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"""
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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