1 Logic Programming School of Informatics, University of Edinburgh Transformations Specification-Program An introduction to moving between Prolog and First Order Predicate Logic.

2 Logic Programming School of Informatics, University of Edinburgh Why Worry About Specifications? Equivalence p(X)  (  Y.(q(X)  r(Y))) p(X)   (  Y.(q(X)   r(Y))) p(X) :- \+ (q(X), \+ r(Y)) is logically equivalent to may be equivalent to

3 Logic Programming School of Informatics, University of Edinburgh Why Worry About Specifications? Refinement subset([], S2) subset([H|T], S2) :- member(H,S2), subset(T, S2) subset(S1, S2)  list(S1)  list(S2)   E.(member(E,S1)  member(E,S2)) subset(S1, S2) :- list(S1),list(S2), \+ (member(E,S1), \+ member(E,S2))

4 Logic Programming School of Informatics, University of Edinburgh Why Worry About Specifications? Desirable properties (e.g. termination) subset([], S2) subset([H|T], S2)  member(H,S2)  subset(T, S2) size(subset([H|T], S2)) = len([H|T]) + len(S2) = len(T) len(S2) > len(S2) = size(member(H,S2)) size(subset ([H|T], S2)) = len([H|T]) + len(S2) = len(T) len(S2) > len(T) + len(S2) = size(subset(T,S2)) size(member(H,[H|T])) = len([H|T]) = len(T) + 1 > len(T) = size(member(H,T)) len([H|T]) = len(T) + 1 size(subset(S1,S2)) = len(S1) + len(S2) size(member(E,S)) = len(S) Given:

5 Logic Programming School of Informatics, University of Edinburgh Pure Prolog l Negation l Control predicates l Meta-logical predicates l External state predicates l Clause-base altering predicates Without: p(X) :- not(q(X)), r(X). q(2). r(1). p(X) :- q(X), !, r(X). q(2). r(1). r(2). p(X) :- var(X), q(X). q(1). p(X) :- read(X). p(X) :- assert(r(X)). q(X) :- r(X).

6 Logic Programming School of Informatics, University of Edinburgh Pure Prolog to Horn Clauses p(X,Y)  (q(X)  r(Y))  s(X, Y) p(X,Y) :- (q(X), r(Y)) ; s(X, Y).

7 Logic Programming School of Informatics, University of Edinburgh Clark Completion p(a 1,…,a N )  body 1 p(b 1,…,b N )  body 2 p(X 1,…,X N )  X 1 = a 1  … X N = a N  body 1 p(Y 1,…,Y N )  Y 1 = b 1  … Y N = b N  body 2 Make head arguments into distinct variables Existentially quantify any variable in the body that is not in the head. p(X 1,…,X N )   V 1 …V i.X 1 = a 1  … X N = a N  body 1 p(Y 1,…,Y N )   W 1 …W j.Y 1 = b 1  … Y N = b N  body 2 Combine clauses p(X 1,…,X N )  (  V 1 …V i.X 1 = a 1  … X N = a N  body 1 )  (  W 1 …W j.Y 1 = b 1  … Y N = b N  body 2 )

8 Logic Programming School of Informatics, University of Edinburgh Clark Completion (Example) subset([], S2) subset([H|T], S2)  member(H,S2)  subset(T, S2) subset(S1, S2)  S1 = [] subset([H|T], S2)  S1 = [H|T]  member(H,S2)  subset(T, S2) Make head arguments into distinct variables Existentially quantify any variable in the body that is not in the head. Combine clauses subset(S1, S2)  S1 = [] subset(S1, S2)  S1 = [H|T]  member(H,S2)  subset(T, S2) subset(S1, S2)  (S1 = [])  (S1 = [H|T]  member(H,S2)  subset(T, S2))

9 Logic Programming School of Informatics, University of Edinburgh Lloyd-Topor Transformation (Simplified) H   1  V  2 H   1   2 H   1 H   2 H   1  2 H   1 (A  B)  2 H   1 A  2 H   1 B  2

10 Logic Programming School of Informatics, University of Edinburgh Lloyd-Topor Transformation (Example) p(X)  a(X)  (  Y.q(X,Y)  r(X,Y)  s(X,Y)) p(X)  a(X)   ((q(X,Y)  r(X,Y))   s(X,Y)) p(X) :- a(X), \+ ((q(X,Y) ; r(X,Y)), \+ s(X,Y))