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-# diff-containers
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+# diff-containers
+
+This package implements difference datatypes for container types. Currently,
+there is only one such implemented difference datatype for `Data.Map.Strict`.
+
+## Differences on `Data.Map`s
+
+A difference datatype for maps is defined and implemented in the
+`Data.Map.Diff.Strict` module. A key property of our definition of diffs is that
+diffs can not only be combined/summed, they can also be inverted and summed
+together, which facilitates *subtraction* of diffs. Our definition of diffs is
+not only a [`Semigroup`](https://en.wikipedia.org/wiki/Semigroup#Definition) and
+a [`Monoid`](https://en.wikipedia.org/wiki/Monoid#Definition), it is also a
+[`Group`](https://en.wikipedia.org/wiki/Group_(mathematics)#Definition).
+
+The goals for our definition of diffs are twofold:
+1. Provide definitions that adhere to the type class laws for `Semigroup`,
+   `Monoid` and `Group`.
+2. Applying diffs to maps should have a sound semantics: if we apply a positive
+   diff (see Section [4) Applying diffs with inverses: positivity and
+   normality](#4-applying-diffs-with-inverses-positivity-and-normality)), then
+   the resulting map is correct.
+
+An overview of the sections in this document:
+
+* Section [1) High-level overview](#high-level-overview) provides a high-level
+  overview of diffs, how to create them, and how to apply them.
+* Section [2) A first attempt at implementing
+  diffs](#2-a-first-attempt-at-implementing-diffs) describes a straightforward
+  way to implement diffs, and where this goes wrong if we try to define a
+  `Group` instance.
+* Section [3) Implementing diffs with
+  inverses](#3-implementing-diffs-with-inverses) describes how we implement
+  diffs to facilitate a `Group` instance.
+* Section [4) Applying diffs with inverses: positivity and
+  normality](#4-applying-diffs-with-inverses-positivity-and-normality) describes
+  how applying diffs can fail, but which can be solved by guaranteeing the
+  positivity and normality properties.
+
+### 1) High-level overview
+
+For each of the key-value pairs in a map, the difference type will keep track of
+whether a value is to be inserted or deleted:
+
+```haskell
+newtype Diff k v = Diff {- omitted -}
+
+data DiffEntry v =
+    Insert !v
+  | Delete !v
+  {- omitted for now -}
+```
+
+One method of constructing a diff is to compute the difference between two maps.
+Other methods include creating diffs from maps and lists directly:
+
+```haskell
+diff :: (Ord k, Eq v) => Map k v -> Map k v -> Diff k v
+
+fromMap :: Map k (DiffEntry v) -> Diff k v
+fromList :: Ord k => [(k, DiffEntry v)] -> Diff k v
+```
+
+Diffs can be applied to maps to obtain new maps. For diffs constructed using
+`diff`, the following `property` also holds:
+
+```haskell
+unsafeApplyDiff :: Ord k => Map k v -> Diff k v -> Map k v
+
+property :: Map k v -> Map k v -> Bool
+property = unsafeApplyDiff m1 (diff m1 m2) == m2
+```
+
+### 2) A first attempt at implementing diffs
+
+A straightfoward to represent differences on maps is to keep track of the most
+recent change for each key in the map.
+
+```haskell
+type Diff_ k v = Diff_ (Map k (Delta_ v))
+data Delta_ = Insert_ !v | Delete_ !v
+```
+
+Diffs can then be summed (or combined/`mappend`ed) through a
+`Semigroup`/`Monoid` instance that behaves like a pair-wise `Last` monoid.
+
+```haskell
+   fromList [(1, Insert_ 54), (2, Delete_ 42)                ]
+<> fromList [                 (2, Insert_ 17), (3, Delete_ 1)]
+== fromList [(1, Insert_ 54), (2, Insert_ 17), (3, Delete_ 1)]
+```
+
+What if we want to remove diffs, i.e. through subtraction provided by a `Group`
+instance? In the current definition, there is no way to recover information
+about previous changes, since we only have the last/most recent change. In
+particular, we would like to be able to have the following, where `(~~)` is a
+subtraction operator provided by `Data.Group`.
+
+```haskell
+  (fromList [(1, Insert_ 54), (2, Delete_ 42)                ]
+<> fromList [                 (2, Insert_ 17), (3, Delete_ 1)])
+~~ fromList [                 (2, Insert_ 17), (3, Delete_ 1)]
+== fromList [(1, Insert_ 54), (2, Insert_ 17), (3, Delete_ 1)]
+~~ fromList [                 (2, Insert_ 17), (3, Delete_ 1)]
+== fromList [(1, Insert_ 54), (2, Delete_ 42)                ]
+```
+
+If we only consider the diff with respect to key `2`, how could we achieve
+`Insert_ 17 ~~ Insert_ 17 == Delete_ 42` if we have no idea that `Delete_ 42`
+was the last delta before we summed with `Insert_ 17`? We throw away this
+information, so there's no way to define subtraction correctly.
+
+Please note that the parenthesisation should not matter for a correct
+implementation of subtraction: `(~~)` is an alias for `\x y -> x <> invert y`,
+where `invert` is provided by `Data.Group` as well, and `(<>)` *should be*
+associative.
+
+### 3) Diffs with inverses
+
+The trick is to store a *history* of deltas for each key. When we subtract diffs
+from one another, we remove deltas from histories. How exactly deltas are
+removed, we will touch upon later. For now, these are the (simplified) types
+that make up a diff:
+
+```haskell
+-- Note: this is a simplified representation. In reality, we enforce statically
+-- that diff histories in a @'Diff'@ are non-empty.
+newtype Diff k v = Diff (Map k (DiffHistory v))
+
+newtype DiffHistory v = DiffHistory { getDiffHistory :: Seq (DiffEntry v) }
+
+data DiffEntry v =
+    Insert !v
+  | Delete !v
+  {- omitted for now -}
+```
+
+The way in which subtraction is facilitated is by implementing the `invert`
+method in a `Group` instance. As a consequence, not only do we have to add
+inverses for diffs, diff histories and diff entries, we must also define how
+these inverses are handled by summing.
+
+First, we start by adding inverse elements to the definition of `DiffEntry`. One
+may think `Insert` and `Delete` are already inverses of one another, and that
+would be correct in terms of adhering to the type class laws, but it would have
+adverse effects on the semantics of applying diffs to maps. Consider what would
+happen if we were to consider `Insert` and `Delete` inverses in the following
+example. Ignore diff histories for now:
+
+```haskell
+d1 = fromList [(0, Delete 17)]
+d2 = fromList [(0, Insert 42)]
+d3 = d1 <> d2 = fromList []
+```
+
+Applying `d3` to a map would behave like the identity function. Applying `d1`
+and `d2` consecutively would not behave like the identity function, as it would
+map `0` to `42`. Through some abuse of the definition of distributivity, we can
+say that applying diffs does not distribute over the summing of diffs.
+
+Instead, we add explicit inverse constructors for both `Insert` and `Delete`. We
+do not make `Semigroup`/`Monoid`/`Group` instances for diff entries, however. We
+shall see that we only need these 4 constructors in order to provide the
+instances for diff histories.
+
+```haskell
+data DiffEntry v =
+    Insert !v
+  | Delete !v
+  | UnsafeAntiInsert !v -- ^ Unsafe!
+  | UnsafeAntiDelete !v -- ^ Unsafe!
+```
+
+The inverse constructors are deemed unsafe in the sense that they should not be
+explicitly constructed or matched on without dire consequences, because it will
+easily violate the positivity and normality properties that are discussed in
+[section 4](#4-applying-diffs-with-inverses-positivity-and-normality).
+
+Inverse diff entries will *cancel out* when we sum diff histories. For example:
+
+```haskell
+h1 = fromList [Insert 1, UnsafeAntiDelete 2, UnsafeAntiInsert 3]
+h2 = fromList [Delete 2, Insert 3, Insert 3]
+h3 = h1 <> h2 = [Insert 1, Delete 2, Insert 3]
+```
+
+How do we invert diff histories? The type class laws that a `Group` instance
+should adhere to give us a clue:
+
+```haskell
+-- left inverse
+a <> invert a == mempty
+-- right inverse
+invert a <> a == mempty
+```
+
+Given that inverse diff entries cancel each other out, inverting a diff history should invert all diff entries and reverse their order. See this example:
+
+```haskell
+h        = fromList [Insert 1, Delete 2, Delete 3, Insert 4]
+invert h = fromList [UnsafeAntiInsert 4, UnsafeAntiDelete 3, UnsafeAntiDelete 2, UnsafeAntiInsert 1]
+
+invert h <> h == mempty
+h <> invert h == mempty
+```
+
+Summing of diffs follows from pairwise summing the inner diff histories.
+Inverting diffs follows from inverting each inner diff history.
+
+### 4) Applying diffs with inverses: positivity and normality
+
+It is an unfortunate side effect of facilitating a `Group` instance for diffs
+that we can get into situations where applying diffs will fail or produce wrong
+results due to diffs containing internally unresolved sums and subtractions. The
+responsibility of downstream code is to ensure that diffs that are applied are
+both *positive* and *normal*. If that is the case, then applying diffs will
+never go wrong.
+
+* Positivity: a positive diff contains only `Insert`s and `Delete`s.
+* Normality: a normal diff has resolved all sums and subtrations internally,
+  i.e., there are no consecutive diff entries in a diff history that can still
+  be cancelled out.
+
+Please note: a positive diff is by definition also a normal diff. Normality is a
+weaker property than positivity. However, normality is a sufficient precondition
+for some definitions to work as expected (see the next paragraph), in which
+cases positivity would be an unnecessarily strong requirement.
+
+Normality is also a precondition for the type class laws to hold. This is an
+optimisation: we could normalise diff histories exhaustively in each sum or
+inversion operation, but that could be costly.
+
+If positivity and normality are guaranteed, then `applyDiff` will always return a `Right` result, and `unsafeApplyDiff` will never throw an error.
+
+```haskell
+applyDiff :: Ord k => Map k v -> Diff k v -> Either () (Map k v)
+unsafeApplyDiff :: Ord k => Map k v -> Diff k v -> Map k v
+```
+
+A number of definitions in the Haskell modules are annotated with PRECONDITION,
+INVARIANT and POSTCONDITION. Use these and the type class laws for `Semigroup`,
+`Monoid` and `Group` (which hold given preconditions) to ensure that applied
+diffs are always both positive and normal. This should not be too complex
+
+* Diffing maps creates diffs that are both positive and normal.
+* Summing preverses normality and positivity.
+* Empty diffs are both normal and positive.
+* Inversion preserves normality.
+* Subtraction preserves positivity if we only subtract diffs that we know are in
+  a sum. For example:
+
+  ```haskell
+  (d1 <> ... <> di <> dj <> ... <> dn) <> invert (dj <> ... <> dn) == d1 <> ... <> di
+  invert (d1 <> ... <> di) <> (d1 <> ... <> di <> dj <> ... <> dn) == dj <> ... <> dn
+  ```
+
+It is important to remember that these sum operations are not commutative.
+Summing/subtracting on the left is not the same as summing/subtracting on the
+right.
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