On the relation between structured $d$-DNNFs and SDDs
This resolves a theoretical question in knowledge compilation for AI and logic, but it is incremental as it builds on prior work without broad practical impact.
The paper addresses whether structured d-DNNFs are more general than SDDs in terms of polynomial-size representations, proving that a function has a polynomial-size SDD if both the function and its complement have polynomial-size structured d-DNNFs sharing the same vtree.
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.