[Paper Review] Circuit Treewidth, Sentential Decision, and Query Compilation

Simone Bova, Stefan Szeider|arXiv (Cornell University)|Jan 17, 2017

Bayesian Modeling and Causal Inference32 references4 citations

TL;DR

This paper establishes that bounded circuit treewidth in Boolean functions is equivalent to bounded SDD (Sentential Decision Diagram) width, providing a tight characterization analogous to the known equivalence between bounded circuit pathwidth and bounded OBDD width. The key contribution is a polynomial-size SDD compilation for queries with bounded treewidth, enabling efficient probabilistic query evaluation via tractable knowledge compilation.

ABSTRACT

The evaluation of a query over a probabilistic database boils down to computing the probability of a suitable Boolean function, the lineage of the query over the database. The method of query compilation approaches the task in two stages: first, the query lineage is implemented (compiled) in a circuit form where probability computation is tractable; and second, the desired probability is computed over the compiled circuit. A basic theoretical quest in query compilation is that of identifying pertinent classes of queries whose lineages admit compact representations over increasingly succinct, tractable circuit classes. Fostering previous work by Jha and Suciu (2012) and Petke and Razgon (2013), we focus on queries whose lineages admit circuit implementations with small treewidth, and investigate their compilability within tame classes of decision diagrams. In perfect analogy with the characterization of bounded circuit pathwidth by bounded OBDD width, we show that a class of Boolean functions has bounded circuit treewidth if and only if it has bounded SDD width. Sentential decision diagrams (SDDs) are central in knowledge compilation, being essentially as tractable as OBDDs but exponentially more succinct. By incorporating constant width SDDs and polynomial size SDDs, we refine the panorama of query compilation for unions of conjunctive queries with and without inequalities.

Motivation & Objective

• To resolve the open question of whether bounded circuit treewidth implies bounded OBDD width for unions of conjunctive queries with inequalities.
• To identify a knowledge compilation target class that is as tractable as OBDDs but exponentially more succinct, suitable for query lineage compilation.
• To extend the known characterization of bounded pathwidth via OBDD width to treewidth using SDDs as the canonical form.
• To refine the landscape of query compilation by incorporating constant-width and polynomial-size SDDs for UCQs with and without inequalities.
• To investigate the relative succinctness of SDDs compared to other structured deterministic forms and OBDDs, particularly in the context of probabilistic database query evaluation.

Proposed method

• Introduces a novel characterization linking circuit treewidth to SDD width using structural properties of Boolean functions and their circuit representations.
• Employs a rectangle cover argument based on Theorem 1 to bound the size of SDD components derived from structured NNFs, leveraging Jensen’s inequality for convex functions.
• Analyzes two complementary cases in the proof: when a subset of variables is fully covered in the SDD structure, and when partial coverage leads to exponential lower bounds.
• Uses a partitioning technique on variable pairs to define regions $ R_p $ where SDD components must grow exponentially with the size of the partition, leading to lower bounds on component size.
• Applies known results on deterministic decomposable circuits and structured NNFs to show that SDDs can simulate such forms with size at most linear in the original circuit.
• Leverages the fact that SDDs are deterministic, decomposable, and structured, enabling efficient probability computation and polynomial-size compilation for bounded-treewidth lineages.

Experimental results

Research questions

• RQ1 Does bounded circuit treewidth imply bounded OBDD width for unions of conjunctive queries with inequalities, as conjectured by Jha and Suciu?
• RQ2 Can SDDs provide a polynomial-size compilation for Boolean functions with bounded circuit treewidth, analogous to the pathwidth-OBDD width equivalence?
• RQ3 What is the relative succinctness of SDDs compared to OBDDs and other structured deterministic forms in the context of query lineage compilation?
• RQ4 Do SDDs with bounded fan-in OR gates admit quasipolynomial simulation by OBDDs, as suggested by recent conjectures?
• RQ5 Can the canonical structured deterministic forms based on factorized implicants be meaningfully compared to SDDs and data structures in factorized databases?

Key findings

• A class of Boolean functions has bounded circuit treewidth if and only if it has bounded SDD width, establishing a tight equivalence analogous to the pathwidth–OBDD width characterization.
• For any Boolean function with circuit treewidth $ k $, there exists an SDD representation of size $ O(f(k)n) $, where $ f(k) $ is a fast-growing function, matching the efficiency of OBDDs for pathwidth.
• The paper proves that SDDs can achieve polynomial-size compilation for bounded-treewidth lineages, avoiding the double-exponential dependence on treewidth seen in OBDDs.
• A lower bound of $ 2^{ heta(n/k)} $ is established for the size of at least one SDD component in a $ k $-partitioned structure, showing inherent exponential growth when coverage is incomplete.
• The indirect access storage (ISA) function, previously considered a candidate for exponential separation between SDDs and OBDDs, is shown to have small SDD size, closing a potential gap in succinctness comparisons.
• Constant-width SDDs and polynomial-size SDDs are incorporated into the query compilation framework for UCQs with and without inequalities, refining the known tractable cases of probabilistic query evaluation.