# Theory/GaneshPatchAlgebra

This page is to describe Ganesh Sittampalam’s formalisation of patch algebra, as discussed on the (now defunct) darcs-conflicts mailing list beginning with this post.

**Nothing here is “official” darcs theory or endorsed by David or anyone else (unless you can deduce otherwise from mailing list posts).**

There should be a strong connection with existing patch theory, and with Marnix Klooster’s patch calculus. In particular my hope is that this view of the world will provide a more low-level foundation which leads to clear justifications of higher-level rules.

Feel free to edit it to add comments etc, but please don’t change the core notation without discussing it on the mailing list first.

A darcs patch has three components: an *identity*, a *context* and an *effect*.

The identity of a patch is invariant under patch commutation. However, commutation always changes the context of a patch, and this may lead to changes in the effect of the patch.

If `identity(A) = identity(B)`

and `context(A) = context(B)`

then `effect(A) = effect(B)`

and we can write `A=B`

.

A *primitive* patch is one whose effect is a single hunk/addfile/replace etc. Compound patches can be made by putting other patches together in sequence, subject to certain rules on the contexts fitting that will be specified later.

A primitive patch (conceptually) has a globally unique identifier associated with it at record time, known as a *ppid*.

[can anyone think of a better name than ppid?]

The identity of a patch is defined as two sets of ppids. One of these sets is the *positive* set and one is the *negative* set. The identity of a primitive patch has a singleton set containing its ppid as the positive set, and the empty set as its negative set. The positive and negative sets of a patch identity must be disjoint.

The context of a patch is defined as a single set of ppids.

```
context(A) /\ positive(identity(A)) = {}
context(A) >= negative(identity(A))
```

[/\ is set intersection and >= is non-strict subset inclusion (and > would be strict subset inclusion)]

Note that this implies `positive(identity(A)) /\ negative(identity(A)) = {`

}

A context is *reachable* if there is some sequence of patches that results in that context (sequences of patches will be defined precisely later). A reachable context has a tree associated with it, that is the same no matter what sequence of patches is used to reach it.

The effect of a patch is given as a pair of trees, the *starting* tree and the *ending* tree. An effect will actually have an internal representation consisting of a sequence of darcs actions (hunk/replace/addfile etc), but two effects are considered equal if they have the same starting and ending trees.

It is always true that:

`starting(effect(A)) = tree(context(A))`

Note that `tree(context(A)) = tree(context(B))`

does **not** imply that `A=B`

or even that `context(A) = context(B)`

.

Two effects can be composed in sequence iff the ending and starting trees match up, i.e. we require that

`ending(effect(A)) = starting(effect(B))`

and then

```
starting(effect(A);effect(B)) = starting(effect(A))
ending(effect(A);effect(B)) = ending(effect(B))
inv(A)
```

exists for any `A`

and is defined by:

```
positive(identity(inv(A)) = negative(identity(A))
negative(identity(inv(A)) = positive(identity(A))
context(inv(A)) = context(A) u positive(identity(A)) \ negative(identity(A))
starting(effect(inv(A)) = ending(effect(A))
ending(effect(inv(A)) = starting(effect(A))
```

Two patches A and B can be composed in sequence iff:

```
context(B) = context(A) u positive(identity(A)) \ negative(identity(A))
positive(identity(A)) /\ positive(identity(B)) = {}
negative(identity(A)) /\ negative(identity(B)) = {}
```

Then (note that the conditions on contexts and the rules about the same context always having the same tree guarantees that the conditions for `effect(A);effect(B)`

to exist are satisfied):

```
context(AB) = context(A)
positive(identity(AB)) = (positive(identity(A)) \ negative(identity(B))) u positive(identity(B))
negative(identity(AB)) = (negative(identity(A)) \ positive(identity(B)) u negative(identity(B))
effect(AB) = effect(A);effect(B)
```

This may all seem a bit complicated but the basic idea is just that to change a context for AB it is first changed for A and then for B.

`A || B`

iff `context(A) = context(B)`

.

`A ~~ B`

iff `identity(A) = identity(B)`

.

The rule that contexts should have identical trees no matter how they are reached leads to the definition of commutation.

`AB <-> B'A'`

iff:

```
AB and B'A' are valid sequential compositions
identity(A) = identity(A')
identity(B) = identity(B')
context(A) = context(B')
effect(AB) = effect(B'A')
<B,A>
```

is defined iff: [need to check this]

```
context(B) = context(A) (i.e. A || B)
positive(identity(B)) /\ positive(identity(A)) = {}
negative(identity(B)) /\ negative(identity(A)) = {}
```

And the properties:

```
identity(<B,A>) = identity(A)
context(<B,A>) = context(B) u positive(identity(B)) / negative(identity(B))
```

Also, `<A,B>`

is defined iff `<B,A>`

is defined (this is a consequence of the rules above), and

`A<A,B> <-> B<B,A>`

Conflicts are always between primitive patches. If a conflict arises during composition of patches, the approach taken is to remove the effects of both conflicting patches from the tree (the tree presented to the user uses conflict markers, but that’s a different matter).

A conflict only makes sense between `A`

and `B`

when `A||B`

.

Since we want the same tree for the same context reached from different patch orderings, this suggests that it must be the case that if two patches conflict, then any two patches with the same ppids will conflict. If this property doesn’t hold, I think all kinds of bad things might happen - it’s probably something we should try to prove, or at least quickcheck, about commutation and conflict detection.

Assuming that this property holds, it makes sense to talk about ppids conflicting.

Then, given a context, the corresponding tree should contain the semantic effects of all the ppids in the context, apart from those that clash with another ppid in the context.

Using the old notation for conflictors, `[B,A]`

has the same pre-conditions and properties as `<B,A>`

, and also:

```
effect([B,A]) = effect(inv(B))
A+B
```

is defined under the same conditions as `<B,A>`

(which are the same as for `<A,B>`

). Properties:

```
positive(identity(A+B)) = positive(identity(A)) u positive(identity(B))
negative(identity(A+B)) = negative(identity(A)) u negative(identity(B))
context(A+B) = context(A) = context(B)
```

if A and B conflict then

`starting(effect(A+B)) = ending(effect(A+B))`

The next bit is a work in progress and mostly wrong:

Moving onto `A+B+C`

, with all of A,B and C conflicting, we want:

```
starting(effect(A+B+C)) = ending(effect(A+B+C))
A+B+C = A[A,BC] = B[B,AC] = C[C,AB]
context([A,BC]) = {a}
So [A,BC] || [A,B] || [A,C]
Suppose [A,BC] = [A,B][,C] = [A,C][,B]
context([,C]) = {a,b}
```