The Equivalence Theorem: First-Class Relationships for Structurally Complete Database Systems
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910052750721024 |
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| author | Alford, Matthew |
| author_facet | Alford, Matthew |
| contents | We prove The Equivalence Theorem: structurally complete knowledge representation requires exactly four mutually entailing capabilities -- n-ary relationships with attributes, temporal validity, uncertainty quantification, and causal relationships between relationships -- collectively equivalent to treating relationships as first-class objects. Any system implementing one capability necessarily requires all four; any system missing one cannot achieve structural completeness. This result is constructive: we exhibit an Attributed Temporal Causal Hypergraph (ATCH) framework satisfying all four conditions simultaneously. The theorem yields a strict expressiveness hierarchy -- SQL < LPG < TypeDB < ATCH -- with witness queries that are structurally inexpressible at each lower level. We establish computational complexity bounds showing NP-completeness for general queries but polynomial-time tractability for practical query classes (acyclic patterns, bounded-depth causal chains, windowed temporal queries). As direct corollaries, we derive solutions to classical AI problems: the Frame Problem (persistence by default from temporal validity), conflict resolution (contradictions as unresolved metadata with hidden variable discovery), and common sense reasoning (defaults with causal inhibitors). A prototype PostgreSQL extension in C validates practical feasibility within the established complexity bounds. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_13603 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Equivalence Theorem: First-Class Relationships for Structurally Complete Database Systems Alford, Matthew Databases Artificial Intelligence H.2.1; H.2.4; F.4.1 We prove The Equivalence Theorem: structurally complete knowledge representation requires exactly four mutually entailing capabilities -- n-ary relationships with attributes, temporal validity, uncertainty quantification, and causal relationships between relationships -- collectively equivalent to treating relationships as first-class objects. Any system implementing one capability necessarily requires all four; any system missing one cannot achieve structural completeness. This result is constructive: we exhibit an Attributed Temporal Causal Hypergraph (ATCH) framework satisfying all four conditions simultaneously. The theorem yields a strict expressiveness hierarchy -- SQL < LPG < TypeDB < ATCH -- with witness queries that are structurally inexpressible at each lower level. We establish computational complexity bounds showing NP-completeness for general queries but polynomial-time tractability for practical query classes (acyclic patterns, bounded-depth causal chains, windowed temporal queries). As direct corollaries, we derive solutions to classical AI problems: the Frame Problem (persistence by default from temporal validity), conflict resolution (contradictions as unresolved metadata with hidden variable discovery), and common sense reasoning (defaults with causal inhibitors). A prototype PostgreSQL extension in C validates practical feasibility within the established complexity bounds. |
| title | The Equivalence Theorem: First-Class Relationships for Structurally Complete Database Systems |
| topic | Databases Artificial Intelligence H.2.1; H.2.4; F.4.1 |
| url | https://arxiv.org/abs/2603.13603 |