Execution of Prolog Programs by gqe14638

VIEWS: 74 PAGES: 7

									                                                                                                             Outline
                                                                                  • Prolog
                                                                                     – Basic data types
        CSC326 Programming Languages                                                 – Lists
              (Week 13, Monday)                                                      – Unification
                                                                                     – Prolog search tree
                                                                                     – Negation as failure
                                                                                     – Some small but useful things in Prolog
                             Yilan Gu                                                – Cut!
                      guyilan@ecf.toronto.edu
              http://www.cs.toronto.edu/~yilan/326f09




         Execution of Prolog Programs                                                                          Cut !
                                                                              •     The goal “!”, pronounced “cut” always succeeds immediately.
•      Unification: variable bindings.
                                                                              •     It has an important side effect: Once it is satisfied, it disallows
•      Backward Chaining/                                                           either:
       Top-Down Reasoning/                                                             –    backtracking back over the cut, or
       Goal-Directed Reasoning:                                                       –    backtracking and applying a different clause of the same
         Reduces a goal to one or more subgoals.                                           predicate to satisfy the present goal.
•      Backtracking:                                                          •     You can think of satisfying cut as making a commitment both
    Systematically searches for all possible solutions that can be obtained          –    to the variable bindings we’ve made during the application
      via unification and backchaining.                                                   of this rule, and
                                                                                     –    to this particular rule itself.
                                    Cut !                                                                               Use of Cut
•      The cut goal succeeds whenever it is the current goal, and the derivation      •      Cut can be used to improve the efficiency of search by reducing Prolog’s
       tree is trimmed of all other choices on the way back to and including the             search space. E.g.,
       point in the derivation tree where the cut was introduced into the sequence    When two predicates are mutually exclusive.
       of goals.
                                                                                      q(X) :- even(X), a(X).
•      Cut tells us: ”Do not pass back through this point when looking for
       alternative solutions. ! acts as a marker back beyond which Prolog will        q(X) :- odd(X), b(X).
       not go. All the choices made prior to the cut are set, and are treated as      With cut
       though they were the only possible choices.                                    q(X) :- even(X), !, a(X).
•      You can think of Cut as telling the interpreter: ”Trust me – if you get this   q(X) :- odd(X), b(X).
       far in the clause, there’s no need to backtrack and try another choice for
       proving this goal,or to try another way of satisfying any of the subgoals
       that were already proved for this goal.”The goal “!”, pronounced “cut”
       always succeeds immediately.




                               Use of Cut                                                                               Use of Cut
                                                                                      •      Cut can remove unwanted answers, but ...
•      Cut can remove unwanted answers.
                                                                                      •      Consider the Family Database:
•      Consider the Family Database:
                                                                                      male(albert).
male(albert).
                                                              
                       male(bob).                                              

male(bob).                                                                                                                               

 
                                                                                      male(edward).
male(edward).                                        

                                                                             

                                                                                     female(alice).
female(alice).                                       

                                                                                
                                                                                      female(victoria).
female(victoria).                                       
                                                                                  parent(albert,edward).
parent(albert,edward).                                                               parent(victoria,edward).
parent(victoria,edward).
                                                                                      parent(albert,alice).
parent(albert,alice).
                                                                                      parent(victoria,alice).
parent(victoria,alice).
                                                                                      parent(albert,bob).
parent(albert,bob).
                                                                                      parent(victoria,bob).
parent(victoria,bob).
                                                                                      % son(?X) iff X is a son
% son(?X) iff X is a son
                                                                                      son(X):-parent(_,X),male(X), !.
son(X):-parent(_,X),male(X).
                                   Use of Cut                                                             Use of Cut
•      Cut is dangerous                                                    Think:
male(albert).                                                                  any male is a potentially good answer, so we want to try all of them: can’t
male(bob).                                                                      put “cut” after “male” in the same rule.
male(edward).                             
                                • if a male has 2 parents, we only want to list him once as the answer: want to
female(alice).                        

                                  put “cut” after “parent”.
                                     

female(victoria).
parent(albert,edward).                  
             Result:
                                        
     
parent(victoria,edward).                                                   son(X):-male(X), isason(X).
parent(albert,alice).                                                      isason(X):-parent(_,X),!.
parent(victoria,alice).                                                    ?- son(X).
parent(albert,bob).                                                        X = bob ;
parent(victoria,bob).
                                                                           X = edward ;
% son(?X) iff X is a son
                                                                           No
son(X):-parent(_,X), !, male(X).




                                   Use of Cut                                                             Use of Cut
                                                              •        Try to use cut:
sibling(X,Y):-parent(P,X),parent(P,Y).                                     siblings(X,Y):-parent(P,X), !, parent(P,Y).
?- sibling(alice,Asib).                                                    ?- sibling(alice,Asib).
Asib = edward ;                                                            Asib = edward ;
Asib = alice ;                                                             Asib = alice ;
Asib = bob ;                                                               Asib = bob ;
Asib = edward ;                                                            No
Asib = alice ;                                                             So far so good.
Asib = bob ;                                                               •        But if we ask:
no.                                                                        siblings(X,Y):-parent(P,X), !, parent(P,Y).
                                                                           ?- sibling(Asib,alice).
                                                                           Asib = edward ;
                                                                           No
                                                                           Not so good.
                              Use of Cut                                                                         Use of Cut
Think:
                                                                                  Finally, we don’t want X to be a sibling on X.
• any two people in the database are potentially good answers, so we want to
    try all of them: can’t put “cut” in a rule after X and/or Y is instantiated
• if 2 people share 2 parents, we only want to list them once as the answer:      sibling(X,Y):-not(X=Y), person(X),person(Y),
     want to put “cut” after 2 “parent” rules.                                                   commmonparent(X,Y).
Result:                                                                           person(X):-male(X).
sibling(X,Y):-person(X),person(Y), commonparent(X,Y).                             person(X):-female(X).
person(X):-male(X).                                                               commonparent(X,Y):-parent(P,X),parent(P,Y),!.
person(X):-female(X).                                                             ?- sibling(alice,Asib).
commonparent(X,Y):-parent(P,X),parent(P,Y),!.                                     No
?- sibling(alice,Asib).                           ?- sibling(Asib,alice).
Asib = bob ;                                      Asib = bob ;                    What went wrong? How to fix? (see next page)
                                                  Asib = edward ;
Asib = edward ;
                                                  Asib = alice ;
Asib = alice ;                                    No
No



                              Use of Cut                                                                    When to use cut?
sibling(X,Y):- person(X),person(Y), not(X=Y),                                     •    Common uses of the !:
                 commmonparent(X,Y).                                                   –   Tell the Prolog system that it has found the right rule for a particular goal:
person(X):-male(X).                                                                         If you get this far, you have picked the correct rule for this goal.
                                                                                           e.g., P(N,X):-odd(N), !, X is N+3. % assume that odd() is defined.
person(X):-female(X).                                                                            P(N,X):-even(N), X is N / 2.
commonparent(X,Y):-parent(P,X),parent(P,Y),!.                                          –   Tell the Prolog system to fail a particular goal immediately without trying
                                                                                           for alternate solutions:
                                                                                           If you get to here, you should stop trying to satisfy the goal.
?- sibling(Asib,alice).
                                                                                           e.g., not(G):- G,!, fail.
Asib = bob ;                                                                                     not(G).
Asib = edward ;                                                                        –   Terminate the generation of alternative solutions:
No                                                                                         If you get to here, you have found the only solution to this problem,
                                                                                           no point in looking for alternatives.
?- sibling(alice,Asib).
                                                                                           e.g., factorial(N,X):- N>0, N1 is N-1, factorial(N1, X1), X is X1*N.
Asib = bob ;                                                                                     factorial(0,1):-!.
Asib = edward ;
No
             Prolog: cut ! - examples                                                                                Prolog: cut ! – example
•   Double-step function:                                                                            •   Double-step function - cont’d:
          if X < 3 then Y = 0                                                                                if X < 3 then Y = 0
          if 3 =< X and X < 6 then Y = 2                                                                          if 3 =<X and X < 6 then Y = 2
          if 6 =< X then Y = 4                                                                                    if 6 =< X then Y = 4

                                                                                                             What do we know about this function that Prolog doesn’t?
     % In Prolog
     f(X,0) :- X < 3.               %rule 1                                                                  % same relations with !
     f(X,2) :- 3 =< X, X < 6.       %rule 2                                                                  f(X,0) :- X < 3,!.     %rule 1                                      f(1,Y)
                                                                                                                                                                                 2<Y
     f(X,4) :- 6 =< X.              %rule 3                         f(1,Y)                                   f(X,2) :- 3 =< X, X < 6,!. %rule 2                                               CUT
                                                                    2<Y                                                                                                                 CUT
                                                                                                             f(X,4) :- 6 =< X.      %rule 3                   rule 1           rule 2           rule 3

    ?- f(4,Y).                 %query         rule 1              rule 2                rule 3                                                                         1<3
    Y=2                                                                                                      ?- f(1,Y),2<Y.          %query
                                                       1<3           3 =< 1,       6 =< 1                                                                                yes
                                                                     1<6                                     no
                                                            yes        no              no
     ?- f(1,Y),2<Y.            %query
                                                                                                                                                                       2<0
     no
                                                       2<0                                                                                                              no

                                                       no                                                        In this example, we changed the procedural
                                                                                                                meaning of the program, but not the declarative meaning



              Prolog: cut ! – example                                                                                      cut ! classification
•   Double-step cont’d:
                                                                                                     •   Green cuts:
     % same relations with !                                        f(5,Y)

     f(X,0) :- X < 3,!.        %rule1                                                                    –      Affect procedural meaning of program but has no effect on the
                                                                                 CUT
     f(X,2) :- 3 =< X, X < 6,!. %rule2         rule 1             rule 2                rule 3
                                                                                                                declarative meaning.
     f(X,4) :- 6 =< X.         %rule3                                                                    –      Do not affect readability of programs.
                                                       5<3           3 =< 5
     | ?- f(5,Y).              %query                                5<6                                 –      Used mainly to avoid wasted computations.
     Y=2                                                no             yes


     Can we come up with a more efficient version?                    f(5,Y)                         •   Red cuts:
     f(X,0) :- X < 3,!.       %rule1                                                                     –      Affect declarative meaning of program as well as procedural meaning.
                                                                                  CUT
     f(X,2) :- X < 6,!.       %rule2            rule 1             rule 2                   rule 3       –      Affect readability:
     f(X,4).                  %rule3                                                                            •   Similar to unconditional jump (goto) in imperative PLs
                                                        5<3           5<6
     | ?- f(5,Y).             %query                                                                     –      Used to avoid wasted computations and also introduce semantics
     Y=2                                                    no             yes
                                                                                                         –      If not used cautiously, can affect result in arbitrary way.
     % What if we removed the cuts?           | ?- f(1,Y).% query
     f(X,0) :- X < 3.     %rule1              Y = 0;        % right answer
     f(X,2) :- X < 6.         %rule2          Y = 2;        % wrong answer,why?
     f(X,4).                  %rule3          Y = 4;        % wrong answer,why?
     Here, we changed the procedural and also the declarative meaning
                            cut ! - examples                                                                            cut ! – examples
    •   Member function:                                                                  •        Bubble-sort without !:
                                                                                                      % Find 2 adjacent elements X and Y in List such that X > Y        Bubblesort without !
        member(Element, [Element | _]).                                                                                                                                 (1) List=[10,3,1]
        member(Element, [_ | Rest]) :- member(Element, Rest).                                         % and swap X and Y to get List1, then sort List1. If there is     --------------------
                                                                                                      % no pair of adjacent X and Y where X > Y, we are done.           (3) List=[3,10,1]
                                                                                                                                                                        --------------------   Correct
        % What’s the problem in the above function?                                                   bubblesort(List,Sorted):-swap(List,List1),                                               answer
                                                                                                                                                                        (8) List=[3,1,10]
                                                                                                              bubblesort(List1,Sorted).                                 --------------------
        member(Element, [Element | _]) :- !.                                                          bubblesort(Sorted,Sorted).                                        (10) List=[1,3,10]
                                                                                                                                                                                                Why
                                                                                                                                                                                                didn’t
                                                                                                                                                                        --------------------    it stop?
    •   Sum function:                                                                                 swap([X,Y | Rest],[Y,X | Rest]) :- X > Y.     % rule 1
                                                                                                                                                                        …(17) List=[3,1,10]
         – Without !:                                                                                 swap([Z|Rest],[Z|Rest1]) :- swap(Rest,Rest1). % rule 2
                                                                                                                                                      rule 1                                      rule 2
        sum_to(1,1).                                                                                  ?- bubblesort([10,3,1],Res). %query               swap                       swap
        sum_to(N, Res) :- N1 is N-1, sum_to(N1,TRes), Res is TRes + N.                                Res = [1,3,10];                                   (2) List=10,3,1            (5) List=3,10,1
                                                                                              ……                        % more output ?                 X=10,Y=3, Rest=1           Z=3, Rest=10,1
        ?- sum_to(1,X).                                                                                                                                     List1=3,10,1            -------------------
        X=1;                                                                                                                                                -------------------     (7) Rest1=1,10
        %Infinite loop………………….                                                            •        Bubble-sort with !:                                      (4) List=3,10,1         Z=3
                                                                                                   bubblesort(List,Sorted):-swap(List,List1),!,             X=3,Y=10,Rest=1         List1=3,1,10
                                                                                                              bubblesort(List1,Sorted).                     ------------------      ------------------
            – With !:                                                                                                                                       (6) List=10,1           (12)Z=1,Rest=3,10
        sum_to(1,1) :- !.                                                                          bubblesort(Sorted,Sorted).                               X=10, Y=1,Rest=         etc..
        sum_to(N, Res) :- N1 is N-1, sum_to(N1,TRes), Res is TRes + N.                                 swap([X,Y | Rest],[Y,X | Rest]) :- X > Y.            List1=1,10
        ?- sum_to(1,X).                                                                                                                                     ------------------
                                                                                                       swap([Z|Rest],[Z|Rest1]) :- swap(Rest,Rest1).
                                                                                                                                                            (9) X=3,Y=1,Rest=10
        X=1;                                                                                       ?- bubblesort([10,3,1],X). %query                        List1=1,3,10
                                                                                                      X = [1,3,10];                                         ------------------
                                                                                                                                                            (11) X=1,Y=3,Rest=10



                           cut ! Pros & Cons                                                                         Prolog: pros & cons
•       Pros:                                                                                  • Cons:
                                                                                                     –   Horn clauses have limited expressive power
        –     More efficient: interpreter will not process unnecessary branches in the
                                                                                                     –   Closed world assumptions (anything not mentioned is false)
              unification tree
                                                                                                     –   Ordering of clauses change the result
        –     Save memory: interpreter will not allocate space for tree nodes that will
              not be unified.                                                                        –   Because horn clause is the basic construct, you must program carefully to
                                                                                                         avoid infinite loops and incorrect negation.
                                                                                                           • There is no 1-solution-fits-all for these problems…
•       Cons:
        –     Creates side effects that change the way backtracking works (not really
              logic programming…)                                                              • Pros:
        –     Makes the place markers of certain goals inaccessible.                                 –   Pattern matching
                                                                                                     –   Backtracking
                                                                                                     –   Unification
                                                                                                     –   Rules and goals are also data (dynamic programming).
                  Die the death of a thousand cuts… Richard O’Kaffe – Craft of Prolog
                                                                                                     –   The logical model is powerful
            Logic vs. Imperative Languages
                              
  
          
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