View the step-by-step solution to:

3 Towers of Hanoi puzzle This object of this famous puzzle is to move N disks from the left peg to the right peg using the center peg as an auxiliary...

Draw a program clause tree for the goal 'move(3,left,right,center)' show that it is a consequence of the program. How is this clause tree related to the substitution process explained above?
2.3 Towers of Hanoi puzzle This object of this famous puzzle is to move N disks from the left peg to the right peg using the center peg as an auxiliary holding peg. At no time can a larger disk be placed upon a smaller disk. The following diagram depicts the starting setup for N=3 disks. Fig. 2.3 Here is a recursive Prolog program that solves the puzzle. It consists of two clauses. move(1,X,Y,_) :- write('Move top disk from '), write(X), write(' to '), write(Y), nl. move(N,X,Y,Z) :- N>1, M is N-1, move(M,X,Z,Y), move(1,X,Y,_), move(M,Z,Y,X). The variables filled in by '_' (or any variables beginning with underscore) are 'don't-care' variables. Prolog allows these variables to freely match any structure, but no variable binding results from this gratuitous matching. Here is what happens when Prolog solves the case N=3. ?- move(3,left,right,center). Move top disk from left to right Move top disk from left to center Move top disk from right to center Move top disk from left to right Move top disk from center to left Move top disk from center to right Move top disk from left to right yes The first clause in the program describes the move of a single disk. The second clause declares how a solution could be obtained, recursively. For example, a declarative reading of the second clause for N=3, X=left, Y=right, and Z=center amounts to the following: move(3,left,right,center) if move(2,left,center,right) and ] * move(1,left,right,center) and move(2,center,right,left). ] ** This declarative reading of the clause is obviously correct. The procedural reading is closely related to the declarative interpretation of the recursive clause. The procedural interpretation would go something like this: In order to satisfy the goal ?- move(3,left,right,center) do this :
Background image of page 1
satisfy the goal ?-move(2,left,center,right), and then satisfy the goal ?-move(1,left,right,center), and then satisfy the goal ?-move(2,center,right,left). Also, we could write the declarative readings for N=2: move(2,left,center,right) if ] * move(1,left,right,center) and move(1,left,center,right) and move(1,right,center,left). move(2,center,right,left) if ] ** move(1,center,left,right) and move(1,center,right,left) and move(1,left,right,center). Now substitute the bodies of these last two implications for the heads and one can "see" the solution that the prolog goal generates. move(3,left,right,center) if move(1,left,right,center) and move(1,left,center,right) and * move(1,right,center,left) and --------------------------- move(1,left,right,center) and --------------------------- move(1,center,left,right) and move(1,center,right,left) and ** move(1,left,right,center). A procedural reading for this last big implication should be obvious. This example illustrates well three major operations of Prolog: 1) Goals are matched against the head of a rule, and 2) the body of the rule (with variables appropriately bound) becomes a new sequence of goals, repeatedly, until 3) some base goal or condition is satisfied, or some simple action is taken (like printing something). The variable matching process is called unification. Exercise 2.3.1 Draw a program clause tree for the goal 'move(3,left,right,center)' show that it is a consequence of the program. How is this clause tree related to the substitution process explained above? Exercise 2.3.2 Try the Prolog goal ?-move(3,left,right,left). What's wrong? Suggest a way to fix this and follow through to see that the fix works.
Background image of page 2
Show entire document

Recently Asked Questions

Why Join Course Hero?

Course Hero has all the homework and study help you need to succeed! We’ve got course-specific notes, study guides, and practice tests along with expert tutors.

-

Educational Resources
  • -

    Study Documents

    Find the best study resources around, tagged to your specific courses. Share your own to gain free Course Hero access.

    Browse Documents
  • -

    Question & Answers

    Get one-on-one homework help from our expert tutors—available online 24/7. Ask your own questions or browse existing Q&A threads. Satisfaction guaranteed!

    Ask a Question