| The Theory of Joins in Relational Databases | Aho, Beeri & Ullman | [dl.acm.org/doi/10.1145/320083.320091](https://dl.acm.org/doi/10.1145/320083.320091) |
| Testing Implications of Data Dependencies | Maier, Mendelzon & Sagiv | [dl.acm.org/doi/10.1145/320107.320115](https://dl.acm.org/doi/10.1145/320107.320115) |
| Oblivious | Largest | Fastest | None | — | Prototyping; when output size doesn't matter |
| Skolem | Medium | Fast | Implicit (Skolem terms) | VLog, Graal | Parallelism; data exchange; deterministic runs |
| Restricted | Smallest | Slower | Explicit (homomorphism check) | RDFox, VLog | Production use; query performance over large KGs |
| Core | Most compact | Slowest | Full core computation | — (theoretical) | Theoretical analysis; minimum representations |
#### Variant Notes
**Oblivious Chase** fires every applicable rule without checking whether the conclusion already holds. Simple to implement but produces redundant facts and is the hardest to guarantee termination for.
**Skolem Chase** uses deterministic Skolem functions to generate nulls — the same rule trigger always produces the same null. This makes it idempotent and easy to parallelize, with no need to track which rules have already fired.
**Restricted Chase** checks before firing whether the rule head is already satisfied via a homomorphism into the current database. Never adds redundant facts, producing the smallest universal solution. The homomorphism check is NP-hard in general but fast in practice with good indexing.
**Core Chase** goes further than the restricted chase, computing the logical core — the most compressed possible universal solution. Theoretically important but computationally expensive; not used in production engines.
