coxeter_automaton/coxeter_automaton.py

import numpy as np
import math
from copy import copy
from collections import deque

class Root:
	def __init__(self, id, rank, depth = 0, v = None, neighbors = None):
		self.id = id
		self.rank = rank
		self.depth = depth
		if v:
			self.v = v
		else:
			self.v = [0] * rank
		if neighbors:
			self.neighbors = neighbors
		else:
			self.neighbors = [None] * rank

	def __copy__(self):
		return Root(self.id, self.rank, self.depth, self.v.copy(), self.neighbors.copy())

class Groupelement:
	def __init__(self, id, rank, word):
		self.id = id
		self.rank = rank
		self.word = word
		self.length = len(word)
		self.left = [None]*rank
		self.right = [None]*rank
		self.node = None
		self.lex_node = None
		self.inverse = None

# compute <alpha_k, beta> where alpha_k is one of the simple roots and beta any root
def form_gen_root(form, k, root):
	rank = len(form)
	return sum([root[i] * form[i][k] for i in range(rank)])

# compute beta - 2<alpha_k, beta>alpha_k, i.e. the reflection of beta along alpha_k
def apply_gen_to_root(form, k, root):
	root[k] -= 2*form_gen_root(form, k, root)

# find a sequence of generators to apply to obtain a negative root, from left to right
# "startwith" argument can be used to force the first entry
def find_word_to_negative(form, root_, startwith = None):
	rank = len(form)
	root = root_.copy()
	word = []
	while not next(filter(lambda x: x < -1e-6, root), None): # while root has no negative entry
		for k in range(rank):
			if startwith and k != startwith:
				continue
			# avoiding 0 might be a problem for reducible groups?
			f = form_gen_root(form, k, root)
			if f > 1e-6:
				apply_gen_to_root(form, k, root)
				word.append(k)
				break
		startwith = None
	return word

# use find_word_to_negative() to find the root, if we already have it
def find_root_from_vector(form, roots, vector):
	rank = len(form)
	for k in range(rank):
		word = find_word_to_negative(form, vector, startwith = k)

		if not word:
			continue

		rootobj = roots[word.pop()]

		while len(word) > 0:
			letter = word.pop()
			if not rootobj.neighbors[letter]:
				rootobj = None
				break
			else:
				rootobj = rootobj.neighbors[letter]

		if rootobj:
			return rootobj
	return None

def find_small_roots(form):
	rank = len(form)
	small_roots = []

	# the simple roots are just the standard basis vectors
	for i in range(rank):
		r = Root(i, rank)
		r.v[i] = 1
		r.depth = 1
		small_roots.append(r)

	# find the other small roots by applying generators to all existing roots
	# and using find_root_from_vector() to see if we already have it
	# then add it if it is a small root = was obtained via a short edge (form between -1 and 0)
	i = 0
	while i < len(small_roots):
		root = small_roots[i]
		for k in range(rank):
			newroot = root.v.copy()
			apply_gen_to_root(form, k, newroot)

			rootobj = find_root_from_vector(form, small_roots, newroot)

			if rootobj:
				root.neighbors[k] = rootobj
			else:
				f = form_gen_root(form, k, root.v)
				if f > -1 + 1e-6 and f < -1e-6:      # root is new and is a small root
					rootobj = Root(len(small_roots), rank, root.depth+1, newroot)
					small_roots.append(rootobj)
					root.neighbors[k] = rootobj
		i = i+1
	return small_roots


def apply_gen_to_node(small_roots, k, node, position, lex_reduced = False):
	# if we want to get the lex reduced langauge
	if lex_reduced:
		for j in range(k):
			if small_roots[j].neighbors[k] and position == small_roots[j].neighbors[k].id:
				return 1

	if position == k:
		return 1
	elif small_roots[position].neighbors[k]:
		swappos = small_roots[position].neighbors[k].id
		return node[swappos]
	else:
		return 0

def generate_automaton(small_roots, lex_reduced = False):
	nroots = len(small_roots)
	rank = small_roots[0].rank
	start = tuple([0]*nroots)
	todo = deque([start])
	nodes = {start: 0}
	levels = {start: 0}
	edges = []
	id = 1

	while todo:
		node = todo.pop()
		for k in range(rank):
			if node[k] == 1:
				continue
			newnode = tuple(
				apply_gen_to_node(small_roots, k, node, i, lex_reduced = lex_reduced)
				for i in range(nroots))
			if not newnode in nodes:
				nodes[newnode] = id
				levels[newnode] = levels[node]+1
				todo.appendleft(newnode)
				id += 1
			edges.append((nodes[node], nodes[newnode], k))

	graph = [[None for i in range(rank)] for j in range(len(nodes))]
	for (fr,to,gen) in edges:
		graph[fr][gen] = to
	return graph


def enumerate_group(graph, graph_lex, max_len):
	rank = len(graph[0])
	group = [Groupelement(0, rank, tuple())]
	group[0].inverse = group[0]
	group[0].node = group[0].lex_node = 0

	i = 0
	size = 1
	while True:
		current = group[i]
		i+=1

		# break if current has the max length we have, as that's when we would start adding elements 1 longer
		if current.length >= max_len:
			break

		for gen, new_lex_node in filter(lambda x: x[1], enumerate(graph_lex[current.lex_node])):
			new_element = Groupelement(size, rank, current.word + (gen,))
			new_element.lex_node = new_lex_node
			new_element.node = graph[current.node][gen]
			group.append(new_element)
			size += 1

			# w = w_1 t, w s = w_1
			# right multiplication, if it decreases length
			for k in range(rank):
				if not graph[new_element.node][k]:
					word = list(new_element.word)
					longer_suffix = group[0]
					while len(word) > 0:
						letter = word.pop()
						shorter_suffix = longer_suffix
						longer_suffix = shorter_suffix.left[letter]

						# w = w_1 t w_2, w_2 s_k = t w_2
						# in the case word = [] longer_suffix could be None
						if len(word) == 0 or shorter_suffix.right[k] == longer_suffix:
							# finish word
							while len(word) > 0:
								shorter_suffix = shorter_suffix.left[word.pop()]
							new_element.right[k] = shorter_suffix
							shorter_suffix.right[k] = new_element

			# find inverse and left multiply
			inverse = group[0]
			word = list(new_element.word)
			while len(word) > 0:
				inverse = inverse.right[word.pop()]
				if not inverse:
					break
			if inverse:
				new_element.inverse = inverse
				inverse.inverse = new_element
				for k in range(rank):
					if inverse.right[k]:
						other = inverse.right[k].inverse
						new_element.left[k] = other
						other.left[k] = new_element
					if new_element.right[k]:
						other = new_element.right[k].inverse
						inverse.left[k] = other
						other.left[k] = inverse
	return group

def word(w):
	return ''.join([chr(ord('a')+x) for x in w])

def generate_automaton_coxeter_matrix(coxeter_matrix, lex_reduced = False):
        form = [[-math.cos(math.pi/m) if m > 0 else -1 for m in row] for row in coxeter_matrix]
        rank = len(coxeter_matrix)
        small_roots = find_small_roots(form)
        return generate_automaton(small_roots, lex_reduced)

def even_graph(graph):
	rank = len(graph[0])
	result = []
	for node in graph:
		newnode = {}
		for i in range(rank):
			for j in range(rank):
				if node[i] and graph[node[i]][j]:
					newnode[(i,j)] = graph[node[i]][j]
		result.append(newnode)
	return result