COSC 387: Artificial Intelligence

Project 3
Fall 2004

Due: Nov 5 @ 5 P.M.
13 points

1. Write Lisp functions that convert a set of propositional logic clauses into clausal form. For example:

   >(rewrite-clause '(cond q r))
   ((NOT Q) R)
   
   >(rewrite-clause '(cond (not (and brd (not w))) hf))
   ((BRD HF) ((NOT W) HF))
   
   >(rewrite-clauses '((cond p q) (cond q r) ((not r)) (p)))
   (((NOT P) Q) ((NOT Q) R) ((NOT R)) (P))
   

   Hint: Realize that each clause is a tree with parent nodes of COND, AND, OR, and NOT. First write a Lisp function that traverses this expression tree. Then insert calls to the match function from match.lisp to find the pattern you want to rewrite. Here is a function that rewrites the conditional operators in a clause:

   ;;;
   ;;; rewrite-conds
   ;;;
   ;;; Rewrites (cond p q) as (or (not p) q)
   ;;;
   
   (defun rewrite-conds (clause)
     (cond
       ((atom clause) clause)
       ((= 1 (length clause)) clause)
       (t (let ((bindings (match clause '(cond (? x) (? y)))))
            (cond ((not (null bindings))
                   (let ((phi (rewrite-conds (second (first bindings))))
                         (psi (rewrite-conds (second (second bindings)))))
                     (list 'or (list 'not phi) psi)))
                  ((eq 'not (car clause))
                   (list (car clause) (rewrite-conds (second clause))))
                  (t (list (car clause)
                           (rewrite-conds (second clause))
                           (rewrite-conds (third clause)))))))))
   

   The match function unifies two clauses and returns a binding list Use the question mark as the first element of a list to signify a variable. These should appear in the second clause. For example:

   >(match '(f a) '(f (? x)))
   (((? X) A) (MATCH T))
   
   >(match '(f a) '(g (? x)))
   NIL
   
   >(match '(f a b) '(f (? x) (? y)))
   (((? X) A) ((? Y) B) (MATCH T))
   
   >(match '(f a b) '(f (? x) (? x)))
   NIL
   

2. Write Lisp functions that implement a resolution theorem prover. Given a set of clauses in clausal form, the primary function should return true if it can derive the empty set from the clauses. For example:

   >(prove '(((NOT P) Q) ((NOT Q) R) ((NOT R)) (P)))
   T
   

   Hint: A useful Lisp function that we did not cover in class is the remove function, which simply removes an element from a list. For example:

   >(remove 'a '(b c d a e f g))
   (B C D E F G)
   

   The default test for this function is #'eq, so, in the default mode, you can't remove lists from lists:

   >(remove '(a b) '(b c (a b) e f g))
   (B C (A B) E F G)
   

   You can, however, supply another test for the remove function. Recall that we can use #'equal to compare lists, so we can type:

   >(remove '(a b) '(b c (a b) e f g) :test #'equal)
   (B C E F G)
   

3. Use the following two functions to test your code. They are for the two propositional logic problems presented in lecture. Include them when you submit your project.

   (defun example1 ()
     (let* ((wffs '((cond p q)
                    (cond q r)
                    ((not r))
                    (p)))       ; negated conclusion
            (clauses (rewrite-clauses wffs)))
       (format t "Original statements: ~a~%" wffs)
       (format t "Clauses in CNF: ~a~%" clauses)
       (prove clauses)))
   
   (defun example2 ()
     (let* ((wffs '((cond (not hf) w)
                    (cond (not w) (not brd))
                    (cond (not (and brd (not w))) hf)
                    (cond (not (or (not hf) (not brd))) (not brch))
                    (brd)
                    (brch)))    ; negated conclusion
            (clauses (rewrite-clauses wffs)))
       (format t "Original clauses: ~a~%" wffs)
       (format t "Clauses in CNF: ~a~%" clauses)
       (prove clauses)))
   

Instructions for Submission

;;;;
;;;; COSC [187|387] Project 3
;;;; Name
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;;;; I certify that, with the exceptions of the course materials and those
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;;;; on this project.
;;;;

You must submit your project before 5 PM on the due date.

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