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Embedded Languages via Overloading

Suppose you are creating a Python library for building logical circuits. You would like programmers to be able to build circuits that correspond to more complex, high-level functions (e.g., from equality of bit vectors to entire hashing functions). How should you approach this task? What are the right abstractions? How can you minimize the amount of extra effort programmers must put into learning your library and its various interfaces?

Python’s operator overloading features are not just a way to add syntactic convenience to your data structures. By leveraging this feature in the right way, you can greatly reduce the conceptual complexity of a solution by allowing programmers to use the skills they already possess. This article reviews a use case that demonstrates the value of recasting a solution as an embedded language within the Python syntax.

Operator Overloading

Python allows programmers to change the meaning of some of its syntactic constructs when these constructs are applied to objects of programmer-defined classes (the definition of Python’s syntax is discussed in more detail in a previous article). Programmers can do so by defining special methods within a class definition. When the Python interpreter sees a certain operator (such as +) being applied to objects of a class that has such a special method defined (such as __add__), the interpreter will invoke that method.

In the example below, the class vector allows users to build objects that represent two-dimensional vectors. The class definition has a method __eq__.

class vector():
def __init__(self, *coordinates):
self.coordinates = coordinates
def __eq__(self, other):
return self.coordinates == other.coordinates

The method __eq__ is invoked when two vector objects are supplied to the infix== operator.

>>> v = vector(1, 2)
>>> w = vector(3, 4)
>>> (v == w, v == v)
(False, True)

Embedded Language for Boolean Circuits

An embedded language is a specified subset of an existing programming language (which can be called the host language) that serves some distinct purpose. An important benefit of embedded languages is that they reuse the existing features of the host language, which can save both implementation effort and the barrier to entry for users of the embedded language.

Returning to the motivating use case of implementing a library for logical circuits, you might consider that a Python programmer using your library already knows how to use Python to implement a logical circuit. Therefore, any effort the programmer spends learning to use your library is potentially wasted. How might such a programmer implement a circuit they want to evaluate on inputs?

For the purposes of this example, assume that the programmer represents bits using the Python integers 0 and 1. One approach the programmer might use is to define a function that operates on integers and uses Python's built-in logical operators on integer values.

def equal(x, y):
return (x & y) | ((1 - x) & (1 - y))

Below are two approaches a Python programmer might use (the first being an imperative style and the second being a functional style) to implement an equality function equals for bit vectors using the equality function equal for bits that is defined above. You would like your circuit library to accommodate all of these different approaches (because these approaches reflect different trade-offs that might be suitable for different use cases, or simply because programmers might have different preferences or levels of experience with Python).

def equals(xs, ys):
z = 1
for i in range(len(xs)):
z = z & equal(xs[i], ys[i])
return z
def equals(xs, ys):
from functools import reduce
es = [equal(x, y) for (x, y) in zip(xs, ys)]
return reduce((lambda e0, e1: e0 & e1), es)

Under its usual interpretation in Python (i.e., when it is applied to lists of integers), the equals function can be used to determine the equality of bit vectors of equal length.

>>> equals([0,1,1], [0,1,1])

Using overloading, it is possible to provide new definitions for the built-in &, | and - operators that programmers already used to define the equal and equals functions. The circuit class definition below includes the __or__, __and__, and __rsub__ methods that correspond to the way in which these three infix operators are used.

class circuit():
def __init__(self, op='bit', args=None):
self.op = op
self.args = [] if args is None else args
def __or__(self, other):
return circuit('or', [self, other])
def __and__(self, other):
return circuit('and', [self, other])
def __rsub__(self, other):
if not isinstance(other, int) or other != 1:
raise ValueError('can only subtract from 1')
return circuit('not', [self])
def __repr__(self):
return 'circuit("' + self.op + '", ' + str(self.args) + ')'
def __str__(self):
return self.__repr__()

To make it easier to see the circuits after they have been constructed, the definition above also includes a recursive __repr__ method and an equivalent __str__ method. The first method is invoked when Python attempts to display an object of the class (e.g., in an interactive prompt) and the second is invoked when the built-in Python str function is applied to an object of the class circuit.

A programmer can now build a circuit using familiar infix operators. The only difference is that instead of using Python integers to represent the bits that constitute the inputs (i.e., base cases), programmers can use circuit objects that represent bit constants.

c0 = circuit("bit", 0)
c1 = circuit("bit", 1)
c2 = 1 — (c0 & c1)

More importantly, programmers can now reuse without modification the equality functions that have already been defined. However, those same functions now generate entire equality circuits (rather than only computing the result of evaluating those circuits).

>>> equal(c1, c2)
circuit("or", [circuit("and", [circuit("bit", 1), circuit("not", [circuit("and", [circuit("bit", 0), circuit("bit", 1)])])]), circuit("and", [circuit("not", [circuit("bit", 1)]), circuit("not", [circuit("not", [circuit("and", [circuit("bit", 0), circuit("bit", 1)])])])])])

Even the more complex equals method works seamlessly whether it is given lists of bit vectors or lists of base case circuits.

>>> equals([c0, c1], [c0, c1])
circuit("and", [circuit("or", [circuit("and", [circuit("bit", 0), circuit("bit", 0)]), circuit("and", [circuit("not", [circuit("bit", 0)]), circuit("not", [circuit("bit", 0)])])]), circuit("or", [circuit("and", [circuit("bit", 1), circuit("bit", 1)]), circuit("and", [circuit("not", [circuit("bit", 1)]), circuit("not", [circuit("bit", 1)])])])])

Portable or Specialized Embeddings via Inheritance

The advantage of the definition of circuit in the section above is that programmers can build new circuits using existing built-in Python operators. However, in order to inspect or traverse the circuit data structure, programmers must become familiar with your class definition. Using inheritance, you can address this second obstacle.

The variant of the circuit class below represents circuits as dictionaries (or, more precisely, as objects of classes that inherit all the features of dictionaries).

class circuit(dict):
def __or__(self, other):
return circuit({'or': [self, other]})
def __and__(self, other):
return circuit({'and': [self, other]})
def __rsub__(self, other):
if not isinstance(other, int) or other != 1:
raise ValueError('can only subtract from 1')
return circuit({'not': [self]})

Because circuits built by programmers are now also dictionaries, it is easier to use existing built-in libraries to perform common operations. For example, a formatted JSON string representation of a circuit can be emitted using the json library. In the example below, a JSON representation of the bit vector equality circuit is constructed.

>>> c0 = circuit({"bit": 0})
>>> c1 = circuit({"bit": 1})
>>> import json
>>> print(json.dumps(equals([c0, c1], [c0, c1])))
{"and": [{"or": [{"and": [{"bit": 0}, {"bit": 0}]}, {"and": [{"not": [{"bit": 0}]}, {"not": [{"bit": 0}]}]}]}, {"or": [{"and": [{"bit": 1}, {"bit": 1}]}, {"and": [{"not": [{"bit": 1}]}, {"not": [{"bit": 1}]}]}]}]}

For another example of this approach that is more specialized, consider the circuit class below that is derived from the Graph class in the NetworkX library. In this implementation, each circuit object is a graph in which nodes correspond to base cases or operations and edges connect operation nodes with their operands. The method corresponding to a binary operator such as & performs the following steps:

  • it accepts two input graphs (each representing an entire circuit, with the output node as the first entry in its node list),
  • it combines these two graphs into a new graph in which each node’s name is extended to reflect the path to that node,
  • it adds a new node corresponding to the binary operation, and
  • it adds edges between this new output node and the root nodes of the two input graphs.

Note that in order to have mathematical notation within the node labels in the final rendering of the circuit, the LaTeX notation is used for the operator name strings.

from networkx import Graph, unionclass circuit(Graph):
def __init__(self, arg=None):
if arg is not None:
def __or__(self, other):
c = union(
union(self, other, rename=("l-", "r-"))
("$\\vee$", "l-" + list(self.nodes)[0]),
("$\\vee$", "r-" + list(other.nodes)[0])
return c
def __and__(self, other):
c = union(
union(self, other, rename=("l-", "r-"))
("$\\wedge$", "l-" + list(self.nodes)[0]),
("$\\wedge$", "r-" + list(other.nodes)[0])
return c
def __rsub__(self, other):
if not isinstance(other, int) or other != 1:
raise ValueError('can only subtract from 1')
c = union(circuit("$\\neg$"), self, rename=("", "n-"))
c.add_edges_from([("$\\neg$", "n-"+list(self.nodes)[0])])
return c

This approach makes it possible to render a graph diagram directly from a circuit object using the NetworkX library. To demonstrate how a programmer can accomplish this, consider the following circuit c built using the original equals method (note that it has been unmodified and yet works with objects of the newest variant of circuit).

c = equals(
[circuit('$x_0$'), circuit('$x_1$')],
[circuit('$y_0$'), circuit('$y_1$')]

Because c inherits all the features of a NetworkX Graph object, a programmer can inspect its nodes and edges. In the example below, the last three nodes and the number of edges in the circuit graph are extracted. Notice that each node name is a string that describes the path to that node from the circuit's overall output node (with 'l' and 'r' representing left-hand and right-hand subgraphs, respectively).

>>> (list(c.nodes)[-3:], len(c.edges))
(['r-r-l-n-$x_1$', 'r-r-r-$\\neg$', 'r-r-r-n-$y_1$'], 18)

The following function is convenient for rendering the graph. It takes a node name string, splits it into its constituent parts (representing the path from the output node), and calculates a horizontal and vertical coordinate for that node.

def position(n):
# Generate coordinates based on the path
# encoded in the node name.

x = sum(
{'l':-1, 'r':1}.get(c, 0)/(2**(i+3))
for (i, c) in enumerate(n.split('-'))
y = n.count('-')/8
return (x, y)

With the help of the position function, rendering the circuit c as a graph is straightforward using the draw_networkx function. The positions for the rendered nodes are specified by building a dictionary using position and the labels for the rendered nodes are obtained by taking the last component of the node names in c.

from networkx import draw_networkx
pos=dict((n, position(n)) for n in c.nodes),
labels=dict((n, n.split('-')[-1]) for n in c.nodes),
An equality circuit on two input bit vectors (𝑥₀, 𝑥₁) and (𝑦₀, 𝑦₁) that is defined using Python syntax and rendered as a graph using the NetworkX library.

Further Reading

This article presented a brief overview of operator overloading in Python, with a motivating example involving a number of embedded domain-specific languages for defining circuits. A Python library that addresses exactly the same circuit construction use case considered in this article is circuitry, which defines an embedded language for constructing circuits as defined in the circuit library. Other examples of feature-rich and widely used domain-specific languages embedded inside Python include the Django web framework and the pandas data analysis library. However, even small built-in libraries such as typing use operator overloading to provide a more streamlined experience for programmers.

Many other programming languages support some form of operator overloading, as well as the introduction of custom infix, prefix, and even postfix operators. Some languages have both of these features. A more general term for the operator overloading capability is ad hoc polymorphism: the ability to provide multiple definitions for the same syntactic construct (with either the interpreter or compiler determining which definition to use based on the types of the inputs). Other, distinct types of polymorphism exist as well, such as parametric polymorphism: the ability to apply exactly the same function (having the same syntactic definition) to arguments of different types. You might notice that the definitions of equal and equals in this article are actually examples of parametric polymorphism.

This article is also available as a Jupyter Notebook and in HTML form on GitHub.



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