Local Computation of Simultaneous Fixed-Points

Abstract

We present a very simple, yet general algorithm for computing simultaneous, minimum fixed-points of monotonic functions, or turning the newpoint slightly, an algorithm for computing minimum solutions to a system of monotonic equations. The algorithm is local (demand-driven, lazy, ... ), i.e. it will try to determine the value of a single component in the simultaneous fixed-point by investigating only certain necessary parts of the description of the monotonic function, or in terms of the equational presentation, it will determine the value of a single variable by investigating only a part of the equational system.

In the worst-case this involves inspecting the complete system, and the algorithm will be a logarithmic factor worse than a global algorithm (computing the values of all variables simultaneously). But despite its simplicity the local algorithm has some advantages which promise much better performance on typical cases. The algorithm should be seen as a schema that for any particular application needs to be refined to achieve better efficiency, but the general mechanism remains the same. As such it seems to achieve performance comparable to, and for some examples improving upon, carefully designed ad hoc algorithms, still maintaining the benefits of being local.

We illustrate this point by tailoring the general algorithm to concrete examples in such (apparently) diverse areas as type inference, model checking, and strictness analysis. Especially in connection with the last example, strictness analysis, and more generally abstract interpretation, it is illustrated how the local algorithm provides a very minimal approach when determining the fixed-points, reminiscent of, but improving upon, what is known as Pending Analysis. In the case of model checking a specialised version of the algorithm has already improved on earlier known local algorithms.