Obfuscation of Abstract Data-Types - Rowan

CHAPTER 4. SPLITTING HEADACHES 88
We can define a split function as follows:
split B k [ ] = 〈k, [ ], [ ]〉 B
split B 0 (x : xs) = 〈0, [ ],x : xs〉 B
split B (k + 1) (x : xs) = 〈k + 1,x : l,r〉 B
where 〈m,l,r〉 B = split B k xs
The unsplitting function is the same as the k-block split:
unsplit B 〈n,l,r〉 B = l + r
The representation xs ❀ 〈n,l,r〉 B satisfies the invariant
dti ≡ (|r| = 0 ∧ |l| < n) ∨ (|l| = n) (4.27)
By storing the decision value, we will see that the operations that we define
can use this value. Hence, we can choose the decision value randomly and so
on different execution runs, the same list could be split differently. Also, when
performing operation on this split, we can change the decision value as well the
lists themselves.
For instance, we can define a cons operation as follows:
cons ∣ B a 〈k,l,r〉 B
∣ |l| < k = 〈k,a : l,r〉 B
otherwise = 〈k + 1,a : l,r〉 B
The definition for the k-block split required keeping the length of the first list
component the same and so the last of element of the first list was added to
the front of the second list. The version of cons for the augmented block split is
more efficient as we merely increase the decision value by one. However cons B
cannot be used to build up split lists starting from an empty split list as the list
r will remain empty.
The straightforward list operations are defined as follows:
|〈k,l,r〉 B | B
= |l| + |r|
null B 〈k,l,r〉 B = null l ∧ null r
elem B a 〈k,l,r〉 B = elem a l ∨ elem a r
{ head r if k = 0
head B 〈k,l,r〉 B =
head l otherwise
{
〈k,l, tail r〉B if k = 0
tail B 〈k,l,r〉 B =
〈k − 1, tail l,r〉 B otherwise
map B f 〈k,l,r〉 B = 〈k, map f l, map f r〉 B
All these definitions, with the exception of tail, match the definitions for the
k-block split. The definition of tail is more efficient as we just decrease the
decision value instead of joining the head of the second component to the end
of the first component.
Here is a possible definition for a concatenation operation:
〈k, [ ], [ ]〉 B + B 〈k ′ ,l ′ ,r ′ 〉 B = 〈k ′ ,l ′ ,r ′ 〉 B
〈k,l,r〉 B + B 〈k ′ ,l ′ ,r ′ 〉 B = 〈|l + r|,l + r,l ′ + r ′ 〉 B