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Selectors are the most important component in a Style program, since they actually describe how to draw elements of a diagram. The syntax for Selectors is as follows:

forall list_object_declarations
where list_relations
with list_object_declarations {
    ... (Selector Body)


  • list_object_declarations is a semicolon-separated list of object declarations, similar to the object declarations in the Substance program. Each object declaration has syntax type_name object_name. The names declared in list_object_declarations are referred to as style variables.
  • list_relations is a semicolon-separated list of constraints (about objects in list_object_declaration) that must be satisfied in order for this style block to be triggered.

The forall, where, and with clauses form the Selector Block Header. One might observe that both the forall clause and the with clause take in list of object declarations. There are no semantic differences between objects declared in the forall clause and those declared in the with clause. Variables declared in these clauses can be accessed within the Selector Block Body.

The first clause of a Selector Header must be forall; the other clauses where and with can be written in any order. The header does not allow for empty lists - that is, list_object_declarations and list_relations in the where and with clauses must not be empty. If it is desirable to not have any elements in a clause, the entire clause must be omitted.

In the set-theory example, a style block may look like

forall Set x {


forall Set x; Set y
where IsSubset (x, y) {


forall Set x
where IsSubst(x, y)
with Set y {

Matching style block against substance program in general

Penrose functions by matching a Selector Header against a Substance program. In a nutshell, given a style block

forall Set x; Set y
where IsSubset (x, y)

the Penrose compiler searches through the Substance program to find sets of objects consistent with Set x; Set y such that IsSubset(x, y) is satisfied. This is done through generating mappings from style variables to substance variables, which are the objects in the Substance program.

For instance, consider a simple set-theory Substance program that works with the previous style block:

Set A, B, C
IsSubset (A, B)
IsSubset (B, C)

By matching the style block against the Substance program, we essentially consider six possible mappings (note that repeated elements are, by default, disallowed; see next section), some of which are valid and some are invalid:

MappingIsSubset(x, y) becomesSatisfied by Substance
x -> A; y -> BIsSubset(A, B)Yes
x -> A; y -> CIsSubset(A, C)No
x -> B; y -> AIsSubset(B, A)No
x -> B; y -> CIsSubset(B, C)Yes
x -> C; y -> AIsSubset(C, A)No
x -> C; y -> BIsSubset(C, B)No

Here, Penrose filters out mappings which do not satisfy the constraints listed in the Style block, and keeps a list of good mappings (in this example, two mappings are kept). For each good mapping, the body of the Style block (list_body_expressions) is executed, where each instance of the Style variables (x and y) is substituted with the corresponding Substance variables (once with A and B, once with B and C).

Repeatable vs Non-Repeatable Matching

Penrose, by default, performs "non-repeatable" matching. That is, if there are two style variables declared in a Selector Header (in with or where clause), then the two style variables must correspond to different substance variables.

As an example, the header

forall Node a; Node b
where Edge(a, b)

does not generate a valid matching against the substance

Node X
Edge(X, X)

This is a design feature because many times, we would want different visualization for self-edges like this. To override this behavior, one can add the repeatable keyword:

forall repeatable Node a; Node b
where Edge(a, b)

and this header will allow a and b to both map to X.

Object Declarations

In the list of object declarations in a Style block, we can declare two types of objects, which are matched differently by the Penrose compiler.

Substance objects

We can declare a Substance object, whose object name is surrounded by backticks. For instance,

forall Set `A` {

can only be mapped to the Substance object with the exact same name (A) provided that the types match (subtyping allowed). In other words, given Substance program

Set A, B, C

matching the Style block against the Substance block yields only one valid mapping: `A` -> A.

Style objects

If an object name is not surrounded by backticks, then this object is a Style object with a Style variable. As seen before, the Penrose compiler will try to map Style variables to any Substance objects, provided that their types match (subtyping allowed).


A Style block supports three types of relations, two of which can also be seen in the Substance program.

Predicate Applications

Just like in the Substance program, each predicate application has syntax

predicate_name (argument_list)

where elements of argument_list can refer to objects declared in list_object_declarations, or be other predicate applications. The types must still match, allowing subtyping.

Optionally, one can give an alias to a predicate application:

predicate_name (argument_list) as alias_name

If such an alias is set, then alias_name will be accessible in the style block body, and it will always refer to the version of the predicate application within the Substance program.


If a predicate is declared as symmetric, then it gets special treatment. Suppose we have the following Domain schema:

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

and the following style block:

forall Hydrogen h; Oxygen o
where Bond (h, o) {

The style block will successfully match the following substance schema:

Hydrogen H
Oxygen O
Bond (O, H)

where Bond (h, o) in the style block matches against Bond (O, H) in the substance schema. Because Bond is declared symmetric, when Penrose looks for Bond (h, o), it also looks for Bond (o, h) and finds a match. In other words, the matching algorithm handles the equivalence between Bond (h, o) and Bond (o, h) correctly.

Function and Constructor Applications

Each function or constructor application has syntax

object_name := function_name (argument_list)

We do not allow aliasing for function and constructor applications. Arguments in argument_list must have types that match the Domain argument types, similar to the substance schema.

Object Property Relations

Aside from predicate applications and function (constructor) applications, Penrose also supports a predicate-like relation that checks whether an object has a certain property, say label. For instance, we may write

forall Set s
where s has label {
    ... some code that uses s.label

If a certain Set A in the Substance program does not have a label (perhaps due to NoLabel declarations), then s will not be mapped to A, thus preventing an access of nonexistent properties.

We can further distinguish between math labels and text labels (see substance labeling): where p has math label matches math labels, whereas where p has text label matches text labels.

Matching Deduplication

The matching algorithm is designed to avoid duplicated mappings. If two mappings give us the same set of matched objects (in the Substance program) and the equivalent set of matched substance relations (predicate applications and function or constructor applications), then the algorithm only triggers on one of them.

For instance, say Penrose tries to match the Style block

forall Set x; Set y {

against Substance program

Set A, B

Then, only one of mappings x -> A; y -> B and x -> B; y -> A triggers the Style block.

Reserved Variables

Within a Style block body, some variable names are reserved for metadata purposes:

  • match_total is an integer that refers to the number of times that this Style blocks will be triggered (or matched) in total; and
  • match_id is the 1-indexed ordinal of this current matching.

These values can directly be read or overwritten within the style block body if needed.

Released under the MIT License.