More Simply Logical lab exercises chapter 3

1. • Describe and explain the two differences in behaviour between the predicates powerset and powerset1.

   • Make powerset tail-recursive by adding an accumulator as extra argument.

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2. The well-known 8-puzzle consists of 8 tiles and an empty space, organised in a square. A move consists in sliding a tile to the empty space. The objective is to reach the following position:

      1 2 3
      8   4
      7 6 5
   

   Such a position will be represented by a list

          [XE/YE,X1/Y1,X2/Y2,X3/Y3,X4/Y4,X5/Y5,X6/Y6,X7/Y7,X8/Y8]
   

   with the first pair denoting the coordinates of the empty space, X1/Y1 denoting the coordinates of tile 1, and so on. Coordinates 1/1 refer to the lower left square. The position above is thus represented by the list

          [2/2,1/3,2/3,3/3,3/2,3/1,2/1,1/1,1/2]
   

   • The 'Manhattan distance' is the sum of the horizontal and vertical distance between two squares. Write a predicate mandist(X1/Y1,X2/Y2,D) for calculating this distance.

   • Write a predicate move_p(P1,P2) indicating whether position P2 can be reached from position P1 in a single move. Use the predicate mandist.

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3. The predicate prove_e(Goal,Explanation) returns an explanation with the answer to a query:

   | ?- prove_e(student_of(S,T),Explanation).
   
   S = paul,
   T = peter,
   Explanation = student_of(paul,peter)because(follows(paul,computer_science)because given)and(teaches(peter,computer_science)because given) ? ;
   
   S = paul,
   T = adrian,
   Explanation = student_of(paul,adrian)because(follows(paul,expert_systems)because given)and(teaches(adrian,expert_systems)because given) ? ;
   
   S = maria,
   T = peter,
   Explanation = student_of(maria,peter)because(follows(maria,ai_techniques)because given)and(teaches(peter,ai_techniques)because given) ? ;
   
   no
   

   Notice that because and and are declared as infix functors. Also notice that any predicate on which we use clause(Head,Body) should be declared as dynamic.
   • Explanations can become long-winded when for instance the append-predicate is involved. Modify prove_e such that only the top-level call to such predicates is included in the explanation:

     | ?- prove_e(sublist1(SL,[a,b]),Explanation).
     
     SL = [],
     Explanation = sublist1([],[a,b])because append([],[a,b],[a,b])and append([],[a,b],[a,b]) ? ;
     
     SL = [a],
     Explanation = sublist1([a],[a,b])because append([],[a,b],[a,b])and append([a],[b],[a,b]) ? ;
     
     SL = [a,b],
     Explanation = sublist1([a,b],[a,b])because append([],[a,b],[a,b])and append([a,b],[],[a,b]) ? ;
     
     SL = [],
     Explanation = sublist1([],[a,b])because append([a],[b],[a,b])and append([],[b],[b]) ? ;
     
     SL = [b],
     Explanation = sublist1([b],[a,b])because append([a],[b],[a,b])and append([b],[],[b]) ? ;
     
     SL = [],
     Explanation = sublist1([],[a,b])because append([a,b],[],[a,b])and append([],[],[]) ? ;
     
     no
     

     Make use of the built-in predicate call(Goal) to prove goals without explaining them.

   • Modify prove_e such that it can handle negation:

     | ?- prove_e(bachelor(X),Explanation).
     X = peter,
     Explanation = bachelor(peter)because(man(peter)because given)and(not married(peter)) ?
     

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