Lesson 12.2: Using type-theoretic relations

In the previous lesson, we learned how to map application types onto database types using the type-theoretic framework of the PERA model. In this lesson, we’ll explore how our models change when we represent application classes with type-theoretic relations.

Modeling type hierarchies

In Lesson 12.1, we encountered the following Review type.

class Review:
    def __init__(
        self,
        id: str,
        reviewed: Book,
        reviewer: User,
        timestamp: datetime,
        score: int
    ):
        self.id = id
        self.reviewed = reviewed
        self.reviewer = reviewer
        self.timestamp = timestamp
        self.score = score

Now, we have decided to make it part of a type hierarchy of user actions instead, represented by the following classes.

class UserAction(ABC):
    def __init__(self, user: User, timestamp: datetime):
        self.user = user
        self.timestamp = timestamp


class Order(UserAction):
    def __init__(
        self,
        id: str,
        user: User,
        timestamp: datetime
    ):
        super().__init__(user, timestamp)
        self.id = id
        self.status = Status.PENDING


class Review(UserAction):
    def __init__(
        self, id: str,
        reviewed: Book,
        user: User,
        timestamp: datetime,
        score: int
    ):
        super().__init__(user, timestamp)
        self.id = id
        self.reviewed = reviewed
        self.score = score


class Login(UserAction):
    def __init__(
        self,
        user: User,
        timestamp: datetime,
        success: bool
    ):
        super().__init__(user, timestamp)
        self.success = success

In Python, the ABC class from the abc (abstract base class) module is used to denote abstract classes. A class that is a direct subtype of ABC is abstract.

Representing these classes in TypeQL is straightforward under the type-theoretic framework. The UserAction type is composed of the composite User type, so user-action should be a relation type with a role user-action:user. That role is then inherited by its subtypes. The Review type is also composed of the composite Book type. In the PERA model, this is represented by adding an extra role review:reviewed to review.

define
user-action sub relation,
    abstract,
    relates user,
    owns timestamp;
order sub user-action,
    owns id @key,
    owns status,
    plays order-line:order,
    plays delivery:delivered;
review sub user-action,
    relates reviewed,
    owns id @key,
    owns score,
    owns verified;
login sub user-action,
    owns success;
user plays user-action:user;
book plays review:reviewed;

Exercise

Write TypeQL type definitions to represent the following Python classes under the type-theoretic PERA framework. Make sure to include any plays statements for roles defined.

class Company(ABC):
    def __init__(self, name: str):
        self.name = name


class Publisher(Company):
    def __init__(self, name: str):
        super().__init__(name)


class Courier(Company):
    def __init__(self, name: str):
        super().__init__(name)


class User:
    def __init__(
        self,
        id: str,
        name: str,
        birth_date: datetime,
        location: City
    ):
        self.id = id
        self.name = name
        self.birth_date = birth_date
        self.location = location


class Delivery:
    def __init__(
        self,
        delivered: Order,
        deliverer: Courier,
        destination: Address
    ):
        self.delivered = delivered
        self.deliverer = deliverer
        self.destination = destination

Sample solution

define
company sub entity,
    abstract,
    owns name;
publisher sub company;
courier sub company,
    plays delivery:deliverer;
user sub relation,
    owns id @key,
    owns name,
    owns birth-date,
    relates location;
delivery sub relation,
    relates delivered,
    relates deliverer,
    relates destination;
city plays user:location;
order plays delivery:delivered;
address plays delivery:destination;

Naming interfaces effectively

Next, let’s consider the following Address class.

class Address:
    def __int__(self, street: str, city: City):
        self.street = street
        self.city = city

    @property
    def state(self) -> State:
        return self.city.state

    @property
    def country(self) -> Country:
        return self.city.country

We could represent this in TypeQL in the following manner. We’ll leave out the properties of Address for now, but we could always replicate their behaviour using rules.

define
address sub relation,
    owns street,
    relates city;
city plays address:city;

If we were to implement this as it is, then there would be a disparity between the way that users (defined in the above exercise) and addresses reference cities. The object type city would play the roles user:location and address:city, and there is no way we can polymorphically query these two roles without explicitly listing them. However, if we renamed the address:city role to address:location, then we can query them polymorphically via their shared role name, as we saw in Lesson 7.2, using the following pattern.

$located (location: $city);

This pattern will match instances of both user and address for $located, as long as we use the following definition for address.

define
address sub relation,
    owns street,
    relates location;
city plays address:location;

For this reason, it is even more important to choose interface names carefully when using the type-theoretic framework of the PERA model.

TypeDB 3.0 will include user-defined functions that allow for ad hoc polymorphism. This will enable a new class of powerful polymorphic queries that do not depend on inheritance hierarchies or interface implementations. By making use of ad hoc polymorphism in TypeDB 3.0, interface names could be chosen freely and the desired polymorphic behaviour could still be implemented!

Handling nullary relations

Finally, we will consider the following classes representing places.

class Place(ABC):
    def __init__(self, name: str):
        self.name = name


class Country(Place):
    def __init__(self, name: str):
        super().__init__(name)


class State(Place):
    def __init__(self, name: str, location: Country):
        super().__init__(name)
        self.location = location

    @property
    def country(self) -> Country:
        return self.location


class City(Place):
    def __init__(self, name: str, location: State | Country):
        super().__init__(name)
        self.location = location

    @property
    def state(self) -> Optional[State]:
        if type(self.location) is State:
            return self.location

    @property
    def country(self) -> Country:
        match self.location:
            case State():
                return self.location.country
            case Country():
                return self.location

Here we run into an interesting problem. The types Place and Country are composed only of primitive types, but State and City are also composed of composite types. The PERA model does not permit the mixing of entity and relation types into a single hierarchy. In the type-theoretic framework, we might consider an implementation where some of the relation types have no roles, as shown here.

define
place sub relation,
    abstract,
    owns name;
country sub place,
    plays city:location,
    plays state:location;
state sub place,
    relates location,
    plays city:location;
city sub place,
    relates location;

However, this schema excerpt will not commit to a database. While abstract relation types can have no roles, all concrete relation types must have at least one role. The type country is concrete but does not have any roles, causing this schema to fail validation. Even if we gave country a dummy role, we could not instantiate it, as any relation instances without roleplayers are erased automatically on commit. We have a few options to deal with this:

Make country an entity type instead, and remove the supertype place altogether.
Add a location role to country, and an entity type world to play it. We will instantiate world once with no attributes, and it will act as the placeholder location for all countries.
Change the supertype place to an entity type, and model the network of locations with locating relations instead of location roles. This is the approach we used in the original bookstore schema under the entity-centric framework.

All of these solutions will allow the schema to pass validation, but none of the resulting schemas will completely match the application model. Here we cannot quite achieve parity with the current model, and we must decide whether to allow the mismatch or modify the application model to match the chosen solution in the database. In this particular case, option (2) lets us come the closest to parity, after which we can make minor changes to the application model to match, if complete parity is required.

define
world sub entity,
    plays country:location;
place sub relation,
    abstract,
    owns name;
country sub place,
    relates location,
    plays city:location,
    plays state:location;
state sub place,
    relates location,
    plays city:location;
city sub place,
    relates location;

class World:
    def __init__(self):
        pass


class Place(ABC):
    def __init__(self, name: str):
        self.name = name


class Country(Place):
    def __init__(self, name: str, location: World):
        super().__init__(name)
        self.location = location


class State(Place):
    def __init__(self, name: str, location: Country):
        super().__init__(name)
        self.location = location

    @property
    def country(self) -> Country:
        return self.location


class City(Place):
    def __init__(self, name: str, location: State | Country):
        super().__init__(name)
        self.location = location

    @property
    def state(self) -> Optional[State]:
        if type(self.location) is State:
            return self.location

    @property
    def country(self) -> Country:
        match self.location:
            case State():
                return self.location.country
            case Country():
                return self.location

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