On Limitations of Structured (Deterministic) DNNFs

• Published in: Theory of computing systems Vol. 64; no. 5; pp. 799 - 825
• Main Authors: Bollig, Beate, Buttkus, Matthias
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.07.2020; Springer Nature B.V
• Subjects: Canonical forms; Circuit diagrams; Computer Science; Equivalence; Knowledge bases (artificial intelligence); Knowledge representation; Polynomials; Simulation; Theory of Computation; Ordered binary decision diagrams; Complexity theory; Decomposable negation normal forms; Knowledge compilation; Sentential decision diagrams
• ISSN: 1432-4350; 1433-0490
• DOI: 10.1007/s00224-019-09960-w

The study of representations for propositional theories has been a central subject in knowledge compilation. Many known representations of propositional knowledge bases are restricted negation normal form circuits (NNFs) or binary decision diagrams. Sentential decision diagrams (SDDs) are one of them, by definition they are restricted structured deterministic DNNFs and more general than ordered binary decision diagrams (OBDDs). In this paper limitations of structured (deterministic) DNNFs and in particular SDDs are considered. The main result in the paper is a quasipolynomial simulation of structured (deterministic) DNNFs by equivalent (unambiguous) nondeterministic OBDDs. The result is tight in the sense that there cannot be a polynomial simulation due to a known separation result. One may ask whether there are simulations in smaller size for restricted structured (deterministic) DNNFs. Answering a question posed by Bova and Szeider ( 2017 ) in the negative it is shown that not every SDD with disjunctions of constant fan-in can be transformed into equivalent OBDDs of polynomial size.