LibraryReExportIsNPComplete
From APIDesign
This page describes a way to convert any wikipedia::3SAT problem to a solution of finding the right configuration from conflicting libraries in a system that can re-export APIs. Thus proving that the later problem is wikipedia::NP-complete.
wikipedia::3SAT
The problem of satisfying a logic formula remains NP-complete even if all expressions are written in wikipedia::conjunctive normal form with 3 variables per clause (3-CNF), yielding the 3SAT problem. This means the expression has the form:
- ...
where each xab is a variable vi or a negation of a variable . Each variable vi can appear multiple times in the expression.
Module Dependencies Problem
Let A,B,C,... denote an API.
Let A1,A1.1,A1.7,A1.11 denote compatible versions of API A.
Let A1,A2.0,A3.1 denote incompatible versions of API A.
Let Ax.y > Bu.v denote the fact that version x.y of API A depends on version u.v of API B.
Let denote the fact that version x.y of API A depends on version u.v of API B and that B re-exports its elements.
Let Repository R = (M,D) be any set of modules with their various versions and their dependencies on other modules with or without re-export.
Let C be a Configuration in a repository R = (M,D), if , where following is satisfied:
- - each re-exported dependency is satisfied with some compatible version
- - each dependency is satisfied with some compatible version
- Let there be two chains of re-exported dependencies and then - this guarantees that each class has just one, exact meaning for each importer
Module Dependency Problem: Let there be a repository R = (M,D) and a module . Does there exist a configuration C in the repository R, such that the module , e.g. the module can be enabled?
Converstion of wikipedia::3SAT to Module Dependencies Problem
Let there be wikipedia::3SAT formula with with variables v1,...,vm as defined above.
For each variable vi let's create two modules and , which are mutually incompatible.
For each formula let's create a module Fi that will have three compatible versions. Each of them will depend on one variable's module. In case the variable is used with negation, it will depend on version 2.0, otherwise on version 1.0. So for the formula
we will get:
Now we will create a module T1.0 that depends all formulas:
...
The claim is that iff there is a way to satisfy all dependencies of module T1.0, then there is a solution to the wikipedia::3SAT formula.