def deltaPrefix(self, MGen, algGens):
# Preliminary tests
if len(algGens) == 0:
return True
if algGens[0].left_idem != MGen.idem2:
return False
if any([algGens[i].right_idem != algGens[i+1].left_idem
for i in range(len(algGens)-1)]):
return False
assignment, algs_local, prod_d = self.getAssignments(MGen, algGens)
if assignment is None:
return E0
local_MGen = MGen.local_gen
dbls = [self.single_idems2[i] for i in range(self.num_singles)
if assignment[i] == self.DOUBLE]
for to_remove in subset(dbls):
if len(to_remove) != 0:
cur_algs_local = tuple([alg.removeSingleHor(to_remove)
for alg in algs_local])
else:
cur_algs_local = algs_local
if self.testPrefix(local_MGen, cur_algs_local):
return True
return False
def search(cur_strands):
"""Search starting with the given list of strands. May only add
strands after the end position of the last strand.
"""
# First, check if the current list of strands is valid.
left_occupied = [0] * self.num_pair
right_occupied = [0] * self.num_pair
for start, end in cur_strands:
start_id, end_id = self.pairid[start], self.pairid[end]
if start_id != -1:
left_occupied[start_id] += 1
if end_id != -1:
right_occupied[end_id] += 1
if any([n >= 2 for n in left_occupied + right_occupied]):
# There should not be two strands starting or ending at points
# in the same pair.
return
# Enumerate all possible ways of adding idempotents.
empty_idems = [i for i in range(self.num_pair)
if left_occupied[i] == 0 and right_occupied[i] == 0]
left_idem = [i for i in range(self.num_pair)
if left_occupied[i] > 0]
for idems_to_add in subset(empty_idems):
result.append(LocalStrandDiagram(
algebra, left_idem + list(idems_to_add), cur_strands))
# Now enumerate all ways of adding more strands.
last_end = 0
if len(cur_strands) > 0:
last_end = cur_strands[-1][1]
for start in range(last_end, self.n):
for end in range(start + 1, self.n):
if self.pairid[start] == -1 and self.pairid[end] == -1 and \
end == start + 1:
# Exclude cases where a strand goes from an
# end-boundary-point to a start-boundary-point
break
search(cur_strands + [(start, end)])
if self.pairid[end] == -1:
# No strand should go beyond an end-boundary-point.
break
def getLocalDAStructure(self):
"""Returns the local type DA structure associated to the anti-braid
resolution.
"""
if self.is_degenerate:
if self.c1 == 0:
patterns_raw = self._get_patterns_bottom()
else:
assert self.c2 == self.n - 1
patterns_raw = self._get_patterns_top()
else:
patterns_raw = self._get_patterns_middle()
arrow_patterns = {}
for pattern in patterns_raw:
start_class, end_class = pattern[0], pattern[1]
coeffs_a = []
for i in range(2, len(pattern)-1):
coeffs_a.append(self.local_pmc.sd(pattern[i]))
key = (start_class, end_class, tuple(coeffs_a))
if key not in arrow_patterns:
arrow_patterns[key] = []
arrow_patterns[key].append(self.local_pmc.sd(pattern[-1]))
# Now start construction of the local DA structure.
alg = LocalStrandAlgebra(F2, self.local_pmc)
local_c_pair = 0
# Compute the set of local generators. Generators of class 0 has
# idempotents (l_idem, r_idem) where l_idem has the c_pair and
# r_idem has one of the u or d pairs (with the rest being the same).
# Generators of class 1 and 2 has l_idem = r_idem such that c_pair
# is in both.
if self.is_degenerate:
single_idems = [1] # local p_pair
da_idems_0 = [([0], [1])]
else: # Non-degenerate case
single_idems = [1, 2] # local d_pair and local u_pair
da_idems_0 = [([0], [1]), ([0], [2]),
([0, 1], [1, 2]), ([0, 2], [1, 2])] # class 0
local_da = LocalDAStructure(
F2, alg, alg, single_idems1 = single_idems,
single_idems2 = single_idems)
for i in range(len(da_idems_0)): # class 0
l_idem, r_idem = da_idems_0[i]
local_da.addGenerator(SimpleDAGenerator(
local_da, LocalIdempotent(self.local_pmc, l_idem),
LocalIdempotent(self.local_pmc, r_idem), "0_%d" % i))
all_idems = subset(range(self.local_pmc.num_pair)) # class 1 and 2
for i in range(len(all_idems)):
idem = LocalIdempotent(self.local_pmc, all_idems[i])
if local_c_pair in idem:
local_da.addGenerator(
SimpleDAGenerator(local_da, idem, idem, "1_%d" % i))
local_da.addGenerator(
SimpleDAGenerator(local_da, idem, idem, "2_%d" % i))
mod_gens = local_da.getGenerators()
# Have to take care of u_map. It is sufficient to know that u_map musst
# preserve class of generators.
for i in range(len(single_idems)):
idem = single_idems[i]
for local_gen in mod_gens:
idem1, idem2 = local_gen.idem1, local_gen.idem2
if idem in idem1 and idem in idem2:
# local_gen is eligible for u_maps[i]
target_idem1 = idem1.removeSingleHor([idem])
target_idem2 = idem2.removeSingleHor([idem])
target_gen = [target for target in mod_gens
if target.idem1 == target_idem1 and
target.idem2 == target_idem2 and
target.name[0] == local_gen.name[0]]
assert len(target_gen) == 1
local_da.add_u_map(i, local_gen, target_gen[0])
# Check all u_map has been filled.
local_da.auto_u_map()
# Add arrows according to arrow_pattern.
for key in arrow_patterns.keys():
start_class, end_class, coeffs_a = key
if len(coeffs_a) == 1 and coeffs_a[0].isIdempotent():
continue
for coeff_d in arrow_patterns[key]:
used = False
for x, y in itertools.product(mod_gens, mod_gens):
if x.name[0] == "%d" % start_class and \
y.name[0] == "%d" % end_class and \
DAStructure.idemMatchDA(x, y, coeff_d, coeffs_a):
local_da.addDelta(x, y, coeff_d, coeffs_a, 1)
used = True
if not used:
print "Warning: unused arrow: %s %s" % (coeffs_a, coeff_d)
return local_da
def sGOfTGrad(t):
return -gOfTGrad(subset(L, sampleSize),
alpha * sampleSize / n,
t, eta)
def getLocalDAStructure(self, seeds_only = False):
"""Returns the local type DA structure associated to slide. If
seeds_only is set to True, get a local DA structure with incomplete
da_action, that can be completed using the autocompleteda module.
"""
# Compute the set of arrow patterns
patterns_raw = self.patterns_fun(seeds_only = seeds_only)
if self.translator is not None:
patterns_raw = ArcslideDA._restrict_local_arrows(
patterns_raw, self.translator[0], self.translator[1])
arrow_patterns = {}
for pattern in patterns_raw:
coeffs_a = []
for i in range(len(pattern)-1):
coeffs_a.append(self.local_pmc2.sd(pattern[i]))
coeffs_a = tuple(coeffs_a)
if coeffs_a not in arrow_patterns:
arrow_patterns[coeffs_a] = []
arrow_patterns[coeffs_a].append(self.local_pmc1.sd(pattern[-1]))
# Now start construction of the local DA structure.
alg1 = LocalStrandAlgebra(F2, self.local_pmc1)
alg2 = LocalStrandAlgebra(F2, self.local_pmc2)
local_dastr = LocalDAStructure(F2, alg1, alg2)
# Mappings between local starting and ending PMC.
slide = self.slide
local_to_r = dict()
for i in range(slide.start_pmc.n):
if i in self.mapping1:
# to_r[i] must be in mapping2
local_to_r[self.mapping1[i]] = self.mapping2[slide.to_r[i]]
local_pair_to_r = dict()
for i in range(self.local_pmc1.n):
if i not in self.local_pmc1.endpoints:
local_pair_to_r[self.local_pmc1.pairid[i]] \
= self.local_pmc2.pairid[local_to_r[i]]
b1, c1 = self.slide.b1, self.slide.c1
local_b1, local_c1 = self.mapping1[b1], self.mapping1[c1]
b_pair1 = self.local_pmc1.pairid[local_b1]
c_pair1 = self.local_pmc1.pairid[local_c1]
# Compute the set of local generators. This includes all
# (l_idem, r_idem) where l_idem = r_idem (under the usual identification
# of pairs), or where l_idem has the c_pair and r_idem has the b_pair.
da_idems = []
num_pair = self.local_pmc1.num_pair
for idem in subset(range(num_pair)):
da_idems.append((list(idem), [local_pair_to_r[p] for p in idem]))
for idem in subset([p for p in range(num_pair)
if p != b_pair1 and p != c_pair1]):
da_idems.append((list(idem) + [c_pair1],
[local_pair_to_r[p]
for p in (list(idem) + [b_pair1])]))
for i in range(len(da_idems)):
l_idem, r_idem = da_idems[i]
local_dastr.addGenerator(SimpleDAGenerator(
local_dastr, LocalIdempotent(self.local_pmc1, l_idem),
LocalIdempotent(self.local_pmc2, r_idem), "%d" % i))
mod_gens = local_dastr.getGenerators()
# After having added all generators, create u_map:
local_dastr.auto_u_map()
# Add arrows according to arrow_pattern.
for coeffs_a in arrow_patterns.keys():
if len(coeffs_a) == 1 and coeffs_a[0].isIdempotent():
continue
for coeff_d in arrow_patterns[coeffs_a]:
for x, y in itertools.product(mod_gens, mod_gens):
if DAStructure.idemMatchDA(x, y, coeff_d, coeffs_a):
local_dastr.addDelta(x, y, coeff_d, coeffs_a, 1)
return local_dastr
def search(start_gen, cur_dgen, algs, last_assign, algs_local,
last_prod_d):
"""Searching for an arrow in the box tensor product.
- start_gen: starting generator in the box tensor product. The
resulting arrow will start from here.
- cur_dgen: current location in the type D structure.
- algs: current list of A-side inputs to the type DA structure (or
alternatively, list of algebra outputs produced by the existing
path through the type D structure).
- algs_local: current list of local restrictions of algs.
- last_assign: a list of length self.num_singles. For each split
idempotent, specify the single assignments at the last algebra
input.
- prod_d: product of the outer restrictions, except for the last
algebra input.
"""
start_dagen, start_dgen = start_gen
local_MGen = start_dagen.local_gen
# Preliminary tests
if len(algs) > 0:
assert algs[0].left_idem == start_dagen.idem2
for i in range(len(algs)-1):
assert algs[i].right_idem == algs[i+1].left_idem
if any(alg.isIdempotent() for alg in algs):
return
# First, adjust local module generator, and check for delta.
if len(algs_local) > 0:
local_MGen = self.adjustLocalMGen(local_MGen, algs_local[0])
if local_MGen is None:
return
local_delta = self.local_da.delta(local_MGen, tuple(algs_local))
has_delta = (local_delta != E0)
# Second, check for delta prefix.
has_delta_prefix = False
if len(algs) == 0:
has_delta_prefix = True
else:
dbls = [self.single_idems2[i] for i in range(self.num_singles)
if last_assign[i] == self.DOUBLE]
for to_remove in subset(dbls):
if len(to_remove) != 0:
cur_algs_local = tuple([alg.removeSingleHor(to_remove)
for alg in algs_local])
else:
cur_algs_local = algs_local
if self.testPrefix(local_MGen, cur_algs_local):
has_delta_prefix = True
break
if (not has_delta) and (not has_delta_prefix):
return
# Now, compute new prod_d.
if len(algs) > 0:
prod_d = self.getNewProdD(last_assign, algs[-1], last_prod_d)
else:
prod_d = last_prod_d
if prod_d is None:
return
# If has_delta is True, add to delta
for (local_d, local_y), ring_coeff in local_delta.items():
alg_d, y = self.joinOutput(local_d, local_y, prod_d)
if alg_d is not None:
dstr_result.addDelta(start_gen, DATensorDGenerator(
dstr_result, y, cur_dgen), alg_d, 1)
if not has_delta_prefix:
return
for (new_alg, dgen_to), ring_coeff in dstr.delta(cur_dgen).items():
new_assign, new_local, last_prod_d = self.extendRestrictions(
last_assign, algs_local, prod_d, new_alg)
if new_assign is not None:
search(start_gen, dgen_to, algs + [new_alg],
new_assign, new_local, last_prod_d)
