Fixpoint computation with widenings using weak topological orderings as defined by François Bourdoncle and implemented in WeakTopological.
Fixpoint is another (simpler) fixpoint computation module, with general references.
The general idea of fixpoint computation is to iteratively compute the result of the analysis a vertex from the results of its predecessors, until stabilisation is achieved on every vertex. The way to determine, at each step, the next vertex to analyse is called a chaotic iteration strategy. A good strategy can make the analysis much faster. To enforce the termination of the analyse and speed it up when it terminates in too many steps, one can also use a widening, to ensure that there is no infinite (nor too big) sequence of intermediary results for a given vertex. However, it usually results in a loss of precision, which is why choosing a good widening set (the set of points on which the widening will be performed) is mandatory.
This module computes a fixpoint over a graph using weak topological ordering, which can be used to get both the iteration strategy and the widening set. The module WeakTopological aims to compute weak topological orderings which are known to be excellent decompositions w.r.t these two critical points.
, François Bourdoncle, Formal Methods in Programming and their Applications, Springer Berlin Heidelberg, 1993
How to determine which vertices are to be considered as widening points.
FromWto indicates to use as widening points the heads of the weak topological ordering given as a parameter of the analysis function. This will always be a safe choice, and in most cases it will also be a good one with respect to the precision of the analysis.Predicate f indicates to use f as the characteristic function of the widening set. Predicate (fun _ -> false) can be used if a widening is not needed. This variant can be used when there is a special knowledge of the graph to achieve a better precision of the analysis. For instance, if the graph happens to be the flow graph of a program, the predicate should be true for control structures heads. In any case, a condition for a safe widening predicate is that every cycle of the graph should go through at least one widening point. Otherwise, the analysis may not terminate. Note that even with a safe predicate, ensuring the termination does still require a correct widening definition.module type G = sig ... endMinimal graph signature for the algorithm. Sub-signature of Traverse.G.
module type Data = sig ... endParameters of the analysis.