On the Role of Canonicity in Bottom-up Knowledge Compilation
This addresses a theoretical problem in knowledge compilation for AI and logic, with incremental implications for understanding language properties.
The paper resolves the open question of whether a polytime Apply function exists for reduced Sentential Decision Diagrams (SDDs), finding that it does not, and reports findings that challenge common assumptions about bottom-up compilation, canonicity, and Apply function complexity.
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.