Disjunctive Query

The core of a disjunctive query is the logical OR operator. It combines multiple filter predicates such that a document is included in the result set if at least one of the conditions evaluates to true. This is formally expressed as predicate_A OR predicate_B OR .... For example, a query for documents where category = 'news' OR author = 'Jane Doe' returns all news articles plus all articles written by Jane Doe, including any that satisfy both conditions.

Disjunctive queries inherently increase recall by expanding the candidate pool. Unlike a conjunctive query (AND) which narrows results, an OR condition aggregates matches from multiple criteria. This is essential for exploratory search, faceted navigation where users select multiple options within a facet (e.g., 'show me shirts in red OR blue'), or building inclusive candidate sets for downstream reranking stages. The cardinality of the result set is at least as large as the largest individual predicate's result and at most the union of all.

Disjunctive logic is a first-class citizen in search and database Query Domain-Specific Languages (DSLs). It is expressed using explicit operators:

• Elasticsearch / OpenSearch: {"bool": {"should": [ {...}, {...} ]}}
• SQL: WHERE category = 'A' OR tag = 'B'
• Pinecone / Weaviate: Nested filter objects with an "Or" operator at the root. These DSLs allow nesting disjunctions within conjunctions (e.g., (A OR B) AND C) to create complex, precise logical trees for retrieval.

In hybrid search systems, disjunctive filters are typically applied to the metadata before or during the vector similarity search. A query like (vector_similarity(query)) FILTER BY (department = 'Sales' OR department = 'Marketing') ensures results are semantically relevant and belong to either department. Efficient execution requires index structures like HNSW with filters or IVF with IDSelector that can prune graph traversal or cluster probing based on OR conditions, avoiding a full post-filter that could discard all vector results.

Disjunctive queries can be computationally more expensive than conjunctive ones for filtering systems. Key performance factors include:

• Index Support: Systems use bitmap indexes or inverted indexes to perform fast union operations for OR clauses.
• Selectivity: Low-selectivity (broad) OR predicates can generate large intermediate result sets, stressing memory. Optimizers may reorder predicates or choose different index access paths.
• Pushdown Efficiency: For ANN with filters, the ability to 'push' the OR condition into the vector index traversal is critical for performance. A naive post-filter after a vector search may yield zero results if the filter is too restrictive, wasting the initial ANN computation.

Disjunctive (OR) and conjunctive (AND) queries are logical duals. Their core difference is in result set inclusion:

• Disjunctive (OR): A OR B → Includes union of sets satisfying A and B.
• Conjunctive (AND): A AND B → Includes intersection of sets satisfying A and B. A conjunctive query narrows and specifies ('this AND that'), while a disjunctive query broadens and aggregates ('this OR that'). Most complex business logic queries, expressed in a Query DSL, are compositions of both AND and OR operators within a single execution plan.

A conjunctive query uses the logical AND operator to combine multiple filter conditions. A record must satisfy all specified predicates to be included in the result set. This is the inverse of a disjunctive query and is used for narrowing results.

• Example: category = 'legal' AND date >= '2024-01-01' AND author = 'counsel'
• In vector search, a conjunctive filter might be applied before an ANN search to ensure all retrieved documents meet strict business rules.

A Boolean filter is a logical expression constructed using AND, OR, and NOT operators applied to metadata fields. Disjunctive and conjunctive queries are specific types of Boolean filters.

• Key Use: Creating complex, nested logic for precise document inclusion/exclusion.
• Example: (department = 'sales' OR department = 'marketing') AND NOT status = 'archived'
• In a Query DSL, Boolean filters define the hard constraints for hybrid or filtered search.

Filtered search is a retrieval process where metadata-based constraints narrow the candidate set before or during a similarity or keyword search. Disjunctive queries are a primary method for expressing these constraints.

• Architecture: Critical for enterprise RAG and e-commerce, where results must obey business rules (e.g., region = 'EU' OR license = 'global').
• Performance Impact: Applying filters pre-search (pre-filtering) reduces the index search space, while post-search (post-filtering) can cause result starvation if filters are too restrictive.

Filter selectivity estimates the proportion of records that will satisfy a filter predicate. It is a crucial metric for database query optimizers when planning the execution of disjunctive or conjunctive queries.

• Calculation: Expressed as a fraction (e.g., 0.05 for 5%). A low-selectivity filter returns few rows.
• Optimization Use: The optimizer may decide to evaluate a high-selectivity disjunctive clause (e.g., status = 'active') later in the plan, as it eliminates few rows. Understanding selectivity helps predict query performance and avoid bottlenecks.

ANN with Filters refers to algorithms for Approximate Nearest Neighbor search that are modified to respect hard metadata constraints during graph or tree traversal. This solves the challenge of efficiently executing a vector similarity search under a disjunctive filter.

• Core Problem: Applying a filter after a pure vector search (post-filtering) can discard all relevant results.
• Solutions: Algorithms like HNSW with Filters or DiskANN integrate filtering logic into the graph traversal, pruning paths that lead to nodes violating the filter (e.g., category = 'A' OR category = 'B').

A Query Domain-Specific Language (DSL) is a specialized syntax for expressing complex search requests. It provides a declarative way to combine vector similarity, keyword matching (BM25), and Boolean filters (including disjunctive queries) in a single statement.

• Example Clause: { vector: [0.1, 0.2,...], filter: { or: [ { author: 'Smith' }, { department: 'R&D' } ] } }
• Purpose: Abstracts complexity for developers and allows the database optimizer to generate the most efficient execution plan for the combined query.