# Key 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`

Datalog is limited to Horn clauses (single atom in head). Geolog supports geometric sequents.

---

## Chase Algorithm

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

### Building Blocks

1. **Axioms** — 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
6. **Congruence closure** — tracks element equality via union-find

### Loop

```
repeat until no changes:
    for each axiom:
        find violations (tensor system)
        fire conclusion for each violation
    propagate equations (congruence closure)
```

---

## Union-Find

Data structure for tracking equivalence classes. Inputs are elements (IDs), not sets.

```
union(3, 7)   // "3 and 7 are now equivalent"
union(7, 12)  // "7 and 12 are now equivalent"
find(3)       // Returns same representative as find(12)
```

### Why Not Dictionary + Sets?

| Approach | Union Cost |
|----------|------------|
| Dict + Sets | O(n) — must update all elements in merged set |
| Union-Find | O(1) amortized — just one pointer update |

Used for congruence closure when axioms conclude `x = y`.

---

## Language Syntax

Not based on any single existing language. Influences:

| Feature | Source |
|---------|--------|
| Postfix application (`x f` not `f(x)`) | Category theory |
| Sequent notation (`premises \|- conclusion`) | Proof theory |
| Path separators (`/` not `.`) | Unique to Geolog |
| Theory/Instance structure | Algebraic specification languages |

---

## Geometric Logic Restrictions

By design, Geolog excludes:
- Negation (no `not R(x)`)
- Classical implication (no `if A then B`)
- Arithmetic (no `x + 1 = y`)

These are intentional limits for topos-theoretic semantics, not missing features.