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Intensional sets in Python

Project Description

Intensional (rule-defined) sets for Python.


There are two ways of defining a set: intensional and extensional. Extensional sets like set([1,3,5,'daisy']) enumerate every member of the set explicitly.

Intensional sets, in contrast, are defined by rules. For example “the set of all prime numbers” or “every word beginning with 'a' and ending with 't'. Intensional sets are often infinite. It may be possible to generate a list of their members, but it’s not as simple as a “give me everything you’ve got!” for loop.

Once you know what you’re looking for, intensional sets are everywhere. Python doesn’t represent them directly, but regular expressions, many list comprehensions, and all manner of testing and filtering operations are really faces of the intensional set concept. Many functions test whether something ‘qualifies’. os.path.isdir(d) for example, tests whether d is in the set of legitimate directories, and isinstance(s, str) tests whether s is a member of the set of str objects. Even the core if conditional can be construed as testing for membership in an intensional set–the set of all items that pass the test.

Many such tests have a temporal aspect–they determine whether a value is a member right now. The answer may change in the future, if conditions change. Others tests are invariant over time. %%734 will never be a valid Python identifier, no matter how many times it’s tested–unless the rules of the overall Python universe change, that is.

Intensional sets are part and parcel of all programming, even if they’re not explicitly represented or called by that name.“intensional“ helps Python programs represent intensional sets directly.


intensional defines several set IntensionalSet subclasses such as Any, Every, ButNot, and EitherOr. These correspond roughly to set operations union, intersection, difference, and symmetric difference (aka xor). Of these, Any is the most useful:

from intensional import *

name = 'Stephen'
if name in Any('Steve', 'Steven', 'Stephen'):
    print 'Good name!'

So far, there’s nothing here you couldn’t do with standard Python set data types. So let’s broaden out to more generic intensional sets:

if name in Test("x.startswith('S')"):
    print "Your name starts with S!"

Test takes a lambda expression or string in its constructor. If it’s a string, Test assumes the interesting variable name is x and compiles the string expression with an automatically provided lambda x: prefix. This makes code a bit terser and cleaner. Now the sets start getting more interesting.:

starts_with_S =  Test("x.startswith('S')")
ends_with_n   =  Test("x.endswith('n')")

if name in Every(starts_with_S, ends_with_n):
    ...  # Stephen and Steven pass, but Steve does not

Of course, this could also be rendered as:

if name in Test("x.startswith('S') and x.endswith('n')"):
    ...  # Stephen and Steven pass, but Steve does not

Or even:

S_something_n = starts_with_S & ends_with_n
if name in S_something_n:

Type Membership

if x in Instances(int):

is identical to:

if isinstance(x, int):

An alias IsInstance exists for Instances for cases where the singular construction is more linguistically natural. A second alias Type is also available.

Set Operations

intensional supports some, but not all, of Python’s classic set operations. There are two primary rules:

  • IntensionalSet attempts to supports all of the collections.Set methods like union() and intersection(). But IntensionalSet objects are immutable, so they do not support self-mutating operations like add(), pop(), and |= defined by collections.MutableSet.
  • Because they are defined by rules rather than explicit lists of members, it is not (in general) possible to determine the cardinality (i.e. len()) of an IntensionalSet, nor to iterate through all of its members, nor to test equality. IntensionalSet objects are used primarily for determining membership.

Because of an implementation detail, IntensionalSet classes are parallel to, but are not true subclasses of, collections.Set.


It’s easy to define new IntensionalSet subclasses that define other kinds of logical tests in generalized, linguistically “clean” ways that make code more readable. As an example, the Instances intensional set is defined like this:

class Instances(with_metaclass(MementoMetaclass, IntensionalSet)):
    An object is in an IsInstance if it is an instance of the given types.
    def __init__(self, *args):
        self.types = tuple(args)

    def __contains__(self, item):
        return isinstance(item, self.types)

__init__() simply remembers what arguments the set is constructed with, while __contains__() implements the test, answering: Does the given item belong in a set constructed with these arguments?

The only complexity here is the with_metaclass(MementoMetaclass, IntensionalSet) phrase, which is simply a compatibility mechanism to be able to define a class in either Python 2 or Python 3 with a given metaclass.

MementoMetaclass is used so that once constructed, a set object is fetched from cache rather than redundantly reconstructed if any subsequent mentions are made. This is a useful performance tweak. For regular expressions, for example, it allows the Re.__init__() set constructor to compile the regular expression just once, even if a program contains many mentions of Re(<some regular exprssion>). Even higher-performance is to assign constructed sets to a name/variable and refer to them via that name. This:

integers = Instances(int)

if x in integers:

requires less work than:

if x in Instances(int):

and is preferred if the test is to be executed frequently. But this pre-naming is just a tweak, and not a requirement.


  • Commenced automated multi-version testing with pytest and tox.
  • Now successfully packaged for, and tested against, all late-model versions of Python: 2.6, 2.7, 3.2, and 3.3 plus one (2.5) that isn’t so very recent, and one (PyPy 1.9, based on Python 2.7.2) that is differently implemented.
  • intensional is just one facet of a larger project to rethink how items are tested for membership and/or chosen from collections. Stay tuned!
  • The author, Jonathan Eunice or @jeunice on Twitter welcomes your comments and suggestions.


To install the latest version:

pip install -U intensional

To easy_install under a specific Python version (3.3 in this example):

python3.3 -m easy_install --upgrade intensional

(You may need to prefix these with “sudo ” to authorize installation. If they’re already installed, the --upgrade flag will be helpful; add it right before the package name.)

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