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An implementation of the Todd-Coxeter Algorithm for Semigroups and Monoids in python3
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#!/usr/bin/env python3 | |
class ToddCoxeter: | |
def __init__(self): | |
self.nodes = [0] | |
self.edges = None | |
self.kappa = [] | |
self.next_node = 1 | |
self.R = [] | |
def set_alphabet(self, A: str) -> None: | |
self.A = A | |
self.edges = [[None] * len(A)] | |
def add_relation(self, u: int, v: int) -> None: | |
u = [self.A.index(x) for x in u] | |
v = [self.A.index(x) for x in v] | |
self.R.append((u, v)) | |
def path(self, c: int, w: list) -> int: | |
w = [w] if not isinstance(w, list) else w | |
for a in w: | |
c = self.edges[c][a] | |
if c is None: | |
return None | |
return c | |
def tc1(self, c: int, a: int) -> int: | |
if self.edges[c][a] is None: | |
self.nodes.append(self.next_node) | |
self.edges[c][a] = self.next_node | |
self.edges.append([None] * len(self.A)) | |
self.next_node += 1 | |
return self.edges[c][a] | |
def tc2(self, c: int, u: list, v: list) -> None: | |
u_1, v_1 = u[:-1], v[:-1] | |
a, b = u[-1] if len(u) > 0 else [], v[-1] if len(v) > 0 else [] | |
if ( | |
self.path(c, u) is None | |
and self.path(c, v) is not None | |
and self.path(c, u_1) is not None | |
): | |
self.edges[self.path(c, u_1)][a] = self.path(c, v) | |
elif ( | |
self.path(c, u) is not None | |
and self.path(c, v) is None | |
and self.path(c, v_1) is not None | |
): | |
self.edges[self.path(c, v_1)][b] = self.path(c, u) | |
elif ( | |
self.path(c, u) is not None | |
and self.path(c, v) is not None | |
and self.path(c, u) != self.path(c, v) | |
): | |
self.kappa.append((self.path(c, u), self.path(c, v))) | |
def tc3(self, i: int, j: int) -> bool: | |
if i == j: | |
return False | |
if i > j: | |
i, j = j, i | |
for a in self.A: | |
if self.path(j, a) is not None: | |
if self.path(i, a) is None: | |
self.edges[i, a] = self.path(j, a) | |
else: | |
self.kappa.append((self.path(i, a), self.path(j, a))) | |
for c in self.nodes: | |
for a in self.A: | |
if self.path(c, a) == j: | |
self.edges[c][a] = i | |
self.kappa = [[i, l] if k == j else [k, l] for k, l in self.kappa] | |
self.kappa = [[k, i] if l == j else [k, l] for k, l in self.kappa] | |
self.nodes.remove(j) | |
return True | |
def felsch(self) -> int: | |
self.A = range(len(self.A)) | |
while True: | |
d, a = next( | |
( | |
(d, a) | |
for d in self.nodes | |
for a in self.A | |
if self.path(d, a) is None | |
), | |
(None, None), | |
) | |
if d is None: | |
return self.size() | |
self.tc1(d, a) | |
for c in self.nodes: | |
for u, v in self.R: | |
self.tc2(c, u, v) | |
while len(self.kappa) != 0: | |
self.tc3(*self.kappa.pop()) | |
def hlt(self) -> int: | |
self.A = range(len(self.A)) | |
seen = set() | |
while True: | |
current = next((c for c in self.nodes if c not in seen), None) | |
if current is None: | |
return self.size() | |
for u, v in self.R: | |
c = current | |
for a in u: | |
c = self.tc1(c, a) | |
c = current | |
for a in v[:-1]: | |
c = self.tc1(c, a) | |
self.tc2(current, u, v) | |
while len(self.kappa) != 0: | |
self.tc3(*self.kappa.pop()) | |
seen.add(current) | |
def size(self) -> int: | |
if any((True for u, v in self.R if len(u) == 0 or len(v) == 0)): | |
return len(self.nodes) | |
else: | |
return len(self.nodes) - 1 | |
C = ToddCoxeter() | |
C.set_alphabet("ab") | |
C.add_relation("aaa", "a") | |
C.add_relation("bbb", "b") | |
C.add_relation("abab", "aa") | |
assert C.felsch() == 14 | |
assert C.hlt() == 14 | |
C = ToddCoxeter() | |
C.set_alphabet("ab") | |
C.add_relation("aaa", "a") | |
C.add_relation("bb", "b") | |
C.add_relation("abab", "aa") | |
assert C.felsch() == 5 | |
assert C.hlt() == 5 | |
C = ToddCoxeter() | |
C.set_alphabet("abc") | |
C.add_relation("ac", "aa") | |
C.add_relation("bb", "b") | |
C.add_relation("ca", "aa") | |
C.add_relation("bc", "cb") | |
C.add_relation("cc", "aa") | |
C.add_relation("aaa", "aa") | |
C.add_relation("aba", "aa") | |
assert C.felsch() == 8 | |
assert C.hlt() == 8 |
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