1. Simply Logical – Chapter 3© Peter Flach, 2000 Luc De Raedt and Andreas Karwath (original slides by Peter Flach) Fortgeschrittene Programmierung PROLOG Teil 3

2. Simply Logical – Chapter 3© Peter Flach, 2000 SLD Resolution and SLD Trees

3. Simply Logical – Chapter 3© Peter Flach, 2000 SLD resolution  Resolution in Prolog based on definite clause logic  definite clauses: one atom in the head  Resolution strategy: which literal and which input clause is chosen for resolution?  Strategy in Prolog is called SLD resolution: - S selection rule (in Prolog from left to right) - L linear resolution - D definite clauses  Leftmost literal in query is selected; clauses are selected in textual order  SLD trees: different from proof trees: only show resolvents, but all possible resolution steps  Prolog searches SLD tree depth-first !

4. Simply Logical – Chapter 3© Peter Flach, 2000 :-teaches(peter,ai_techniques) :-teaches(peter,expert_systems) :-teaches(peter,computer_science) ?-student_of(S,peter) SLD-tree :-follows(S,C),teaches(peter,C) :-teaches(peter,expert_systems) :-teaches(peter,computer_science) :-teaches(peter,ai_techniques) ?-student_of(S,peter) student_of(X,T):-follows(X,C),teaches(T,C). follows(paul,computer_science). follows(paul,expert_systems). follows(maria,ai_techniques). teaches(adrian,expert_systems). teaches(peter,ai_techniques). teaches(peter,computer_science). p.44-5 []

5. Simply Logical – Chapter 3© Peter Flach, 2000 Infinite SLD-trees brother_of(X,Y):-brother_of(Y,X). brother_of(paul,peter). brother_of(paul,peter). brother_of(peter,adrian). brother_of(X,Y):-brother_of(X,Z), brother_of(Z,Y). ?-brother_of(paul,B) [] :-brother_of(paul,Z),brother_of(Z,B) :-brother_of(paul,Z1),brother_of(Z1,Z),brother_of(Z,B):-brother_of(peter,B) [] :-brother_of(peter,Z),brother_of(Z,B) :-brother_of(B,peter) [] ?-brother_of(peter,B) :-brother_of(peter,B) :-brother_of(B,peter) [] p.45-6

6. Simply Logical – Chapter 3© Peter Flach, 2000 Depth-first search ?-plist(L) plist([]). plist([H|T]):- p(H),plist(T). p(1).p(2). [] L = [] p.47 :-p(H1),plist(T1) :-plist(T1) [] L = [1] :-p(H1),plist(T1) :-plist(T1) [] L = [1,1] :-plist(T1) [] L = [1,2] [] L = [2] :-p(H1),plist(T1) :-plist(T1) [] L = [2,1] [] L = [2,2] ?-plist(L). L=[]; L=[1]; L=[1,1]; …

7. Simply Logical – Chapter 3© Peter Flach, 2000 Summary  First problem:  trapped in infinite subtrees  Prolog is incomplete by the way the SLD tree is searched (depth-first; a deliberate design decision for memory efficiency)  if breadth-first search were used, then it would be refutation complete!  Second problem:  looping if no (further) solution for any infinite SLD tree  due to semi-decidability of full clausal logic (infinity of Herbrand base)

8. Simply Logical – Chapter 3© Peter Flach, 2000 Programming Example: Monkey and banana

9. Simply Logical – Chapter 3© Peter Flach, 2000 Problem: Monkey and Banana  The Problem  There is a monkey at the door into a room. In the middle of the room a banana is hanging from the ceiling. The monkey is hungry and wants to get the banana, but he cannot stretch high enough from the floor. At the window of the room there is a box that the monkey can use. The monkey can perform the following actions: walk on the floor, climb the box, push the box around (if he is already at it), and grasp the banana if he is standing on the box and directly underneath the banana.  Can the monkey grasp the banana?

10. Simply Logical – Chapter 3© Peter Flach, 2000 banana Problem: Monkey and Banana door window box mokey middle

11. Simply Logical – Chapter 3© Peter Flach, 2000 Monkey and Banana: States state( MR, MB, P, B ) Monkey‘s Room Position at_door middle at_window Monkey‘s Box Position on_floor on_box Box Position at_door middle at_window Banana has has_not

12. Simply Logical – Chapter 3© Peter Flach, 2000 Monkey and Banana: Actions I Possible actions: walk, push, climb, grasp Defined using: move(OldState, Action, NewState) OldStateNewState Action

13. Simply Logical – Chapter 3© Peter Flach, 2000 Monkey and Banana: Actions II  Grasp Banana: move(state(middle, on_box, middle, has_not), grasp, state(middle, on_box, middle, has) ). move(state(P, on_floor, P, H), climb, state(P, on_box, P, H)).  Climb Box:

14. Simply Logical – Chapter 3© Peter Flach, 2000 Monkey and Banana: Actions III  Push Box: move(state(P1, on_floor, P1, H), push(P1, P2), state(P2, on_floor, P2, H)). move(state(P1, on_floor, P, H), walk(P1, P2), state(P2, on_floor, P, H)).  Walk:

15. Simply Logical – Chapter 3© Peter Flach, 2000 Monkey and Banana: Actions Summary move(state(middle, on_box, middle, has_not), grasp, state(middle, on_box, middle, has)). move(state(P, on_floor, P, H), climb, state(P, on_box, P, H)). move(state(P1, on_floor, P1, H), push(P1, P2), state(P2, on_floor, P2, H)). move(state(P1, on_floor, P, H), walk(P1, P2), state(P2, on_floor, P, H)).

16. Simply Logical – Chapter 3© Peter Flach, 2000 Monkey and Banana: Goals  How can the monkey get the banana? canget(state(_, _, _, has)). canget(OldState) :- move(OldState, Move, NewState), canget(NewState). Can the monkey get the banana from the initial state? ?- canget( state(at_door, on_floor,at_window, has_not)).

17. Simply Logical – Chapter 3© Peter Flach, 2000 Monkey and Banana: Summary move(state(middle, on_box, middle, has_not), grasp, state(middle, on_box, middle, has)). move(state(P, on_floor, P, H), climb, state(P, on_box, P, H)). move(state(P1, on_floor, P1, H), push(P1, P2), state(P2, on_floor, P2, H)). move(state(P1, on_floor, P, H), walk(P1, P2), state(P2, on_floor, P, H)). canget(state(_, _, _, has)). canget(OldState) :- move(OldState, Move, NewState), canget(NewState).

18. Simply Logical – Chapter 3© Peter Flach, 2000 Monkey and Banana: Trace | ?- canget(state(at_door,on_floor,at_window,hasnot)). Call: canget(state(at_door,on_floor,at_window,hasnot)) ? Call: move(state(at_door,on_floor,at_window,hasnot),_989,_990) ? Exit: move(state(at_door,on_floor,at_window,hasnot),walk(at_door,_1476),state(_1476,on_floor,at_window,hasnot)) ? Call: canget(state(_1476,on_floor,at_window,hasnot)) ? Call: move(state(_1476,on_floor,at_window,hasnot),_2415,_2416) ? Exit: move(state(at_window,on_floor,at_window,hasnot),climb,state(at_window,on_box,at_window,hasnot)) ? Call: canget(state(at_window,on_box,at_window,hasnot)) ? Call: move(state(at_window,on_box,at_window,hasnot),_3841,_3842) ? Fail: move(state(at_window,on_box,at_window,hasnot),_3841,_3842) ? Fail: canget(state(at_window,on_box,at_window,hasnot)) ? Redo: move(state(at_window,on_floor,at_window,hasnot),climb,state(at_window,on_box,at_window,hasnot)) ? Exit: move(state(at_window,on_floor,at_window,hasnot),push(at_window,_2902),state(_2902,on_floor,_2902,hasnot)) ? Call: canget(state(_2902,on_floor,_2902,hasnot)) ? Call: move(state(_2902,on_floor,_2902,hasnot),_3844,_3845) ? Exit: move(state(_2902,on_floor,_2902,hasnot),climb,state(_2902,on_box,_2902,hasnot)) ? Call: canget(state(_2902,on_box,_2902,hasnot)) ? Call: move(state(_2902,on_box,_2902,hasnot),_5270,_5271) ? Exit: move(state(middle,on_box,middle,hasnot),grasp,state(middle,on_box,middle,has)) ? Call: canget(state(middle,on_box,middle,has)) ? Exit: canget(state(middle,on_box,middle,has)) ? Exit: canget(state(middle,on_box,middle,hasnot)) ? Exit: canget(state(middle,on_floor,middle,hasnot)) ? Exit: canget(state(at_window,on_floor,at_window,hasnot)) ? Exit: canget(state(at_door,on_floor,at_window,hasnot)) ? yes

19. Simply Logical – Chapter 3© Peter Flach, 2000 Monkey and Banana: Notes  Order of clauses in monkey-banana example: 1. grasp 2. climb 3. push 4. walk  Changing this order can cause Prolog to loop indefinitely !