Paragraph

A pure Python micro-framework supporting seamless lazy and concurrent evaluation of computation graphs.

Introduction

Paragraph adds the functional programming paradigm to Python in a minimal fashion. One additional class, Variable, and a function decorator, op, is all it takes to turn regular Python code into a computation graph, i.e. a computer representation of a system of equations:

>>> import paragraph as pg
>>> import operator
>>> x, y = pg.Variable("x"), pg.Variable("y")
>>> add = pg.op(operator.add)
>>> s = add.op(x, y)

The few lines above fully instantiate a computation graph, here in its simplest form with just one equation relating x, y and s via the function add. Given values for the input variables x and y, the value of s is resolved as follows:

>>> pg.evaluate([s], {x: 5, y: 10})
[15]

Key features

The main benefits of using paragraph stem from the following features of pg.session.evaluate:

Lazy evaluation

Irrespective of the size of the computation graph, only the operations required to evaluate the output variables are executed. Consider the following extension of the above graph:

>>> z = pg.Variable("z")
>>> t = add.op(y, z)

Then the statement:

>>> pg.evaluate([t], {y: 10, z: 50})
[60]

just ignores the variables s and x altogether, since they do not contribute to the evaluation of t. In particular, the operation add(x, y) is not executed.

Eager currying

Invoking an op with invariable arguments (that is, arguments that are not of type Variable) just returns an invariable value: evaluation is eager whenever possible. If invariable arguments are provided for a subset of the input variables, the computation graph can be simplified using solve, which returns a new variable:

>>> u_x = pg.solve([u], {y: 10, z: 50})[0]

Here, u_x is a different variable from u: it now depends on a single input variable (x), and it knows nothing about a variable y or z, instead storing a reference to the value of their sum t, i.e. 60.

Thus, pg.session.solve acts much as functools.partial, except it simplifies the system of equations where possible by executing dependent operations whose arguments are invariable.

Graph composition

Assume a variable y depends on a number of input variables x_1,…, x_p, and another variable v on u_1,…,``u_q`` (not necessarily different), and v should be identified to x_p. The following statement:

>>> y_v = pg.solve([y], args={x_p: v})[0]

returns a new variable y_v that depends on x_1,…, x_{p-1} as well as on u_1,…, u_q, as if the two computation graphs defining y and v had been pieced together.

Note that the respective input variables may overlap, with the restriction that v should not depend on x_p as that would result in a circular dependency. Also, additional arguments may be added to args in the statement above to set further values of the input variables x_1,…, x_{p-1}. However, values cannot be set for u_1,…, u_q here, since they are not dependencies of y, but of y_v.

Transparent multithreading

Invoking evaluate or solve with an instance of concurrent.ThreadPoolExecutor will allow independent blocks of the computation graph to run in separate threads:

>>> with ThreadPoolExecutor as ex:
...     res = pg.evaluate([z_t], {t: 5}, ex)

This is particularly beneficial if large subsets of the graph are independent.

Constraints

Side-effects

The features listed above come at some price, essentially because the order in which operations are actually executed generally differs from the order of their invocations. For paragraph to guarantee that a variable always evaluates to the same value given the same inputs, as in a system of mathematical equations, it is paramount that operations remain free of side-effects, i.e. they never mutate an object they received as an argument, or store as an attribute. The state sequence of the object would be, by definition, out of the control of the programmer.

There is close to nothing paragraph can do to prevent such a thing happening. When in doubt, make sure to operate on a copy of the argument.

Typing

Variables do not carry any information regarding the type of the value they represent, which precludes binding a method of the underlying value to an instance of Variable: such instructions can appear only within the code of an op. Since binary operators are implemented using special methods in Python, this also precludes such statements as:

>>> s = x + y

for this would be resolved by the Python interpreter into s = x.__add__(y), then s = y.__radd__(x), yet none of these methods is defined by Variable.

For more information please consult the documentation.

Going further

Partial evaluation

When the arguments passed to paragraph.session.evaluate are insufficient to resolve fully an output variable (that is, at least one transitive dependency of the output variable is left uninitialized), the value returned for the output variable is simply another variable. This new variable has in general fewer dependencies, for dependencies fully resolved upon evaluation are replaced by their values.

Note

The ambiguity on the type returned by paragraph.session.evaluate will be lifted in version 2.0. From there on, paragraph.session.evaluate will raise in situations such as described above. The support for partial evaluation will however be continued using the new function paragraph.session.solve.

Mapping over inputs

The function paragraph.session.apply extends paragraph.session.evaluate to take, in addition, an iterator over input arguments to the computation graph. It takes advantage of partial evaluation to reduce the number of operations evaluated at each iteration.

Concurrency

Building upon the guarantees granted by a forward traversal, concurrent execution of ops comes at no additional cost. This feature relies on the concurrent package from the Python standard library: ops are simply submitted to an instance of concurrent.futures.Executor for evaluation. The executor should be provided externally and allow calling concurrent.futures.Future methods from a submitted callable (in particular, this excludes concurrent.futures.ProcessPoolExecutor). The responsibility for shutting down the executor properly lies on the user. In absence of an executor, variables are evaluated in a sequential manner, yet still lazily.

Should an operation be executed in the main process, it can be marked as such by setting the attribute Op.thread_safe to False.

Example usage:

>>> ...graph definition...
>>> with ThreadPoolExecutor() as ex:
...     res = evaluate([output], args={input: input_value}, executor=ex)

Note

Argument values passed to paragraph.session.evaluate can be of type concurrent.futures.Future, in which case the consuming operations will simply block until the result is available.

Note

Similarly, an executor can be passed to the function paragraph.session.apply.

Eager mode

Within the context manager paragraph.session.eager_mode, ops are executed eagerly: the underlying _run method is invoked directly rather than returning an instance of Variable. In this mode, arguments of type Variable are generally not accepted. No concurrent evaluation occurs in eager mode.

This mode is particularly useful when testing or debugging a computation graph without modifying the code defining it, by simply bypassing the machinery set up by the framework.

Backward propagation

Conversely, information can be backward propagated through the computation graph using Requirements. Where applicable, an op can implement the arg_requirements method that resolves the requirement bearing on each of its arguments given this bearing on its ouput. This comes in handy e.g. when a particular time range should be available from the output, while rolling operations (such as sum, average,…) are performed in the graph (or any operation requiring a additional “prefetch” operations from the past).

The arg_requirements method receives the requirements bearing on the output variable and the name of a variable argument of the operation, and returns the requirements that should bear on the said variable argument.

Requirements are substantiated by mixin classes, which add attributes and assume full responsibility for their proper aggregation. They are usually defined in the same module as the operations using them. Then, a compound requirements class is simply defined by:

>>> @attr.s
... class MyRequirements(DateRangeRequirement, DatasetContentsRequirement):
...     pass

A requirement class must define the method merge(self, other) that aggregates requirements (more accurately, the requirement attributes it defines) arising from multiple usages of the same variable. This method should fulfill a small number of properties documented in the base class.

Once all components are in place, requirements can be backpropagated:

>>> reqs = solve_requirements(output=v2, output_requirements=MyRequirements(date_range=ExactRange("2001-01-01", "2001-02-01")))
>>> reqs[v1].date_range  # Holds the backpropagated required date_range

Caveats

Side effects

The order in which variables are evaluated should not be expected to match the order in which they are defined. As a consequence, it is not safe for operations to change variable arguments in place (aka side effects). As Python offers no mechanism to prevent side-effects, it is the responsibility of the user to ensure that copies are returned instead.

For the very same reasons, operations and graphs should be stateless, as their state sequence would otherwise lie outside of the control of the author of a computation graph.

Glossary

variable

Throughout this module, the term _variable_ should be understood in its mathematical sense. A variable can be unbound, and serve as an input placeholder, or bound, and symbolize the result of a certain operation applied to a certain set of arguments, at least one of which is also a variable.

operation

An operation (or simply op) relates variables together.

transitive dependency

A dependency of a variable is any other variable related to it by an operation. The transitive dependencies of a variable are the variables whose values enter its own evaluation, i.e. all variables in the union of its dependencies, their own dependencies, and so on until no more dependency is found. Together with the initial dependent variable, they form the computation graph spanned by the latter.

boundary

A boundary is an arbitrary list of variables whose dependencies are excluded from the transitive dependency. The set of unbound variables is a canonical boundary associated to the transitive dependencies of all its variables. In the context of this module, it essentially allows to prune computation branches whose evaluation is not required.

traversal

An ordering of the variables resulting from following the dependency relationships (the edges) of a computation graph. Dependency relationships can be excluded by setting a boundary to the traversal.

forward traversal

Depth-first traversal of a computation graph, where every dependent variable occurs after all its dependencies. In this order, variables can be evaluated in turn, as the values of their dependencies are resolved before their own resolution occurs.

backward traversal

Breadth-first traversal of a computation graph, where a dependency occurs after all the variables depending on it, directly or transitively. In this order, information can be backward propagated through the graph.