Lower Bounds on Intermediate Results in Bottom-Up Knowledge Compilation

TL;DR

This paper proves that bottom-up knowledge compilation into structured DNNF faces exponential complexity due to large intermediate results, even for inputs with small representations.

Contribution

It establishes a fundamental lower bound showing exponential time and space complexity in bottom-up compilation for certain CNF formulas, highlighting inherent limitations.

Findings

Bottom-up compilation produces large intermediate results.

Exponential time and space are unavoidable in general settings.

The inefficiency is inherent to the bottom-up paradigm.

Abstract

Bottom-up knowledge compilation is a paradigm for generating representations of functions by iteratively conjoining constraints using a so-called apply function. When the input is not efficiently compilable into a language - generally a class of circuits - because optimal compiled representations are provably large, the problem is not the compilation algorithm as much as the choice of a language too restrictive for the input. In contrast, in this paper, we look at CNF formulas for which very small circuits exists and look at the efficiency of their bottom-up compilation in one of the most general languages, namely that of structured decomposable negation normal forms (str-DNNF). We prove that, while the inputs have constant size representations as str-DNNF, any bottom-up compilation in the general setting where conjunction and structure modification are allowed takes exponential time and space, since large intermediate results have to be produced. This unconditionally proves that the inefficiency of bottom-up compilation resides in the bottom-up paradigm itself.