CS 641 Lecture -*- Outline -*-

* Coercion Operators (9.3)

  The coercion operators map from predicates and functions to relations.

  Q: Does coercion from relations to predicates and functions work?
     not in general, because relations can be partial and nondeterministic

  Q: So which is the supertype of the other?
     relations are a supertype of functions in this sense

** definition

  Q: How could we regard a function of type S -> G as a relation in S <-> G?

------------------------------------------
       COERCION OPERATORS (9.3)

Mapping
 |.| : (S -> G) -> (S <-> G)
         ^
 |f|.s.g = f.s == g

Test
 |.| : (S -> Bool) -> (S <-> S)
         ^
 |p|.s.g = s == g /\ p.s
------------------------------------------

        Q: What do these mean?

        Q: For example, what is |(x := 1)|.s, where s maps x to 0?
           {s'}, where s' maps x to 1 and is otherwise like s

        Q: What is |x >= 3|.s.g ?
                s == g /\ (x >= 3).s
           which means ...

        Q: Is |x >= 3| like "assert x >= 3" or "assume x >= 3"?
           Think of s as pre-state, and g as post-state...
           Rather like both when p.s is true, since then |p|.s == {s}.
           Also like both when p.s is false, since then |p|.s == {}.
           The question is, how we interpret {}
               (total vs. partial correctness, us vs. them,
                see section 9.5)

** utility

   these are useful for expressing assertions at the relational level
   which is convenient because of the richness of relational algebra

   Q: What's |true|? |false|?
             Id (not True!)
        calculate |true|.s.g, and compare to True.s.g.
 
             |false|.s.g
           == { def }
             s == g /\ false.s
           == { def of false }
             s == g /\ F
           == { F /\ rule }
             F

       So |false| == False

   Q: How would you express conjunction of predicates using test?
      |p \intersect q| == |p|;|q|
      prove this

   Q: So can one express Q.s.g /\ p.s without using /\ ?

** properties

        Q: Is test, |.|: (S -> Bool) -> (S <-> S) monotonic?
           Yes, also a bottom, postive meet, and universal join homomorphism
           (prove one of these?, we did bottom above!)

        Q: Does test, |.|: (S -> Bool) -> (S <-> S) preserve top? why?
           see the previous calculations!

        Q: Does test, |.|: (S -> Bool) -> (S <-> S) preserve negation? why?
           |!p|.s.g
        ==  {definition}
           (s == g) /\ (!p).s
        ==  {pointwise extension}
           (s == g) /\ !(p.s)
       This isn't the same as
           !(|p|.s.g)
        ==  {definition}
           !((s == g) /\ p.s)
        ==  {de Morgan}
           (s != g) \/ p.s

           for example, take p.s == val.x.s > 3, s maps x to 4, and s == g

------------------------------------------
         DOMAIN AND RANGE

dom.R.s = (\exists g :: R.s.g)
ran.R.g = (\exists s :: R.s.g)

What are the types of these?



------------------------------------------

** partial functions
------------------------------------------
        PARTIAL FUNCTIONS

Modeled as deterministic relations.

Def: R: S <-> G is deterministic iff
    (\forall s g g' : R.s.g /\ R.s.g' :
                      g' == g)
------------------------------------------

        Q: Is False deterministic?  Is True deterministic?  Are these partial?

        Q: When is |p| partial?