nondeterminism

A Python 3 library for writing nondeterministic algorithms, as in nondeterministic Turing machines.

This library is mostly designed for teaching purposes (providing executable pseudocode for nondeterministic algorithms), and is not suitable for production code (the implementation is based on fork(2), so this code currently only runs on Unix, and is very slow for any nontrivial practical purpose).

Here is, as an exemple, a nondeterministic algorithm for checking if a graph admits a Hamiltonian cycle.

from nondeterminism import nondeterministic, guess


@nondeterministic
def hamiltonian(vertices, edges):
    vertices = vertices.copy()
    n = len(vertices)
    perm = []
    while vertices:
        v = guess(vertices)
        perm.append(v)
        vertices.remove(v)
    for i in range(n):
        if (perm[i], perm[(i + 1) % n]) not in edges:
            return False
    return True


vertices = {1, 2, 3}
edges = {(1, 3), (3, 1), (3, 2), (2, 1)}
result = hamiltonian(vertices, edges)
print('Does the graph have a Hamiltonian cycle?', result)

Each function using nondeterminism (i.e., calling guess()) must be decorated with the @nondeterministic decorator.

The function guess() takes as an optional argument an iterable object, such as a list or a set, and defaults to returning either True or False, as shown by the following example (solving SAT over three variables):

from nondeterminism import nondeterministic, guess


@nondeterministic
def satisfiable(formula):
    x = guess()
    y = guess()
    z = guess()
    return formula(x, y, z)


def formula(x, y, z):
    return ((x or y) and
            (y or z) and
            (not x or z))


result = satisfiable(formula)
print('Is the formula satisfiable?', result)

Notice that only nondeterministic functions need to be decorated with @nondeterministic, as shown by the following code for checking the primality of natural numbers:

from nondeterminism import nondeterministic, guess


@nondeterministic
def composite(n):
    if n < 3:
        return False
    d = guess(range(2, n))
    return n % d == 0


def prime(n):
    return n > 1 and not composite(n)


maximum = 100
print(f'The primes below {maximum} are:')
for n in range(maximum):
    if prime(n):
        print(n)

You can also return non-Boolean values. In that case, the first result different from False and None is returned. Here we search for a subset of the values having sum target, and return the subset itself rather than just True if it exists:

from nondeterminism import nondeterministic, guess


@nondeterministic
def subset_sum(values, target):
    subset = set()
    for item in values:
        if guess():
            subset.add(item)
    if sum(subset) == target:
        return subset


values = {1, 4, 5}
for target in range(sum(values) + 1):
    result = subset_sum(values, target)
    print(f'A subset of {values} having sum {target} is: {result}')