TRANSITIONS (Table 2.6)

[asg] ([x := a]^l, s) --> s[x |-> A[[a]]s]

[skip] ([skip]^l, s) --> s

          (S1, s) --> (S1', s')
[seq1] --------------------------
       (S1;S2, s) --> (S1';S2, s')

          (S1, s) --> s'
[seq2] --------------------------
       (S1;S2, s) --> (S2, s')

[if1] (if [b]^l then S1 else S2, s)
         --> (S1, s)
                       if B[[b]]s = true

[if2] (if [b]^l then S1 else S2, s)
         --> (S2, s)
                       if B[[b]]s = false


[wh1] (while [b]^l do S, s)
         --> (S; while [b]^l do S, s)
                       if B[[b]]s = true

[wh2] (while [b]^l do S, s)
         --> s
                       if B[[b]]s = false


           EXAMPLE

Program

   [q := 0]^1;
   [r := x]^2;
   while [r >= y]^3 do ([r := r-y]^4;
                        [q := q+1]^5)

Name each statement S_i if it has label i
Let S_123 be the whole program (a sequence)
Let S_3 be the while loop
Let S_23 be the sequential composition of S_2 and S_3
Let S_45 be the body of the while loop

Let sqrxy be the state in which
    q has value q, ..., y has value y
E.g, s7062(q) = 7, s0062(x) = 6.


  (S_123, s7062)
--> {by [seq2]}
   * ([q := 0]^1, s7062)
   --> {by [asg]}
      s0062
. (S_23, s0062)
--> {by [seq2]}
   * <[r := x]^2, s0062)
   --> {by [asg]}
     s0662
. (S_3, s0662)
--> {by [wh1], since B[[r >= y]](s0662) = true}
  (S_45; S_3, s0662)
--> {by [seq1]}
   * (S_45, s0662)
   --> {by [seq2]}
      * <[r := r-y]^4, s0662)
      --> {by [asg]}
         s0462
   . (S_5, s0462)
. (S_5; S_3, s0462)
--> 


      PROPERTIES OF WHILE's SEMANTICS

How does the flow graph change
as the configurations change?

Case 1:  (S,s) --> (S',s')

  Compare           vs.
           final(S)      final(S')
           flow(S)       flow(S')
           blocks(S)     blocks(S')

Case 2:  (S,s) --> s'

  what can we say about the graph of S?


          EQUATION SYSTEM

LV^=(S*) defined by:

LVexit(l) = 
  if l \in final(S*) then {}
  else \bigcup { LVentry(l') |
                 (l', l) \in flow^R(S*) }
LVentry(l) =
 (LVexit(l) \ killLV(B^l)) \cup genLV(B^l)
   where B^l \in blocks(S*)

Functional form, for a given S*:

F^S*_LV: ({entry,exit} -> Lab* -> P(Var*))
      -> ({entry,exit} -> Lab* -> P(Var*))

F^S*_LV(F)(exit)(l) = 
  if l \in final(S*) then {}
  else \bigcup { F(entry)(l') |
                 (l', l) \in flow^R(S*) }
F^S*_LV(F)(entry)(l) =
 (F(exit)(l) - killLV(B^l)) \cup genLV(B^l)
   where B^l \in blocks(S*)

Solutions:

  live: {entry,exit} -> Lab* -> P(Var*)

def: live solves LV^=(S*),
     written live |= LV^=(S*),
     iff live is a fixpoint of F^S*_LV.


              EXAMPLE

Q: What is F^S*_LV for:

      [x := 3]^1;
      [y := x+2]^2;
      [y := y+1]^3










   DEFINING LIVE VARIABLES SEMANTICALLY

What should a solution to LV mean?

  At a given point,
  only the live variables matter.

def: States s1 and s2 are *similar with
     respect to a set of variables V *,
     written s1 ~_V s2, iff
      (\forall x \in V :: s1(x) = s2(x)).

def: functional
       liv: {entry,exit} -> Lab -> State
   *solves a set LV^=(S)* iff
    for all d in {entry,exit}, l in Lab:
    if V = liv(d)(l),
    then liv(d)(l) satisfies LV^=(S)(d)(l)
      
Conjecture: Let S be label consistent.
Suppose live |= LV^=(S) and
        s1 ~_{live(entry)(init(S))} s2.
Then:

(1) if (S, s1) --> (S', s1'),
    then there is some s2' such that
        (S, s2) --> (S', s2')
    and s1' ~_{live(entry)(init(S'))} s2'.

(2) if (S, s1) --> s1'
    then there is some s2' such that
        (S, s2) --> s2'
    and s1' ~_{live(exit)(init(S))} s2'.


           CONSTRAINT SYSTEM

LV^{\subseteq}(S*) defined by:

LVexit(l) \supseteq 
  if l \in final(S*) then {}
  else \bigcup { LVentry(l') |
                 (l', l) \in flow^R(S*) }
LVentry(l) \supseteq 
 (LVexit(l) \ killLV(B^l)) \cup genLV(B^l)
   where B^l \in blocks(S*)

Functional form, for a given S*, as above

F^S*_LV: ({entry,exit} -> Lab* -> P(Var*))
      -> ({entry,exit} -> Lab* -> P(Var*))

F^S*_LV(F)(exit)(l) =
  if l \in final(S*) then {}
  else \bigcup { F(entry)(l') |
                 (l', l) \in flow^R(S*) }
F^S*_LV(F)(entry)(l) =
 (F(exit)(l) \ killLV(B^l)) \cup genLV(B^l)
   where B^l \in blocks(S*)


Solutions:

  live: {entry,exit} -> Lab* -> P(Var*)

def: live solves LV^{\subseteq}(S*),
     written live |= LV^{\subseteq}(S*),
     iff live \sqsupseteq F^S*_LV(live).

def: lv1 \sqsupseteq lv2 iff
   (\forall e \in {entry,exit} ::
      (\forall l in Lab* ::
         lv1(e)(l) \supseteq lv2(e)(l)))

Lemma 2.15: If S* is label consistent,
 then live |= LV^=(S*)
      implies
      live |= LV^{\subseteq}(S*).
Furthermore, the least solutions coincide.

    SOLUTIONS WORK FOR SUBSTATEMENTS

Lemma 2.16:
Suppose S1 is label consistent,
and live |= LV^{\subseteq}(S1).
If flow(S1) \supseteq flow(S2) and
   blocks(S1) \supseteq blocks(S2),
then S2 is label consistent and
     live |= LV^{\subseteq}(S2).


        SOLUTIONS PRESERVED

Corollary 2.17:
Suppose S is label consistent,
and live |= LV^{\subseteq}(S).
If (S,s) --> (S',s'),
then live |= LV^{\subseteq}(S').


     SOLUTION CAN ONLY SHRINK FORWARD

Lemma 2.18:
Suppose S is label consistent,
and live |= LV^{\subseteq}(S).
Then for all (l,l') \in flow(S),

  live(exit)(l) \supseteq live(entry)(l').

Proof: construction of LV^{\subseteq}(S).

        GETTING CORRECTNESS RIGHT

Theorem 2.21: Let S be label consistent.
Suppose live |= LV^{\subseteq}(S) and
        s1 ~_{live(entry)(init(S))} s2.
Then:

(1) if (S, s1) --> (S', s1'),
    then there is some s2' such that
        (S, s2) --> (S', s2')
    and s1' ~_{live(entry)(init(S'))} s2'.

(2) if (S, s1) --> s1'
    then there is some s2' such that
        (S, s2) --> s2'
    and s1' ~_{live(exit)(init(S))} s2'.