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KnuthBendix.py

"""
    An implementation of the Knuth-Bendix algorithm,
    as described in (1), p. 143.
    For determining if two paths in a category are equal.

The algorithm as given here,
takes a set of equations in the form of a sequence:

E = [(a, b), (c, d) ...]

where a, b, c, d are 'paths'.

Paths are given as strings, for example:

E = [ ('fhk', 'gh'), ('m', 'kkm') ]

means that the path 'fhk' equals 'gh' and 'm' equals 'kkm'.

Each arrow in the path is here a single character.  If longer arrow
names are required, a delimiter string can be specified as in:

kb(E, delim='.')

The paths must then be given by the delimiter between each arrow;

E = [ ('h_arrow.g_arrow', 'g_arrow.k_arrow') ... ]

The function kb(E) returns an object, say A, which is

o callable: A(a, b)->boolean determines if two paths given by a, b are equal.
o has a method A.reduce(a)->pathstring, which reduces a path to normal form.

An optional parameter to kb, max_iterations, determines the maximum
number of iterations the algorithm should try making the reduction
system 'confluent'. The algorithm is not guaranteed to terminate
with a confluent system in a finite number of iterations, so if the
number of iterations needed exceeds max_iterations an exception
(ValueError) will be raised. The default is 100.

References

(1)
@book{walters91categories,
    title={Categories and Computer Science},
    author={R. F. C. Walters},
    publisher={Cambridge University Press},
    location={Cambridge},
    year=1991}

(2)
@book{grimaldi94discrete,
author="Ralph P. Grimaldi".
title="Discrete and Combinatorial Mathematics: An Applied Introduction",
publisher="Addison-Wesley",
location="Readin, Massachusetts",
year=1994
}

"""

import functools

class KnuthBendix:
    def __init__(self, E, delim='', max_iterations=100):
        self.reductions = []
        self.delim = delim
        for a, b in E:
            if delim:
                a = self.wrap_delim(a)
                b = self.wrap_delim(b)
            if self.gt(b, a):
                a, b = b, a
            self.reductions.append((a, b))
        self.make_confluent(max_iterations)
        self.sort()

def __call__(self, x, y):
        return self.reduce(x) == self.reduce(y)

def gt(self, a, b):
        delim = self.delim
        if delim:
            la = len(a)
            lb = len(b)
        else:
            la = a.count(delim)
            lb = b.count(delim)
        if la > lb:
            return 1
        if la < lb:
            return 0
        return a > b

def make_confluent(self, max_iterations):
        def add_reduction(p, q):
            if p != q:
                if self.gt(p, q):
                    self.reductions.append((p, q))
                else:
                    self.reductions.append((q, p))
                self.confluent = 0

reds_tested = {}
        for i in range(max_iterations):
            self.confluent = 1
            reds = list(self.reductions)
            for u1, v1 in reds:
                for u2, v2 in reds:
                    red = (u1, u2, u2, v2)
                    if red in reds_tested:
                        continue
                    reds_tested[red] = 1
                    if u2 in u1:
                        p = self.freduce(v1)
                        i = u1.index(u2)
                        while i >= 0:
                            uuu = u1[:i]+v2+u1[i+len(u2):]
                            q = self.freduce(uuu)
                            add_reduction(p, q)
                            i = u1.find(u2, i+1)
                    lu1 = len(u1)
                    for i in range(1, lu1-len(self.delim)):
                        if u2[:lu1-i] == u1[i:]:
                            p = self.freduce(v1 + u2[lu1-i:])
                            q = self.freduce(u1[:i] + v2)
                            add_reduction(p, q)

assert ('', '') not in reds
            # Remove redundant reductions
            newr = []
            nullred = (self.delim, self.delim)
            for i, uv in enumerate(self.reductions):
                u, v = uv
                self.reductions[i] = nullred
                ru = self.freduce(u)
                rv = self.freduce(v)
                if ru != v and ru != rv:
                    urv = (u, rv)
                    newr.append(urv)
                    self.reductions[i] = urv
                else:
                    pass
            if len(newr) != self.reductions:
                assert ('', '') not in newr
                self.reductions = newr
            assert ('', '') not in self.reductions
            if self.confluent:
                break

else:
            raise ValueError("""\
KnuthBendix.make_confluent did not terminate in %d iterations.
Check your equations or specify an higher max_iterations value.'
""" % max_iterations)

def freduce(self, p):
        # This (internal) variant of reduce:
        # Uses the internal representaion:
        # Assumes p is .surrounded. by the delimiter
        # and returns the reduced value .surrounded. by it.
        # This is primarily for internal use by make_confluent

while 1:
            q = p
            for uv in self.reductions:
                p = p.replace(*uv)
            if q == p:
                break
        return p

def reduce(self, p):
        # This (external) variant of reduce:
        # will add delim if not .surrounded. by delim
        # but the return value will not be surrounded by it.

if self.delim:
            p = self.wrap_delim(p)
        p = self.freduce(p)
        if self.delim:
            p = p.strip(self.delim)
        return p

def sort(self, reds=None):
        if reds is None:
            reds = self.reductions

def cmp(xxx_todo_changeme, xxx_todo_changeme1):
            (x, _) = xxx_todo_changeme
            (y, __) = xxx_todo_changeme1
            if self.gt(x, y):
                return 1
            if x == y:
                return 0
            return -1
        reds.sort(key=functools.cmp_to_key(cmp))

def pp(self):
        printreds(self.reductions)

def wrap_delim(self, p):
        if not p.startswith(self.delim):
            p = self.delim + p
        if not p.endswith(self.delim):
            p = p + self.delim
        return p

def printreds(reds):
    for i, uv in enumerate(reds):
        print('%s\t' % (uv,), end=' ')
        if (i + 1) % 4 == 0:
            print()
    if (i + 1) % 4 != 0:
        print()

def kb(E, *a, **k):
    return KnuthBendix(E, *a, **k)

class _GLUECLAMP_:
    pass

def test2():
    #
    # The group of complex numbers {1, -1, i, -i} under multiplication;
    # generators and table from Example 16.13 in (2).

G = ['1', '-1', 'i', '-i']
    E = [('1.i',         'i'),
         ('i.i',         '-1'),
         ('i.i.i',       '-i'),
         ('i.i.i.i',     '1'),
         ]
    R = kb(E, delim='.')
    T = [['.']+G] + [[y]+[R.reduce('%s.%s' % (y, x)) for x in G] for y in G]

assert T == [
        ['.', '1', '-1', 'i', '-i'],
        ['1', '1', '-1', 'i', '-i'],
        ['-1', '-1', '1', '-i', 'i'],
        ['i', 'i', '-i', '-1', '1'],
        ['-i', '-i', 'i', '1', '-1']]

return R

def test():
    E = [('.a.', '.b.')]
    a = kb(E, delim='.')
    assert a('.a.', '.b.')
    E = [('fhk', 'gh'), ('m', 'kkm')]
    a = kb(E)
    p = a.reduce('fffghkkkm')
    q = a.reduce('ffghkm')
    assert p == 'ffffhm'
    assert q == 'fffhm'
    assert not a(p, q)

E = [('.a.', '.b.')]
    a = kb(E, delim='.')
    p = a.reduce('aa')
    assert p == 'aa'
    p = a.reduce('.bb.')
    assert p == 'bb'
    p = a.reduce('b')
    assert p == 'a'

E = [('.f.h.k.', '.g.h.'), ('.m.', '.k.k.m.')]
    a = kb(E, delim='.')
    p = a.reduce('.f.f.f.g.h.k.k.k.m.')
    q = a.reduce('.f.f.g.h.k.m.')
    assert p, q == ('.f.f.f.f.h.m.', '.f.f.f.h.m.')
    assert p == 'f.f.f.f.h.m'
    assert q == 'f.f.f.h.m'

E = [('.f.ff.fff.', '.ffff.ff.'), ('.fffff.', '.fff.fff.fffff.')]

a = kb(E, delim='.')

p = a.reduce('.f.f.f.ffff.ff.fff.fff.fff.fffff.')
    q = a.reduce('.f.f.ffff.ff.fff.fffff.')

assert p == 'f.f.f.f.ff.fffff'
    assert q == 'f.f.f.ff.fffff'

def test3():
    # From 9.3 in 251
    E = [('Hcc', 'H'),
         ('aab', 'ba'),
         ('aac', 'ca'),
         ('cccb', 'abc'),
         ('caca', 'b')]

a = kb(E)

canon = [
        ('Hb', 'Ha'),    ('Haa', 'Ha'),   ('Hab', 'Ha'),   ('Hca', 'Hac'),
        ('Hcb', 'Hac'),  ('Hcc', 'H'),    ('aab', 'ba'),   ('aac', 'ca'),
        ('abb', 'bb'),   ('abc', 'cb'),   ('acb', 'cb'),   ('baa', 'ba'),
        ('bab', 'bb'),   ('bac', 'cb'),   ('bba', 'bb'),   ('bca', 'cb'),
        ('bcb', 'bbc'),  ('cab', 'cb'),   ('cba', 'cb'),   ('cbb', 'bbc'),
        ('cbc', 'bb'),   ('ccb', 'bb'),   ('Haca', 'Hac'), ('Hacc', 'Ha'),
        ('bbbb', 'bb'),  ('bbbc', 'cb'),  ('bbcc', 'bbb'), ('bcca', 'bb'),
        ('caca', 'b'),   ('ccaa', 'ba'),  ('ccca', 'cb'),  ('cacca', 'cb')
    ]

a.canon = canon

return a

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