Source-linked AI summary
Provenance for Aggregate Queries
Yael Amsterdamer, Daniel Deutch, Val Tannen
TL;DR
Aggregate queries create provenance challenges that tuple-only commutative-semiring annotations do not resolve. The paper annotates both tuples and value computations, yielding polynomial-size constructions for aggregation while addressing compatibility and robustness.
Problem
Aggregate queries pose novel provenance challenges, including interpreting comparisons over symbolic aggregate values and non-monotonic result membership.
Method
The paper annotates aggregate value computations alongside tuples, using semimodules over commutative semirings and a tensor product construction for annotated aggregation.
Results
The constructions provide faithful aggregation semantics for compatible semiring–monoid pairs and support arbitrary aggregate queries with polynomial representation size.
Takeaways & Limitations
Idempotent aggregates such as MIN and MAX align with set semantics, whereas SUM and PROD require bag-oriented semirings.
Takeaways & Limitations
Approaches adding negative operations still require exponentially many separate tuples to represent all possible summation results.
Abstract
from arXiv · showhide
We study in this paper provenance information for queries with aggregation. Provenance information was studied in the context of various query languages that do not allow for aggregation, and recent work has suggested to capture provenance by annotating the different database tuples with elements of a commutative semiring and propagating the annotations through query evaluation. We show that aggregate queries pose novel challenges rendering this approach inapplicable. Consequently, we propose a new approach, where we annotate with provenance information not just tuples but also the individual values within tuples, using provenance to describe the values computation. We realize this approach in a concrete construction, first for "simple" queries where the aggregation operator is the last one applied, and then for arbitrary (positive) relational algebra queries with aggregation; the latter queries are shown to be more challenging in this context. Finally, we use aggregation to encode queries with difference, and study the semantics obtained for such queries on provenance annotated databases.
1. INTRODUCTION
The paper extends semiring-based provenance to aggregate queries, showing that tuple-only annotations are insufficient and proposing provenance annotations for aggregate-value computations. It develops polynomial-size semantics satisfying standard set/bag behavior and homomorphism commutation, while also introducing a provenance-aware semantics for difference.
- Motivation: Semiring provenance previously handled positive relational algebra, but this paper’s main goal is extending the framework to aggregation.The prior framework also supports applications including trust, security, multiplicity, and deletion propagation.
- Motivation: Tuple-only provenance cannot satisfy both homomorphism commutation and ordinary set or bag semantics for aggregation.The paper proves this incompatibility and rejects extensions that require enumerating exponentially many aggregate outcomes.
- Approach: The proposed approach annotates aggregate values and their computation, representing sums and minima as formal expressions such as p1 ⊗ 20 + p2 ⊗ 10 + p3 ⊗ 15.This captures how aggregate results depend on the provenance-annotated input values rather than treating results as ordinary numbers.
- Results: Compatible semirings and aggregation monoids faithfully recover aggregate results; idempotent functions such as MIN and MAX suit sets, whereas SUM and PROD require bags.Non-idempotent monoids are compatible only with bag semirings admitting a homomorphism to N.
- Results: Formal comparisons between symbolic aggregate values are added as provenance expressions, enabling arbitrary positive queries with aggregation while retaining polynomial representation size.Comparison tokens can continue through later query operators as ordinary provenance tokens.
- Results: The paper’s aggregation semantics matches usual min/max/sum/prod set/bag behavior, commutes with semiring homomorphisms, and has only polynomial-size overhead; it also defines provenance-aware difference.The difference semantics removes tuples present in the subtrahend while preserving annotations for tuples absent from it.
2. PRELIMINARIES
The preliminaries recall semiring-based provenance for positive relational algebra and develop semimodule and tensor constructions to extend aggregation to annotated data.
- Semiring provenance: The paper recalls semiring annotations as the algebraic foundation for provenance in positive relational algebra queries.K-relations use commutative-semiring elements as tuple annotations, with union, projection, and join defined through semiring operations.
- Semimodules and aggregates: K-relations form a K-semimodule, allowing semiring annotations to interact with aggregate values through scalar multiplication.The construction satisfies distributivity, zero, associativity of scalar multiplication, and identity laws.
- Semimodules and aggregates: For a K-set of values, aggregation computes the sum of each value scaled by its annotation in the chosen K-semimodule.SetAgg_W(S) = k1 ∗W w1 +W · · · +W kn ∗W wn.
- Aggregation queries: SetAgg initially operates on sets rather than relations, motivating its extension to arbitrary relations and grouped aggregation.Grouped aggregation separates tuples by group-by attributes and computes each group’s aggregate similarly to AGG.
- Tensor construction: The tensor construction embeds a commutative monoid M into a K-semimodule capable of representing K-annotated aggregations.It forms equivalence classes of finite bags of simple tensors k⊗m under distributive congruence rules, and the resulting K ⊗ M is a K-semimodule.
- Aggregation queries: The framework supports lifting semiring homomorphisms to tensor semimodules and restricts SPJU-A queries to one final aggregation operation.SPJU-A corresponds to an SPJU expression followed by a single AGG application, without grouping.
3. SIMPLE AGGREGATION QUERIES
Simple aggregation queries place aggregation last, enabling a provenance semantics based on tensor-product expressions that annotates aggregate-value computations. This construction satisfies the stated compatibility, homomorphism, and polynomial-overhead requirements for SPJU-A queries.
- Scope: Simple aggregation queries apply aggregation last, avoiding comparisons between annotated aggregate values and simplifying provenance treatment.The framework initially targets queries such as un-nested SELECT FROM WHERE GROUP BY queries, with broader queries handled later.
- Construction: K ⊗ M combines tuple annotations from a commutative semiring K with aggregate values from a commutative monoid M.The construction represents aggregate computations as formal tensor-product expressions, with examples including SUM, PROD, MAX, and MIN.
- Properties: The SPJU-A semantics satisfies set/bag compatibility, commutation with semiring homomorphisms, and polynomial-size overhead.The main theorem characterizes homomorphism commutation for SPJU-A queries, while the representation avoids enumerating exponentially many possible aggregate answers.
- Behavior: Aggregate results are represented by a single tensor expression whose provenance valuations recover possible aggregate values and support deletion propagation.For SUM, valuations can produce concrete results such as 80, while changing an annotation to zero removes the corresponding input contribution.
- Compatibility: For B ⊗ MAX, B ⊗ MIN, and N ⊗ SUM, the canonical embedding from M into K ⊗ M is a monoid isomorphism, yielding familiar aggregate results.More generally, N[X] is compatible with every commutative monoid, whereas arbitrary semirings need not preserve this property.
4. NESTED AGGREGATION QUERIES
Nested aggregation introduces non-monotonic comparisons that defeat the restricted construction, so the paper extends the semiring with symbolic equality expressions. The resulting semantics handles arbitrary aggregation queries while preserving compatibility, homomorphism commutation, and polynomial-size representations.
- 4.1 Difficulties: Aggregate comparisons can depend non-monotonically on which input tuples exist, making their truth values undecidable from basic symbolic aggregate expressions.For example, mapping r1 to 1 and r2 to 0 includes a tuple, whereas mapping both to 1 excludes it.
- 4.1 Difficulties: No (M, K)-relation semantics for nested MAX/MIN aggregation is both set-compatible and homomorphism-commuting; SUM has the analogous bag limitation.
- 4.3 The Extended Semantics: The extended construction adds symbolic equality elements [x = y] to the annotation semiring when aggregate values must be compared.Equality expressions remain symbolic until a homomorphism provides values in the aggregation monoid and the equality can be resolved.
- 4.3 The Extended Semantics: The semantics represents comparison-dependent tuple existence by multiplying tuple annotations by corresponding equation annotations.This applies to selections and joins involving aggregate values.
- 4.3 The Extended Semantics: The construction handles arbitrary aggregation queries with set/bag compatibility, polynomial-size overhead, and proved commutation with homomorphisms.
- 4.3 The Extended Semantics: Nested aggregation can produce non-monotone aggregate values: changing r2 from 0 to 1 changes the example’s result from 1 ⊗40 to 1 ⊗20.
5. DIFFERENCE
The paper encodes difference using aggregation over a Boolean monoid, yielding provenance semantics in which the right-hand relation acts as a Boolean exclusion condition. It characterizes resulting equivalences and undecidability across annotation semirings.
- 5. DIFFERENCE: Aggregation over the Boolean monoid bB = ({⊥, ⊤}, ∨, ⊥) provides a semantics for arbitrary queries with difference on K-relations.
- 5.1 Semantics for Difference: Under the obtained semantics, a tuple appears in R − S exactly when it appears in R but not in S, retaining R’s original annotation.Existence in S is treated as a Boolean condition rather than as multiplicity subtraction.
- 5.2 Equivalence: For K = B, equivalence under the paper’s semantics coincides with equivalence under set semantics.
- 5.2 Equivalence: For K = N, the semantics differs from bag equivalence because the right-hand side of difference acts as a Boolean condition instead of reducing multiplicity.In the example, a department disappears under the paper’s semantics but remains with multiplicity 1 under bag semantics.
- 5.2 Equivalence: The paper’s equivalence notion and Z-semantics equivalence do not imply one another.
- 5.3 Deciding Query Equivalence: For semirings such as B and N whose results can be interpreted in bB, query equivalence is undecidable.
6. RELATED WORK
The paper extends established semiring-based provenance research to aggregate queries and relates its framework to applications including scientific workflows. It also distinguishes its difference semantics from earlier annotated-relation approaches.
- Provenance and aggregation: The paper presents itself as the first study of aggregate queries in the context of provenance semirings.
- Provenance applications: Its framework is intended to support provenance applications such as trust, deletion propagation, security, and fine-grained scientific-workflow provenance.
- Applications: Aggregate-query provenance is positioned as relevant to applications including OLAP, mobile computing, and streaming-data analysis.
- Difference semantics: The paper’s semantics for difference is different from previously proposed semantics based on c-tables, monus, Z, and Z[X].
- Future work: Optimizing aggregate-query evaluation on probabilistic databases through provenance management is identified as a future research challenge.
7. CONCLUSION
The paper develops provenance semantics for aggregation by annotating aggregate-value computation, then extends the construction to comparisons, nested aggregation, and difference. The authors report that the framework satisfies their stated desiderata and supports applications such as deletion propagation and security annotations.
- 7. CONCLUSION: The paper evaluates provenance approaches using set/bag compatibility, commutation with semiring homomorphisms, and polynomial-size overhead.
- 7. CONCLUSION: Annotating aggregate-value computation leads to a tensor-product semantics for SPJU queries followed by aggregation or group-by aggregation.
- 7. CONCLUSION: An adapted semimodule framework extends the semantics to queries comparing aggregation results while satisfying the stated desiderata.
- 7. CONCLUSION: The aggregation semantics also yields a semantics for arbitrary queries with difference on K-relations.
- 7. CONCLUSION: The approach is exemplified for deletion propagation and security annotations, with broader application work left for future research.
A. SPJU ALGEBRA FOR K-RELATIONS
The SPJU algebra for K-relations defines standard relational operations over K-annotated relations, with annotations combined according to semiring operations.
- An empty relation assigns annotation 0 to every tuple.
- Selection multiplies each tuple's annotation by a predicate value of 0 or 1.
- Projection sums annotations of input tuples that agree on the retained attributes.
- Natural join multiplies the annotations of the two joined tuples.
- Union combines two K-relations using the semiring addition operation.
B. PROPERTIES OF K ⊗M
The construction K ⊗M is a K-semimodule generated by M, with a universal extension property for monoid homomorphisms.
- K ⊗M forms a K-semimodule satisfying the six semimodule axioms.
- The embedding ι maps each m in M to the simple tensor 1K ⊗m.
- Every tensor is a K-linear combination of simple tensors from the image of ι.
- For any K-semimodule W and monoid homomorphism f from M to W, there is a unique K-semimodule homomorphism f ∗ extending f.
- K ⊗M is therefore characterized as the K-semimodule freely generated by M.
- The construction supports lifting a semiring homomorphism h from K to K′ to a monoid homomorphism hM from K⊗M to K′⊗M.
C. ADDITIONAL PROOFS
The additional proofs establish homomorphism compatibility for aggregation semantics and show why nested aggregation requires an extended treatment.
- The homomorphism characterization is proved using abstractly tagged N[X]-databases with distinct provenance indeterminates.
- Restricted aggregation commutes with homomorphisms because tuple annotations change while tuple values and aggregate contributions remain aligned.
- If all annotations in a group become zero, its δ expression becomes zero and the group is effectively deleted.
- For SUM, a contradiction example shows that no provenance polynomial can simultaneously represent an aggregate result of 20 under the required homomorphisms.
- The extended nested-aggregation semantics also satisfies homomorphism commutation, proved by induction over query structure.
- Before equation values can be evaluated, the extended semantics may retain as many groups as the original support, even when homomorphism application would reduce it.