# Geolog Summary

*Date: 2026-03-19*

## What Is Geolog?

Geolog is a **Geometric Logic REPL** — a reasoning engine written in Rust that implements type theory with semantics in topoi. It was AI-generated by Claude Opus 4.5.

## Core Concepts

### Axioms

An axiom is an "if-then" rule: "If premises are true, then conclusion must be true."

```
forall x, y, z. friends(x,y), friends(y,z) |- knows(x,z)
```

Similar to Datalog rules, but extended with:
- Existentials in conclusions (`|- exists x. R(x)`)
- Disjunctions (`|- A \/ B`)
- Equality conclusions (`|- x = y`)

### Chase Algorithm

Repeatedly applies axioms until no new facts can be derived (fixpoint). Basic blocks:

1. **Axioms** — the rules to apply
2. **Structure** — carriers (element sets), functions, relations
3. **Bindings** — variable-to-element mappings
4. **Tensor system** — finds violations (premises true, conclusion false)
5. **Fire conclusion** — adds relations, defines functions, creates elements, adds equations
6. **Congruence closure** — union-find for tracking element equality

### Union-Find

Data structure for tracking equivalence classes. Used for congruence closure when axioms conclude `x = y`. More efficient than naive dictionary + sets approach:
- Naive: O(n) per union
- Union-find: O(1) amortized per union

### Language Syntax

Geolog's syntax is not based on any single existing language. Influences:
- **Postfix application** (`x f` not `f(x)`) — category theory
- **Sequent notation** (`premises |- conclusion`) — proof theory
- **Path separators** (`/` not `.`) — unique to Geolog

## Skeptical Assessment of Applications

### Claimed Use Cases
- Business process workflow orchestration
- Formal verification
- Database query design
- Petri net reachability

### Reality

| Claim | Assessment |
|-------|------------|
| Workflow orchestration | No APIs, no integrations, no distributed execution — just a REPL |
| Formal verification | Lean4 proofs "in progress"; not comparable to Coq/Lean/TLA+ |
| Database query design | Not a database — no persistence at scale, no SQL, no transactions |
| Petri net reachability | Toy examples only, not industrial-scale |

### What It Actually Is

A **research/educational tool** for:
- Learning geometric logic and chase algorithms
- Experimenting with category-theoretic ideas
- Prototyping constraint systems
- Academic exploration

**Not suitable for:** Production systems, real workflow orchestration, industrial formal verification, or database replacement.

## Project Status

- Version: 0.1.0
- 30,000+ lines of Rust
- 168 tests (all passing)
- 30 example files
- Well-documented but unproven in real-world use