Expressing Linear Orders Requires Exponential-Size DNNFs

TL;DR

This paper proves that representing the set of all linear orders over n candidates with DNNF circuits inherently requires exponential size, establishing both lower and upper bounds.

Contribution

It demonstrates the exponential size requirement for DNNF circuits expressing linear orders and constructs circuits matching this bound.

Findings

Any DNNF circuit for linear orders over n candidates has size at least 2^{Ω(n)}.

There exist DNNF circuits of size 2^{O(n)} expressing linear orders.

The results establish tight bounds on the complexity of representing linear orders with DNNF circuits.

Abstract

We show that any DNNF circuit that expresses the set of linear orders over a set of $n$ candidates must be of size $2^{\Omega(n)}$. Moreover, we show that there exist DNNF circuits of size $2^{O(n)}$ expressing linear orders over $n$ candidates.