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

Axioms — rules to apply
Structure — carriers (element sets), functions, relations
Bindings — variable-to-element mappings
Tensor system — finds violations (premises true, conclusion false)
Fire conclusion — adds relations, defines functions, creates elements
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.

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