fact is defining it. The scope is the fact itself. 18

19. Relational LP - Syntax % Signature: loves(Someone, Somebody)/2 % Purpose: Someone loves Somebody loves(rina,Y). % rina loves everybody. loves(moshe, rachel). loves(moshe, rina). loves(Y,Y). % everybody loves himself Queries: ?- loves(rina,moshe). true ; false. 19

20. Relational LP - Syntax % Signature: loves(Someone, Somebody)/2 % Purpose: Someone loves Somebody loves(rina,Y). % rina loves everybody. loves(moshe, rachel). loves(moshe, rina). loves(Y,Y). % everybody loves himself Queries: ?- loves(rina,X). true ; X = rina. 20

21. Relational LP - Syntax % Signature: loves(Someone, Somebody)/2 % Purpose: Someone loves Somebody loves(rina,Y). % rina loves everybody. loves(moshe, rachel). loves(moshe, rina). loves(Y,Y). % everybody loves himself Queries: ?- loves(X,rina). X = rina ; X = moshe ; X = rina. 21

22. Relational LP - Syntax % Signature: loves(Someone, Somebody)/2 % Purpose: Someone loves Somebody loves(rina,Y). % rina loves everybody. loves(moshe, rachel). loves(moshe, rina). loves(Y,Y). % everybody loves himself Queries: ?- loves(X,X). X = rina ; true. this query has two answers. 22

23. Relational LP Syntax - Rules Syntax: H :−B 1,..., B n. is an assertion of an implication statement. The conjunction of B 1,.., B n implies the head H. B i 's and H are atomic formulas. % Signature: mother(Mum, Child), % Purpose: Mum is a mother of Child mother(Mum, Child) :- parent(Mum, Child), female(Mum). Variables occurring in rule heads are universally quantified. The lexical scope of the variable is the rule. A variable can appear multiple times in the head. variables are bound within a rule. 23

24. Relational LP Syntax - Rules % Signature: mother(Mum, Child), % Purpose: Mum is a mother of Child mother(Mum, Child) :- parent(Mum, Child), female(Mum). ?- mother(M,C). M = rina, C = moshe ; M = rina, C = rachel ; M = rachel, C = yossi ; false. 24

25. Relational LP Syntax - Rules % Signature: mother(Mum, Child), % Purpose: Mum is a mother of Child mother(Mum, Child) :- parent(Mum, Child), female(Mum). “Find a two-different-kids mother” ?- mother(M,C1),mother(M,C2),C1\=C2. M = rina, C1 = moshe, C2 = rachel ; M = rina, C1 = rachel, C2 = moshe ; false. 25

26. Relational LP Syntax - Rules the ancestor relationship - a recursive rule that computes the transitive closure of the parent relationship. % Signature: ancestor(Ancestor, Descendant)/2 % Purpose: Ancestor is an ancestor of Descendant. ancestor(Ancestor, Descendant) :- parent(Ancestor, Descendant). ancestor(Ancestor, Descendant) :- parent(Ancestor, Person), ancestor(Person, Descendant). Variables occurring in the rule body and not in the head are existentially quantified. "for all Ancestor and for all Descendant, ancestor(Ancestor, Descendant) if there exists some Person such that parent(Ancestor, Person) and ancestor(Person, Descendant)." 26

27. Relational LP Syntax - Rules the ancestor relationship - a recursive rule that computes the transitive closure of the parent relationship. % Signature: ancestor(Ancestor, Descendant)/2 % Purpose: Ancestor is an ancestor of Descendant. ancestor(Ancestor, Descendant) :- parent(Ancestor, Descendant). ancestor(Ancestor, Descendant) :- parent(Ancestor, Person), ancestor(Person, Descendant). ?- ancestor(rina,D). D = moshe ; D = rachel ; D = yossi ; false. 27

28. Relational LP Syntax - Rules the ancestor relationship - a recursive rule that computes the transitive closure of the parent relationship. % Signature: ancestor(Ancestor, Descendant)/2 % Purpose: Ancestor is an ancestor of Descendant. ancestor(Ancestor, Descendant) :- parent(Ancestor, Descendant). ancestor(Ancestor, Descendant) :- parent(Ancestor, Person), ancestor(Person, Descendant). ?- ancestor(A,yossi). A = rachel ; A = rina ; false. The reported result/functionality depends on the variables and their location in the query. 28

29. Relational LP Syntax - Rules ancestor1(Ancestor, Descendant) :- parent(Ancestor, Descendant). ancestor1(Ancestor, Descendant) :- ancestor1(Person, Descendant), parent(Ancestor, Person). ?- ancestor1(A,yossi). A = rachel ; A = rina ; ERROR: Out of local stack ?- ancestor1(rina,yossi). true ; ERROR: Out of local stack This procedure is not tail recursive. Since this query cannot be answered using the base case, new similar queries are infinitely created. 29

30. Note Facts can be considered as rules with an empty body. For example, parent(rina, moshe). parent(rina, moshe):- true. have equivalent meaning. true - is the zero-arity predicate. 30

31. Concrete syntax of Relational Logic Programming -> + -> ( | )+ with identical predicate and arity -> ’: -’ ’.’ -> ’.’ -> -> ( ’,’)* -> | ’(’( ’,’)* ’)’ -> -> | -> A string starting with a lower case letter. -> A string starting with an upper case letter. -> ’?-’ ( ’,’)* ’.’ 31

32. Summary - RLP Semantic and syntax parent(rina, moshe). parent(rina, rachel). parent(rachel, yossi). parent(reuven, moshe). ancestor(Ancestor, Descendant) :- parent(Ancestor, Descendant). ancestor(Ancestor, Descendant) :- parent(Ancestor, Person), ancestor(Person, Descendant). ?- ancestor(A,yossi). A = rachel ; A = rina ; false. Concepts: predicate symbol, terms, facts, rules, query. Semantics: Quantification of variables (universal/existential) Answers are partial substitutions to query variables (or true/false indications). 32

33. Operational Semantics for LP Input: a program P and a query Q Interpreter of LP: 1.Unify - pattern matching between an atomic formula from Q and a head of some rule/fact from P. 2.Answer-query (proof-tree)- Create a proof tree. Back track from a leaf if it is a "dead end" fail leaf, or if it is a success leaf and there may be additional answers to the query. 33

34. Unification The unification operation: two atomic formulas ==> substitution p(3, X), p(Y, 4) ==> {X = 4, Y = 3} p(X, 3, X), p(Y, Z, 4) ==> {X = 4, Z = 3, Y = 4} substitution - a finite mapping, s, from variables to terms, such that s(X)≠X. Examples:s={X=4, Y=4, Z=3} {X = 4, Z = 3, U = X}, {X = 4, Z = 3, U = V } Not substitutions: {X = 4, Z = 3, Y = Y }, {X = 4, Z = 3, X = Y } 34

35. Application of Substitution atomic formula ◦ substitution ==> atomic formula' p(X, 3, X, W ) ◦ {X = 4, Y = 4} = p(4, 3, 4, W) p(X, 3, X, W ) ◦ {X = 4, W = 5} = p(4, 3, 4, 5) p(X, 3, X, W ) ◦ {X = W, W = X} = p(W, 3, W, X) 35

36. Unifier A unifier of atomic formulas A and B is a substitution s, such that A◦s = B◦s. For example, the following substitutions are unifiers of p(X,3,X,W) and p(Y,Z,4,W): -{X = 4,Z = 3,Y = 4} -{X = 4,Z = 3,Y = 4,W = 5} -{X = 4,Z = 3,Y = 4,W = 0} 36

37. Instantiation and Generalization An atomic formula A’ is an instance of an atomic formula A if there is a substitution s such that A◦s = A’ A is more general than A’ if A’ is an instance of A 37

38. Most General Unifier (MGU) mgu of atomic formulas A and B is a unifier s of A and B such that A◦s = B◦s is more general than all other instances of A and B obtained by applying a unifier 38

39. Unification Goal of Unify(A,B): find the most general unifier. Unify( p(X, 3, X, W), p(Y, Z, 4, W ) ) ==> {X=4, Y=4, Z=3} p(X, 3, X, W ) ◦ {X=4, Y=4, Z=3} = p(4, 3, 4, W ) p(Y, Z, 4, W ) ◦ {X=4, Y=4, Z=3} = p(4, 3, 4, W) Less general unifiers: – {X = 4, Z = 3, Y = 4, W = 5} – {X = 4, Z = 3, Y = 4, W = 0, V=6} 39

40. Combination of substitutions s ◦ s' 1. s' is applied to the terms of s 2.A variable X for which s(X) is defined, is removed from the domain of s' 3.The modified s' is added to s. 4.Identity bindings are removed. {X = Y, Z = 3, U = V } ◦ {Y = 4, W = 5, V = U, Z = X} = {X = 4, Z = 3, Y = 4, W = 5, V = U}. 40

41. Disagreement Set The disagreement set of atomic formulas is the set of left most symbols on which the formulas disagree. disagreement-set(p(X, 3, X, W ), p(Y, Z, 4, W )) = {X, Y}. disagreement-set(p(5, 3, X, W ), p(5, 3, 4, W )) = {X, 4}. 41

42. Unify - A unification algorithm Signature: unify(A, B) Type: atomic-formula*atomic-formula -> a substitution or FAIL Post-condition: result = mgu(A, B) if A and B are unifiable or FAIL, otherwise 42

43. Unify - examples 43

44. Properties of unify(A, B) algorithm : The algorithm always terminates. If only one atomic formula includes variables, it’s a pattern matching. If B does not include variables and A does not include repeated variable occurrences, the time complexity can be linear. 44

45. Operational Semantics for LP Interpreter of LP: 1.Unify - pattern matching between an atomic formula from Q and a head of some rule/fact from P. 2.Answer-query (proof-tree)- Create a proof tree. Back track from a leaf if it is a "dead end" fail leaf, or if it is a success leaf and there may be additional answers to the query. 45

46. answer-query: an interpretation algorithm for LP Input: A query: Q = ?- Q 1,..., Q n. Each component is called goal A program P, with numbered rules (denoted by number(R)) A goal selection policy G sel A rule selection policy R sel Output: A set of (possibly partial) substitutions for variables of Q. General Idea: Repeated effort to select a goal using roles. 46

47. answer-query Backtracking: if selected rule does not lead to proof: next rule is tried If no rule leads to proof: fail Rule selection is performed by unification between goal and head of rule Rules and goals selection: – Leftmost goal – Next rule in file 47

48. What about facts? Facts are rules with body true 48

49. Answer Query Algorithm Iterator – We keep a pointer to the rules for the backtracking Two versions: – Build proof tree – Scan proof tree 49

50. Proof Tree Tree with labeled nodes and edges Nodes labeled by queries and marked goals Edges labeled by substitutions and rule numbers Root node is input query Children on node are all successful rules for the goal 50

51. answer-query Algorithm 1.PT := proof-tree(make_node(Q)) 2.Return {s | s ∈ labels(Success(P T ))/Q }, where Success(PT) is the set of Success nodes of P T, and labels/Q is the restriction of the substitutions in labels to the variables of Q. 51

52. Possible Answers Empty set (fail of query with variables) Empty set (success for grounded query) Answer 52

53. Tree Operations make_node(label) add_child(parent, edge, child) label(node) – selector set_label!(node, new_label) 53

54. 54

55. Comments about answer-query Variable renaming according to depth in tree. X i at depth i. Unify(A,G), where G is the selected goal and A the head of the selected rule. – Let X G be a variable from G and X A a variable of A – Selecting X A =X G or X G =X A does not change query results. – Selecting X A =X G leaves the query variables in the tree. The goal and rule selection decisions can affect the performance of the interpreter. 55

56. Example % Signature: father(F,C)/2 parent(abraham,isaac).%1 parent(isaac, jacob).%2 parent(haran,lot).%3 parent(haran,yiscah).%4 parent(haran,milcah).%5 % Signature: male(P)/1 male(isaac).%1 male(lot).%2 % Signature: son(C, P)/2 son(X, Y) - parent(Y, X), male(X).%1 % Signature: ancestor(Ancestor, Descendant)/2 anc(Anc, Des) :- parent(Anc, Des). %1 anc(Anc, Des) :- parent(Anc, Person), %2 anc(Person, Des). 56

57. 57 ?- son(S, haran).

58. ?- anc(abraham, D). 58 What happens if rules 1 and 2 of 'anc' are switched?

59. What happens if bodies 1 and 2 in rule 2 of 'anc' are switched?

60. Significant kinds of proof trees: Finite success proof tree: A finite tree with a successful path. Finite failure proof tree: A finite tree with no successful path. Infinite success proof tree: An infinite tree with a successful path. In this case it is important not to explore an infinite path. For Prolog: Tail recursion is safe. Infinite failure proof tree: An infinite tree with no successful path. 60

61. 6.1.6 Relational logic programming and Structured Query Language (SQL) operations RLP naturally represents structured databases (tables with static columns). A procedure consisting of facts can represent a table in the database. Often databases are access via elementary SQL operations such as: select, project, union, Cartesian product and join. Select (rows from table r, fitting some criteria): r1(X1, X2, X3) :- r(X1, X2, X3), X2 \= X3. Project (some columns from table r): r1(X1, X3) :- r(X1, X2, X3). 61

62. 6.1.6 Relational logic programming and Structured Query Language (SQL) operations Union (unite tables r and s, with identical columns): r_union_s(X1,..., Xn) :- r(X1,..., Xn). r_union_s(X1,..., Xn) :- s(X1,..., Xn). Cartesian product (all combinations of rows from r and s): r_X_s(X1,..., Xn, Y1,..., Ym) :- r(X1,..., Xn ), s(Y1,..., Ym). Natural Join (join tables r and s, with mutual column X): r_join_s(X1,..., Xn, X, Y1,..., Ym) :- r(X1,..., Xn, X ), s(X, Y1,..., Ym). 62

63. 6.1.5.3 Halting problem, RLP decidability L RLP = {(P,Q) | P|- Q in RLP syntax} Claim: Given a program P and a query Q, the problem "Is Q provable from P ", denoted P|- Q, is decidable. Proof: The number of terms(constants/variables) and predicates appearing in P and Q is finite. Thus, the number of possible atomic formula (i.e. goals appearing in a node of the proof tree) is finite (except for renaming). Let N(P,Q) be that number. Then, any path that is longer than N(P,Q) is infinite. QED Most programming languages are only partially decidable (recursively enumerable/TM recognizable) 63

64. Prolog LP Logic Programming 64 Relational LP typeless atomic terms, type safe decidable multi-directional functors, typeless composite terms, type safe partially decidable, multi-directional arithmetics, uni-directional dynamically typed, not type safe, system predicates (e.g. !)

65. Logic Programming A functor symbol is added to the syntax, to represent data structures. Terms (definition): constant individual symbols variables f(t 1,..., t n ) for terms t 1,..., t n and a functor f. Implications additional expressiveness (composite data structures) the LP language is partially decidable (recursively enumerable/TM recognizable) (in the same rank as other programming languages). 65

66. Atomic formula in FLP - examples parent(rina, Child) p(f(f(f(g(a,g(b,c)))))) member(cube(red(X)), Lst) 66 predicate terms (a constant and a variable) predicate a term (functors f,g combining constants a,b,c) predicate terms (functors cube and red applied to variable X)

67. Formalizing the syntax extension -> | | -> ’(’ ( ’,’)* ’)’ -> 67

68. Unification for terms that include functors A substitution s is a finite mapping from variables to terms, such that s(X) does not include X. Unify remains the same, except for two points: 1.Disagreement set can happen within a term - unify(member(X,tree(X,Left,Right)), member(Y,tree(9,void,tree(3,void,void)))) ==> {Y=9, X=9, Left=void, Right=tree(3,void,void)} − unify(t(X, f(a),X), t(g(U),U,W)) ==> {X=g(f(a)), U=f(a), W=g(f(a))} 68

69. Unification for terms that include functors A substitution s is a finite mapping from variables to terms, such that s(X) does not include X. Unify remains the same, except for two points: 1. Disagreement set can happen within a term 2. Validation of occur check error (i.e. s(X) includes X). − unify(t(X,f(X),X), t(g(U),U,W)) ==> fails due to occur check error! Expansion is infinite Unify algorithm for LP is modified so that it fails if occur check error is found in the {X=t} substitution at the disagreement-set. 69

70. Unification with Functors 70

71. Example: Natural Numbers Arithmetic Natural numbers can be represented by Church numerals: The constant zero denotes the number 0, s(0) - denotes 1, s(...s(s(0))...), n times - denotes natural number n, where s is a functor. % Signature: natural_number(N)/1 % Purpose: N is a natural number. natural_number(zero).%1 natural_number(s(X)) :- natural_number(X).%2 % Signature: le(X,Y)/2 % Purpose: X is less or equal Y. le(zero, X) :- natural_number(X).%1 le(s(X), s(Z)) :- le(X, Z).%2 71

72. ?- le(s(s(zero)), s(s(s(s(zero))))). true. ?- le(s(s(X)), s(s(s(s(zero))))). X = zero ; X = s(zero) ; X = s(s(zero)) ; false. ?- le(s(s(X)), s(s(s(s(Y))))). X = Y, Y = zero ; X = zero, Y = s(zero) ; X = zero, Y = s(s(zero)) ; X = zero, Y = s(s(s(zero)))... 72 le(zero, X) :- natural_number(X). le(s(X), s(Z)) :- le(X, Z).

73. 73 substitution for first leaf to the left {X1=zero, N=s(Z1)}  {X2=Z1}  {Z1=zero}= {X1=zero, N=s(zero), X2=zero, Z1=zero} substitution for second leaf: {X1=zero, N=s(Z1)}  {X2=Z1}  {Z1=s(X3)}  {X3=zero}= {X1=zero, N=s(s(zero)), X2=s(zero), Z1=s(zero), X3=zero}

74. Example: Binary Tree Data Structure % Signature: binary_tree(T)/1 % Purpose: T is a binary tree. binary_tree(void). binary_tree(tree(Element,Left,Right)) :- binary_tree(Left),binary_tree(Right). % Signature: tree_member(X, T)/2 % Purpose: X is a member of T. tree_member(X, tree(X, _, _)). tree_member(X, tree(Y,Left, _)):- tree_member(X,Left). tree_member(X, tree(Y, _, Right)):- tree_member(X,Right). 74

75. Summary Proof tree types LP with functors Unify + occur check error 75

76. Lists in LP Lists are a primitive composite data structure. Unlike non-primitive data-structure prefix notation f(t1,..., tn) list functor appears in infix notation. Syntax [ ] - a 0-arity functor representing the empty list. [Head|Tail] - a 2-arity functor representing a list that is constructed from its head and its tail, where the tail is also a list. 76

77. Examples ?- X=[]. X=[]. ?- X=[a|[ ]]. X = [a]. ?- X = [a]. X = [a]. ?- [a|[ ]] = [a]. true. ?- X= [a | [ b | [] ]]. X = [a, b]. 77 ?- Y = [1,2,3]. Y = [1, 2, 3]. ?- Y=[1,2,3], X= [a,b| Y]. Y = [1, 2, 3], X = [a, b, 1, 2, 3]. ?- X = [a, b, c|[d,e,f]]. X = [a,b,c,d,e,f]. ?- X=[1|t]. /* not a list */ X = [1|t].

78. LP lists - List membership % Signature: member(X, List)/2 % Purpose: X is a member of List. member(X, [X|Xs]). member(X, [Y|Ys]) :- member(X, Ys). % checks membership ?- member(a, [b,c,a,d]). % takes an element from a list ?- member(X, [b,c,a,d]). % generates a list containing b ?- member(b, Z). 78

79. LP lists - List concatenation % Signature: append(List1, List2, List3)/3 % Purpose: List3 is the concatenation of List1 and List2. append([], Xs, Xs). append([X|Xs], Ys, [X|Zs]) :- append(Xs, Ys, Zs). /* addition of two lists */ ?- append([a,b], [c], X). /* finds a difference between lists */ ?- append(Xs, [a,d], [b,c,a,d]). /* divides a list into two lists */ ?- append(Xs, Ys, [a,b,c,d]). 79

80. append([], Xs, Xs).%1 append([X|Xs], Ys, [X|Zs] ) :- append(Xs, Ys, Zs).%2 80 append(Xs, [a,d], [b,c,a,d]) append(Xs1, [a,d], [c,a,d]) 2 {Xs=[X1|Xs1], Ys1=[a,d], X1=b Zs1=[c,a,d]} true 2 {Xs1=[X2|Xs2], Ys2=[a,d], X2=c Zs2=[a,d]} append(Xs2, [a,d], [a,d]) 2 {Xs2=[X3|Xs3], Ys3=[a,d], X3=a Zs3=[d]} 1 {Xs2=[], Xs3=[a,d]} append(Xs3, [a,d], [d]) 2 {Xs3=[X4|Xs4], Ys4=[a,d], X4=d Zs4=[]} append(Xs4, [a,d], []) fail

81. LP lists - List concatenation % Signature: append(List1, List2, List3)/3 % Purpose: List3 is the concatenation of List1 and List2. append([], Xs, Xs). append([X|Xs], Ys, [X|Zs]) :- append(Xs, Ys, Zs). List prefix and suffix: prefix(Xs, Ys) :- append(Xs, Zs, Ys). suffix(Xs, Ys) :- append(Zs, Xs, Ys). Redefine member: member(X, Ys) :- append(Zs, [X|Xs], Ys). Adjacent list elements: adjacent(X, Y, Zs) :- append(Ws, [X,Y|Ys], Zs). Last element of a list: last(X, Ys) :- append(Xs, [X], Ys). 81

82. LP lists - List Reverse % Signature: reverse(List1, List2)/2 % Purpose: List2 is the reverse of List1. reverse([], []). reverse([H|T], R) :- reverse(T, S), append(S, [H], R). ?- reverse([a,b,c,d],R). R=[d,c,b,a] ?- reverse(R,[a,b,c,d]). Rule body ordering impacts the performance in various directions. 82

83. 83

84. reverse([], []). %1 reverse([H|T], R) :- reverse(T, S), append(S, [H], R). %2 append([], Xs, Xs).%1 append([X|Xs], Y, [X|Zs] ) :- append(Xs, Y, Zs).%2 84 reverse(RList,[a,b,c]) reverse(T1,S1), append(S1, [H1], [a,b,c]) 2 {Rlist=[H1|T1], R1=[a,b,c]} 2 {T1=[H2|T2], R2=S1} fail 1 {T2=[], S2=[]} reverse(T3, S3) append(S3, [H3], S2) append(S2, [H2], S1) append(S1, [H1], [a,b,c]) 1 {T1=[], S1=[]} append([], [H1], [a,b,c]) reverse(T2, S2) append(S2, [H2], S1) append(S1, [H1], [a,b,c]) append([], [H2], S1) append(S1, [H1], [a,b,c]) 1 {Xs1=[H2], S1=[H2]} append([H2], [H1], [a,b,c]) fail 2 {T2=[H3|T3], R3=S2} 1 {T3=[],S3=[]} append([], [H3], S2) append(S2, [H2], S1) append(S1, [H1], [a,b,c]) 1 {Xs1=[H3], S2=[H3]} append([H3], [H2], S1) append(S1, [H1], [a,b,c]) true append([H3,H2], [H1], [a,b,c])... S1=[H3,H2]... H3=a, H2=b, H1=c Rlist=[c,b,a]...

85. lists - List Reverse An iterative version: uses the unification mechanism to accumulate the result in the second parameter which is returned in the base case. The help procedure is global. In Prolog all procedures are global. % Signature: reverse(List1, List2)/2 reverse(Xs, Ys):- reverse_help(Xs,[],Ys). % Signature: reverse_help(List1, AccReversed, RevList)/2 reverse_help([ ],Ys,Ys ). reverse_help([X|Xs], Acc, Ys ) :- reverse_help(Xs,[X|Acc],Ys). ?- reverse([a,b,c],R). R=[c,b,a] ?- reverse(R,[a,b,c]). ERROR: Out of local stack 85

86. Prolog LP Prolog 86 RLP typeless atomic terms, type safe decidable multi-directional definitions functors, typeless composite terms, type safe partially decidable, multi-directional definitions arithmetics, uni-directional procedures dynamically typed, not type safe, system predicates (e.g. !)

87. The Cut Operator - Pruning Trees The cut system predicate, denoted !, is a Prolog built-in predicate, for pruning proof trees. avoiding traversing failed sub-trees. eliminates wrong answers or infinite branches 87

88. The Cut Operator For a node v, in which rule H :- B 1,...B i, !, B i +1,..., B n is applied, and having a branch to a node u, in which the current goal is !, all alternative branches splitting from nodes in the path between v (including) and node u are trimmed. 88 v u

89. Example: trimming unwanted answers Problem domain: colored pieces, each piece has one color. color(P, red) :- red(P). color(P, black) :- black(P). color(P, unknown). red(a). black(b). The queries have to return a single solution ?- color(a, Color). 89

90. Example: trimming unwanted answers Eliminates wrong answers color(P, red) :- red(P),!. color(P, black) :- black(P),!. color(P, unknown). red(a). black(b). 90

91. Example: avoiding unnecessary searches (duplicate answers) member(X,[X|Ys]). member(X,[Y|Zs]) :- member(X, Zs). Adding cut: member(X,[X|Ys]) :- !. member(X,[Y|Zs]) :- member(X, Zs). ?- member(5, [5, 9, 24, 17, 5, 2])). % After one application the proof tree is complete. ?- member(9, [5, 9, 24, 17, 5, 2])). % After 2 applications the proof tree is complete. ?- member(X, [5, 9, 24, 17, 5, 2]). % Only one answer will be returned. 91

92. Red & Green Cuts Green: Only optimize the search Red: Does not compute the intended relation (Don’t look for color in the code… it’s a matter of semantics) 92

93. Meta-circular interpreters for LP Based on unification and backtracking. Two points of selection: (a) Goal selection - leftmost for Prolog. (b) Rule selection - top-to-bottom for Prolog, with backtracking to the following rules, in case of a failure. 93

94. univ Predicate Denoted ‘ =.. ’ Turns term into a list. Examples: ?- f(a,X,g(Y))=..Term_list. Term_list = [f, a, X, g(Y)]. ?- Term=..[f, a, X, g(Y)]. Term = f(a, X, g(Y)). 94

95. Unify (recall) 95

96. Unify In order to implement it we need to distinguish between variables and constants. Our implementation: constants denoting variables are distinguished from true constants. For example: c(f(X), G,X), c(G, Y, f(d)) => c(f(x), g, x), c(g, y, f(d)) 96

97. Needed Procedures substitute(Exp, S, ExpS) ?- substitute( p(a, b(x), c(d(y)), x), [[z, 5], [x, 6]], Ans). Ans = p(a, b(6), c(d(y)), 6). 97

98. Needed Procedures disagreement(A, B, Set) ?- disagreement( p(x,y) p(x,5), Set) Set = [y,5] 98

99. Needed Procedures not_occurs(T, X) ?- not_occurs(f(g(x)),x) fail ?- not_occurs(f(g(x)),y) true 99

100. Needed Procedures % Signature: variable(Name)/1 variable(X) :- variable_list(Vars), member(X,Vars). variable_list([x,y,z]). % Signature: constant(Name)/1 constant(C) :- constant_list(Consts), member(C, Consts). constant_list([p,a,b,c,d,[],5,6]). 100

101. Needed Procedures unify(A, B, Mgu)/3 ?- unify(a(x, 5), a(6, y), Mgu). Mgu = [[y, 5], [x, 6]]. ?- unify(c(a(x), z, x), c(z, y, a(d)), Mgu). Mgu = [[x, a(d)], [y, a(a(d))], [z, a(a(d))]] ?- unify(c(a(x), d, x), c(z, y, a(z)), Mgu). false 101

102. Meta Circular Interpreter for LP Based on unification and backtracking Main Procedure is called solve (queries are presented to it) Need pre-processing (next slide) 102

103. Pre-Processing Program P is translated into P’ that has only a single predicate: rule : A :- B 1, B 2,..., B n => rule(A, [B 1, B 2,..., B n ] ). 103

104. Pre-Processing % father(abraham, isaac). % father(haran, lot). % father(haran, milcah). % father(haran, yiscah). % male(isaac). % male(lot). % female(milcah). % female(yiscah). % son(X, Y) :- father(Y, X), male(X). % daughter(X, Y) :- father(Y, X), % female(X). rule(father(abraham, isaac), true). %1 rule(father(haran, lot), true). %2 rule(father(haran, milcah), true). %3 rule(father(haran, yiscah), true). %4 rule(male(isaac), true). %5 rule(male(lot), true). %6 rule(female(milcah), true). %7 rule(female(yiscah), true). %8 rule(son(X, Y), [father(Y, X), male(X)]). %9 rule(daughter(X, Y), [father(Y, X), female(X)]). %10 104

105. Vanilla (Basic) Interpreter solve(true). %1 solve([]). %2 solve([A|B]) :- solve(A), solve(B). %3 solve(A) :- rule(A,B), solve(B). %4 105

106. Vanilla (Basic) Interpreter rule(father(abraham, isaac), true). rule(father(haran, lot), true). rule(father(haran, milcah), true). rule(father(haran, yiscah), true). rule(male(isaac), true). rule(male(lot), true). rule(son(X, Y), [father(Y, X), male(X)]). solve(true). %1 solve([]). %2 solve([A|B]) :- solve(A), solve(B). %3 solve(A) :- rule(A,B), solve(B). %4 ?- solve(son(X, Y)). X = isaac, Y = abraham ; X = lot, Y = haran ; false. 106

107. Interpreter with Associated Proofs solve(true, true). %1 solve([], []). %2 solve([A|B],[ProofA|ProofB]) :- solve(A, ProofA), solve(B, ProofB). %3 solve(A, node(A,Proof)) :- rule(A,B), solve(B,Proof). %4 107

108. Interpreter with Associated Proofs rule(father(abraham, isaac), true). rule(father(haran, lot), true). rule(father(haran, milcah), true). rule(father(haran, yiscah), true). rule(male(isaac), true). rule(male(lot), true). rule(son(X, Y), [father(Y, X), male(X)]). solve(true, true). %1 solve([], []). %2 solve([A|B],[ProofA|ProofB]) :- solve(A, ProofA), solve(B, ProofB). %3 solve(A, node(A,Proof)) :- rule(A,B), solve(B,Proof). %4 ?- solve(son(lot,haran), Proof). Proof = node( son(lot,haran), [node(father(haran,lot),true), node(male(lot),true)]); 108

109. Logic Programming Summary Pure(relational) LP: typeless atomic terms, atomic formula program axioms, queries lexical scoping, global definitions unification, build proof tree (backtracking) decidability FLP: functors, composite terms lists Prolog: cut! meta-circular interpreters (clause, tuples/list) 109