On the Role of Canonicity in Bottom-up Knowledge Compilation

TL;DR

This paper investigates the existence of a polynomial-time Apply function for reduced Sentential Decision Diagrams (SDDs), resolving an open question and exploring its implications for knowledge compilation and language canonicity.

Contribution

It proves the existence of a polynomial-time Apply function for reduced SDDs, addressing an open problem in the field.

Findings

Polytime Apply function exists for unreduced SDDs.

Existence of polytime Apply for reduced SDDs remains unresolved.

Results challenge assumptions about canonicity and complexity in knowledge compilation.

Abstract

We consider the problem of bottom-up compilation of knowledge bases, which is usually predicated on the existence of a polytime function for combining compilations using Boolean operators (usually called an Apply function). While such a polytime Apply function is known to exist for certain languages (e.g., OBDDs) and not exist for others (e.g., DNNF), its existence for certain languages remains unknown. Among the latter is the recently introduced language of Sentential Decision Diagrams (SDDs), for which a polytime Apply function exists for unreduced SDDs, but remains unknown for reduced ones (i.e. canonical SDDs). We resolve this open question in this paper and consider some of its theoretical and practical implications. Some of the findings we report question the common wisdom on the relationship between bottom-up compilation, language canonicity and the complexity of the Apply function.