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Tractable Circuits in Database Theory

Antoine Amarilli, Florent Capelli

TL;DR

This paper surveys how tractable circuit classes from knowledge compilation can be leveraged to solve a broad range of database tasks. It shows how Boolean provenance and answer functions for MSO and CQ/UCQ queries can be represented by DNNF/SDNNF circuits under determinism and structuredness, enabling PTIME probabilistic query evaluation, approximate PQE via FPRAS, and efficient enumeration and ranking. It also connects these Boolean representations to multivalued representations via factorized databases, providing circuit-based encodings of query answers and provenance, and describes practical toolchains and algorithmic strategies (DPLL-based and bottom-up compilers). The work highlights two main directions: using circuits to obtain a unified, modular tractability framework across diverse tasks, and employing factorized circuits to succinctly represent large answer sets; it concludes with open questions such as the intensional-extensional conjecture for UCQs, incremental maintenance, and extending circuit methods to new database paradigms.

Abstract

This work reviews how database theory uses tractable circuit classes from knowledge compilation. We present relevant query evaluation tasks, and notions of tractable circuits. We then show how these tractable circuits can be used to address database tasks. We first focus on Boolean provenance and its applications for aggregation tasks, in particular probabilistic query evaluation. We study these for Monadic Second Order (MSO) queries on trees, and for safe Conjunctive Queries (CQs) and Union of Conjunctive Queries (UCQs). We also study circuit representations of query answers, and their applications to enumeration tasks: both in the Boolean setting (for MSO) and the multivalued setting (for CQs and UCQs).

Tractable Circuits in Database Theory

TL;DR

This paper surveys how tractable circuit classes from knowledge compilation can be leveraged to solve a broad range of database tasks. It shows how Boolean provenance and answer functions for MSO and CQ/UCQ queries can be represented by DNNF/SDNNF circuits under determinism and structuredness, enabling PTIME probabilistic query evaluation, approximate PQE via FPRAS, and efficient enumeration and ranking. It also connects these Boolean representations to multivalued representations via factorized databases, providing circuit-based encodings of query answers and provenance, and describes practical toolchains and algorithmic strategies (DPLL-based and bottom-up compilers). The work highlights two main directions: using circuits to obtain a unified, modular tractability framework across diverse tasks, and employing factorized circuits to succinctly represent large answer sets; it concludes with open questions such as the intensional-extensional conjecture for UCQs, incremental maintenance, and extending circuit methods to new database paradigms.

Abstract

This work reviews how database theory uses tractable circuit classes from knowledge compilation. We present relevant query evaluation tasks, and notions of tractable circuits. We then show how these tractable circuits can be used to address database tasks. We first focus on Boolean provenance and its applications for aggregation tasks, in particular probabilistic query evaluation. We study these for Monadic Second Order (MSO) queries on trees, and for safe Conjunctive Queries (CQs) and Union of Conjunctive Queries (UCQs). We also study circuit representations of query answers, and their applications to enumeration tasks: both in the Boolean setting (for MSO) and the multivalued setting (for CQs and UCQs).
Paper Structure (19 sections, 1 theorem, 2 figures, 2 tables)

This paper contains 19 sections, 1 theorem, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Database Tasks and Boolean Functions
  3. Boolean Provenance
  4. Boolean Answer Functions
  5. Other Tasks via Boolean Functions
  6. Knowledge Compilation
  7. DNNF Circuits
  8. Restrictions on DNNF Circuits
  9. Determinism.
  10. Structuredness.
  11. Tools
  12. Circuits for MSO Queries over Trees
  13. Aggregative Tasks for CQs and UCQs
  14. Circuits to Represent Query Answers
  15. Representing Relations as Circuits.
  16. ...and 4 more sections

Key Result

Theorem 6.1

Given a free-connex acyclic conjunctive query $Q$ and a database $\mathbb{D}$, we can build a structured $\{\mathsf{dec},\times\}$-circuit in time $O(\mathrm{poly\,} |Q|\cdot|\mathbb{D}|)$ computing $Q(\mathbb{D})$. The circuit is of size $O(\mathrm{poly\,} |Q|\cdot|\mathbb{D}|)$.

Figures (2)

  • Figure 1: A DNNF and its satisfying valuations
  • Figure 2: Inclusions between circuit classes. Arrows denote inclusion (i.e., linear-time transformations); they are all known to be strict in the sense that reverse arrows do not exist, except inclusions involving d-Ds which are not separated from d-DNNF monet2020solving and only conditionally separated from NNF. Most results are in amarilli2020connectingamarilli2024circus.

Theorems & Definitions (9)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Example 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 6.1: olteanu2015size