A toy regex engine based on backtracking. It is probably buggy. The first variant uses recursion and parsing with derivatives. Due to Python's lack of tail recursion it can only be used for simple regexes. The second variant avoids recursion and makes the backtracking explicit through a stack. It can be extended with more operators.

toy_regex_engine.py

#!/usr/bin/env python3
"""
A toy regex engine.
Copyright 2018 Lukas Atkinson

Permission is hereby granted, free of charge, to any person obtaining
a copy of this software and associated documentation files (the "Software"),
to deal in the Software without restriction, including without limitation
the rights to use, copy, modify, merge, publish, distribute, sublicense,
and/or sell copies of the Software, and to permit persons to whom the
Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included
in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM,
DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE
OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
"""


def Sequence(*steps):
    steps = list(steps)
    if not steps:
        return ''
    seq = steps.pop()
    while len(steps):
        seq = _Cons(steps.pop(), seq)
    return seq


class _Cons(object):
    def __init__(self, left, right):
        self.left = left
        self.right = right

    def __str__(self):
        return str(self.left) + str(self.right)


def Choice(*choices):
    choices = list(choices)
    if not choices:
        return ''
    choice = choices.pop()
    while len(choices):
        choice = _Alt(choices.pop(), choice)
    return choice


class _Alt(object):
    def __init__(self, left, right):
        self.left = left
        self.right = right

    def __str__(self):
        return '({}|{})'.format(self.left, self.right)


class Star(object):
    def __init__(self, pattern):
        self.pattern = pattern

    def __str__(self):
        return '({})*'.format(self.pattern)


class Maybe(object):
    def __init__(self, pattern):
        self.pattern = pattern

    def __str__(self):
        return '({})?'.format(self.pattern)


def match(regex, string, cont=None, depth=0):
    """Match a regex AST against a string.

    Returns True if the regex matches at the beginning."""

    depth = depth + 1
    print("{}match state: {!r} against {}, continue {}".format(
          ' ' * (depth - 1), string, regex, cont))

    if regex is None:
        return True

    if isinstance(regex, str):
        if string.startswith(regex):
            return match(cont, string[len(regex):], depth=depth)
        return False

    if isinstance(regex, _Cons):
        return match(regex.left, string,
                     cont=(_Cons(regex.right, cont) if cont else regex.right),
                     depth=depth)

    if isinstance(regex, _Alt):
        if match(regex.left, string, cont=cont, depth=depth):
            return True
        return match(regex.right, string, cont=cont, depth=depth)

    if isinstance(regex, Star):
        if match(regex.pattern, string, cont=_Cons(regex, cont), depth=depth):
            return True
        return match(cont, string, depth=depth)

    if isinstance(regex, Maybe):
        if match(regex.pattern, string, cont=cont, depth=depth):
            return True
        return match(cont, string, depth=depth)

    assert False, "unknown regex operator"


if __name__ == "__main__":
    def check(regex, string, fail=False):
        result = match(regex, string)
        print(string, result)
        if fail:
            assert not result
        else:
            assert result

    # (a|b)*c
    regex = Sequence(Star(Choice('a', 'b')), 'c')
    check(regex, 'abababaca')
    check(regex, 'abababa', fail=True)
    check(regex, 'cab')
    check(regex, 'ababada', fail=True)

    regex = Sequence(Star('a'), Star('a'), Star('a'), 'b')
    check(regex, 'aaaac', fail=True)

toy_regex_engine_without_recursion.py

#!/usr/bin/env python3
"""
A toy regex engine.
Copyright 2018 Lukas Atkinson

Permission is hereby granted, free of charge, to any person obtaining
a copy of this software and associated documentation files (the "Software"),
to deal in the Software without restriction, including without limitation
the rights to use, copy, modify, merge, publish, distribute, sublicense,
and/or sell copies of the Software, and to permit persons to whom the
Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included
in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES
OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM,
DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE
OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
"""


class Sequence(object):
    def __init__(self, *patterns):
        self.patterns = patterns

    def __str__(self):
        return ''.join(str(pattern) for pattern in self.patterns)


def Choice(*choices):
    choices = list(choices)
    if not choices:
        return Sequence()
    choice = choices.pop()
    while len(choices):
        choice = _Alt(choices.pop(), choice)
    return choice


class _Alt(object):
    def __init__(self, left, right):
        self.left = left
        self.right = right

    def __str__(self):
        return '({}|{})'.format(self.left, self.right)


class Star(object):
    def __init__(self, pattern):
        self.pattern = pattern

    def __str__(self):
        return '({})*'.format(self.pattern)


class Maybe(object):
    def __init__(self, pattern):
        self.pattern = pattern

    def __str__(self):
        return '({})?'.format(self.pattern)


class Fail(object):
    def __str__(self):
        return '(*FAIL)'


def match(regex, string):
    op_stack = [regex]
    backtrack_points = []  # contains (op_stack, string) tuples
    # fail if we backtrack into the end of the stack
    backtrack_points.append(([Fail()], string))

    def save_backtrack_point(alt=None):
        saved_op_stack = op_stack[:]
        if alt is not None:
            saved_op_stack.append(alt)
        backtrack_points.append((saved_op_stack, string))

    while op_stack:
        op = op_stack.pop()

        depth = len(backtrack_points)
        print("{}match state: {!r} against {}, continue {}".format(
              ' ' * (depth - 1), string, op, Sequence(*reversed(op_stack))))

        if isinstance(op, str):
            if string.startswith(op):
                string = string[len(op):]
            else:
                # restore the last backtrack point
                op_stack, string = backtrack_points.pop()

        elif isinstance(op, Sequence):
            op_stack.extend(reversed(op.patterns))

        elif isinstance(op, _Alt):
            save_backtrack_point(op.right)
            op_stack.append(op.left)

        elif isinstance(op, Star):
            save_backtrack_point()
            op_stack.append(op)
            op_stack.append(op.pattern)

        elif isinstance(op, Maybe):
            save_backtrack_point()
            op_stack.append(op.pattern)

        elif isinstance(op, Fail):
            return False

        else:
            assert False, "unknown regex operator"

    return True


if __name__ == "__main__":
    def check(regex, string, fail=False):
        result = match(regex, string)
        print(string, result)
        if fail:
            assert not result
        else:
            assert result

    # (a|b)*c
    regex = Sequence(Star(Choice('a', 'b')), 'c')
    check(regex, 'abababaca')
    check(regex, 'abababa', fail=True)
    check(regex, 'cab')
    check(regex, 'ababada', fail=True)

    regex = Sequence(Star('a'), Star('a'), Star('a'), 'b')
    check(regex, 'aaaac', fail=True)