CS 641 meeting -*- Outline -*-

* fixpoints in complete lattices
	(Some prefer "fixedpoint" to "fixpoint".)

** motivation

	This is the key concept both in the denotational approach
		and in pred. transformer semantics a la Hesselink

	Also found in math and psychology (informally): love, religion, ...

--------------------------------
        FIXPOINTS

def: Let W be a complete lattice.
Let D : W -> W.
Then v \in W is a *fixpoint of D*
iff D.v = v.
---------------------------------
	note D has the same domain and range.  Why?
	Q: how would you express a fixpoint in terms of the ordering on W
		not equality?
	v is a fixpoint iff D.v <= v and v <= D.v

---------------------------------
def: v is a *least fixpoint* of D
iff v is a fixpoint
    and for every fixpoint w, v <= w.

def: v is a *greatest fixpoint* of D
iff v is a fixpoint
    and for every fixpoint w, v >= w.
--------------------------------
	Note that the second line is the "universal property"
		of the lfp and gfp.

	Q: is the lfp unique?  The gfp?

	In denotational semantics, typically only concerned with lfp.

	Q: Suppose f \in MT.  What do these definitions mean?
		Recall MT = {f | f : |P -> |P, f monotone}
		p is a fixpoint iff f.p equiv p
		p is a least fixpoint iff for all other fixpoints q,
			[p ==> q]

	Q: Suppose f \in (Command -> MT) -> (Command -> MT).  Then what?
		w is a fixpoint iff f.w = w
		What's the induced ordering?
			w1 <= w2 iff (all c :: w1.c <= w2.c)
				 iff (all c,p :: w1.c.p <= w2.c.p)
				 iff (all c,p :: [w1.c.p ==> w2.c.p])
		So w is a least fixpoint iff for all other fixpoints w',
			(all c,p :: [w.c.p ==> w'.c.p])

---------------------------
Theorem (Knaster-Tarski):
Let W be a complete lattice.
Let D : W -> W be monotonic.
Then D has a least fixpoint wa
and a greatest fixpoint wb.

Let V \subseteq W be D-invariant;
i.e., (all v \in V :: D.v \in V).
Then (a) if V is sup-closed, wa \in V.
     (b) if V is inf-closed, wb \in V.
---------------------------
	Remark, sup-closed or inf-closed sets contain the sup or inf
		of the empty set, and thus cannot be empty.

	Pf: Note that W itself is D-invariant by the type of D.
		Also, as W is a complete lattice, it is sup-closed.
	Hence D-invariant, sup-closed subsets of W exist.
	Let V be such a subset.
	To construct wa, let U be defined by

		u \in U  equiv  u \in V /\ (all w \in W : D.w <= w : u <= w).

	Q: how would you describe or picture this?

Draw first W, then V, then where stuff goes in V
	then the subset of V where w <= D.w, then U, then wa.
------------------------------------------
                     ^
                    / \ W
        	   /   \
        	  /    	\
        	 /   ^	 \
        	/   / \ V \
               /   /   \   \
              /	  /     \   \
             /	 /       \   \
            /	/         \   \
           /   /     	   \   \
          /   /      	    \  	\
         /   /       	     \   \
        /   /   |------\      \   \
       /   /    |    w  \      \   \
      /	  /     /     \  \      \   \
     /	 /     /       v  \      \   \
    /	/      [      D.w ]       \   \
   /   / D.v   -----*-----         \   \
  /   /  ^    /    wa     \ U       \  	\
 /   /  /    /    = (sup U)\         \   \
<   /  v    /               \  	      \   >
 \ /_______/_________________\_________\ /
  \                   	        	/
   \                   	   	       /
    \                      	      /
     \                 	   	     /
      \              	   	    /
       \             	   	   /
       	\            	   	  /
	 \           		 /
	  \	     		/
	   \	     	       /
	    \	     	      /
	     \	     	     /
	      \	     	    /
	       \     	   /
	       	\    	  /
		 \   	 /
		  \  	/
		   \   /
		    \ /
		     v
------------------------------------------

	Let wa = (sup U).
	Since U \subseteq V and V is sup-closed,
		wa \in V.

	Now we show wa \in U.

	      wa \in U
	equiv	{def of U, wa \in V}
	      (all w \in W : D.w <= w : wa <= w)
	equiv	{wa = (sup U) and def of (sup U)}
	      (all w \in W : D.w <= w : (all u \in U :: u <= w))
	equiv	{def of U}
	      (all w \in W : D.w <= w : true)
	equiv	{calc}
	      true

	It will be enough to show wa is a fixpoint:
		i.e., that D.wa <= wa and wa <= D.wa.
	This will be done using the above.

	      D.wa <= wa
	<==	{wa = (sup U), so wa is an upper bound of U}
	      D.wa \in U
	equiv	{def of U}
	      D.wa \in V /\ (all w \in W :: D.w <= w : D.wa <= w)
	<==	{V is D-invariant, and transitivity of <= (read from below)}
	      wa \in V /\ (all w \in W :: D.w <= w : D.wa <= D.w)
	<==	{D is monotone}
	      wa \in V /\ (all w \in W :: D.w <= w : wa <= w)
	equiv	{wa \in U and def of U}
	      true

	(so this means wa is in the intersection of U and the set of
		(all w \in V :: D.w <= w)

	      wa <= D.wa
	<==	{def of U, with u := wa and w := D.wa}
	      wa \in U /\ D.(D.wa) <= D.wa
	<==	{wa \in U; monotony of D}
	      true /\ D.wa <= wa
	<==	{calculus and above calculation that D.wa <= wa}
	      true

	So wa is a fixpoint.
	Remains to show it is least.
	But wa \in U, and the stuff in U is no larger than the fixpoints of D.
	QED
 
	Remark, in effect, the fixpoints of D form a complete lattice.