# Module Fixpoint

`module Fixpoint: `sig` .. `end``
Fixpoint computation implemented using the work list algorithm. This module makes writing data-flow analysis easy.

One of the simplest fixpoint analysis is that of reachability. Given a directed graph module `G`, its analysis can be implemented as follows:

``````      module Reachability = Graph.Fixpoint.Make(G)
(struct
type vertex = G.E.vertex
type edge = G.E.t
type g = G.t
type data = bool
let direction = Graph.Fixpoint.Forward
let equal = (=)
let join = (||)
let analyze _ = (fun x -> x)
end)
``````

The types for `vertex`, `edge` and `g` are those of the graph to be analyzed. The `data` type is `bool`: It will tell if the vertex is reachable from the start vertex. The `equal` operation for `bool` is simply structural equality; the `join` operation is logical or. The `analyze` function is very simple, too: If the predecessor vertex is reachable, so is the successor vertex of the edge.

To use the analysis, an instance of a graph `g` is required. For this analysis a predicate `is_root_vertex : G.E.vertex -> bool` is required to initialize the reachability of the root vertex to `true` and of all other vertices to `false`.

``````      let g = ...
let result = Reachability.analyze is_root_vertex g
``````

The `result` is a map of type `G.E.vertex -> bool` that can be queried for every vertex to tell if the vertex is reachable from the root vertex.
Author(s): Markus W. Weissmann
`module type G = `sig` .. `end``
``type direction = ``
 `|` `Forward` `|` `Backward` `(*` Type of an analysis `*)`
`module type Analysis = `sig` .. `end``
```module Make: `functor (``G`` : ``G``) -> ``functor (``A`` : ``Analysis``  with type g = G.t``  with type edge = G.E.t````
with type vertex = G.V.t````) -> ``sig` .. `end````