Connecting Width and Structure in Knowledge Compilation (Extended Version)

TL;DR

This paper establishes a direct connection between the width parameters of knowledge compilation circuits and the size of their structured representations, providing both constructive upper bounds and exponential lower bounds.

Contribution

It introduces a linear-time, singly exponential upper bound for converting bounded-treewidth circuits to structured forms and proves exponential lower bounds for monotone DNF/CNF formulas, linking width to compilation complexity.

Findings

Bounded-treewidth circuits can be converted to d-SDNNF in linear time.

Exponential lower bounds for d-SDNNF and OBDD sizes based on width parameters.

Compilation complexity is characterized by pathwidth and treewidth for DNF and CNF formulas.

Abstract

Several query evaluation tasks can be done via knowledge compilation: the query result is compiled as a lineage circuit from which the answer can be determined. For such tasks, it is important to leverage some width parameters of the circuit, such as bounded treewidth or pathwidth, to convert the circuit to structured classes, e.g., deterministic structured NNFs (d-SDNNFs) or OBDDs. In this work, we show how to connect the width of circuits to the size of their structured representation, through upper and lower bounds. For the upper bound, we show how bounded-treewidth circuits can be converted to a d-SDNNF, in time linear in the circuit size. Our bound, unlike existing results, is constructive and only singly exponential in the treewidth. We show a related lower bound on monotone DNF or CNF formulas, assuming a constant bound on the arity (size of clauses) and degree (number of occurrences of each variable). Specifically, any d-SDNNF (resp., SDNNF) for such a DNF (resp., CNF) must be of exponential size in its treewidth; and the same holds for pathwidth when compiling to OBDDs. Our lower bounds, in contrast with most previous work, apply to any formula of this class, not just a well-chosen family. Hence, for our language of DNF and CNF, pathwidth and treewidth respectively characterize the efficiency of compiling to OBDDs and (d-)SDNNFs, that is, compilation is singly exponential in the width parameter. We conclude by applying our lower bound results to the task of query evaluation.