CSC195, Class 30: Semantics of Assignment

Overview:
* Background: Weakest Precondition, Substitution
* Defining simple assignment.
* Some variations.
* Defining multiple assignment

Notes:
* 100 Grand, Baby Ruth, Nestles Crunch, Fast Break,
  3 Musketeers, Milky Way, Snickers
* Where is BurtonE?
* Extra credit for attending Zuk talks.
* The muddy-children problem.
  Extra credit for Lindsey and Peter
* Questions on exam 1?

Goal for today: Formal definition of various kinds of 
  assignment
Reminder: 
  What is wp?
    The most general form of *something*
    A shorthand for "weakest precondition"
    It takes a statement and a postcondition and
      gives you a set of states
    That is, it's a FUNCTION from statement x predicate to
      predicate
    From statment and postcondition to precondition
      Whenever the precondition holds before the statement,
      the postcondition holds afterwards
  How do we use wp?
    (1) Generate preconditions
    (2) Define meaning of statements

Substitution notation

   x
  E
   e

Replaces "all" instances of x in E with e.

  substitute e for x in E

  substitute 1 for x in x => 1
   x
  x   => 1
   1

  substitute 3+y for x in x*x => (3+y)*(3+y)
        x
    x*x
        3+y

Question: What about parenthesization?  It's there implicitly.

  substitute e for x in x => e
  subsittute e for x in y => y, y is a variable and is not x
  substitute e for x in f*g => (substitute e for x in f) *
                               (substitute e for x in g)
    similar for 
    Mathematical: +,-,/
    function application,
      substitute e for x in f(exp) => f(substitute ...)
    Logical: and, or, not, cand, cor, equals, implies
    set operations: element-of, subset, superset,
      intersection, ...
    Numeric comparators: < <= > >= =
    Quantifiers: For all, There exists, Count

Potential problems:
  * Does this work for cand and cor?
  * Quantifiers

"for all x, x > 5"

Substitute 2 for x in "for all x, x > 5"
  Naive textual subsitution: for all 2, 2 > 5

Substitution warning 1: Don't substitute for *bound* variables.
  (Alternate: Only substitute for unbound variables.)

   
substitute e for x in "for all y, exp" =>
  for all y, (substitute e for x in exp)

"exists y, y > x"

Substitute y for x in "exists y, y > x"

  exists y, (substitute y for x in "y > x")
  exists y, (sub y for x in "y") > (sub y for x in "x")
  exists y, y > y

Substitution warning 2: Don't include variables with the 
  same name as bound variables in your substitution

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Define simple assignment

  wp("x := e"; P) = domain(e) cand substitute e for x in P

Why cand?  To handle some stuff being potentially undefined.
Why "backwards" (why does the assignment of e to x seem to
  modify the precondition and not the postcondition)?

Normal way of thinking about things
  You begin with some understanding of the world
  You do an assignment
  You therefore come up with a new understanding of the world
How does the new understanding relate?
  It's based on some substitution in the prior understanding

Consider P, the postcondition
  Any characteristic we expect of x afterwards must have
    held for e beforehand
  Hence, we can reflect that need by doing substitution

wp("x := 5"; x > 2) => sub 5 for x in x > 2 => 5 >2 => true

wp("x := 1"; x > 2) => sub 1 for x in x > 2 => 1 > 2 => false

wp("x := y"; x > 2) => sub y for x in x > 2 => y > 2

wp("x := x*x"; x > 4) => sub x*x for x in x > 4 => x*x > 4
   => (x > 2 or x < -2)
     The "x" here is the *old* x (the one before the assignment)

wp("x := a[i]", x > 2) => i is a valid index and a[i] > 2

wp("x := a[x]", x = a[x]) => ???

  Suppose a[0] is 3; a[1] is 4; a[2] is 1; and a[3] is 1

  x := 0
  x := a[x]