Cliquewidth and Knowledge Compilation

TL;DR

This paper reveals that cliquewidth and treewidth are closely related in Boolean circuit representation, enabling efficient knowledge compilation into DNNF with bounds depending on cliquewidth.

Contribution

It establishes a surprising equivalence between cliquewidth and treewidth for Boolean circuits and applies this to achieve the first known efficient knowledge compilation parameterized by cliquewidth.

Findings

Boolean circuits of bounded cliquewidth can be transformed into circuits with bounded treewidth.

Such circuits can be compiled into DNNF with size exponential in cliquewidth.

This is the first result linking cliquewidth to knowledge compilation efficiency.

Abstract

In this paper we study the role of cliquewidth in succinct representation of Boolean functions. Our main statement is the following: Let $Z$ be a Boolean circuit having cliquewidth $k$. Then there is another circuit $Z^*$ computing the same function as $Z$ having treewidth at most $18k+2$ and which has at most $4|Z|$ gates where $|Z|$ is the number of gates of $Z$. In this sense, cliquewidth is not more `powerful' than treewidth for the purpose of representation of Boolean functions. We believe this is quite a surprising fact because it contrasts the situation with graphs where an upper bound on the treewidth implies an upper bound on the cliquewidth but not vice versa. We demonstrate the usefulness of the new theorem for knowledge compilation. In particular, we show that a circuit $Z$ of cliquewidth $k$ can be compiled into a Decomposable Negation Normal Form ({\sc dnnf}) of size $O(9^{18k}k^2|Z|)$ and the same runtime. To the best of our knowledge, this is the first result on efficient knowledge compilation parameterized by cliquewidth of a Boolean circuit.