1. 1 Logic Programming Robert Kowalski’s equation: Algorithm = Logic + Control Definite logic program: A  B 1, B 2, …, B m (program clause) headbody  A 1, A 2, …, A n (goal clause)

2. 2 Logic Programming SLD-resolution:  Resolvent:  A 1, …, A k-1, A k, A k+1, …, A n A  B 1, B 2, …, B m   (A 1, …, A k-1, B 1, B 2, …, B m, A k+1, …, A n )  A k =  A SL-resolution for Definite clauses - SL = Linear resolution with Selection function

3. 3 Logic Programming SLD-resolution:  SLD-derivation: G 0 = G, G 1, G 2, …, G n C 1, C 2, …, C n  1,  2, …,  n (mgu’s)  i+1 C i+1 =  i+1 G i  SLD-refutation: G n = empty  Computed answer:  n …  2  1

4. 4 Logic Programming SLD-resolution:  Soundness: if  n …  2  1 is a computed answer, then P   n …  2  1 G  Completeness: if P   G, then there exists a computed answer  such that  =  for some  Example: p(X,Z)  q(X,Y), p(Y,Z) p(X,X) q(a,b)

5. 5 Logic Programming  p(X,b)  q(X,Y), p(Y,b) {X/b} success  p(b,b)  q(b,U), p(U,b) failure {X/b} success SLD-tree: Each branch is an SLD-derivation

6. 6 Logic Programming PROLOG (Alain Colmerauer 1972):  Only Horn sentences are acceptable  The occur-check is omitted from the unification  unsound Example: test  p(X,X) p(X,f(X))  Backward chaining with depth-first search  incomplete Example: p(X,Y)  q(X,Y) p(X,X) q(X,Y)  q(Y,X)

7. 7 Logic Programming  p(X,b)  q(X,b) {X/b} success  q(b,X)  q(X,b) Infinite SLD-tree:

8. 8 Logic Programming PROLOG (Alain Colmerauer 1972):  Unsafe cut  incomplete Example: A  B, C  A B  D, !, E D   B, C  D, !, E, C  !, E, C  Negation as failure:  P if fails to prove P

9. 9 Theorem Provers Accepts full first-order logic Keeps control distinct from logic A  B  C A  C  B

10. 10 Production Systems Working memory: contains a set of positive literals with no variables Rule memory: contains a set of reference rules of the form p 1  p 2 …  act 1  act 2 … p i are literals act i are actions to delete or add elements to the working memory

11. 11 Production Systems Forwarding chaining in each cycle:  Match phase:computes the subset of rules whose left-hand side is satisfied by the current contents of the working memory  Conflict resolution phase: decides which of the rules to be executed  Act phase: executes the actions in the chosen rules

12. 12 Production Systems Example:  Working memory: {A(1), A(2), B(2), B(3), B(4), C(5)}  Rule memory: A(x)  B(x)  C(y)  add D(x) A(x)  B(y)  D(x)  add E(x) A(x)  B(x)  E(x)  delete A(x)

13. 13 Production Systems AB D A=BC E A=Dadd E add D delete A RETE network A(1), A(2)B(2), B(3), B(4) A(2) B(2) C(5)D(2)

14. 14 Exercises In Russell & Norvig’s AIMA (Chapter 10): 10.2, 10.4