acsets-relational-thinking

SKILL: ACSets Relational Thinking

Version: 2.0.0 Trit: 0 (ERGODIC) Domain: database, category-theory, rewriting Source: Topos Institute RelationalThinking Course + AlgebraicJulia

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Overview

ACSets (Attributed C-Sets) are functors X: C → Set where C is a small category (schema). This skill integrates:

1. RelationalThinking Course - Topos Institute's pedagogical approach
2. DPO Rewriting - Double Pushout graph transformation
3. Self-Play Loop - Query → Execute → Evaluate → Refine
4. GF(3) Conservation - Triadic skill composition

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Core Concept: C-Set as Functor

Schema (Category C)          Instance (Functor X: C → Set)
┌─────────────────┐          ┌─────────────────────────────┐
│  E ──src──→ V   │    X     │  X(E) = {e1, e2, e3}        │
│    ←──tgt──     │   ───→   │  X(V) = {v1, v2}            │
└─────────────────┘          │  X(src): e1↦v1, e2↦v1, e3↦v2│
                             │  X(tgt): e1↦v2, e2↦v2, e3↦v1│
                             └─────────────────────────────┘

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Schema Definition

Basic Graph Schema

using Catlab.CategoricalAlgebra

@present SchGraph(FreeSchema) begin
  V::Ob                    # Vertices (object)
  E::Ob                    # Edges (object)
  src::Hom(E, V)           # Source morphism
  tgt::Hom(E, V)           # Target morphism
end

@acset_type Graph(SchGraph, index=[:src, :tgt])

Entity-Subtype Hierarchy (from RelationalThinking Ch8)

@present SchKitchen(FreeSchema) begin
  Entity::Ob
  
  Food::Ob
  food_in_on::Hom(Food, Entity)
  food_is_entity::Hom(Food, Entity)
  
  Kitchenware::Ob
  ware_in_on::Hom(Kitchenware, Entity)
  ware_is_entity::Hom(Kitchenware, Entity)
  
  BreadLoaf::Ob
  bread_loaf_is_food::Hom(BreadLoaf, Food)
  Knife::Ob
  knife_is_ware::Hom(Knife, Kitchenware)
end

@acset_type Kitchen(SchKitchen)

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DPO Rewriting (Double Pushout)

The Pattern: L ← K → R

     L ←──l── K ──r──→ R
     │        │        │
match│        │        │
     ↓        ↓        ↓
     G ←───── D ──────→ H
       pushout      pushout
       complement