LibraryReExportIsNPComplete

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This page describes a way to convert any [[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]].
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Reusing libraries produce by others is essential aspect of [[DistributedDevelopment]]. It simplifies [[Time To Market]], it reduces long term cost of ownership and leads to creation of [[Good Technology|good technologies]]. However it does not comes for free. Read details or directly jump to the [[LibraryReExportIsNPComplete#Implications|implications]] that shall improve your daily development habits.
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This page starts by describing a way to convert any [[3SAT]] problem to a solution of finding the right configuration from conflicting libraries in a system that can re-export [[API]]s. Thus proving that the later problem is [[wikipedia::NP-complete|NP-Complete]]. Then it describes the importance of such observations on our [[DistributedDevelopment|development practices]].
== [[3SAT]] ==
== [[3SAT]] ==

Revision as of 07:58, 25 August 2009

Reusing libraries produce by others is essential aspect of DistributedDevelopment. It simplifies Time To Market, it reduces long term cost of ownership and leads to creation of good technologies. However it does not comes for free. Read details or directly jump to the implications that shall improve your daily development habits.

This page starts by describing a way to convert any 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 NP-Complete. Then it describes the importance of such observations on our development practices.

Contents

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:

(x_{11} \vee x_{12} \vee x_{13}) \wedge
(x_{21} \vee x_{22} \vee x_{23}) \wedge
(x_{31} \vee x_{32} \vee x_{33}) \wedge
...
(x_{n1} \vee x_{n2} \vee x_{n3})

where each xab is a variable vi or a negation of a variable \neg v_i. Each variable vi can appear multiple times in the expression.

Module Dependencies Problem

Let A,B,C,... denote various APIs.

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 A_{x.y} \gg B_{u.v} denote the fact that version x.y of API A depends on version u.v of API B and that it re-exports B's 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 C \subseteq M, where following is satisfied:

\forall A_{x.y} \in C, \forall A_{x.y} \gg B_{u.v} \in D \Rightarrow \exists w >= v \wedge B_{u.w} \in C - each re-exported dependency is satisfied with some compatible version
\forall A_{x.y} \in C, \forall A_{x.y} > B_{u.v} \in D \Rightarrow \exists w >= v \wedge B_{u.w} \in C - each dependency is satisfied with some compatible version
Let there be two chains of re-exported dependencies A_{p.q} \gg ... \gg B_{x.y} and A_{p.q} \gg ... \gg B_{u.v} then x = u \wedge y = v - 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 A \in M. Does there exist a configuration C in the repository R, such that the module A \in C, e.g. the module can be enabled?

Conversion of 3SAT to Module Dependencies Problem

Let there be 3SAT formula with with variables v1,...,vm as defined above.

Let's create a repository of modules R. For each variable vi let's create two modules M^i_{1.0} and M^i_{2.0}, which are mutually incompatible and put them into repository R.

For each formula (x_{i1} \vee x_{i2} \vee x_{i3}) 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

v_a \vee \neg v_b \vee \neg v_c

we will get:

F^i_{1.1} \gg M^a_{1.0}
F^i_{1.2} \gg M^b_{2.0}
F^i_{1.3} \gg M^c_{2.0}

All these modules and dependencies add into repository R

Now we will create a module T1.0 that depends all formulas:

T_{1.0} \gg F^1_{1.0}
T_{1.0} \gg F^2_{1.0}
...
T_{1.0} \gg F^n_{1.0}

and add this module as well as its dependencies into repository R.

Claim: There \exists C (a configuration) of repository R and T_{1.0} \in C \Longleftrightarrow there is a solution to the 3SAT formula.

Proof

"\Leftarrow": Let's have an evaluation of each variable to either true or false that evaluates the whole 3SAT formula to true. Then

C = \{ T_{1.0} \} \bigcup
\{ M^i_{1.0} : v_i \} \bigcup \{M^i_{2.0} : \neg v_i \} \bigcup
\{ F^i_{1.1} : x_{i1} \} \bigcup \{ F^i_{1.2} : \neg x_{i1} \wedge x_{i2} \} \bigcup \{ F^i_{1.3} : \neg x_{i1} \wedge \neg x_{i2} \wedge x_{i3} \}

It is clear from the definition that each Mi and Fi can be in the C just in one version. Now it is important to ensure that each module is present always at least in one version. This is easy for Mi as its vi needs to be true or false, and that means one of M^i_{1.0} or M^i_{2.0} will be included. Can there be a Fi which is not included? Only if \neg x_{i1} \wedge \neg x_{i2} \wedge \neg x_{i3} but that would mean the whole 3-or would evaluate to false and as a result also the 3SAT formula would evaluate to false. This means that dependencies of T1.0 on Fi modules are satisfied. Are also dependencies of every F^i_{1.q} satisfied? From all the three versions, there is just one F^i_{1.q}, the one its xiq evaluates to true. However xiq can either be without negation, and as such F^i_{1.q} depends on M^j_{1.0} which is included as vj is true. Or xiq contains negation, and as such F^i_{1.q} depends on M^j_{2.0} which is included as vj is false.

"\Rightarrow": Let's have a C configuration satisfies all dependencies of T1.0. Can we also find positive valuation of 3SAT formula?

For i-th 3-or there is T_{1.0} \gg F^i_{1.0} dependency which is satisfied. At least by one from F^i_{1.0} or F^i_{1.1} or F^i_{1.2}. The one Fi that has the satisfied dependency reexports M^j_{1.0} (which means vj = true) or M^j_{2.0} (which means vj = false). Anyway each i 3-or evaluates to true.

The only remaining question is whether a C configuration can force truth variable vj to be true in one 3-or and false in another. However that would mean that there is re-export via T_{1.0} \gg F^i_{1.x} \gg M^j_{1.0} and also another one via T_{1.0} \gg F^p_{1.u} \gg M^j_{2.0}. However those two chain of dependencies ending in different versions of Mj cannot be in one C as that breaks the last condition of configuration definition. Thus each Mj is represented only by one version and each vj is evaluated either to true or false, but never both.

The 3SAT formula's evaluation based on the configuration C is consistent and satisfies the formula.

qed.

Implications

If there is a repository of modules in various (incompatible) versions, with mutual dependencies and re-export of their APIs, then deciding whether some of them can be enabled is NP-complete problem.

As NP-complete problems are hard to solve, it is usually our best desire to avoid them in real life situations. What does that mean in case one decides to practise DistributedDevelopment (and it is inevitable that software for 21st century needs this development style)? If you want to avoid headaches with finding the right configuration of various version of the libraries that allows to execute them together, then stick with following simple rules.

Be Compatible!

If you develop your own libraries in backward compatible way, you can always select the most recent version of each library. That is the configuration you are looking for. It is easy to find (obviously) and also it is the most desirable, as it delivers the most modern features and bugfixes that users of such libraries want.

Reuse with Care!

If you happen to reuse libraries (and you should because reuse lowers Time_To_Market, just like it did for me when I was publishing my first animated movie), then choose those libraries that can be trusted to evolve compatibly.

Hide Incompatibilities!

If you happen to reuse a library that cannot be trusted to keep its BackwardCompatibility, then do whatever you can to not re-export its APIs! This has been discussed in Chapter 10, Cooperating with Other APIs, but in short: If you hide such library for internal use and do not export any of its interfaces, you can use whatever version of library you want (even few years older) and nobody shall notice. Moreover in many module systems there can even be multiple versions of the same library in case they are not re-exported.


Conclusion

Avoid complexities and NP-complete problems. Learn to develop in backward compatible way. Reading TheAPIBook is a perfect entry point into the software design for 21st century.

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