def expr_count_of(self, char):
"""Counts char in string, returns expression.
Returns a z3 expression representing the number of times char appears in
the string.
"""
return (z3.Sum([z3.If(bv == ord(char), 1, 0) for bv in self.bitvecs]))
def applications_to_solver(applications):
# try building applications one by one
solver = z3.Solver()
solver_list = []
for application in applications:
l, m, r = application.split(' ')
l_c = z3.Sum( [z3.Int(elem) for elem in l.split('+')] )
try:
r_c = float(r)
r_c_lb = r_c - 1
r_c_ub = r_c + 1
except:
r_c = z3.Int(r)
# all of type elem_1 <= elem_2
if m == '<=':
solver.add(l_c <= r_c)
solver_list.append(l_c <= r_c)
# all of type elem == int
elif m == '==':
# NOTE: works only if we allow a margin of error -- is this coming from the conversion from mean to sum?
solver.add(l_c >= r_c_lb)
solver.add(l_c <= r_c_ub)
solver_list.append(l_c >= r_c_lb)
solver_list.append(l_c <= r_c_ub)
# solver.add(l_c == r_c)
# solver_list.append(l_c == r_c)
# all of type elem >= 0
elif m == '>=':
solver.add(l_c >= r_c)
solver_list.append(l_c >= r_c)
return(solver, solver_list)
def main(nanobots):
solver = z3.Optimize()
bx = z3.Int("x")
by = z3.Int("y")
bz = z3.Int("z")
d = z3.Int("d")
bots = []
for i, bot in enumerate(nanobots):
print("adding bot {} to z3".format(i))
b = z3.Int("b{:04}".format(i))
solver.add(b == z3.If(manhattan_distance((bx, by, bz), (bot.x, bot.y, bot.z)) <= bot.radius, 1, 0))
bots.append(b)
print("adding constraints")
solver.add(d == manhattan_distance((bx, by, bz), (0, 0, 0)))
solver.maximize(z3.Sum(*bots))
solver.minimize(d)
print("solving...")
print(solver.check())
model = solver.model()
print("answer:")
print(model[d])
def getNearestFeasible(self, coord):
# get the variables corresponding to the axis names
for i in self.axis_names:
locals()[i] = Globals.z3variables[i]
# Convert into parameters
rpp = self.search.coordToPerfParams(coord)
for i, name in enumerate(self.axis_names):
if not self.z3IsNumeric(i):
rpp[name] = Globals.z3types[name].index(rpp[name])
# create a new scope
self.optim.push()
# Get a possible point that minimizes the 1-norm distance to this random point
# forget the booleans
l = []
for name in self.axis_names:
if not z3.is_bool(locals()[name]):
l.append(self.z3abs(locals()[name] - rpp[name]))
definition = z3.Sum(l)
self.optim.minimize(definition)
# For some reason, it does not work with a list comprehension.
# self.optim.minimize( z3.Sum( [ self.z3abs( locals()[name] - rpp[name] )for name in self.axis_names if not z3.is_bool( locals()[name] ) ] ) )
if z3.unsat == self.optim.check():
return None
model = self.optim.model()
# restore the state (exit the scope)
self.optim.pop()
return self.z3ToPoint(model)
def Sum(*args: BitVec) -> BitVec:
"""Create sum expression.
:return:
"""
raw = z3.Sum([a.raw for a in args])
annotations = [] # type: Annotations
bitvecfuncs = []
for bv in args:
annotations += bv.annotations
if isinstance(bv, BitVecFunc):
bitvecfuncs.append(bv)
if len(bitvecfuncs) >= 2:
return BitVecFunc(raw=raw,
func_name=None,
input_=None,
annotations=annotations)
elif len(bitvecfuncs) == 1:
return BitVecFunc(
raw=raw,
func_name=bitvecfuncs[0].func_name,
input_=bitvecfuncs[0].input_,
annotations=annotations,
)
return BitVec(raw, annotations)
def _link_affine(self, phi, include_bterm=True):
assert isinstance(phi, amnet.Affine) or \
isinstance(phi, amnet.Linear)
m, n = phi.w.shape
assert m >= 1 and n >= 1
# extract children
yvar = self.var_of(phi)
xvar = self.var_of(phi.x)
assert (len(yvar) == m) and (len(xvar) == n)
# go row-by-row of w
for i in range(m):
rowi = phi.w[i, :]
assert len(rowi) == n
rowsum = z3.Sum(
[wij * xj for wij, xj in izip(rowi, xvar) if wij != 0])
if include_bterm:
assert isinstance(phi, amnet.Affine), \
'Warning: tried to encode bterm in non-Affine'
self.solver.add(yvar[i] == (rowsum + phi.b[i]))
else:
self.solver.add(yvar[i] == rowsum)
def solve(grid):
"""
. . .
.1.1.
. .3.
"""
s = z3.Solver()
# Construct atoms to represent the dots
Dot = z3.Datatype("Dot")
for d in grid.dots():
Dot.declare("dot_{}".format(d))
Dot = Dot.create()
dot_atom = {d: getattr(Dot, "dot_{}".format(d)) for d in grid.dots()}
# Booleans for each of the points
dot = {d: z3.Bool("dot-{}".format(d)) for d in grid.dots()}
# Booleans for each of the connectors
line = {l: z3.Bool("line-{}".format(l)) for l in grid.lines()}
# For each point: if it is on, it must have precisely two adjoining lines activated
# If it's off, then there are zero adjoining lines activated
bool_to_int = z3.Function("bool_to_int", z3.BoolSort(), z3.IntSort())
s.add(bool_to_int(True) == 1)
s.add(bool_to_int(True) != 0)
s.add(bool_to_int(False) == 0)
s.add(bool_to_int(False) != 1)
for d in grid.dots():
# Get all lines coming out of this dot
ls = [line[l] for l in d.lines()]
sm = z3.Sum(*(bool_to_int(l) for l in ls))
s.add(z3.Implies(dot[d], sm == 2))
s.add(z3.Implies(z3.Not(dot[d]), sm == 0))
# For each line: if it is activated, then the points at both ends are activated
for l in line:
d1, d2 = l.ends()
s.add(z3.Implies(line[l], z3.And(dot[d1], dot[d2])))
# "Is connected to" relationship
connected = z3.Function("connected", Dot, Dot, z3.BoolSort())
# For each line:
# The Dot at each end is connected to the other iff the line is activated
for d1 in grid.dots():
for d2 in grid.dots():
a = dot_atom[d1]
b = dot_atom[d2]
if (l := grid.line(d1, d2)) is None:
# These dots are never connected
s.add(connected(a, b) != True)
else:
s.add(z3.Implies(line[l], connected(a, b)))
s.add(z3.Implies(z3.Not(line[l]), z3.Not(connected(a, b))))
def incidence_sums(petrinet,
variables,
subnet_transitions="all",
reverse=False):
if subnet_transitions == "all":
subnet_transitions = set(range(petrinet.num_transitions))
sums = []
for p in range(petrinet.num_places()):
nonzero = petrinet.transitions_set({p}, reverse)
incidence_row = [(petrinet.get_post(p, t) - petrinet.get_pre(p, t),
variables[t]) for t in nonzero
if t in subnet_transitions]
sum_elements = [
expression(coeff, var) for coeff, var in incidence_row
if coeff != 0
]
curr_sum = z3.Sum(
sum_elements) if len(sum_elements) > 0 else z3.IntVal(0)
sums.append(curr_sum)
return sums
def add_cage_constraints(self, cage):
cage_sum = int(cage[:2])
cage_vars = []
for r, c in zip(cage[2::2], cage[3::2]):
cage_vars.append(self._grid[int(r) - 1][int(c) - 1])
self._solver.add(z3.Distinct(cage_vars))
self._solver.add(z3.Sum(cage_vars) == cage_sum)
def add_killer_cage(s, cage_coords, cage_sum):
'''Add a single killer sudoku cage.
Takes a list of (row, col) coordinates and the sum
'''
cagecells = [X[i][j] for i, j in cage_coords]
# Digits dont repeat within cages
s.add(z3.Distinct(cagecells))
s.add(z3.Sum(cagecells) == cage_sum)
def do_POPCOUNT(op, stack, state):
b, = pop_values(stack, state)
n = b.size()
bits = [z3.Extract(i, i, b) for i in range(n)]
bvs = [z3.Concat(z3.BitVecVal(0, n - 1), b) for b in bits]
nb = z3.Sum(*bvs)
stack.append(z3.simplify(nb))
def Sum(*args: BitVec) -> BitVec:
"""Create sum expression.
:return:
"""
nraw = z3.Sum([a.raw for a in args])
annotations = [] # type: Annotations
for bv in args:
annotations += bv.annotations
return BitVec(nraw, annotations)
def _bin_packing_constraints(manifest, storage, color, volume):
constraint = []
# heuristic = []
limits = [z3.And(0 <= v, v <= s['volume_left']) for v, s in zip(volume, storage)]
for i, stor in enumerate(storage):
summation = z3.Sum([z3.If(col == i, vol, 0) for col, vol in zip(color, volume)])
constraint.append(summation <= stor['volume_left'])
# heuristic.append( z3.Sum([z3.If(z3.Or(summation==0, summation==stor['volume_left']), 1, 0)]) )
return limits + constraint # , z3.Sum(heuristic)
def setup_line(self, x0, y0, dx, dy, sum):
if sum == 0:
return
vs = []
x, y = x0 + dx, y0 + dy
while (x, y) in self.cells:
vs.append(self.cells[x, y])
x += dx
y += dy
self.solver.add(z3.Distinct(vs))
self.solver.add(z3.Sum(vs) == sum)
def make_hash_equation(pj: int, var_b: Tuple[int], b1: int):
domain_bit_count = int(ceil(log2(pj)))
bc = int(ceil(log2(pj + 1)))
slices = get_slices(domain_bit_count)
return z3.URem(
z3.Sum([
z3.ZeroExt(bc - s.size(), s) * z3.BitVecVal(var_b[i], bc)
for i, s in enumerate(slices)
]) + z3.BitVecVal(b1, bc),
z3.BitVecVal(pj, bc)) == z3.BitVecVal(0, bc)
def agreesAtLeastNTimes(inputOutputList, expressionValues, gene, aVars, rVars,
n):
both = [
totalAgreement(inputOutputList[p], gene, aVars, rVars,
expressionValues, p)
for p in range(len(inputOutputList))
]
encodings = [both[i][0] for i in range(len(both))]
scoreValues = [both[i][1] for i in range(len(both))]
return z3.And(z3.And(encodings), z3.Sum(scoreValues) >= n)
def largest_clique(self, timeout=120):
if z3 is None:
raise ImportError()
solver = z3.Optimize()
solver.set("timeout", timeout)
vars = {}
# rev = {}
start = time.clock()
m = z3.Int(name="cliquesize")
for v in self.nodes_iter():
vars[v] = z3.Int(name="node{0}".format(v))
# rev[vars[v]] = v
solver.add(vars[v] <= 1)
solver.add(vars[v] >= 0)
solver.add(z3.Sum([vars[v] for v in self.nodes_iter()]) >= m)
# solver.add(z3.Or([vars[v] == 1 for v in self.nodes_iter()]))
adj = self.adj
for u in self.nodes_iter():
for v in self.nodes_iter():
if u < v and u not in adj[v]:
solver.add(z3.Or(vars[u] == 0, vars[v] == 0))
if timeout != 0 and time.clock() - start >= timeout:
return None
r = None
try:
r = solver.maximize(m)
solver.check()
except z3.Z3Exception as e:
logging.error(e.message)
if r is None:
return None
res = solver.lower(r)
# assert(str(res) != 'epsilon' and str(res) != 'unsat' and isinstance(res, z3.IntNumRef) and res.as_long() >= 1)
if str(res) == 'epsilon' or str(res) == 'unsat':
logging.error(res)
elif not isinstance(res, z3.IntNumRef):
logging.error("not an int")
elif res.as_long() < 1:
logging.error("clique result < 1")
else:
cl = [
k for (k, v) in vars.items()
if solver.model()[v].as_long() == 1
]
if len(cl) != res.as_long():
logging.error("{0} vs. {1}".format(len(cl), res.as_long()))
# assert(len(cl) == res.as_long())
return None
# print cl
return cl
return None
def solve_part_2(nanobots):
s = z3.Optimize()
x = z3.Int('x')
y = z3.Int('y')
z = z3.Int('z')
d = z3.Int('d')
dists = [z3.Int(i) for i in range(len(nanobots))]
for i, bot in enumerate(nanobots):
dists[i] = z3.If(dist2(bot['pos'], (x, y, z)) <= bot['r'], 1, 0)
s.maximize(z3.Sum(*dists))
s.add(d == dist2((x, y, z), (0, 0, 0)))
s.minimize(d)
s.check()
return s.model()[d]
def _solve(
solver: z3.Optimize, channels: List[Channel], devices: List[Device]
) -> Tuple[Problem, z3.Model]:
problem = _problem(solver, freqs=[c.frequency for c in channels], devices=devices)
if solver.check() != z3.sat:
# TODO: consider getting an unsat core
raise ValueError(f"No valid assignment possible, add more devices")
# find the minimal number of devices that cover all frequencies
# it is faster to do this iteratively than to offload these to
# smt constraints. do this in reverse so we end up assigning
# the lowest numbered devices
for r in problem.ranges[::-1]:
# to control the device placements adjust the order they're eliminated from
# the solution. for example, this will prefer devices that have a smaller
# minimum sample rate over devices with a larger one:
# for r, _ in zip(problem.ranges, problem.devices), key=lambda x: -x[1].min_sample_rate
solver.push()
solver.add(r == 0)
if solver.check() != z3.sat:
solver.pop()
break
devices_required = z3.Sum([z3.If(r > 0, 1, 0) for r in problem.ranges])
# minimize the sum of frequency ranges
solver.minimize(z3.Sum(*problem.ranges))
# and the frequency, just to produce deterministic results
solver.minimize(z3.Sum(*problem.lower_freq))
assert solver.check() == z3.sat
model = solver.model()
print(f"Devices required: {model.eval(devices_required)}", file=sys.stderr)
return problem, model
def generate_constraints(capacity, sizes, endpoints, caches):
print('Generate constraints...')
print('SERVE_SUM')
SERVE_SUM = SERVE == z3.Sum([
find_max(e.L.items(), v, e.L_D) * ratio
for e, _ in ((e, print('.', end='', flush=True)) for e in endpoints)
for v, ratio in e.requests.items()
])
print()
print('CAPACITY')
CAPACITY = [
z3.Sum([z3.If(has_video(v, c), sizes[v], 0)
for v in videos]) == capacity for c, videos in caches.items()
]
VIDEO = z3.Int('VIDEO')
print('IMP')
IMP = [
z3.ForAll([VIDEO],
z3.Implies(has_video(VIDEO, c),
z3.Or([VIDEO == v for v in videos])))
for c, videos in caches.items()
]
return [CAPACITY, SERVE_SUM, IMP]
def find_sum_combo_with_z3(numbers, n, target=2020):
solver = z3.Solver()
symvars = [z3.Int(f'sv_{i}') for i in range(n)]
solver.add(z3.Sum(symvars) == target)
solver.add(z3.Distinct(symvars))
for sv in symvars:
solver.add(z3.Or([sv == number for number in numbers]))
if str(solver.check()) == 'sat':
print(f'Model = {solver.model()}')
return mult(solver.model()[sv].as_long() for sv in symvars)
else:
print('No solution')
return -1
def prepare_solver(instance: Instance) -> Tuple[z3.Solver, Dict[int, str]]:
solver = z3.Solver()
variables = []
ingredient_indices: Dict[str, int] = {}
ingredient_names: Dict[int, str] = {}
num_of_meals = len(MEAL_NAMES)
# create a list of variables for each ingredient
for ingredient_id in range(len(instance.ingredients)):
meal_vars = []
name = instance.ingredients[ingredient_id].name
ingredient_indices[name] = ingredient_id
ingredient_names[ingredient_id] = name
for meal_id in range(num_of_meals):
var_name = make_variable_name(meal_id, ingredient_id)
var = z3.Int(var_name)
solver.add(var >= 0)
meal_vars.append(z3.ToReal(var))
variables.append(meal_vars)
# add constraints for food conflicts
for food1, food2 in instance.conflicts:
food1_id = ingredient_indices[food1]
food2_id = ingredient_indices[food2]
food1_vars = variables[food1_id]
food2_vars = variables[food2_id]
for var1, var2 in zip(food1_vars, food2_vars):
solver.add(z3.Or(var1 == 0, var2 == 0))
# add constraints for total amount of substances
macro_sums = {}
for macro in instance.macros:
macro_sums[macro] = 0.0
for ingredient_vars, ingredient in zip(variables, instance.ingredients):
ingredient_sum = z3.Sum(ingredient_vars)
for macro in instance.macros:
macro_sums[macro] += ingredient_sum * ingredient.macros[macro]
for target in instance.targets:
macro_sum = macro_sums[target.macro]
solver.add(macro_sum >= target.min)
solver.add(macro_sum <= target.max)
# ensure that meals are not empty
num_of_ingredients = len(variables)
for meal_id in range(num_of_meals):
meal_conditions = []
for ingredient_id in range(num_of_ingredients):
meal_conditions.append(variables[ingredient_id][meal_id] > 0)
solver.add(z3.Or(meal_conditions))
return solver, ingredient_names
def __init__(self, variable_names: Sequence[str],
qualities: Sequence[float]):
assert len(variable_names) == len(qualities)
# Improve optimizer performance by sorting qualities decreasingly; adapt order of variable names accordingly:
qualities, variable_names = zip(
*sorted(zip(qualities, variable_names), key=lambda x: -x[0]))
self.qualities = qualities
self.variables = [expr.Variable(name=x) for x in variable_names]
self.constraints = []
self.optimizer = z3.Optimize()
# Direct multiplication between bool var and real quality returns wrong type (BoolRef) if quality is 1,
# so we use "If" instead (multiplication is transformed to such an expression anyway):
objective = z3.Sum([
z3.If(var.get_z3(), q, 0)
for (q, var) in zip(qualities, self.get_variables())
])
self.objective = self.optimizer.maximize(objective)
self.optimizer.push() # restore point for state without constraints
def objective_function(self):
"""
Objective function is a weighted combination of gate errors and decoherence errors
"""
import z3
self.fidelity_terms = [self.gate_fidelity[gate] for gate in self.gate_fidelity]
self.coherence_terms = []
for q in self.qubit_lifetime:
val = -self.qubit_lifetime[q] / min(self.bp_t1_time[q], self.bp_t2_time[q])
self.coherence_terms.append(val)
all_terms = []
for item in self.fidelity_terms:
all_terms.append(self.weight_factor * item)
for item in self.coherence_terms:
all_terms.append((1 - self.weight_factor) * item)
self.opt.maximize(z3.Sum(all_terms))
def orderG(G, offset):
solver = z3.Solver()
solver.set('timeout', 1000) # Timeout in one minute
V, E = G.nodes, G.edges
# Assignment: uniquely give vertical positions ([0, 1, ..., N-1]) to the vertices in G.
Positions = [z3.Int('v' + n) for n in V] # positions
assignment = dict(zip(V, Assignment)) # vertex -> position
solver.add(z3.Distinct(Positions)) # Uniqueness condition on Positions
for pos in Positions:
solver.add(position >= 0) # Two range constraints on Positions
solver.add(position < len(V))
# Objective function CumLengths: A weighted sum of the edge lengths
WeightedLengths = []
for n1, n2 in E:
pos1, pos2 = assignment[n1], assignment[n2]
weight = E[n1, n2]['connections']
WeightedLengths.append(weight * Abs(pos1 - pos2))
Objective = z3.Sum(*WeightedLengths) # Objective formula
# Optimization
length_contraint = sys.maxsize
current_best = dict(zip(Positions,
range(len(Positions)))) # Default assignment
history = [current_best.evaluate(Objective).as_long()]
while True:
solver.add(Objective < length_constraint)
result = solver.check()
if result != z3.sat: break
current_best = solver.model()
length_constraint = current_best.evaluate(
Objective).as_long() # Tighter constraint
history.append(length_constraint)
Solution = [(v, current_best[v].as_long() + offset) for v in Assignment]
print(f'Solution: {Solution}')
print(f'Is optimum: {result == z3.unsat}')
print(f'Optimum CumLength: {length_constraint}')
print('Reduction history:', *history)
return Solution
def encode_network(weights, biases, z3_input):
"""
Returns a z3 expression representing the output of the dnn when applied to the input represented
by the list of z3 RealSort constants `z3_input`. The dnn is characterised by the given array of
weights and biases, where `weights` is a list of weight matrices for each non-input layer, and
`biases` is a list of corresponding bias vectors. The matrices are represented as lists of
lists in a column first manner, vecorts as lists. Returns a list of z3 expressions representing
the output.
"""
# z3 consts for values of each layer
z3_nodes = [z3_input]
for w, b in zip(weights, biases):
z3_nodes.append([
z3.Sum([w[j][i] * z3_nodes[-1][j]
for j in range(len(w[i]))]) + b[i] for i in range(len(b))
])
z3_nodes[-1] = [z3.If(z >= 0, z, 0) for z in z3_nodes[-1]]
return z3_nodes[-1]
def _coalesing_constraints(manifest, storage, color, volume):
constraint = []
# heuristic = z3.Int('coalesing_heuristic')
chemical_dict = {}
for s in storage:
for c in s['chemicals']:
name = c['chemical']
if not name in chemical_dict:
chemical_dict[name] = 0
chemical_dict[name] += c['total_volume'] - c['current_volume']
for key, value in chemical_dict.items():
# value=0 means no coalesing, therefore no need to have a constraint.
if value != 0:
summation = z3.Sum([m['volume'] - v for m, v in zip(manifest, volume) if m['chemical'] == key])
constraint.append(summation <= value)
# heuristic = heuristic + summation
return constraint # , heuristic
def _optimize(self):
"""
The current implementation of the optimization is based on minimizing the number of selected cpvs, by asking the SAT solver to minimize that number.
"""
optimize = z3.Optimize()
cpvs = set()
packages = set()
# add the constraint
for k, v in self._loaded_map.items():
if (v[0] is dep_solver.kind.cpv):
p = self._repo.get_package(k)
cpvs.add(k)
packages.add(p)
optimize.add(self._z3_translator.visit(p.get_spc()))
else:
optimize.add(self._z3_translator.visit(self._repo.get_cp(k)))
# define the optimization criteria
optimize.minimize(
z3.Sum(
tuple(
z3.If(self._z3_translator.visit(cpv), 1, 0)
for cpv in cpvs)))
# forbid to use non loaded packages
simplify = gzl.substitutionVisitor({}).visit
for p in packages:
to_disable = p._dep_package.difference(cpvs)
if (len(to_disable) is not 0):
optimize.add(
self._z3_translator.visit(
simplify(gzl.Not(gzl.Or(*to_disable)))))
cpvs.update(to_disable)
# get the model
optimize.check()
self._solution = optimize.model()
self._solution = {
str(var)
for var in self._z3_translator._var_map.values()
if (self._solution[var])
}
def solve_weights_problem(N):
solver = z3.Solver()
weights = [z3.Int("w" + str(i)) for i in range(N)]
for i in range(N):
# Weights positive
solver.add(weights[i] > 0)
for j in range(i + 1, N):
# Weights not equal
solver.add(weights[i] != weights[j])
# Sum of these is equal to the sum of some other subset
others = weights[:i] + weights[i + 1:j] + weights[j + 1:]
sums = [z3.Sum(otherset) for otherset in powerset(others)]
solver.add(z3.Or([weights[i] + weights[j] == sum for sum in sums]))
# Solve the constraints.
## Uncomment to print assertions
# print(f"Constraints: {solver.assertions()}")
result = str(solver.check())
print(f"Result: {result}")
if result == 'sat':
print(f"Model: {solver.model()}")
return result
def incidence_sums(petrinet,
variables,
subnet_transitions='all',
reverse=False):
pre_matrix, post_matrix = petrinet
num_places, num_transitions = pre_matrix.shape
if subnet_transitions == 'all':
subnet_transitions = set(range(num_transitions))
if reverse:
pre_matrix, post_matrix = post_matrix, pre_matrix
sums = []
for p in range(num_places):
if config.representation_mode == config.DENSE:
nonzero = np.union1d(pre_matrix[p].getA1().nonzero()[0],
post_matrix[p].getA1().nonzero()[0])
elif config.representation_mode == config.SPARSE:
nonzero = np.union1d(
pre_matrix.getrow(p).nonzero()[1],
post_matrix.getrow(p).nonzero()[1])
incidence_row = [(post_matrix[p, t] - pre_matrix[p, t], variables[t])
for t in nonzero if t in subnet_transitions]
sum_elements = [
_expression(coeff, var) for coeff, var in incidence_row
if coeff != 0
]
curr_sum = z3.Sum(
sum_elements) if len(sum_elements) > 0 else z3.IntVal(0)
sums.append(curr_sum)
return sums