1. Totalization
To define refinement for partial relations, we consider their totalised forms:
where
X = X ⋃ {⊥}
Y = Y ⋃ {⊥}

2. Complement of s

3. Example

4. Example a b c d ⊥ a b c d ⊥

5. Example a b c d ⊥ a b c d ⊥

6. Refinement
Having decided upon totalization using ⊥, we may derive the conditions for one partial relation to be a correct refinement of another.
If σ and ρ are two partial relations of the same type then σ refines ρ precisely when is a subset of .
This is true if and only if the domain of σ is at least as big as that of ρ and σ agrees with ρ on dom ρ.

7. Refinement

8. Refinement a a b b c ρ a a
Provided some additional information by extending the domain and removed some non-determinism b b c c σ

9. Not a Refinement

10. Using Totalization

11. Refinement
If σ and ρ are two partial relations of the same type then σ refines ρ precisely when is a subset of .

12. Example
We may corrupt a bit – an element of set {0, 1} – by changing its value.

13. The relation corruptsto associates two sequences of bits if the second is no longer than the first, and no two adjacent bits have been corrupted.
_corruptsto_ : seq Bit ↔ seq Bit
∀ bs, bs’ : seq Bit ∙
bs corruptsto bs’ ⇔
#bs’ ≤ #bs ∧ ∀ i : 1 .. #bs’ – 1 ∙
bs i ≠ bs’ i ⇒ bs(i + 1) = bs’(i + 1)
For example,
<1, 0, 0, 0, 1, 1> corruptsto <1, 0, 1, 0, 0, 1>
<1, 1, 0, 1, 1, 1, 0, 0> corruptsto <0, 1, 0, 0, 1>

14. The relation changesto associates two sequences of bits if the second is no longer than the first, and every bit with an odd index has been corrupted.
_changesto_ : seq Bit ↔ seq Bit
∀ bs, bs’ : seq Bit ∙
bs changesto bs’ ⇔
#bs’ ≤ #bs ∧ ∀ i : 1 .. (#bs’ – 1) ∙
i ∈ {n : N1 ∙ 2 * n} ⇒ bs i = bs’ i ∧
i ∈ {n : N1 ∙ 2 * n + 1} ⇒ bs i ≠ bs’ i
For example,
<1, 1, 0, 1, 1, 1, 0, 0> changesto <0, 1, 1, 1, 0>
<1, 0, 0, 0, 1, 1> changesto <0, 0, 1, 0, 0, 1>

15. The second relation is a refinement of the first:
both are total relations on seq Bit
changesto resolves all the non-determinism present in the definition of corruptsto.
If we are content with the behavior of corruptsto, then we will be content with changesto.