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Syntax and Semantics

Given a domain schema (specifying the domain of the diagram) and a substance program (specifying what to draw), the style schema describes the recipe of drawing the objects and relations on a canvas.

A style schema is composed of blocks, of which there are two types:

  • namespaces, which can be used to help program organization;
  • style blocks, which carry out the meat of the diagram generation.


A namespace is a collection of value assignments. The syntax for its declarations is as follows:

namespace_name {
    field_1 = expr_1
    field_2 = expr_2

Values declared within a namespace can be read outside of the namespace using the "dot" operator:


Hence they are also called global variables. Overwriting these values is not allowed.

Canvas Preamble Block

Each style schema must contain a canvas preamble block, a special type of namespace which describes the width and height of the canvas. For example, preamble block

canvas {
    width = 800
    height = 700

tells Penrose that the drawing canvas should have a width of 800 pixels and a height of 700 pixels.

Style Blocks

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

forall list_object_declarations
where list_relations {


  • list_object_declarations is a semicolon-separated list of object declarations, similar to the object declarations in the substance schema. 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.
  • list_body_expressions are statements that represent the computational and graphical aspects of the diagrams that are triggered when this style block is triggered.

If list_relations is empty, then the clause where ... needs to 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) {

Matching style block against substance program in general

Penrose functions by matching a style block 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 not allowed), 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).

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).

Allowed Relations

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.

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.

Style Block Body

The body of a style block lists out computations, shape declarations, and constraint / objective definitions that define how to draw elements of a diagram.


In style block bodies, we can assign an expression to a field:

field = expression

where field can either be

  • A single identifier, which denotes a local assignment, not accessible outside of this matching; or
  • An object name (defined in list_object_declarations) or predicate application alias, followed by a dot operator and an identifier, which denotes an assignment bound to a substance instance of object or predicate application after we substitute in the mapping. These assignments are accessible if the same substance object or predicate application is matched again.

For example, consider the following style block:

forall MyType t1; MyType t2
where MyPredicate (t1, t2) as r1 {
	x = ... // this is a local assignment
	t1.a = ... // this is bound to the substance instance of `MyType t1`
	r1.c = ... // this is bound to the substance instance of `MyPredicate (t1, t2)`

On the other hand, expression is either a mathematical expression, or a shape definition, both of which can refer to global, local, and other variables bound to substance instance of objects and predicate applications. Shape declarations have syntax

shape_name {
    property_name_1 : value_1
    property_name_2 : value_2

A full list of available shapes and their properties can be found in the Shape Library.

Constraints and Objectives

A good diagram must satisfy some basic constraints, while trying to optimize upon some objectives (specifying diagram beauty). We declare these constraints and objectives within the style blocks. A constraint declaration has syntax

ensure constraint_name (argument_list)

and an objective declaration has syntax

encourage objective_name (argument_list)

where argument_list may refer to constant values, global / local variables, and other variables bound to substnace instances of objects and predicate applications. A full list of available constraints and objectives can be found here.

We also provide syntax sugar expressions for some commonly-used objectives and constraints. In particular,

  • a > b is the syntax sugar for the constraint / objective greaterThan(a, b),
  • a == b is the syntax sugar for the constraint / objective equal(a, b), and
  • a < b is the syntax sugar for the constraint / objective lessThan(a, b).


We can specify the layering between two shapes (particularly useful when two shapes overlap) using layering statements: either

layer shape_1 above shape_2


layer shape_1 below shape_2

where shape_1 and shape_2 can be variables assigned to shapes.

Reserved Variables

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

  • match_count 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, or overwritten, if needed.