Pizza Ontology

Pizza Ontology in OWL

The Full OWL/Turtle Serialization

Below is the complete pizza ontology in Turtle syntax (a readable OWL serialization), with every construct labeled:

@prefix rdf:  <http://www.w3.org/1999/02/22-rdf-syntax-ns#> .
@prefix rdfs: <http://www.w3.org/2000/01/rdf-schema#> .
@prefix owl:  <http://www.w3.org/2002/07/owl#> .
@prefix xsd:  <http://www.w3.org/2001/XMLSchema#> .
@prefix pizza: <http://example.org/pizza#> .

# ─────────────────────────────────────────────
#  ONTOLOGY DECLARATION
# ─────────────────────────────────────────────
<http://example.org/pizza> a owl:Ontology .

# ─────────────────────────────────────────────
#  TBOX — Terminological Knowledge (Classes & Properties)
# ─────────────────────────────────────────────

# Top-level classes
pizza:Pizza          a owl:Class .
pizza:PizzaBase      a owl:Class .
pizza:PizzaTopping   a owl:Class .

# Topping subclasses (disjoint!)
pizza:CheeseTopping  a owl:Class ; rdfs:subClassOf pizza:PizzaTopping .
pizza:VegTopping     a owl:Class ; rdfs:subClassOf pizza:PizzaTopping .
pizza:MeatTopping    a owl:Class ; rdfs:subClassOf pizza:PizzaTopping .

# More specific toppings
pizza:MozzarellaTopping a owl:Class ; rdfs:subClassOf pizza:CheeseTopping .
pizza:TomatoTopping     a owl:Class ; rdfs:subClassOf pizza:VegTopping .
pizza:PepperoniTopping  a owl:Class ; rdfs:subClassOf pizza:MeatTopping .
pizza:MushroomTopping   a owl:Class ; rdfs:subClassOf pizza:VegTopping .

# Base subclasses
pizza:ThinBase         a owl:Class ; rdfs:subClassOf pizza:PizzaBase .
pizza:ThickBase        a owl:Class ; rdfs:subClassOf pizza:PizzaBase .

# Disjointness — ensures a MeatTopping cannot also be a VegTopping
[] a owl:AllDisjointClasses ;
   owl:members ( pizza:CheeseTopping pizza:VegTopping pizza:MeatTopping ) .

[] a owl:AllDisjointClasses ;
   owl:members ( pizza:ThinBase pizza:ThickBase ) .

# ── Object Properties ──
pizza:hasTopping a owl:ObjectProperty ;
    rdfs:domain pizza:Pizza ;
    rdfs:range  pizza:PizzaTopping .

pizza:hasBase a owl:ObjectProperty ;
    rdfs:domain pizza:Pizza ;
    rdfs:range  pizza:PizzaBase ;
    a owl:FunctionalProperty .         # Each pizza has exactly one base

# ── Named Pizza Types (defined classes) ──
pizza:MargheritaPizza a owl:Class ;
    rdfs:subClassOf pizza:Pizza ;
    rdfs:subClassOf
        [ a owl:Restriction ;
          owl:onProperty pizza:hasTopping ;
          owl:someValuesFrom pizza:TomatoTopping ] ;
    rdfs:subClassOf
        [ a owl:Restriction ;
          owl:onProperty pizza:hasTopping ;
          owl:someValuesFrom pizza:MozzarellaTopping ] .

pizza:VegetarianPizza a owl:Class ;
    owl:equivalentClass
        [ a owl:Class ;
          owl:intersectionOf (
              pizza:Pizza
              [ a owl:Restriction ;
                owl:onProperty pizza:hasTopping ;
                owl:allValuesFrom
                    [ owl:unionOf ( pizza:CheeseTopping pizza:VegTopping ) ] ]
          ) ] .

pizza:PepperoniPizza a owl:Class ;
    rdfs:subClassOf pizza:Pizza ;
    rdfs:subClassOf
        [ a owl:Restriction ;
          owl:onProperty pizza:hasTopping ;
          owl:someValuesFrom pizza:PepperoniTopping ] .

# ─────────────────────────────────────────────
#  ABOX — Assertional Knowledge (Individuals)
# ─────────────────────────────────────────────

pizza:myMargherita a pizza:MargheritaPizza ;
    pizza:hasBase    pizza:myThinBase ;
    pizza:hasTopping pizza:myMozzarella, pizza:myTomato .

pizza:myThinBase   a pizza:ThinBase .
pizza:myMozzarella a pizza:MozzarellaTopping .
pizza:myTomato     a pizza:TomatoTopping .

pizza:studentPizza a pizza:Pizza ;
    pizza:hasBase    pizza:myThickBase ;
    pizza:hasTopping pizza:myMozzarella, pizza:myMushroom .

pizza:myThickBase  a pizza:ThickBase .
pizza:myMushroom   a pizza:MushroomTopping .

OWL Expression Glossary

Here is what each major construct means:12

ExpressionMeaning
a owl:ClassDeclares a named class
rdfs:subClassOfSubsumption: every member of A is also in B
owl:equivalentClassTwo class expressions have the same extension
owl:AllDisjointClassesNo individual can belong to more than one listed class
owl:ObjectPropertyA binary relation between individuals
owl:FunctionalPropertyEach subject has at most one value for this property
rdfs:domain / rdfs:rangeType constraints on a property’s subject/object
owl:RestrictionAnonymous class defined by a property constraint
owl:someValuesFromExistential restriction — ∃ p . C (at least one filler of type C)
owl:allValuesFromUniversal restriction — ∀ p . C (all fillers must be type C)
owl:intersectionOfClass intersection (logical AND)
owl:unionOfClass union (logical OR)

TBox vs ABox

These two components together form the Knowledge Base in Description Logic:34

  • TBox (Terminological Box) — the schema. It contains class declarations, subclass axioms, property definitions, restrictions, and equivalences. Everything above the ABox section in the Turtle above is TBox. Think of it as “what kinds of things exist and how are they related.”
  • ABox (Assertional Box) — the data. It contains named individuals (pizza:myMargherita, pizza:myMozzarella) and assertions about them (their types and property values). It answers “what specific things exist.”

A key insight: reasoning bridges the two. The ABox says pizza:studentPizza hasTopping myMozzarella, myMushroom. The TBox says VegetarianPizza ≡ Pizza ∧ ∀hasTopping.(CheeseTopping ⊔ VegTopping). A reasoner can then infer that studentPizza is a VegetarianPizza — even though you never stated it explicitly.1

Common Reasoning Tasks

OWL reasoners (HermiT, Pellet, Openllet) perform several standard tasks:52

1. Subsumption / Class Hierarchy

“Is MargheritaPizza a subtype of Pizza?” The reasoner checks whether every model satisfying MargheritaPizza also satisfies Pizza. This is used to auto-classify the hierarchy.

2. Instance Classification (ABox Reasoning)

“What type is studentPizza?” Given its toppings are MozzarellaTopping (a CheeseTopping) and MushroomTopping (a VegTopping), the allValuesFrom restriction on VegetarianPizza is satisfied → studentPizza is inferred to be a VegetarianPizza.

3. Consistency Checking

“Can a pizza have a PepperoniTopping AND be a VegetarianPizza?” Since MeatTopping and VegTopping are disjoint, and PepperoniTopping ⊑ MeatTopping, the allValuesFrom restriction on VegetarianPizza would be violated → the ontology becomes inconsistent for such an individual.

4. Realization

“What is the most specific type of myMargherita?” The reasoner finds the most specific named class that myMargherita belongs to: MargheritaPizza.

SPARQL Queries on the Pizza Ontology

These queries run against an Apache Jena Fuseki or similar SPARQL endpoint loaded with the ontology above.61

Q1 — Find all pizzas and their toppings:

PREFIX pizza: <http://example.org/pizza#>

SELECT ?pizza ?topping WHERE {
  ?pizza a pizza:Pizza .
  ?pizza pizza:hasTopping ?topping .
}

Q2 — Find all subclasses of PizzaTopping:

PREFIX pizza: <http://example.org/pizza#>
PREFIX rdfs:  <http://www.w3.org/2000/01/rdf-schema#>

SELECT ?toppingType WHERE {
  ?toppingType rdfs:subClassOf+ pizza:PizzaTopping .
}

The + operator follows the subclass chain transitively, so this returns CheeseTopping, VegTopping, MozzarellaTopping, TomatoTopping, etc.1

Q3 — Find all pizzas that have a TomatoTopping (schema-level):

PREFIX pizza: <http://example.org/pizza#>
PREFIX owl:   <http://www.w3.org/2002/07/owl#>
PREFIX rdfs:  <http://www.w3.org/2000/01/rdf-schema#>

SELECT ?pizzaClass WHERE {
  ?pizzaClass rdfs:subClassOf+ pizza:Pizza .
  ?pizzaClass rdfs:subClassOf [
      a owl:Restriction ;
      owl:onProperty pizza:hasTopping ;
      owl:someValuesFrom pizza:TomatoTopping
  ] .
}

This returns MargheritaPizza at the TBox level.1

Q4 — Inferred vegetarian pizzas (requires reasoner active):

PREFIX pizza: <http://example.org/pizza#>

SELECT ?p WHERE {
  ?p a pizza:VegetarianPizza .
}

Without a reasoner, this returns nothing (no pizza was asserted as VegetarianPizza). With HermiT/Pellet running, it infers and returns pizza:studentPizza automatically — this is the power of OWL reasoning.5

Inductive Logic Programming (ILP) with the Pizza Ontology

ILP learns general rules from specific examples and background knowledge. Given the pizza ontology as background, the goal is to induce rules like “what makes a pizza vegetarian?”7

Background Knowledge (Prolog/Datalog facts from the ABox)

% ABox as Prolog facts
hasTopping(myMargherita, myMozzarella).
hasTopping(myMargherita, myTomato).
hasTopping(studentPizza, myMozzarella).
hasTopping(studentPizza, myMushroom).

% TBox encoded as type hierarchy
mozzarellaTopping(myMozzarella).
tomatoTopping(myTomato).
mushroomTopping(myMushroom).

cheeseTopping(X) :- mozzarellaTopping(X).
vegTopping(X)    :- tomatoTopping(X).
vegTopping(X)    :- mushroomTopping(X).

Positive and Negative Examples

% Positive: these ARE vegetarian
pos(vegetarianPizza(myMargherita)).
pos(vegetarianPizza(studentPizza)).

% Negative: these are NOT vegetarian (a pepperoni pizza)
neg(vegetarianPizza(myPepperoni)).

ILP Learning Task

An ILP system (e.g., FOIL, Aleph, or Popper) is given $B, E^+, E^-$ and must find hypothesis $H$ such that $B \cup H \models E^+$ and $B \cup H \not\models E^-$.7

Induced Rule (what the ILP system learns):

vegetarianPizza(P) :-
    pizza(P),
    hasTopping(P, T),
    \+ meatTopping(T).

This reads: “A pizza is vegetarian if it is a pizza and none of its toppings are meat toppings.” The system generalized this rule from the examples — it was never told the definition, only given instances.87

Why This Connects to the Ontology

The TBox’s owl:equivalentClass definition of VegetarianPizza using allValuesFrom is essentially the same semantics as the induced Prolog rule — ILP re-discovers ontological definitions from data. This makes ILP a powerful complement to ontology learning: you can use ILP to mine rules from ABox data and promote them to TBox axioms.97 10111213141516

⁂

  1. https://csiro-enviro-informatics.github.io/info-engineering/tutorials/tutorial-intro-to-rdf-and-owl.html ↩︎ ↩︎ ↩︎ ↩︎ ↩︎

  2. https://cs.uwaterloo.ca/~a78khan/docs/PizzaOntologyReview.pdf ↩︎ ↩︎

  3. https://www.michaeldebellis.com/post/using-sparql-subgraphs-to-segment-tbox-and-abox ↩︎

  4. https://stackoverflow.com/questions/36276986/get-tbox-axioms-with-owl-api ↩︎

  5. https://www.uio.no/studier/emner/matnat/ifi/IN3060/v20/exercises/owl_solutions.pdf ↩︎ ↩︎

  6. https://stackoverflow.com/questions/45153600/sparql-query-on-pizza-ontology ↩︎

  7. https://en.wikipedia.org/wiki/Inductive_logic_programming ↩︎ ↩︎ ↩︎ ↩︎

  8. https://swi-prolog.discourse.group/t/popper-inductive-logic-programming-ilp-and-my-popper-page/3929 ↩︎

  9. https://www.cs.nmsu.edu/~ipivkina/Spring05cs579/StudentPres/inductiveLP.pdf ↩︎

  10. https://protege.stanford.edu/conference/2009/tutorials.html ↩︎

  11. https://people.cs.vt.edu/~kafura/ComputationalThinking/Class-Notes/Tutorial-Highlighted-Day1.pdf ↩︎

  12. https://www.youtube.com/watch?v=kK5iLNNqZnc ↩︎

  13. https://aispace.org/exercises/exercise13-a-1.shtml ↩︎

  14. https://www.youtube.com/watch?v=VFbVYioceLI ↩︎

  15. https://www.youtube.com/watch?v=TKMW5udKzIM ↩︎

  16. https://owlready2.readthedocs.io/en/latest/sparql.html ↩︎