quiz 3 next time:
    search algorithms   and results (see prior Lecture notes).
    np-completeness
    def:  deterministic
	  stochastic
	  random
	  closed system
	  open system

       closed => deterministic

     SAT matrices
     3 Lisp questions.


given a set of logic sentences:
   they are:
       a "tautology" if true in all worlds

       a "contradiction" or absurdity, if true of no worlds

       satisfiable, if true of at least one world.

(a "world" is a a binding of true false to the variables).


SATISFIABILITY
===========================================================================
Problem 1:
~(E implies F) and (F or (D implies ~E)). Is this satisfiable?
In conjunctive normal form:
(E) and (~F) and (F or ~D or ~E).  (use relationship p implies q to reduce to.
                              ~p or q, use De Morgans laws and others.)

In boolean matrix (with *s) form: (rows are clauses, columns are variables)
    DEF
    ---
    *1*
    **0
    001

Problem 2:
 ~((D&E) implies F) and (F or (D implies ~E)). Is this satisfiable?
    
In conjunctive normal form:
    (D) and (E) and (~F) and F or ~D or ~E.

    DEF
    ---
    1**
    *1*
    **0
    001

Satisfiability question is whether or not we can find a row of 0 and
1 that matches every other row in one or more places.

In problem 1, YES:010. In problem 2: NO.

--------------------------------------------------------
   Satisfibility in Lisp. 

assumes input is in form ( (0 0 * 1) (1 1 0 0) (* * 0 0) (* * 1 0)) etc.
and goal is to construct a list  of 0 and 1 that matches each input list
in at least 1 place.

(defun snoc (x l) (reverse (cons x (reverse l))))

(defun sat (l sofar)
;assumes l is not initially nil
    (cond
	 ((null l) sofar)
	 ((null (car l)) nil)
	 (t (or (sat (bind 0 l) (snoc 0 sofar))
		(sat (bind 1 l) (snoc 1 sofar))))))

(defun bind (x l)
     (remove 'hit (mapcar #'(lambda (y)
		      (if (equal (car y) x) 'hit (cdr y)))
			    l)))

Above function is correct! But, alas, is exponential in worst case.

 Complete means: If a proof exists we can find it.


KNOWLEDGE REPRESENTATION:
Variable-Free   Propositional Logic     Decidable but NP-complete
                                        Complete: Resolution, Truth
					Tables

Variable-Based  Predicate Logic         Semi-Decidable
                 = first order logic
		 = predicate calculus
				       (also Godel-Incomplete may be
				       things that are obviously true
						      but not provable)
                                         Can't decide for an arbitrary
					 statement whether it is true,
					 false or undeterminable.
                                       (but if I tell you 
				       that it is "T or F" you can
				       determine which and prove it.)
                                       Complexity is exponential
				       or higher.

 
 we will be learning 3 proof methods:
     resolution (see text)
     sat matrices (as above)
     egs (the wild thing we did in class - will be discussed in future)