Conjunctive Metrics: Theory & Applications

• Conjunctive metrics are quantitative measures that assess multi-way conjunctions in queries, algebraic structures, and probabilistic settings, highlighting key properties like expressiveness and tractability.
• They are constructed using methods such as shortest path proofs in containment graphs, WL-dimension analysis, and linear equation systems to capture correlation patterns and leakage degrees.
• Applications range from optimizing database queries and evaluating secure search protocols to informing machine learning models, making these metrics essential for advanced system analysis.

A conjunctive metric is a quantitative or structural measure associated with conjunctive queries, algebraic structures, probability/correlation patterns, or secure search protocols exhibiting conjunctive behavior. In contemporary research, conjunctive metrics have emerged as critical analytical tools in areas spanning logic and database theory, algebra, machine learning, information security, and causal inference. They serve as both technical invariants and evaluative standards, capturing expressiveness, similarity, tractability, information leakage, or algebraic regularity in systems governed by conjunction-like operations.

1. Formal Definitions and Structural Invariants

The notion of a conjunctive metric is domain-dependent but consistently involves quantifying, comparing, or characterizing entities formed by conjunction or multi-way conjunction-like structured interactions.

Database and Query Languages

• Semantic Query Metrics: A semantic query metric δ: L×L → ℕ on a query language L defines δ(Q₁, Q₂) as the length of a shortest path in a “maximal containment graph” of queries, where adjacency is governed by immediate maximal containment. Maximal containment here is defined such that Q₁ is contained in Q₂ and there are no strictly intermediate queries except for those equivalent to the endpoints (Fletcher et al., 23 Mar 2025).
  • For conjunctive queries (CQs), standard semantic metrics are generally infeasible due to the possibility of infinite chains in the containment lattice. However, for subclasses such as 2CQs (where each relation appears at most twice), δ can be shown to satisfy metric axioms and is computationally tractable (Fletcher et al., 23 Mar 2025).
• Quantum Conjunctive Queries and WL-Dimension: The Weisfeiler-Leman (WL) dimension of a conjunctive query φ, which is fundamental for understanding the complexity of counting φ’s answers in a graph, is shown to be the “semantic extension width” $\mathsf{sew}(\varphi)$, a parameter combining treewidth and quantified star size (the latter measures how existential variables interact with free variables). For a linear combination (quantum query) $Q = \sum_{i=1}^\ell c_i \cdot (H_i, X_i)$, the WL-dimension is the maximum semantic extension width of its constituents (Göbel et al., 2023).

Algebra and Lattices

• Ideal Conjunctivity: Within a join-semilattice L, a conjunctive metric can emerge from the property that every principal ideal is the intersection of maximal ideals. This is equivalent to being “ideally conjunctive.” In the context of algebraic frames (as per Martinez and Zenk), this property aligns with the Yosida condition, providing a metric-like separation between elements via their maximal ideal spectra (Delzell et al., 2020).

Probabilistic and Causal Settings

• Additive Metric Structure in Probability: In the analysis of conjunctive forks, a metric $x_{\{A,B\}} = -\ln(\mathrm{corr}(A,B))$ is assigned to pairs of events, with the key additivity relation $x_{\{A,C\}} = x_{\{A,B\}} + x_{\{B,C\}}$ for any ternary fork (A,B,C). Existence of such a metric is tantamount to representability of the ternary relation by conjunctive forks, and recognition can be performed efficiently via solving systems of linear equations (Chvátal et al., 2016).

Security and Leakage

• Metrics for Leakage-abuse in Encrypted Search: In conjunctive searchable symmetric encryption (CSSE), several “conjunctive metrics” (s-acc, f-acc, l-acc, and CAD) quantify the extent to which multi-keyword queries can be reconstructed by an adversary:

  Metric	Definition
  s-acc	Fraction of queries with s-term (least-frequent keyword) correctly identified
  f-acc	Fraction of queries with full keyword set recovered
  l-acc	Fraction of individual keywords successfully recovered across all queries
  CADₓ	Fraction of queries with at least x keywords recovered

These metrics offer fine-grained perspectives on both partial and total query recovery in practical attack scenarios (Su et al., 5 Jul 2025).

2. Characterizations, Feasibility, and Non-Existence Phenomena

A central thread is the feasibility (or impossibility) of robust conjunctive metrics:

• Non-Existence in Full CQs: For general conjunctive queries, the ascending chain condition may fail in the maximal containment lattice, precluding the existence of a well-defined metric structure on the space of queries (Fletcher et al., 23 Mar 2025).

• Feasible Subclasses: For 2CQs and certain “star query” forms—very common in real workloads—semantic containment lattices are well-founded and permit the definition of effective metrics, with maximal containment relations computable in PTIME (Fletcher et al., 23 Mar 2025).

This dichotomy is echoed in algebra and logic: while many algebraic or logical structures lack suitable meets/joins or immediate predecessors necessary for metric construction, certain subclasses (c-acyclic queries, distributive lattices with ideal conjunctivity, special frames) remain amenable.

3. Methods for Construction and Computation

• Containment Graphs and Shortest Paths: For feasible subclasses, the set of queries is structured as a graph whose edges encode maximal containment. The conjunctive metric δ is then the shortest path length, with explicit recursive formula:

$$\delta(Q_1, Q_2) = \begin{cases} 0 & \text{if } Q_1 \equiv Q_2 \ 1 & \text{if } Q_1 \text{ is maximally contained in } Q_2 \ i>1 & \text{if } \exists Q: \delta(Q_1, Q) = i-1, \delta(Q_2, Q) = 1 \end{cases}$$

(Fletcher et al., 23 Mar 2025)

• Probabilistic Fork Patterns: The forkness ternary relations are checked through (i) validation of forkness properties, (ii) computation of equivalence class quotients, (iii) solving the corresponding system of linear equations for the “log-correlation” metric (Chvátal et al., 2016).
• Algebraic and Lattice Methods: In ideal-conjunctive semi-lattices, the intersection patterns of maximal ideals are leveraged to define “distinguishing” elements, and morphisms respect these intersections (Delzell et al., 2020).

4. Applications in Analysis, Comparison, and Optimization

• Workload Analytics, Recommender Systems, and View Selection: Semantic query metrics are leveraged to group or cluster queries with similar input/output behavior, facilitating indexing, materialized view selection, database tuning, benchmark selection, and debugging (Fletcher et al., 23 Mar 2025).
• Attack Evaluation in CSSE: The conjunctive metrics s-acc, l-acc, CAD, etc., are used to quantify the severity and granularity of leakage under multi-keyword attacks, revealing risks that are underestimated by single-keyword metrics (Su et al., 5 Jul 2025).
• Structural Query Characterization: In machine learning, the WL-dimension (semantic extension width) derived as a conjunctive metric directly determines the minimum order of higher-order GNNs necessary for counting or recognizing patterns defined by conjunctive queries (Göbel et al., 2023).
• Probabilistic Causal Inference: The additive log-correlation metric within forkness structures aids recognition and classification of probabilistic phenomena with dependence relations mirroring causal or information flow patterns (Chvátal et al., 2016).

5. Theoretical Impact and Limitations

The existence (or failure) of conjunctive metrics is intimately tied to deep properties of the logical or structural system:

• Metric Axioms and Failure Modes: A metric must satisfy non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. For the space of all conjunctive queries, violations can occur due to infinite anti-chains or infinite descending sequences in the semantic containment order, undermining the metric structure entirely (Fletcher et al., 23 Mar 2025).
• Significance of Feasible Metrics: In settings where a conjunctive metric is defined (notably, for star queries or low-width subclasses), it provides a principled and computable method for capturing semantic distances, which coincides with practical needs in database analytics and information retrieval (Fletcher et al., 23 Mar 2025, Göbel et al., 2023).
• Limitation in Attack Mitigation: In CSSE settings, the metrics reveal that even minimal leakage patterns (such as those from s-term exposure) are sufficient to compromise large portions of query privacy, highlighting that traditional defenses may not shield against the nuanced risks captured by these conjunctive metrics (Su et al., 5 Jul 2025).

6. Future Directions and Broader Relevance

Conjunctive metrics are likely to drive several research and practical directions:

• Refinement of metric-based analytics for large-scale query workloads, exploiting tractable subclasses identified in (Fletcher et al., 23 Mar 2025).
• Integration of conjunctive metrics into graph learning frameworks, aligning model expressive power precisely with the parameters identified, such as semantic extension width (Göbel et al., 2023).
• Extension of CSSE leakage models to account for joint/stochastic dependencies beyond independent keyword accesses, with evaluation grounded in the conjunctive metrics established (Su et al., 5 Jul 2025).
• Further exploration in algebraic logic and information theory linking metric-like invariants with logical or probabilistic structure as seen in (Chvátal et al., 2016, Delzell et al., 2020).

Summary Table: Conjunctive Metric Perspectives

Context	Core Metric/Invariants	Notable Complexity/Implication
Database Queries	Semantic query metric δ(Q₁, Q₂)	Tractable for 2CQs; impossible in full CQs (Fletcher et al., 23 Mar 2025)
Graph Homomorphism	WL-dimension = semantic extension width	Governs GNN/algorithm expressiveness (Göbel et al., 2023)
Probabilistic Forks	Additive log-correlation metric x{A,B}	Recognizable in PTIME via linear equations (Chvátal et al., 2016)
CSSE Leakage	s-acc, f-acc, l-acc, CAD	Reveals partial/full leakage; high-accuracy attacks (Su et al., 5 Jul 2025)
Algebraic Lattices	Ideal conjunctivity	Characterizes spectra/topology (Delzell et al., 2020)

Conjunctive metrics therefore constitute a unifying analytic theme in the evaluation of complex, conjunctive structures in logic, databases, security, and statistical learning, enabling both theoretical generalization and practical algorithmic advances.