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Provenance Analysis and Semiring Semantics for First-Order Logic

Erich Grädel, Val Tannen

TL;DR

This work extends data provenance from positive query languages to full first-order logic by introducing dual-indeterminate polynomials and quotient semirings to capture both positive and negative facts in model checking. It establishes a robust semantic framework where FO model checking is interpreted via commutative semirings, proves a fundamental compositional property, and provides a universal provenance construction with dual tokens that encode all potential proof trees. The authors demonstrate practical uses, including explanations for missing answers, integrity constraint repairs, confidence scoring, and reverse provenance analyses that identify models compatible with provenance assumptions. They also explore incremental provenance maintenance, alternative negation modeling, and broad theoretical developments linking semiring semantics to fixed-point logics, game strategy analysis, and model theory, highlighting both opportunities and limitations across various semiring classes.

Abstract

A provenance analysis for a query evaluation or a model checking computation extracts information on how its result depends on the atomic facts of the model or database. Traditional work on data provenance was, to a large extent, restricted to positive query languages or the negation-free fragment of first-order logic and showed how provenance abstractions can be usefully described as elements of commutative semirings -- most generally as multivariate polynomials with positive integer coefficients. We describe and evaluate here a provenance approach for dealing with negation, based on quotient semirings of polynomials with dual indeterminates. This not only provides a semiring provenance analysis for full first-order logic (and other logics and query languages with negation) but also permits a reverse provenance analysis, i.e., finding models that satisfy various properties under given provenance tracking assumptions. We describe the potential for applications to explaining missing query answers or failures of integrity constraints, and to using these explanations for computing repairs. This approach also is the basis of a systematic study of semiring semantics in a broad logical context.

Provenance Analysis and Semiring Semantics for First-Order Logic

TL;DR

This work extends data provenance from positive query languages to full first-order logic by introducing dual-indeterminate polynomials and quotient semirings to capture both positive and negative facts in model checking. It establishes a robust semantic framework where FO model checking is interpreted via commutative semirings, proves a fundamental compositional property, and provides a universal provenance construction with dual tokens that encode all potential proof trees. The authors demonstrate practical uses, including explanations for missing answers, integrity constraint repairs, confidence scoring, and reverse provenance analyses that identify models compatible with provenance assumptions. They also explore incremental provenance maintenance, alternative negation modeling, and broad theoretical developments linking semiring semantics to fixed-point logics, game strategy analysis, and model theory, highlighting both opportunities and limitations across various semiring classes.

Abstract

A provenance analysis for a query evaluation or a model checking computation extracts information on how its result depends on the atomic facts of the model or database. Traditional work on data provenance was, to a large extent, restricted to positive query languages or the negation-free fragment of first-order logic and showed how provenance abstractions can be usefully described as elements of commutative semirings -- most generally as multivariate polynomials with positive integer coefficients. We describe and evaluate here a provenance approach for dealing with negation, based on quotient semirings of polynomials with dual indeterminates. This not only provides a semiring provenance analysis for full first-order logic (and other logics and query languages with negation) but also permits a reverse provenance analysis, i.e., finding models that satisfy various properties under given provenance tracking assumptions. We describe the potential for applications to explaining missing query answers or failures of integrity constraints, and to using these explanations for computing repairs. This approach also is the basis of a systematic study of semiring semantics in a broad logical context.

Paper Structure

This paper contains 38 sections, 27 theorems, 59 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Commutative semirings
  3. First-Order Logic Interpreted in Commutative Semirings
  4. Semiring interpretations
  5. Intermezzo: Positive Semirings
  6. Sanity Checks
  7. "Consistency" and "completeness" for semiring interpretations
  8. Proof trees
  9. A Provenance Semiring for First-Order Logic
  10. Dual-Indeterminate Polynomials
  11. Provenance Tracking: An Example and a Characterization
  12. The size of provenance
  13. Explanations Obtained from Provenance
  14. From Provenance to Confidence
  15. Detailed Provenance Analysis: Top-Secret Proofs
  16. ...and 23 more sections

Key Result

proposition 1

$\pi[\![ \varphi ]\!] ~=~ \pi[\![ \mathop{\mathrm{\mathrm{nnf}}}\nolimits(\varphi) ]\!]$.

Figures (7)

  • Figure 1: The model $G$
  • Figure 2: Provenance tracking assumptions
  • Figure 3: The model $\mathcal{F}$
  • Figure 4: Maximum confidence model with dominant vertex
  • Figure 5: The model $\mathcal{H}$
  • ...and 2 more figures

Theorems & Definitions (49)

  • definition 1: Semiring
  • definition 2
  • proposition 1
  • proposition 2: Fundamental Property
  • proof
  • proposition 3
  • proposition 4: sanity checks
  • definition 3
  • proposition 5: another sanity check
  • proof
  • ...and 39 more