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

**See also**

*Introduction to Lattices and Order*B. A. Davey and H. A. Priestley, Cambridge University Press, 2002*Fixed Point Theory*Andrzej Granas and James Dugundji, Springer, 2003*Principles of Program Analysis*Flemming Nielson, Hanne Riis Nielson and Chris Hankin, Springer, 2005*Ubersetzerbau 3: Analyse und Transformation*Reinhard Wilhelm and Helmut Seidl, Springer, 2010

module type G =`sig`

..`end`

Minimal graph signature for work list algorithm

`type `

direction =

`|` |
`Forward` |
|||

`|` |
`Backward` |
`(*` |
Type of an analysis
| `*)` |

module type Analysis =`sig`

..`end`

module Make: