# Domain Usage #

A domain schema describes the types of objects, as well as relations between these objects, that Penrose diagrams work with. For example, say we want to use Penrose to draw a Venn diagram that represents the relationship between mathematical sets. We can declare the following domain schema:

type Set

predicate Not (Prop p1)
predicate Intersecting (Set s1, Set s2)
predicate IsSubset (Set s1, Set s2)


This schema tells Penrose that the diagrams works with and illustrates Set objects, relations such as Intersecting and IsSubset, and logical relation Not applied upon other statements.

Notably, the domain schema are not instructions of how to draw the Set objects and relations between them. Such instructions of the specific diagram elements are provided by the style schema, not the domain schema. This allows multiple visual representations to be applied to objects from the same domain.

There are four types of statements that can appear in the domain schema.

## Type Declarations #

A type is a class of objects that a diagram works with. In the set-theory example, Set is a type. The syntax for a type declaration is as follows:

type typename


This statement declares a type with name typename. After declaration, a type can be referred to in other parts of the domain schema, as well as the style and substance schemas.

For example, statement type Set declares a type with name Set. We can then use the name Set to later refer to this type.

### Subtypes #

Conceptually, a type might be a subtype of another type. As an example, in a chemistry domain, types Hydrogen and Oxygen might both be subtypes of the type Atom. Penrose allows us to represent subtyping relations as follows:

type subtype_name <: supertype_name


Take the example of types Hydrogen, Oxygen, and Atom:

type Atom
type Hydrogen <: Atom
type Oxygen <: Atom


Subtyping describes "is-a" relationships between types: a Hydrogen "is-a[n]" Atom; an Oxygen "is-a[n]" Atom. Using the "is-a" interpretation, since an Atom "is-a[n]" Atom, inherently, for all type A, we have that A is a subtype of A. This interpretation of subtyping is consistent with "inheritance" in many object-oriented programming languages.

Penrose adopts this interpretation: if Penrose expects to see an object with type A, then it will also accept an object of type B, as long as B is a subtype of A. We say that "B matches A."

## Predicate Declarations #

A predicate represents mathematical or logical statements regarding zero or more objects or other statements. To declare a predicate, we write

predicate predicate_name (argument_list)


where

• predicate_name is the name of the predicate, which can be referred to in the substance and style schemas; and
• argument_list describes the types of objects that this predicate works with. We will go more into its details below.

The arguments in argument_list contains the types of objects that this predicate accepts, separated by commas. Here is a domain schema that contains a couple of type declarations and four different, valid, predicate declarations:

type FirstType, SecondType

predicate P1 (FirstType, SecondType)
predicate P2 (SecondType a1, SecondType a2)
predicate P3 ()
predicate P4 (FirstType, SecondType, SecondType)


Notice how some declarations have names associated with the argument types (e.g. P2 with argument names a1 and a2), and some don't. These names to argument types are allowed and encouraged to enhance readability, but are not required.

### The Prop Type #

A special type, Prop, is built into Penrose, and is available to be used as predicate arguments. The Prop type represents predicates themselves. For example, the predicate Not is declared in the set-theory example:

predicate Not (Prop p1)


When we use the Not predicate in the substance or style schemas, then, we can pass in any predicates as Prop. For example, we can write Not(Intersecting(s1, s2)) in the substance or style schemas to conceptually denote that s1 and s2 are disjoint.

### Symmetric Predicates #

Some relations between objects are symmetric: for example, in the set-theory domain, the relations "a intersects b" and "b intersects a" are equivalent. As a recent new feature, Penrose supports symmetric predicates:

symmetric predicate predicate_name (argument_type, argument_type)


For now, Penrose only supports symmetry of binary predicates (taking exactly two arguments), and, the two arguments must be of the exact same type in the domain schema (not necessarily when the predicate is used in the substance and style schemas).

As an example, consider the following valid domain schema in the chemistry domain:

type Atom
type Hydrogen <: Atom
type Oxygen <: Atom
symmetric predicate Bond (Atom, Atom)


The predicate Bond is declared symmetric. Then, if H is a Hydrogen and O is an Oxygen, then Bond(H, O) and Bond(O, H) are considered equivalent by the Penrose engine. In other words, if Penrose expects to see Bond(H, O) in the substance or style schemas, then it will also accept Bond(O, H) even though the order of arguments are different.

## Function and Constructor Declarations #

In mathematics, we sometimes define functions that takes some inputs and returns some output. Penrose allows us to define functions that behave similarly:

function function_name (argument_list) -> output_type


where

• function_name declares the name of the function, which can be referred to by the substance and style schemas;
• argument_list is a list of the types of inputs that this function accepts, similar to the argument list in predicate declarations; and
• output_type represents the type of the output of the function. When we assign the result of the function to an object in the substance and style schema, the output of the function must match the type of the assigned object (either the types are exactly the same, or the output type is a subtype of the assigned object type).

For example, in the linear-algebra domain, we can define

type Vector
function addVector (Vector v1, Vector v2) -> Vector


Constructors in Penrose are functionally equivalent to functions, with the same declaration syntax, except for the first word:

constructor constructor_name (argument_list) -> output_type


-- this is a comment