developing I. loops, an introduction (Cohen's chapter 8)
A. before and after
1. other forms of postconditions
2. establishing the invariant
B. steps
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INVARIANCE THEOREM FOR LOOPS
{P} do B -> S od {P /\ !B}
<== {P /\ B} S {P} (Invariance)
/\ {P /\ B /\ t==T} S {t<T} (Progress)
/\ P /\ t<=0 ==> !B (Boundedness)
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1. invariant choice and initialization
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STEPS IN DEVELOPMENT OF LOOP
0. choose invariant P and guard B
so that P /\ !B ==> R
1. Develop S0 to establish P, i.e. find S0
so that {Q} S0 {P}
---------------------------
C. the loop body
what would you choose for t if P is 0 <= i and B is i > 1?
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FINISHING STEPS
2. choose bound function, t, such that
P /\ (t <= 0) ==> !B
3. develop loop body S to satisfy
a. invariance {P /\ B} S {P}
b. progress {P /\ B /\ t == T} S {t < T}
--------------------
1. summary
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ANNOTATED FORM OF TYPICAL LOOP {PROOF}
{Q}
S0
{invariant P, bound t}
; do B -> {P /\ B /\ t = T} S {P /\ t < T} od
{P /\ t <= 0, hence P /\ !B, hence R}
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D. remarks
II. developing loops (Cohen's chapters 9-10)
A. heuristic 1: deleting a conjunct (chapter 9)
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HEURISTIC 1 FOR CHOOSING INVARIANT
(deleting a conjunct):
if postcondition R is a conjunction,
then delete
the most difficult to truthify.
(what's most difficult?):
conjunct that gives enough info
for solution (i.e, part of answer)
--------------
1. integer division example (9.1)
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EXAMPLE (INTEGER DIVISION)
var x,y: int {Q: 0<=x /\ 0<y}
; var q,r:int
; q,r: (R:0<=r /\ r<y /\ q*y+r == x)
-----------------
a. choose invariant
-----------------
0. choose invariant and guard
invariant, P: 0<=r /\ q*y+r == x
guard: r>=y,
------------------
b. establish invariant
----------------
1. establish invariant
{Q: 0<=x /\ 0<y}
q,r := 0,x
{P}
-----------------
check that wp.(q,r := 0,x).P <== Q?
c. choose bound function
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2. choose bound function
let bound function be r
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d. develop loop body
---------------
3. develop loop body
solve for K>0 and E in
{P /\ (r>=y)} := r,q r-K,E {P}
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why does boundness hold?
2. linear search example (9.2)
----------------
EXAMPLE (LINEAR SEARCH)
var x: int {0 <= X /\ f.X}
; x: (R: 0 <= x /\ f.x
/\ (\forall i : 0 <= i /\ i < x : !f.x))
-----------------
can you develop this? What are the steps?
3. avoid avoidable case analysis (9.3) (omit)
4. postpone design decisions (9.4) (omit)
B. replacing constants by fresh variables (chapter 10)
-------------------
HEURISTIC 2 FOR CHOOSING INVARIANT
(replace constants by fresh variables):
If the postcondition has the form
x == f.N,
then rewrite it as
x == f.n /\ n == N
and choose
invariant: x == f.n
guard: !(n == N)
(constrain fresh vars to ensure defined):
If f is partial,
and defined on (i:0<=i<M)
then choose
invariant: x == f.n /\ (0<=n<M)
guard: !(n == N)
----------------------
---------------------------------------------------------
EXAMPLE (10.2) THE MINIMUM VALUE
var b: array of int {#b > 0}
; var x: int
; x : (R: x == (\min i : 0 <= i /\ i < #b : b.i))
where \min is not allowed in the program.
---------------------------------------------------------
what constant can you eliminate?
What constraints are needed?
1. accumulators (10.2, 10.4, 10.5)
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ACCUMULATORS
Assume P0: x == (\min i : 0 <= i /\ i < n : b.i),
...