How to model a RESTful API with inheritance?

But now the relationship between dogs and cats is lost.

Indeed, but keep in mind that URI simply never reflects relations between objects.

I would suggest:

• Using only one URI per resource
• Differentiating between animals solely at the attribute level

Setting up multiple URIs to the same resource is never a good idea because it can cause confusion and unexpected side effects. Given that, your single URI should be based on a generic scheme like /animals.

The next challenge of dealing with the entire collection of dogs and cats at the "base" level is already solved by virtue of the /animals URI approach.

The final challenge of dealing with specialized types like dogs and cats can be easily solved using a combination of query parameters and identification attributes within your media type. For example:

GET /animals (Accept : application/vnd.vet-services.animals+json)

{
   "animals":[
      {
         "link":"/animals/3424",
         "type":"dog",
         "name":"Rex"
      },
      {
         "link":"/animals/7829",
         "type":"cat",
         "name":"Mittens"
      }
   ]
}

• GET /animals - gets all dogs and cats, would return both Rex and Mittens
• GET /animals?type=dog - gets all dogs, would only return Rex
• GET /animals?type=cat - gets all cats, would only Mittens

Then when creating or modifying animals, it would be incumbent on the caller to specify the type of animal involved:

Media Type: application/vnd.vet-services.animal+json

{
   "type":"dog",
   "name":"Fido"
}

The above payload could be sent with a POST or PUT request.

The above scheme gets you the basic similar characteristics as OO inheritance through REST, and with the ability to add further specializations (i.e. more animal types) without major surgery or any changes to your URI scheme.

This question can be better answered with the support of a recent enhancement introduced in the latest version of OpenAPI, v3 at time of writing.

It's been possible to combine schemas using keywords such as oneOf, allOf, anyOf and get a message payload validated since JSON schema v1.0.

https://spacetelescope.github.io/understanding-json-schema/reference/combining.html

However, in OpenAPI (former Swagger), schemas composition has been enhanced by the keywords discriminator (v2.0+) and oneOf (v3.0+) to truly support polymorphism.

https://github.com/OAI/OpenAPI-Specification/blob/master/versions/3.0.0.md#schemaComposition

Your inheritance could be modeled using a combination of oneOf (to choose one of the subtypes) and allOf (to combine the type and one of its subtypes). Below is a sample definition for the POST method.

paths:
  /animals:
    post:
      requestBody:
      content:
        application/json:
          schema:
            oneOf:
              - $ref: '#/components/schemas/Dog'
              - $ref: '#/components/schemas/Cat'
              - $ref: '#/components/schemas/Fish'
            discriminator:
              propertyName: animal_type
     responses:
       '201':
         description: Created

components:
  schemas:
    Animal:
      type: object
      required:
        - animal_type
        - name
      properties:
        animal_type:
          type: string
        name:
          type: string
      discriminator:
        property_name: animal_type
    Dog:
      allOf:
        - $ref: "#/components/schemas/Animal"
        - type: object
          properties:
            playsFetch:
              type: string
    Cat:
      allOf:
        - $ref: "#/components/schemas/Animal"
        - type: object
          properties:
            likesToPurr:
              type: string
    Fish:
      allOf:
        - $ref: "#/components/schemas/Animal"
        - type: object
          properties:
            water-type:
              type: string

I would go for /animals returning a list of both dogs and fishes and what ever else:

<animals>
  <animal type="dog">
    <name>Fido</name>
    <fur-color>White</fur-color>
  </animal>
  <animal type="fish">
    <name>Wanda</name>
    <water-type>Salt</water-type>
  </animal>
</animals>

It should be easy to implement a similar JSON example.

Clients can always rely on the "name" element being there (a common attribute). But depending on the "type" attribute there will be other elements as part of the animal representation.

There is nothing inherently RESTful or unRESTful in returning such a list - REST does not prescribe any specific format for representing data. All it says is that data must have some representation and the format for that representation is identified by the media type (which in HTTP is the Content-Type header).

Think about your use cases - do you need to show a list of mixed animals? Well, then return a list of mixed animal data. Do you need a list of dogs only? Well, make such a list.

Whether you do /animals?type=dog or /dogs is irrelevant with respect to REST which does not prescribe any URL formats - that is left as an implementation detail outside the scope of REST. REST only states that resources should have identifiers - never mind what format.

You should add some hyper media linking to get closer to a RESTful API. For instance by adding references to the animal details:

<animals>
  <animal type="dog" href="/animals/123">
    <name>Fido</name>
    <fur-color>White</fur-color>
  </animal>
  <animal type="fish" href="/animals/321">
    <name>Wanda</name>
    <water-type>Salt</water-type>
  </animal>
</animals>

By adding hyper media linking you reduce client/server coupling - in the above case you take the burden of URL construction away from the client and let the server decide how to construct URLs (which it by definition is the only authority of).