On the relation between structured $d$-DNNFs and SDDs

TL;DR

This paper investigates the relationship between structured d-DNNFs and SDDs, proving that polynomial-size SDDs can be obtained from functions with polynomial-size structured d-DNNFs and their complements sharing the same vtree.

Contribution

It establishes that functions with polynomial-size structured d-DNNFs and their complements can be represented by polynomial-size SDDs when sharing the same vtree.

Findings

Polynomial-size SDDs can be derived from functions with polynomial-size structured d-DNNFs and their complements.

Structured d-DNNFs are not necessarily more general than SDDs under certain conditions.

Shared vtrees are crucial for converting structured d-DNNFs into SDDs of polynomial size.

Abstract

Structured $d$-DNNFs and SDDs are restricted negation normal form circuits used in knowledge compilation as target languages into which propositional theories are compiled. Structuredness is imposed by so-called vtrees. By definition SDDs are restricted structured $d$-DNNFs. Beame and Liew (2015) as well as Bova and Szeider (2017) mentioned the question whether structured $d$-DNNFs are really more general than SDDs w.r.t. polynomial-size representations (w.r.t. the number of Boolean variables the represented functions are defined on.) The main result in the paper is the proof that a function can be represented by SDDs of polynomial size if the function and its complement have polynomial-size structured $d$-DNNFs that respect the same vtree.