Spaces:
Sleeping
Sleeping
File size: 17,095 Bytes
6a86ad5 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 |
from collections import deque
from sympy.combinatorics.rewritingsystem_fsm import StateMachine
class RewritingSystem:
'''
A class implementing rewriting systems for `FpGroup`s.
References
==========
.. [1] Epstein, D., Holt, D. and Rees, S. (1991).
The use of Knuth-Bendix methods to solve the word problem in automatic groups.
Journal of Symbolic Computation, 12(4-5), pp.397-414.
.. [2] GAP's Manual on its KBMAG package
https://www.gap-system.org/Manuals/pkg/kbmag-1.5.3/doc/manual.pdf
'''
def __init__(self, group):
self.group = group
self.alphabet = group.generators
self._is_confluent = None
# these values are taken from [2]
self.maxeqns = 32767 # max rules
self.tidyint = 100 # rules before tidying
# _max_exceeded is True if maxeqns is exceeded
# at any point
self._max_exceeded = False
# Reduction automaton
self.reduction_automaton = None
self._new_rules = {}
# dictionary of reductions
self.rules = {}
self.rules_cache = deque([], 50)
self._init_rules()
# All the transition symbols in the automaton
generators = list(self.alphabet)
generators += [gen**-1 for gen in generators]
# Create a finite state machine as an instance of the StateMachine object
self.reduction_automaton = StateMachine('Reduction automaton for '+ repr(self.group), generators)
self.construct_automaton()
def set_max(self, n):
'''
Set the maximum number of rules that can be defined
'''
if n > self.maxeqns:
self._max_exceeded = False
self.maxeqns = n
return
@property
def is_confluent(self):
'''
Return `True` if the system is confluent
'''
if self._is_confluent is None:
self._is_confluent = self._check_confluence()
return self._is_confluent
def _init_rules(self):
identity = self.group.free_group.identity
for r in self.group.relators:
self.add_rule(r, identity)
self._remove_redundancies()
return
def _add_rule(self, r1, r2):
'''
Add the rule r1 -> r2 with no checking or further
deductions
'''
if len(self.rules) + 1 > self.maxeqns:
self._is_confluent = self._check_confluence()
self._max_exceeded = True
raise RuntimeError("Too many rules were defined.")
self.rules[r1] = r2
# Add the newly added rule to the `new_rules` dictionary.
if self.reduction_automaton:
self._new_rules[r1] = r2
def add_rule(self, w1, w2, check=False):
new_keys = set()
if w1 == w2:
return new_keys
if w1 < w2:
w1, w2 = w2, w1
if (w1, w2) in self.rules_cache:
return new_keys
self.rules_cache.append((w1, w2))
s1, s2 = w1, w2
# The following is the equivalent of checking
# s1 for overlaps with the implicit reductions
# {g*g**-1 -> <identity>} and {g**-1*g -> <identity>}
# for any generator g without installing the
# redundant rules that would result from processing
# the overlaps. See [1], Section 3 for details.
if len(s1) - len(s2) < 3:
if s1 not in self.rules:
new_keys.add(s1)
if not check:
self._add_rule(s1, s2)
if s2**-1 > s1**-1 and s2**-1 not in self.rules:
new_keys.add(s2**-1)
if not check:
self._add_rule(s2**-1, s1**-1)
# overlaps on the right
while len(s1) - len(s2) > -1:
g = s1[len(s1)-1]
s1 = s1.subword(0, len(s1)-1)
s2 = s2*g**-1
if len(s1) - len(s2) < 0:
if s2 not in self.rules:
if not check:
self._add_rule(s2, s1)
new_keys.add(s2)
elif len(s1) - len(s2) < 3:
new = self.add_rule(s1, s2, check)
new_keys.update(new)
# overlaps on the left
while len(w1) - len(w2) > -1:
g = w1[0]
w1 = w1.subword(1, len(w1))
w2 = g**-1*w2
if len(w1) - len(w2) < 0:
if w2 not in self.rules:
if not check:
self._add_rule(w2, w1)
new_keys.add(w2)
elif len(w1) - len(w2) < 3:
new = self.add_rule(w1, w2, check)
new_keys.update(new)
return new_keys
def _remove_redundancies(self, changes=False):
'''
Reduce left- and right-hand sides of reduction rules
and remove redundant equations (i.e. those for which
lhs == rhs). If `changes` is `True`, return a set
containing the removed keys and a set containing the
added keys
'''
removed = set()
added = set()
rules = self.rules.copy()
for r in rules:
v = self.reduce(r, exclude=r)
w = self.reduce(rules[r])
if v != r:
del self.rules[r]
removed.add(r)
if v > w:
added.add(v)
self.rules[v] = w
elif v < w:
added.add(w)
self.rules[w] = v
else:
self.rules[v] = w
if changes:
return removed, added
return
def make_confluent(self, check=False):
'''
Try to make the system confluent using the Knuth-Bendix
completion algorithm
'''
if self._max_exceeded:
return self._is_confluent
lhs = list(self.rules.keys())
def _overlaps(r1, r2):
len1 = len(r1)
len2 = len(r2)
result = []
for j in range(1, len1 + len2):
if (r1.subword(len1 - j, len1 + len2 - j, strict=False)
== r2.subword(j - len1, j, strict=False)):
a = r1.subword(0, len1-j, strict=False)
a = a*r2.subword(0, j-len1, strict=False)
b = r2.subword(j-len1, j, strict=False)
c = r2.subword(j, len2, strict=False)
c = c*r1.subword(len1 + len2 - j, len1, strict=False)
result.append(a*b*c)
return result
def _process_overlap(w, r1, r2, check):
s = w.eliminate_word(r1, self.rules[r1])
s = self.reduce(s)
t = w.eliminate_word(r2, self.rules[r2])
t = self.reduce(t)
if s != t:
if check:
# system not confluent
return [0]
try:
new_keys = self.add_rule(t, s, check)
return new_keys
except RuntimeError:
return False
return
added = 0
i = 0
while i < len(lhs):
r1 = lhs[i]
i += 1
# j could be i+1 to not
# check each pair twice but lhs
# is extended in the loop and the new
# elements have to be checked with the
# preceding ones. there is probably a better way
# to handle this
j = 0
while j < len(lhs):
r2 = lhs[j]
j += 1
if r1 == r2:
continue
overlaps = _overlaps(r1, r2)
overlaps.extend(_overlaps(r1**-1, r2))
if not overlaps:
continue
for w in overlaps:
new_keys = _process_overlap(w, r1, r2, check)
if new_keys:
if check:
return False
lhs.extend(new_keys)
added += len(new_keys)
elif new_keys == False:
# too many rules were added so the process
# couldn't complete
return self._is_confluent
if added > self.tidyint and not check:
# tidy up
r, a = self._remove_redundancies(changes=True)
added = 0
if r:
# reset i since some elements were removed
i = min(lhs.index(s) for s in r)
lhs = [l for l in lhs if l not in r]
lhs.extend(a)
if r1 in r:
# r1 was removed as redundant
break
self._is_confluent = True
if not check:
self._remove_redundancies()
return True
def _check_confluence(self):
return self.make_confluent(check=True)
def reduce(self, word, exclude=None):
'''
Apply reduction rules to `word` excluding the reduction rule
for the lhs equal to `exclude`
'''
rules = {r: self.rules[r] for r in self.rules if r != exclude}
# the following is essentially `eliminate_words()` code from the
# `FreeGroupElement` class, the only difference being the first
# "if" statement
again = True
new = word
while again:
again = False
for r in rules:
prev = new
if rules[r]**-1 > r**-1:
new = new.eliminate_word(r, rules[r], _all=True, inverse=False)
else:
new = new.eliminate_word(r, rules[r], _all=True)
if new != prev:
again = True
return new
def _compute_inverse_rules(self, rules):
'''
Compute the inverse rules for a given set of rules.
The inverse rules are used in the automaton for word reduction.
Arguments:
rules (dictionary): Rules for which the inverse rules are to computed.
Returns:
Dictionary of inverse_rules.
'''
inverse_rules = {}
for r in rules:
rule_key_inverse = r**-1
rule_value_inverse = (rules[r])**-1
if (rule_value_inverse < rule_key_inverse):
inverse_rules[rule_key_inverse] = rule_value_inverse
else:
inverse_rules[rule_value_inverse] = rule_key_inverse
return inverse_rules
def construct_automaton(self):
'''
Construct the automaton based on the set of reduction rules of the system.
Automata Design:
The accept states of the automaton are the proper prefixes of the left hand side of the rules.
The complete left hand side of the rules are the dead states of the automaton.
'''
self._add_to_automaton(self.rules)
def _add_to_automaton(self, rules):
'''
Add new states and transitions to the automaton.
Summary:
States corresponding to the new rules added to the system are computed and added to the automaton.
Transitions in the previously added states are also modified if necessary.
Arguments:
rules (dictionary) -- Dictionary of the newly added rules.
'''
# Automaton variables
automaton_alphabet = []
proper_prefixes = {}
# compute the inverses of all the new rules added
all_rules = rules
inverse_rules = self._compute_inverse_rules(all_rules)
all_rules.update(inverse_rules)
# Keep track of the accept_states.
accept_states = []
for rule in all_rules:
# The symbols present in the new rules are the symbols to be verified at each state.
# computes the automaton_alphabet, as the transitions solely depend upon the new states.
automaton_alphabet += rule.letter_form_elm
# Compute the proper prefixes for every rule.
proper_prefixes[rule] = []
letter_word_array = list(rule.letter_form_elm)
len_letter_word_array = len(letter_word_array)
for i in range (1, len_letter_word_array):
letter_word_array[i] = letter_word_array[i-1]*letter_word_array[i]
# Add accept states.
elem = letter_word_array[i-1]
if elem not in self.reduction_automaton.states:
self.reduction_automaton.add_state(elem, state_type='a')
accept_states.append(elem)
proper_prefixes[rule] = letter_word_array
# Check for overlaps between dead and accept states.
if rule in accept_states:
self.reduction_automaton.states[rule].state_type = 'd'
self.reduction_automaton.states[rule].rh_rule = all_rules[rule]
accept_states.remove(rule)
# Add dead states
if rule not in self.reduction_automaton.states:
self.reduction_automaton.add_state(rule, state_type='d', rh_rule=all_rules[rule])
automaton_alphabet = set(automaton_alphabet)
# Add new transitions for every state.
for state in self.reduction_automaton.states:
current_state_name = state
current_state_type = self.reduction_automaton.states[state].state_type
# Transitions will be modified only when suffixes of the current_state
# belongs to the proper_prefixes of the new rules.
# The rest are ignored if they cannot lead to a dead state after a finite number of transisitons.
if current_state_type == 's':
for letter in automaton_alphabet:
if letter in self.reduction_automaton.states:
self.reduction_automaton.states[state].add_transition(letter, letter)
else:
self.reduction_automaton.states[state].add_transition(letter, current_state_name)
elif current_state_type == 'a':
# Check if the transition to any new state in possible.
for letter in automaton_alphabet:
_next = current_state_name*letter
while len(_next) and _next not in self.reduction_automaton.states:
_next = _next.subword(1, len(_next))
if not len(_next):
_next = 'start'
self.reduction_automaton.states[state].add_transition(letter, _next)
# Add transitions for new states. All symbols used in the automaton are considered here.
# Ignore this if `reduction_automaton.automaton_alphabet` = `automaton_alphabet`.
if len(self.reduction_automaton.automaton_alphabet) != len(automaton_alphabet):
for state in accept_states:
current_state_name = state
for letter in self.reduction_automaton.automaton_alphabet:
_next = current_state_name*letter
while len(_next) and _next not in self.reduction_automaton.states:
_next = _next.subword(1, len(_next))
if not len(_next):
_next = 'start'
self.reduction_automaton.states[state].add_transition(letter, _next)
def reduce_using_automaton(self, word):
'''
Reduce a word using an automaton.
Summary:
All the symbols of the word are stored in an array and are given as the input to the automaton.
If the automaton reaches a dead state that subword is replaced and the automaton is run from the beginning.
The complete word has to be replaced when the word is read and the automaton reaches a dead state.
So, this process is repeated until the word is read completely and the automaton reaches the accept state.
Arguments:
word (instance of FreeGroupElement) -- Word that needs to be reduced.
'''
# Modify the automaton if new rules are found.
if self._new_rules:
self._add_to_automaton(self._new_rules)
self._new_rules = {}
flag = 1
while flag:
flag = 0
current_state = self.reduction_automaton.states['start']
for i, s in enumerate(word.letter_form_elm):
next_state_name = current_state.transitions[s]
next_state = self.reduction_automaton.states[next_state_name]
if next_state.state_type == 'd':
subst = next_state.rh_rule
word = word.substituted_word(i - len(next_state_name) + 1, i+1, subst)
flag = 1
break
current_state = next_state
return word
|