# Can a directed-graph-structured knowledge base give negation a geometric witness? **URL:** https://swi-prolog.discourse.group/t/can-a-directed-graph-structured-knowledge-base-give-negation-a-geometric-witness/9645 **Category:** Discussion **Created:** [June 6, 2026, 12:28pm UTC](https://swi-prolog.discourse.group/t/can-a-directed-graph-structured-knowledge-base-give-negation-a-geometric-witness/9645 "2026-06-06T12:28:43Z") **Posts on this page:** 1 **Page:** 1
### Author: ![EnisOlgac](https://yyz2.discourse-cdn.com/free1/user_avatar/swi-prolog.discourse.group/enisolgac/32/7926_2.png) [@EnisOlgac](https://swi-prolog.discourse.group/u/EnisOlgac) #### Post date: [June 6, 2026, 12:28pm UTC](https://swi-prolog.discourse.group/t/can-a-directed-graph-structured-knowledge-base-give-negation-a-geometric-witness/9645/1 "2026-06-06T12:28:43Z")
I have been working on a framework for encoding directed graphs incrementally with O(1) reachability queries, and a question emerged that I would like to think through with people who know Prolog’s foundations well. Prolog’s negation-as-failure was arguably the first time negation was treated as a prime computational object. The limitation is well known: NAF produces a contentless result. The search tree is exhausted and failure is reported, but the negation carries no witness. In a triordinal-encoded knowledge base—where reachability is confirmed only by unanimous positional consensus across three mutually constraining orderings—a dissent is not a search failure. It is a geometric fact, a positive certificate that the path does not exist, readable in O(1) directly from the encoding. Negation would acquire content. There is a structural consequence beyond performance. A tree-structured knowledge base cannot represent cross-cutting relationships without duplication or loss. A directed-graph encoding handles these natively. I am looking for discussion partners who find this direction interesting. A feature overview of the framework is at: [https://doi.org/10.5281/zenodo.19480474](https://doi.org/10.5281/zenodo.19480474) Enis Olgac [enis@digraphs.info](mailto:enis@digraphs.info)