1. Logic Programming (cont’d): Lists Lists in Prolog are represented by a functor (similar to cons in Scheme): 1.The empty list is represented by the constant - [ ]. 2.The term.(X, Xs) represents a list of which X is the 1 st element and Xs is the rest of its elements. 3.The functor ‘. ’ has an infix representation by the pipe symbol ‘|’ : [X | Xs] For example: [3 | [5 | [] ] ] 4.Lists of known length can be represented as a list of comma-separated expressions: For example: [ X, 2, f(Y) ] or [3, 5]

2. Logic Programming (cont’d): Lists Example 1: Implementing a Context Free Grammar in PROLOG: NP VP Det N V NP | V a | the woman | man saw S NP VP Det N V Q: What language does this CFG represent? s(Z) :- np(X),vp(Y),append(X,Y,Z). np(Z):- det(X),n(Y),append(X,Y,Z). vp(Z):- v(X),np(Y),append(X,Y,Z). vp(Z):- v(Z). det([the]). det([a]). n([woman]). n([man]). v([saw]). s(Z) :- np(X),vp(Y),append(X,Y,Z). np(Z):- det(X),n(Y),append(X,Y,Z). vp(Z):- v(X),np(Y),append(X,Y,Z). vp(Z):- v(Z). det([the]). det([a]). n([woman]). n([man]). v([saw]). Representation in PROLOG: ?- s([a,woman,saw,a,man]). true ?-s(X). X = [the,woman,saw,the,woman]; X = [the,woman,saw,the,man]; X = [the,woman,saw,a,woman]; X = [the,woman,saw,a,man]; X = [the,woman,saw] … ?-s([the,man|X]). X = [saw,the,woman]; X = [saw,the,man]; X = [saw,a,woman] …

3. Logic Programming (cont’d): Lists Example 1: Implementing a Context Free Grammar in PROLOG: NP VP Det N | Det Adj N vicious | marvelous V NP | V a | the woman | man saw S NP Adj VP Det N V Representation in PROLOG: Q: How would we implement the modifications above? s(Z) :- np(X),vp(Y),append(X,Y,Z). np(Z):- det(X),n(Y),append(X,Y,Z). vp(Z):- v(X),np(Y),append(X,Y,Z). vp(Z):- v(Z). det([the]). det([a]). n([woman]). n([man]). v([shoots]). s(Z) :- np(X),vp(Y),append(X,Y,Z). np(Z):- det(X),n(Y),append(X,Y,Z). vp(Z):- v(X),np(Y),append(X,Y,Z). vp(Z):- v(Z). det([the]). det([a]). n([woman]). n([man]). v([shoots]). s(Z) :- np(X),vp(Y),append(X,Y,Z). np(Z):- det(X),n(Y),append(X,Y,Z). np(Z):- det(X),adj(W),n(Y),append([X,W,Y],Z). adj([vicious]). adj([marvelous]). vp(Z):- v(X),np(Y),append(X,Y,Z). vp(Z):- v(Z). det([the]). det([a]). n([woman]). n([man]). v([shoots]). s(Z) :- np(X),vp(Y),append(X,Y,Z). np(Z):- det(X),n(Y),append(X,Y,Z). np(Z):- det(X),adj(W),n(Y),append([X,W,Y],Z). adj([vicious]). adj([marvelous]). vp(Z):- v(X),np(Y),append(X,Y,Z). vp(Z):- v(Z). det([the]). det([a]). n([woman]). n([man]). v([shoots]).

4. Logic Programming (cont’d): Lists Example 2: Determining the order of specific times o In logic programming, symbols have no values: ‘2’ and ‘9’ are names and not numbers. o We cannot use the < relation to compare the order of numbers (it is infinite!). o However, a finite ordered relation may be represented by positions in lists. We would like to implement an ordered relation between pairs of the form (hour, weekday). % Type: Hour. % Signature: h(Hour)/1 % Example: h(18). % Type: Hour. % Signature: h(Hour)/1 % Example: h(18). % Type: Weekday. % Signature: d(Weekday)/1 % Example: d(Tue). % Type: Weekday. % Signature: d(Weekday)/1 % Example: d(Tue).

5. Logic Programming (cont’d): Lists Example 2: Determining the order of specific times o In logic programming, symbols have no values: ‘2’ and ‘9’ are names and not numbers. o We cannot use the < relation to compare the order of numbers (it is infinite!). o However, a finite ordered relation may be represented by positions in lists. We would like to implement an ordered relation between pairs of the form (hour, weekday). % Signature: weekday_list(List)/1 % Purpose: Holds the ordered list of week days. weekday_list([d('Sun'),d('Mon'),d('Tue'),d('Wed'),d('Thu'),d('Fri'),d('Sat')]). % Signature: weekday_list(List)/1 % Purpose: Holds the ordered list of week days. weekday_list([d('Sun'),d('Mon'),d('Tue'),d('Wed'),d('Thu'),d('Fri'),d('Sat')]). % Signature: hour_list(List)/1 % Purpose: Holds the ordered list of days hours. hour_list([h(0),h(1),h(2),h(3),h(4),h(5),h(6),h(7),h(8),h(9),h(10),h(11),h(12), h(13),h(14),h(15),h(16),h(17),h(18),h(19),h(20),h(21),h(22),h(23)]). % Signature: hour_list(List)/1 % Purpose: Holds the ordered list of days hours. hour_list([h(0),h(1),h(2),h(3),h(4),h(5),h(6),h(7),h(8),h(9),h(10),h(11),h(12), h(13),h(14),h(15),h(16),h(17),h(18),h(19),h(20),h(21),h(22),h(23)]).

6. Logic Programming (cont’d): Lists Example 2: Determining the order of specific times weekday_list([d('Sun'),d('Mon'),d('Tue'),d('Wed'),d('Thu'),d('Fri'),d('Sat')]). hour_list([h(0),h(1),h(2),h(3),h(4),h(5),h(6),h(7),h(8),h(9),h(10),h(11),h(12), h(13),h(14),h(15),h(16),h(17),h(18),h(19),h(20),h(21),h(22),h(23)]). % Signature: is_hour?(Hour)/1 % Purpose: Succeeds iff Hour is an hour of the weekday. is_hour(h(H)) :- hour_list(Hour_list), member(h(H),Hour_list). % Signature: is_hour?(Hour)/1 % Purpose: Succeeds iff Hour is an hour of the weekday. is_hour(h(H)) :- hour_list(Hour_list), member(h(H),Hour_list). A predicate to identify hours: % Signature: hour_order(H1,H2)/2 % Purpose: hour H1 precedes the hour H2 in some weekday. hour_order(h(H1),h(H2)) :- is_hour(h(H1)), is_hour(h(H2)), hour_list(Hour_list), precedes(h(H1),h(H2),Hour_list). % Signature: hour_order(H1,H2)/2 % Purpose: hour H1 precedes the hour H2 in some weekday. hour_order(h(H1),h(H2)) :- is_hour(h(H1)), is_hour(h(H2)), hour_list(Hour_list), precedes(h(H1),h(H2),Hour_list). An order relation between hours:

7. Logic Programming (cont’d): Lists Example 2: Determining the order of specific times % Signature: is_weekday?(Day)/1 % Purpose: Success iff Weekday is a weekday of the week. is_weekday?(d(D)) :- weekday_list(Weekday_list), member(d(D),Weekday_list). % Signature: is_weekday?(Day)/1 % Purpose: Success iff Weekday is a weekday of the week. is_weekday?(d(D)) :- weekday_list(Weekday_list), member(d(D),Weekday_list). A predicate to identify weekdays: % Signature: weekday_order(Weekday1,Weekday2)/2 % Purpose: Weekday1 precedes weekday2 % in some week. weekday_LT(d(D1),d(D2)) :- is_weekday(d(D1)), is_weekday(d(D2)), weekday_list(Weekday_list), precedes(d(D1),d(D2),Weekday_list). % Signature: weekday_order(Weekday1,Weekday2)/2 % Purpose: Weekday1 precedes weekday2 % in some week. weekday_LT(d(D1),d(D2)) :- is_weekday(d(D1)), is_weekday(d(D2)), weekday_list(Weekday_list), precedes(d(D1),d(D2),Weekday_list). An order relation between weekdays: weekday_list([d('Sun'),d('Mon'),d('Tue'),d('Wed'),d('Thu'),d('Fri'),d('Sat')]). hour_list([h(0),h(1),h(2),h(3),h(4),h(5),h(6),h(7),h(8),h(9),h(10),h(11),h(12), h(13),h(14),h(15),h(16),h(17),h(18),h(19),h(20),h(21),h(22),h(23)]).

8. Logic Programming (cont’d): Lists Example 2: Determining the order of specific times % Signature: time_LT(T1,T2)/2 % Purpose: The time T1 precedes the time T2 in the week. % Example: ?- time_order(time(h(5),d(‘Mon’)), time(h(1),d(‘Tue’)). % true. time_order(time(h(H1),d(D1)),time(h(H2),d(D2))) :- is_time(time(h(H1),d(D1))), is_time(time(h(H2),d(D2))), weekday_order(d(D1),d(D2)). time_order(time(h(H1),d(D)),time(h(H2),d(D))) :- is_time(time(h(H1),d(D))), is_time(time(h(H2),d(D))), hour_order(h(H1),h(H2)). % Signature: time_LT(T1,T2)/2 % Purpose: The time T1 precedes the time T2 in the week. % Example: ?- time_order(time(h(5),d(‘Mon’)), time(h(1),d(‘Tue’)). % true. time_order(time(h(H1),d(D1)),time(h(H2),d(D2))) :- is_time(time(h(H1),d(D1))), is_time(time(h(H2),d(D2))), weekday_order(d(D1),d(D2)). time_order(time(h(H1),d(D)),time(h(H2),d(D))) :- is_time(time(h(H1),d(D))), is_time(time(h(H2),d(D))), hour_order(h(H1),h(H2)). An order relation between times: % Signature: is_time?(Time)/1 % Purpose: Succeeds iff Time is an hour of some weekday. % Example: ? – is_time? (time(h(1),d(‘Sun’))). true. is_time?(time(h(H),d(D))) :- is_hour?(h(H)), is_weekday?(d(D)). % Signature: is_time?(Time)/1 % Purpose: Succeeds iff Time is an hour of some weekday. % Example: ? – is_time? (time(h(1),d(‘Sun’))). true. is_time?(time(h(H),d(D))) :- is_hour?(h(H)), is_weekday?(d(D)). A predicate to identify times: % Type: Time. % Signature: time(Hour,WeekDay)/2 % Example: time(h(18),d(Tue)). % Type: Time. % Signature: time(Hour,WeekDay)/2 % Example: time(h(18),d(Tue)).

9. Logic Programming (cont’d): Lists Example 3: Deleting one occurrence of an element from a list. % Signature: select(X,HasXs,OneLessXs)/3 % Purpose: The list OneLessXs is the result of removing % one occurrence of X from the list HasXs. select(Element, HasXs, OneLessXs) :- append([Prefix, [Element], Suffix], HasXs), append(Prefix, Suffix, OneLessXs). % Signature: select(X,HasXs,OneLessXs)/3 % Purpose: The list OneLessXs is the result of removing % one occurrence of X from the list HasXs. select(Element, HasXs, OneLessXs) :- append([Prefix, [Element], Suffix], HasXs), append(Prefix, Suffix, OneLessXs). ?- select(4,[2,3,2,4,5,2,4],X). X = [2, 3, 2, 5, 2, 4]; X = [2, 3, 2, 4, 5, 2]; false

10. Logic Programming (cont’d): Lists Example 4: Merging sorted lists % Signature: merge(Xs,Ys,Zs)/3 % Purpose: Zs is the sorted merge of the sorted lists Xs and Ys. Assumes % there a predicate "lt" of order between elements of Xs and Ys. merge([X|Xs],[Y|Ys],[X|Zs]) :- lt(X,Y), merge(Xs,[Y|Ys],Zs).%1 merge([X|Xs],[X|Ys],[X,X|Zs]) :- merge(Xs,Ys,Zs).%2 merge([X|Xs],[Y|Ys],[Y|Zs]) :- lt(Y,X), merge([X|Xs],Ys,Zs).%3 merge(Xs,[],Xs).%4 merge([],Ys,Ys).%5 % Signature: merge(Xs,Ys,Zs)/3 % Purpose: Zs is the sorted merge of the sorted lists Xs and Ys. Assumes % there a predicate "lt" of order between elements of Xs and Ys. merge([X|Xs],[Y|Ys],[X|Zs]) :- lt(X,Y), merge(Xs,[Y|Ys],Zs).%1 merge([X|Xs],[X|Ys],[X,X|Zs]) :- merge(Xs,Ys,Zs).%2 merge([X|Xs],[Y|Ys],[Y|Zs]) :- lt(Y,X), merge([X|Xs],Ys,Zs).%3 merge(Xs,[],Xs).%4 merge([],Ys,Ys).%5 ?- merge([time(h(5),d('Sun')),time(h(5),d('Mon'))], X, [time(h(2),d('Sun')),time(h(5),d('Sun')),time(h(5),d('Mon'))]). X = [time(h(2),d( 'Sun'))] % Signature: lt(Obj1,Obj2)/2 % Purpose: Obj1 precedes Obj2 by some comparison criteria. lt(Time1,Time2) :- time_LT(Time1,Time2). % Signature: lt(Obj1,Obj2)/2 % Purpose: Obj1 precedes Obj2 by some comparison criteria. lt(Time1,Time2) :- time_LT(Time1,Time2).

11. Logic Programming (cont’d): Lists Example 5: Merging sorted lists merge([t1,t3,t5],[t2,t3],Xs) { X_1=t1,Xs_1=[t3,t5], Y_1=t2,Ys_1=[t3], Xs=[t1|Zs_1] } 1 merge([t3,t5],[t2,t3],Zs_1) 2 – failure branch… 1 – failure branch… { X_3=t3,Xs_3=[t5],Y_3=t3,Ys_3=[],Zs_2=[t3,t3|Zs_3] } 2 merge([t3,t5],[t3],Zs_2) 1 – failure branch… merge([t5],[],Zs_3) true { Xs_4=[t5],Ys_4=[], Zs_3=[t5] } 4 merge([X|Xs],[Y|Ys],[X|Zs]) :- lt(X,Y), merge(Xs,[Y|Ys],Zs). merge([X|Xs],[X|Ys],[X,X|Zs]) :- merge(Xs,Ys,Zs). merge([X|Xs],[Y|Ys],[Y|Zs]) :- lt(Y,X), merge([X|Xs],Ys,Zs). merge(Xs,[],Xs). merge([],Ys,Ys). merge([X|Xs],[Y|Ys],[X|Zs]) :- lt(X,Y), merge(Xs,[Y|Ys],Zs). merge([X|Xs],[X|Ys],[X,X|Zs]) :- merge(Xs,Ys,Zs). merge([X|Xs],[Y|Ys],[Y|Zs]) :- lt(Y,X), merge([X|Xs],Ys,Zs). merge(Xs,[],Xs). merge([],Ys,Ys). ?-merge( [ time(h(1),d('Sun')), time(h(3),d('Wed')), time(h(5), d('Sat'))], [ time(h(2), d('Sun')), time(h(3),d('Wed'))], Xs) lt(t1,t2), merge([t3,t5],[t2,t3],Zs_1) * lt(t2,t3), merge([t3,t5],[t3],Zs_2) * 1 – failure branch… { X_2=t3,Xs_2=[t5], Y_2=t2, Ys_2=[t3], Zs_1=[t2|Zs_2] } 3 2 – failure branch… 3 – failure branch…

12. Logic Programming (cont’d): Backtracking optimization merge([t1,t3,t5],[t2,t3],Xs) !, merge([t3,t5],[t2,t3],Zs_1) merge([X|Xs],[Y|Ys],[X|Zs]) :- lt(X,Y), !, merge(Xs,[Y|Ys],Zs). merge([X|Xs],[X|Ys],[X,X|Zs]) :- merge(Xs,Ys,Zs). merge([X|Xs],[Y|Ys],[Y|Zs]) :- lt(Y,X), merge([X|Xs],Ys,Zs). merge([X|Xs],[Y|Ys],[X|Zs]) :- lt(X,Y), !, merge(Xs,[Y|Ys],Zs). merge([X|Xs],[X|Ys],[X,X|Zs]) :- merge(Xs,Ys,Zs). merge([X|Xs],[Y|Ys],[Y|Zs]) :- lt(Y,X), merge([X|Xs],Ys,Zs). lt(t1,t2), !, merge([t3,t5],[t2,t3],Zs_1) Example 7: Optimize the procedure merge. o When none of the first two params is [], only one of the rules can be valid, o Since either lt(Y,X), or lt(X,Y) or X==Y holds, exclusively. o Once one of the above holds, we can skip further rule selections. merge([t3,t5],[t2,t3],Zs_1) 3 – failure branch… 2 – failure branch…

13. Logic Programming (cont’d): Backtracking optimization Rule k:A :- B 1, …B i, !, B i+1, …, B n. Rule k+1:A :- …. … ?- B 1,…, B i, !, B i+1, …B n,Q 1 ’,…Q m ’ Rule k * ?- !, B i+1 ’, …B n ’,Q 1 ’’,…Q n ’’ ?- B i+1 ’, …B n ’,Q 1 ’’,…Q n ’’ … Following selections of A rules are cut. Further substitutions for goals before ‘!’ are cut. The cut operator (!): Used to avoid redundant computations.

14. Logic Programming (cont’d): Backtracking optimization Example 8: red VS green cuts. o Green cut: Helps optimizing the program by avoiding redundant computations. o Red cut: Omits possible solutions. o Red cuts are undesired, unless specifically required (E.g., to provide only the first solution). Q: How many possible solutions does the following query have? ?- merge([],[],X). X = []; X = [] The query matches both facts 4 and 5: merge(Xs,[],Xs). merge([],Ys,Ys). merge(Xs,[],Xs). merge([],Ys,Ys). To avoid this, we add ‘!’ to fact 4, rewriting it as a rule: merge(Xs,[],Xs):- !. merge([],Ys,Ys). merge(Xs,[],Xs):- !. merge([],Ys,Ys). A red cut! The number of solutions has changed!

15. Logic Programming (cont’d): Meta circular interpreter Recall the second version of the meta-circular interpreter (seen in class): o Based on goals reduction, with explicit control of selection order (using a stack of goals). o Operates in two phases: 1.Preprocessing: The given program, P, is translated into a new program P’, which has a single predicate, rule, and consists of facts only. 2.Queries are applied to the new program using the procedure solve.

16. Logic Programming (cont’d): Meta circular interpreter member(X,[X|Xs]). member(X,[Y|Ys]) :- member(X,Ys). member(X,[X|Xs]). member(X,[Y|Ys]) :- member(X,Ys). % Signature: rule(Head,BodyList)/2 rule(member(X,[X|Xs]),[]). rule(member(X,[Y|Ys]),[member(X,Ys)]). % Signature: rule(Head,BodyList)/2 rule(member(X,[X|Xs]),[]). rule(member(X,[Y|Ys]),[member(X,Ys)]). Example 10: Applying the meta circular interpreter. Preprocessing % Signature: solve(Goal)/1 % Purpose: Goal is true if it is true when posed to the original program P. % Precondition: rule/2 is defined for the program. solve(Goal) :- solve(Goal,[]). % Signature: solve(Goal,Rest_of_goals)/2 solve([],[]). % (Nothing to prove, empty stack - done). solve([],[G | Goals]) :- solve(G,Goals). % (Nothing to prove, non-empty stack - pop). solve([A|B],Goals) :- append(B,Goals,Goals1), solve(A,Goals1). % (List to prove – push cdr). solve(A,Goals) :- rule(A,B), solve(B,Goals). % Signature: solve(Goal)/1 % Purpose: Goal is true if it is true when posed to the original program P. % Precondition: rule/2 is defined for the program. solve(Goal) :- solve(Goal,[]). % Signature: solve(Goal,Rest_of_goals)/2 solve([],[]). % (Nothing to prove, empty stack - done). solve([],[G | Goals]) :- solve(G,Goals). % (Nothing to prove, non-empty stack - pop). solve([A|B],Goals) :- append(B,Goals,Goals1), solve(A,Goals1). % (List to prove – push cdr). solve(A,Goals) :- rule(A,B), solve(B,Goals).

17. … {X_3 = X, Y_3 = a, Ys_3 = [b, c], B_2 = [member(X, [b, c])]} Rule 2 rule {A_4 = member(X, [b, c]), B_4 = [], Goals_4 = []} 3 {Goals1_4 = []} Rule of append {A_5 = member(X, [b, c]), Goals_5 = []} 4 {X = b, X_6 = b, Xs_6 = [c], B_5 = []} 1 {Goal_1 = member(X, [a, b, c])} 1 solve(member(X, [a, b, c])) solve(member(X, [a, b, c]), []) {A_2 = member(X, [a, b, c], Goals_1 = []} 4 rule(member(X, [a, b, c], B_2), solve(B_2, [] ) solve([],[]) {X = a, X_3 = a, Xs_3 = [b, c], B_2 = []} 1 true {X = a} 1 solve( [member(X, [b, c])], [] ) append([], [], Goals1_4), solve(member(X, [b, c]), Goals1_4) solve(member(X, [b, c]), []) rule(member(X,[b, c]), B_5), solve(B_5, []) solve([],[]) true {X = b} 1