Prolog for Dummies

1. Prolog for Dummies Ulf Nilsson Dept of Computer and Information Science Linköping University

2. Logic programs A logic program describes individuals and relations between individuals (or properties of individuals). The program is used to answer queries about the world described in the program.

3. Relations • Adam is a parent of Bill • Paris is the capital of France • 5 is greater than 2 plus 2 • X times 1 is equal to X • X is a subset of Y • 5 is the maximumof 2 and 5 • There is an edge from a to b

4. Properties • Adam is a parent • Adam is male • X plus 1 is non-zero • Paris is a capital • Grass is green • The music was loud

5. Queries • Who is the father of Bill? • Is there an edge from a to b? • Which town is a capital? • Who is male?

6. Language primitives • Constantsadam, paris, 5, 3.14, [], ´Adam’, ... • VariablesX, Y, List, _12, _, ... • Function symbolsplus/2, +/2, f/1, ... • Predicate symbolscapital/2, greater/2, non_zero/1, >/2, ...

7. Terms Terms represent individuals • Constants • Variables • Compound terms • E.g. paris, X, plus(2,3), plus(2,plus(3,4)) • Infix notation: 2+3

8. Atomic formulas Atomic formulas describe relations: • If p is a predicate letter of arity n and t1,...,tn are terms then p(t1,...,tn) is an atomic formula. • E.g. capital(paris,france) greater(X,2) • Infix notation: X > 2

9. Logic Programs • A logic program is a set of clauses: • facts • rules • The program is used to answer queries.

10. Facts • A fact is an expression of the form: A. where A is an atomic formula. • Examples: edge(a, X). parent(adam, bill).

11. Interpretation Facts • Consider a fact A. • Declarative (logical) reading: For all variables, A is true. • Procedural (operational) reading: A is solved.

12. Rules • A rule is an expression of the form: A0 :- A1, ... , An. where each Ai is an atomic formula. • Examples: path(X,Y) :- edge(X,Y). father(X,Y) :- parent(X,Y), male(X).

13. Interpretation Rules • Consider a rule A0 :- A1, ... , An. • Declarative (logical) reading: For all variables, A0 if A1 and...and An. • Procedural (operational) reading: To solve A0, first solve A1, then A2 etc.

14. Example gp(X,Y) :- p(X,Z), p(Z,Y). p(X,Y) :- f(X,Y). p(X,Y) :- m(X,Y). f(adam,bill). f(bill,carl). m(anne,bill).

15. Queries • A query is an expression of the form: ?- A1, ..., An. where n=0,1,2,... and A1, ..., An are atomic formulas. • Examples: ?- father(X, bill). ?- parent(X, bill), male(X).

16. Interpretation Queries • Consider a query ?- A1, ... , An. • Declarative (logical) reading: Are there variables such that A1 and...and An? • Procedural (operational) reading: First solve A1, then A2 etc

17. ?- B1,...,Bm,A2,...,An. Ground SLD-Resolution ?- A1,A2,...,An. A1 :- B1,...,Bm. where A1 :- B1,...,Bm is an instantiated program clause.

18. ?- father(adam,bill) ?- true A Derivation parent(X,Y) :- father(X,Y). parent(X,Y) :- mother(X,Y). father(adam,bill). mother(anne,bill). parent(X,Y) :- father(X,Y). parent(X,Y) :- mother(X,Y). father(adam,bill). mother(anne,bill). parent(X,Y) :- father(X,Y). parent(X,Y) :- mother(X,Y). father(adam,bill). mother(anne,bill). ?- parent(adam,bill)

19. ?- mother(anne,bill) ?- true Another Derivation parent(X,Y) :- father(X,Y). parent(X,Y) :- mother(X,Y). father(adam,bill). mother(anne,bill). parent(X,Y) :- father(X,Y). parent(X,Y) :- mother(X,Y). father(adam,bill). mother(anne,bill). parent(X,Y) :- father(X,Y). parent(X,Y) :- mother(X,Y). father(adam,bill). mother(anne,bill). ?- parent(anne,bill)

20. B0 :- B1,...,Bm. ?- A1= B0, B1,...,Bm,A2,...,An. ?- (B1,...,Bm,A2,...,An)q. where: • B0 :- B1,...,Bm is a renamed program clause. • q is a solution to the equation A1 = B0. Full SLD-Resolution ?- A1,A2,...,An.

21. ?- X=X1, bill=Y1, father(X1,Y1). ?- father(X,bill). ?- X=adam, bill=bill. ?- true. Answer: X=adam Yet Another Derivation ?- parent(X,bill). parent(X1,Y1) :- father(X1,Y1). father(adam,bill).

22. X=adam ?- p(bill,Y). ?- f(X,Z1), p(Z1,Y). ?- X=X2, Z1=Y2, f(X2,Y2), p(Z1,Y). ?- bill=X3, Y=Y3, f(X3,Y3). ?- true. ?- X=X1, Y=Y1, p(X1,Z1), p(Z1,Y1). ?- f(bill,Y). ?- bill=bill, Y=carl. ?- X=adam,Z1=bill, p(Z1,Y). ?- p(X,Z1), p(Z1,Y). Y=carl And Another One... ?- gp(X,Y). f(bill,carl). p(X3,Y3) :- f(X3,Y3). p(X2,Y2) :- f(X2,Y2). gp(X1,Y1) :- p(X1,Z1),p(Z1,Y1). f(adam,bill).

23. X=bill ?- p(carl,Y). ?- fail. ?- f(X,Z1), p(Z1,Y). ?- X=X2, Z1=Y2, f(X2,Y2), p(Z1,Y). ?- X=bill,Z1=carl, p(Z1,Y). ?- p(X,Z1), p(Z1,Y). ?- X=X1, Y=Y1, p(X1,Z1), p(Z1,Y1). ?- f(carl,Y). ?- carl=X3, Y=Y3, f(X3,Y3). And a Failed One... ?- gp(X,Y). p(X3,Y3) :- f(X3,Y3). f(bill,carl). p(X2,Y2) :- f(X2,Y2). gp(X1,Y1) :- p(X1,Z1),p(Z1,Y1). FAILURE!!!

24. ?- p(X,Z),p(Z,Y). X=anne X=adam ?- f(X,Z),p(Z,Y). ?- m(X,Z),p(Z,Y). ?- p(bill,Y). ?- p(bill,Y). ?- p(carl,Y). ?- f(carl,Y). ?- m(carl,Y). ?- f(bill,Y). ?- m(bill,Y). ?- f(bill,Y). ?- m(bill,Y). ?- fail. ?- fail. ?- true. Y=carl ?- fail. ?- true. Y=carl ?- fail. SLD-Tree ?- gp(X,Y).

25. Example /* or(In1, In2, Out) */ or(0, 0, 0). or(0, 1, 1). or(1, 0, 1). or(1, 1, 1). /* nand(In1, In2, Out) */ nand(X, Y, Z) :- and(X, Y, Tmp), inv(Tmp, Z). /* inv(In, Out) */ inv(0, 1). inv(1, 0). /* and(In1, In2, Out) */ and(0, 0, 0). and(0, 1, 0). and(1, 0, 0). and(1, 1, 1).

26. Database lecturer(Lecturer,Course) :- course(Course,_,Lecturer,_). duration(Course,Length) :- course(Course,time(_,S,F),_,_), plus(S,Length,F). teaches(Lect,Day) :- course(_, time(Day,_,_), Lect, _). occupied(Room,Day,Time) :- course(_,time(Day,S,F),_,Room), S =< Time, Time =< F. % Database course(logic, time(monday, 8, 10), dave, a12). ...

27. b d f a c e g Recursion edge(a,b). edge(a,c). edge(b,d). edge(c,d). edge(d,e). edge(f,g). path(Node,Node). path(Node1,Node3) :- edge(Node1,Node2), path(Node2,Node3).

28. . . a a . b b c [] c List Notation .(a, .(b, .(c, [])))

29. More On List Notation • The empty list: [] • A non-empty list: .(X,Y) or [X|Y] Syntactic Sugar: • [b] instead of [b|[]] and .(b, []) • [a,b] instead of [a|[b]] and [a|[b|[]]] • [a,b|X] instead of [a|[b|X]]

30. List manipulation list([ ]). list([X|Xs]) :- list(Xs). member(X,[X|Xs]). member(X,[Y|Ys]) :- member(X,Ys). append([ ],Ys,Ys). append([X|Xs],Ys,[X|Zs]) :- append(Xs,Ys,Zs).

31. List Manipulation % reverse(A, B) % B is A in reverse order reverse([ ],[ ]). reverse([X|Xs],Zs) :- reverse(Xs,Ys), append(Ys,[X],Zs). % Alternative version reverse(Xs,Ys) :- reverse(Xs,[ ],Ys). reverse([ ],Ys,Ys). reverse([X|Xs],Acc,Ys) :- reverse(Xs,[X|Acc],Ys).

32. Insertion Sort % sort(A,B) % B is a sorted version of A sort([X|Xs],Ys) :- sort(Xs,Zs), insert(X,Zs,Ys). sort([ ],[ ]). % insert(A,B,C) % if B is a sorted list, then C is sorted % and contains all elements in B plus A insert(X,[ ],[X]). insert(X,[Y|Ys],[Y|Zs]) :- X > Y, insert(X,Ys,Zs). insert(X,[Y|Ys],[X,Y|Ys]) :- X =< Y.

33. Binary Trees % binary_tree(A) % A is a binary tree binary_tree(void). binary_tree(tree(Element,Left,Right)) :- binary_tree(Left), binary_tree(Right). % tree_member(A,B) % A is a node in the tree B tree_member(X,tree(X,_,_)). tree_member(X,tree(_,Left,_)) :- tree_member(X,Left). tree_member(X,tree(_,_,Right)) :- tree_member(X,Right).

34. Built In Predicates • setof(X, p(X), S) ~ S is the set of all X such that p(X) • bagof(X, p(X), B) ~ B is the sequence of all X such that p(X) • findall(X, p(X), B) B is the sequence of all X such that p(X)

35. Negation • Prolog contains a weak form of negation called “negation as failure”. • Written: \+ p(a) • A query ?- \+ p(a) succeeds if the query ?- p(a) fails finitely. • Robust only when the goal contains no variables. (Use only as a test!)

36. a b Example Negation on_top(X) :- \+ blocked(X). blocked(X) :- on(Y, X). on(a, b). %---------------------------- ?- on_top(a). yes