"""CSP (Constraint Satisfaction Problems) problems and solvers. (Chapter 6)"""
import itertools
import random
import re
import string
from collections import defaultdict, Counter
from functools import reduce
from operator import eq, neg
from sortedcontainers import SortedSet
import search
from utils import argmin_random_tie, count, first, extend
[docs]
class CSP(search.Problem):
"""This class describes finite-domain Constraint Satisfaction Problems.
A CSP is specified by the following inputs::
variables A list of variables; each is atomic (e.g. int or string).
domains A dict of {var:[possible_value, ...]} entries.
neighbors A dict of {var:[var,...]} that for each variable lists
the other variables that participate in constraints.
constraints A function f(A, a, B, b) that returns true if neighbors
A, B satisfy the constraint when they have values A=a, B=b
In the textbook and in most mathematical definitions, the
constraints are specified as explicit pairs of allowable values,
but the formulation here is easier to express and more compact for
most cases (for example, the n-Queens problem can be represented
in O(n) space using this notation, instead of O(n^4) for the
explicit representation). In terms of describing the CSP as a
problem, that's all there is.
However, the class also supports data structures and methods that help you
solve CSPs by calling a search function on the CSP. Methods and slots are
as follows, where the argument 'a' represents an assignment, which is a
dict of {var:val} entries::
assign(var, val, a) Assign a[var] = val; do other bookkeeping
unassign(var, a) Do del a[var], plus other bookkeeping
nconflicts(var, val, a) Return the number of other variables that
conflict with var=val
curr_domains[var] Slot: remaining consistent values for var
Used by constraint propagation routines.
The following methods are used only by graph_search and tree_search::
actions(state) Return a list of actions
result(state, action) Return a successor of state
goal_test(state) Return true if all constraints satisfied
The following are just for debugging purposes::
nassigns Slot: tracks the number of assignments made
display(a) Print a human-readable representation
"""
def __init__(self, variables, domains, neighbors, constraints):
"""Construct a CSP problem. If variables is empty, it becomes domains.keys()."""
super().__init__(())
variables = variables or list(domains.keys())
self.variables = variables
self.domains = domains
self.neighbors = neighbors
# store the constraint relation privately and expose it through the
# constraints() method below, which counts every consistency check
self.constraint_relation = constraints
self.curr_domains = None
self.nassigns = 0
self.nchecks = 0
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def constraints(self, A, a, B, b):
"""Return True if neighbors A, B satisfy the constraint when A=a, B=b.
Every call is tallied in self.nchecks, so callers can measure the number
of consistency checks performed (e.g. to reproduce the benchmarks in
Figure 6.1), just as self.nassigns tallies the assignments made."""
self.nchecks += 1
return self.constraint_relation(A, a, B, b)
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def assign(self, var, val, assignment):
"""Add {var: val} to assignment; Discard the old value if any."""
assignment[var] = val
self.nassigns += 1
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def unassign(self, var, assignment):
"""Remove {var: val} from assignment.
DO NOT call this if you are changing a variable to a new value;
just call assign for that."""
if var in assignment:
del assignment[var]
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def nconflicts(self, var, val, assignment):
"""Return the number of conflicts var=val has with other variables."""
# Subclasses may implement this more efficiently
def conflict(var2):
return var2 in assignment and not self.constraints(var, val, var2, assignment[var2])
return count(conflict(v) for v in self.neighbors[var])
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def count_lost_values(self, var, val, assignment):
"""Return how many values would be ruled out in the domains of the
unassigned neighbours of var if var were assigned val (the count used
by the least-constraining-value heuristic)."""
return count(not self.constraints(var, val, neighbor, dval)
for neighbor in self.neighbors[var] if neighbor not in assignment
for dval in self.domains[neighbor])
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def display(self, assignment):
"""Show a human-readable representation of the CSP."""
# Subclasses can print in a prettier way, or display with a GUI
print(assignment)
# These methods are for the tree and graph-search interface:
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def actions(self, state):
"""Return a list of applicable actions: non conflicting
assignments to an unassigned variable."""
if len(state) == len(self.variables):
return []
else:
assignment = dict(state)
var = first([v for v in self.variables if v not in assignment])
return [(var, val) for val in self.domains[var]
if self.nconflicts(var, val, assignment) == 0]
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def result(self, state, action):
"""Perform an action and return the new state."""
(var, val) = action
return state + ((var, val),)
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def goal_test(self, state):
"""The goal is to assign all variables, with all constraints satisfied."""
assignment = dict(state)
return (len(assignment) == len(self.variables)
and all(self.nconflicts(variables, assignment[variables], assignment) == 0
for variables in self.variables))
# These are for constraint propagation
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def support_pruning(self):
"""Make sure we can prune values from domains. (We want to pay
for this only if we use it.)"""
if self.curr_domains is None:
self.curr_domains = {v: list(self.domains[v]) for v in self.variables}
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def suppose(self, var, value):
"""Start accumulating inferences from assuming var=value."""
self.support_pruning()
removals = [(var, a) for a in self.curr_domains[var] if a != value]
self.curr_domains[var] = [value]
return removals
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def prune(self, var, value, removals):
"""Rule out var=value."""
self.curr_domains[var].remove(value)
if removals is not None:
removals.append((var, value))
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def choices(self, var):
"""Return all values for var that aren't currently ruled out."""
return (self.curr_domains or self.domains)[var]
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def infer_assignment(self):
"""Return the partial assignment implied by the current inferences."""
self.support_pruning()
return {v: self.curr_domains[v][0]
for v in self.variables if 1 == len(self.curr_domains[v])}
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def restore(self, removals):
"""Undo a supposition and all inferences from it."""
for B, b in removals:
self.curr_domains[B].append(b)
# This is for min_conflicts search
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def conflicted_vars(self, current):
"""Return a list of variables in current assignment that are in conflict"""
return [var for var in self.variables
if self.nconflicts(var, current[var], current) > 0]
# ______________________________________________________________________________
# Constraint Propagation with AC3
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def no_arc_heuristic(csp, queue):
"""Return the arc queue unchanged (no ordering heuristic for AC3)."""
return queue
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def dom_j_up(csp, queue):
"""Order the arc queue so arcs whose second variable has the smallest current
domain are popped first (a SortedSet keyed by the negated domain size)."""
return SortedSet(queue, key=lambda t: neg(len(csp.curr_domains[t[1]])))
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def AC3(csp, queue=None, removals=None, arc_heuristic=dom_j_up):
"""[Figure 6.3]"""
if queue is None:
queue = {(Xi, Xk) for Xi in csp.variables for Xk in csp.neighbors[Xi]}
csp.support_pruning()
queue = arc_heuristic(csp, queue)
checks = 0
while queue:
(Xi, Xj) = queue.pop()
revised, checks = revise(csp, Xi, Xj, removals, checks)
if revised:
if not csp.curr_domains[Xi]:
return False, checks # CSP is inconsistent
for Xk in csp.neighbors[Xi]:
if Xk != Xj:
queue.add((Xk, Xi))
return True, checks # CSP is arc-consistent
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def revise(csp, Xi, Xj, removals, checks=0):
"""Return true if we remove a value."""
revised = False
for x in csp.curr_domains[Xi][:]:
# If Xi=x conflicts with Xj=y for every possible y, eliminate Xi=x
# if all(not csp.constraints(Xi, x, Xj, y) for y in csp.curr_domains[Xj]):
conflict = True
for y in csp.curr_domains[Xj]:
if csp.constraints(Xi, x, Xj, y):
conflict = False
checks += 1
if not conflict:
break
if conflict:
csp.prune(Xi, x, removals)
revised = True
return revised, checks
# Constraint Propagation with AC3b: an improved version
# of AC3 with double-support domain-heuristic
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def AC3b(csp, queue=None, removals=None, arc_heuristic=dom_j_up):
"""An improved version of AC3 that uses double-support checks to share work
between an arc (Xi, Xj) and its reverse (Xj, Xi). Returns a tuple
(is_consistent, checks): False if a domain is wiped out, True if the CSP is
made arc-consistent, where checks counts the consistency checks performed."""
if queue is None:
queue = {(Xi, Xk) for Xi in csp.variables for Xk in csp.neighbors[Xi]}
csp.support_pruning()
queue = arc_heuristic(csp, queue)
checks = 0
while queue:
(Xi, Xj) = queue.pop()
# Si_p values are all known to be supported by Xj
# Sj_p values are all known to be supported by Xi
# Dj - Sj_p = Sj_u values are unknown, as yet, to be supported by Xi
Si_p, Sj_p, Sj_u, checks = partition(csp, Xi, Xj, checks)
if not Si_p:
return False, checks # CSP is inconsistent
revised = False
for x in set(csp.curr_domains[Xi]) - Si_p:
csp.prune(Xi, x, removals)
revised = True
if revised:
for Xk in csp.neighbors[Xi]:
if Xk != Xj:
queue.add((Xk, Xi))
if (Xj, Xi) in queue:
if isinstance(queue, set):
# or queue -= {(Xj, Xi)} or queue.remove((Xj, Xi))
queue.difference_update({(Xj, Xi)})
else:
queue.difference_update((Xj, Xi))
# the elements in D_j which are supported by Xi are given by the union of Sj_p with the set of those
# elements of Sj_u which further processing will show to be supported by some vi_p in Si_p
for vj_p in Sj_u:
for vi_p in Si_p:
conflict = True
if csp.constraints(Xj, vj_p, Xi, vi_p):
conflict = False
Sj_p.add(vj_p)
checks += 1
if not conflict:
break
revised = False
for x in set(csp.curr_domains[Xj]) - Sj_p:
csp.prune(Xj, x, removals)
revised = True
if revised:
for Xk in csp.neighbors[Xj]:
if Xk != Xi:
queue.add((Xk, Xj))
return True, checks # CSP is arc-consistent
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def partition(csp, Xi, Xj, checks=0):
"""Helper for AC3b: split the domains of Xi and Xj using double-support checks.
Returns (Si_p, Sj_p, Sj_u, checks) where Si_p is the set of Xi values supported
by Xj, Sj_p the set of Xj values already known to be supported by Xi, Sj_u the
remaining (as yet unconfirmed) Xj values, and checks the updated check count."""
Si_p = set()
Sj_p = set()
Sj_u = set(csp.curr_domains[Xj])
for vi_u in csp.curr_domains[Xi]:
conflict = True
# now, in order to establish support for a value vi_u in Di it seems better to try to find a support among
# the values in Sj_u first, because for each vj_u in Sj_u the check (vi_u, vj_u) is a double-support check
# and it is just as likely that any vj_u in Sj_u supports vi_u than it is that any vj_p in Sj_p does...
for vj_u in Sj_u - Sj_p:
# double-support check
if csp.constraints(Xi, vi_u, Xj, vj_u):
conflict = False
Si_p.add(vi_u)
Sj_p.add(vj_u)
checks += 1
if not conflict:
break
# ... and only if no support can be found among the elements in Sj_u, should the elements vj_p in Sj_p be used
# for single-support checks (vi_u, vj_p)
if conflict:
for vj_p in Sj_p:
# single-support check
if csp.constraints(Xi, vi_u, Xj, vj_p):
conflict = False
Si_p.add(vi_u)
checks += 1
if not conflict:
break
return Si_p, Sj_p, Sj_u - Sj_p, checks
# Constraint Propagation with AC4
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def AC4(csp, queue=None, removals=None, arc_heuristic=dom_j_up):
"""Enforce arc consistency using AC4, which keeps per-value support counters so
that pruning a value only triggers re-checks of the values it supported. Returns
a tuple (is_consistent, checks): False if a domain is wiped out, True if the CSP
is made arc-consistent, where checks counts the consistency checks performed."""
if queue is None:
queue = {(Xi, Xk) for Xi in csp.variables for Xk in csp.neighbors[Xi]}
csp.support_pruning()
queue = arc_heuristic(csp, queue)
support_counter = Counter()
variable_value_pairs_supported = defaultdict(set)
unsupported_variable_value_pairs = []
checks = 0
# construction and initialization of support sets
while queue:
(Xi, Xj) = queue.pop()
revised = False
for x in csp.curr_domains[Xi][:]:
for y in csp.curr_domains[Xj]:
if csp.constraints(Xi, x, Xj, y):
support_counter[(Xi, x, Xj)] += 1
variable_value_pairs_supported[(Xj, y)].add((Xi, x))
checks += 1
if support_counter[(Xi, x, Xj)] == 0:
csp.prune(Xi, x, removals)
revised = True
unsupported_variable_value_pairs.append((Xi, x))
if revised:
if not csp.curr_domains[Xi]:
return False, checks # CSP is inconsistent
# propagation of removed values
while unsupported_variable_value_pairs:
Xj, y = unsupported_variable_value_pairs.pop()
for Xi, x in variable_value_pairs_supported[(Xj, y)]:
revised = False
if x in csp.curr_domains[Xi][:]:
support_counter[(Xi, x, Xj)] -= 1
if support_counter[(Xi, x, Xj)] == 0:
csp.prune(Xi, x, removals)
revised = True
unsupported_variable_value_pairs.append((Xi, x))
if revised:
if not csp.curr_domains[Xi]:
return False, checks # CSP is inconsistent
return True, checks # CSP is arc-consistent
# ______________________________________________________________________________
# CSP Backtracking Search
# Variable ordering
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def first_unassigned_variable(assignment, csp):
"""The default variable order."""
return first([var for var in csp.variables if var not in assignment])
[docs]
def mrv(assignment, csp):
"""Minimum-remaining-values heuristic."""
return argmin_random_tie([v for v in csp.variables if v not in assignment],
key=lambda var: num_legal_values(csp, var, assignment))
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def num_legal_values(csp, var, assignment):
"""Return how many values are still legal for var (used by the mrv heuristic):
the size of its current domain if pruning is active, otherwise the number of
domain values that conflict with no other variable in the assignment."""
if csp.curr_domains:
return len(csp.curr_domains[var])
else:
return count(csp.nconflicts(var, val, assignment) == 0 for val in csp.domains[var])
# Value ordering
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def unordered_domain_values(var, assignment, csp):
"""The default value order."""
return csp.choices(var)
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def lcv(var, assignment, csp):
"""Least-constraining-values heuristic."""
return sorted(csp.choices(var), key=lambda val: csp.count_lost_values(var, val, assignment))
# Inference
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def no_inference(csp, var, value, assignment, removals):
"""The default inference step for backtracking: do nothing and always succeed."""
return True
[docs]
def forward_checking(csp, var, value, assignment, removals):
"""Prune neighbor values inconsistent with var=value."""
csp.support_pruning()
for B in csp.neighbors[var]:
if B not in assignment:
for b in csp.curr_domains[B][:]:
if not csp.constraints(var, value, B, b):
csp.prune(B, b, removals)
if not csp.curr_domains[B]:
return False
return True
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def mac(csp, var, value, assignment, removals, constraint_propagation=AC3b):
"""Maintain arc consistency."""
return constraint_propagation(csp, {(X, var) for X in csp.neighbors[var]}, removals)
# The search, proper
[docs]
def backtracking_search(csp, select_unassigned_variable=first_unassigned_variable,
order_domain_values=unordered_domain_values, inference=no_inference):
"""[Figure 6.5]"""
def backtrack(assignment):
if len(assignment) == len(csp.variables):
return assignment
var = select_unassigned_variable(assignment, csp)
for value in order_domain_values(var, assignment, csp):
if 0 == csp.nconflicts(var, value, assignment):
csp.assign(var, value, assignment)
removals = csp.suppose(var, value)
if inference(csp, var, value, assignment, removals):
result = backtrack(assignment)
if result is not None:
return result
csp.restore(removals)
csp.unassign(var, assignment)
return None
result = backtrack({})
assert result is None or csp.goal_test(result)
return result
# ______________________________________________________________________________
# Min-conflicts Hill Climbing search for CSPs
[docs]
def min_conflicts(csp, max_steps=100000):
"""Solve a CSP by stochastic Hill Climbing on the number of conflicts."""
# Generate a complete assignment for all variables (probably with conflicts)
csp.current = current = {}
for var in csp.variables:
val = min_conflicts_value(csp, var, current)
csp.assign(var, val, current)
# Now repeatedly choose a random conflicted variable and change it
for i in range(max_steps):
conflicted = csp.conflicted_vars(current)
if not conflicted:
return current
var = random.choice(conflicted)
val = min_conflicts_value(csp, var, current)
csp.assign(var, val, current)
return None
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def min_conflicts_value(csp, var, current):
"""Return the value that will give var the least number of conflicts.
If there is a tie, choose at random."""
return argmin_random_tie(csp.domains[var], key=lambda val: csp.nconflicts(var, val, current))
# ______________________________________________________________________________
[docs]
def tree_csp_solver(csp):
"""[Figure 6.11]"""
assignment = {}
root = csp.variables[0]
X, parent = topological_sort(csp, root)
csp.support_pruning()
for Xj in reversed(X[1:]):
if not make_arc_consistent(parent[Xj], Xj, csp):
return None
assignment[root] = csp.curr_domains[root][0]
for Xi in X[1:]:
assignment[Xi] = assign_value(parent[Xi], Xi, csp, assignment)
if not assignment[Xi]:
return None
return assignment
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def topological_sort(X, root):
"""Returns the topological sort of X starting from the root.
Input:
X is a list with the nodes of the graph
N is the dictionary with the neighbors of each node
root denotes the root of the graph.
Output:
stack is a list with the nodes topologically sorted
parents is a dictionary pointing to each node's parent
Other:
visited shows the state (visited - not visited) of nodes
"""
neighbors = X.neighbors
visited = defaultdict(lambda: False)
stack = []
parents = {}
build_topological(root, None, neighbors, visited, stack, parents)
return stack, parents
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def build_topological(node, parent, neighbors, visited, stack, parents):
"""Build the topological sort and the parents of each node in the graph."""
visited[node] = True
for n in neighbors[node]:
if not visited[n]:
build_topological(n, node, neighbors, visited, stack, parents)
parents[node] = parent
stack.insert(0, node)
[docs]
def make_arc_consistent(Xj, Xk, csp):
"""Make arc between parent (Xj) and child (Xk) consistent under the csp's constraints,
by removing the possible values of Xj that cause inconsistencies."""
# csp.curr_domains[Xj] = []
for val1 in csp.domains[Xj]:
keep = False # Keep or remove val1
for val2 in csp.domains[Xk]:
if csp.constraints(Xj, val1, Xk, val2):
# Found a consistent assignment for val1, keep it
keep = True
break
if not keep:
# Remove val1
csp.prune(Xj, val1, None)
return csp.curr_domains[Xj]
[docs]
def assign_value(Xj, Xk, csp, assignment):
"""Assign a value to Xk given Xj's (Xk's parent) assignment.
Return the first value that satisfies the constraints."""
parent_assignment = assignment[Xj]
for val in csp.curr_domains[Xk]:
if csp.constraints(Xj, parent_assignment, Xk, val):
return val
# No consistent assignment available
return None
# ______________________________________________________________________________
# Map Coloring CSP Problems
[docs]
class UniversalDict:
"""A universal dict maps any key to the same value. We use it here
as the domains dict for CSPs in which all variables have the same domain.
>>> d = UniversalDict(42)
>>> d['life']
42
"""
def __init__(self, value): self.value = value
def __getitem__(self, key): return self.value
def __repr__(self): return '{{Any: {0!r}}}'.format(self.value)
[docs]
def different_values_constraint(A, a, B, b):
"""A constraint saying two neighboring variables must differ in value."""
return a != b
[docs]
def MapColoringCSP(colors, neighbors):
"""Make a CSP for the problem of coloring a map with different colors
for any two adjacent regions. Arguments are a list of colors, and a
dict of {region: [neighbor,...]} entries. This dict may also be
specified as a string of the form defined by parse_neighbors."""
if isinstance(neighbors, str):
neighbors = parse_neighbors(neighbors)
return CSP(list(neighbors.keys()), UniversalDict(colors), neighbors, different_values_constraint)
[docs]
def parse_neighbors(neighbors):
"""Convert a string of the form 'X: Y Z; Y: Z' into a dict mapping
regions to neighbors. The syntax is a region name followed by a ':'
followed by zero or more region names, followed by ';', repeated for
each region name. If you say 'X: Y' you don't need 'Y: X'.
>>> parse_neighbors('X: Y Z; Y: Z') == {'Y': ['X', 'Z'], 'X': ['Y', 'Z'], 'Z': ['X', 'Y']}
True
"""
dic = defaultdict(list)
specs = [spec.split(':') for spec in neighbors.split(';')]
for (A, Aneighbors) in specs:
A = A.strip()
for B in Aneighbors.split():
dic[A].append(B)
dic[B].append(A)
return dic
australia_csp = MapColoringCSP(list('RGB'), """SA: WA NT Q NSW V; NT: WA Q; NSW: Q V; T: """)
usa_csp = MapColoringCSP(list('RGBY'),
"""WA: OR ID; OR: ID NV CA; CA: NV AZ; NV: ID UT AZ; ID: MT WY UT;
UT: WY CO AZ; MT: ND SD WY; WY: SD NE CO; CO: NE KA OK NM; NM: OK TX AZ;
ND: MN SD; SD: MN IA NE; NE: IA MO KA; KA: MO OK; OK: MO AR TX;
TX: AR LA; MN: WI IA; IA: WI IL MO; MO: IL KY TN AR; AR: MS TN LA;
LA: MS; WI: MI IL; IL: IN KY; IN: OH KY; MS: TN AL; AL: TN GA FL;
MI: OH IN; OH: PA WV KY; KY: WV VA TN; TN: VA NC GA; GA: NC SC FL;
PA: NY NJ DE MD WV; WV: MD VA; VA: MD DC NC; NC: SC; NY: VT MA CT NJ;
NJ: DE; DE: MD; MD: DC; VT: NH MA; MA: NH RI CT; CT: RI; ME: NH;
HI: ; AK: """)
france_csp = MapColoringCSP(list('RGBY'),
"""AL: LO FC; AQ: MP LI PC; AU: LI CE BO RA LR MP; BO: CE IF CA FC RA
AU; BR: NB PL; CA: IF PI LO FC BO; CE: PL NB NH IF BO AU LI PC; FC: BO
CA LO AL RA; IF: NH PI CA BO CE; LI: PC CE AU MP AQ; LO: CA AL FC; LR:
MP AU RA PA; MP: AQ LI AU LR; NB: NH CE PL BR; NH: PI IF CE NB; NO:
PI; PA: LR RA; PC: PL CE LI AQ; PI: NH NO CA IF; PL: BR NB CE PC; RA:
AU BO FC PA LR""")
# ______________________________________________________________________________
# n-Queens Problem
[docs]
def queen_constraint(A, a, B, b):
"""Constraint is satisfied (true) if A, B are really the same variable,
or if they are not in the same row, down diagonal, or up diagonal."""
return A == B or (a != b and A + a != B + b and A - a != B - b)
[docs]
class NQueensCSP(CSP):
r"""
Make a CSP for the nQueens problem for search with min_conflicts.
Suitable for large n, it uses only data structures of size O(n).
Think of placing queens one per column, from left to right.
That means position (x, y) represents (var, val) in the CSP.
The main structures are three arrays to count queens that could conflict::
rows[i] Number of queens in the ith row (i.e. val == i)
downs[i] Number of queens in the \ diagonal
such that their (x, y) coordinates sum to i
ups[i] Number of queens in the / diagonal
such that their (x, y) coordinates have x-y+n-1 = i
We increment/decrement these counts each time a queen is placed/moved from
a row/diagonal. So moving is O(1), as is nconflicts. But choosing
a variable, and a best value for the variable, are each O(n).
If you want, you can keep track of conflicted variables, then variable
selection will also be O(1).
>>> len(backtracking_search(NQueensCSP(8)))
8
"""
def __init__(self, n):
"""Initialize data structures for n Queens."""
CSP.__init__(self, list(range(n)), UniversalDict(list(range(n))),
UniversalDict(list(range(n))), queen_constraint)
self.rows = [0] * n
self.ups = [0] * (2 * n - 1)
self.downs = [0] * (2 * n - 1)
[docs]
def nconflicts(self, var, val, assignment):
"""The number of conflicts, as recorded with each assignment.
Count conflicts in row and in up, down diagonals. If there
is a queen there, it can't conflict with itself, so subtract 3."""
n = len(self.variables)
c = self.rows[val] + self.downs[var + val] + self.ups[var - val + n - 1]
if assignment.get(var, None) == val:
c -= 3
return c
[docs]
def assign(self, var, val, assignment):
"""Assign var, and keep track of conflicts."""
old_val = assignment.get(var, None)
if val != old_val:
if old_val is not None: # Remove old val if there was one
self.record_conflict(assignment, var, old_val, -1)
self.record_conflict(assignment, var, val, +1)
CSP.assign(self, var, val, assignment)
[docs]
def unassign(self, var, assignment):
"""Remove var from assignment (if it is there) and track conflicts."""
if var in assignment:
self.record_conflict(assignment, var, assignment[var], -1)
CSP.unassign(self, var, assignment)
[docs]
def record_conflict(self, assignment, var, val, delta):
"""Record conflicts caused by addition or deletion of a Queen."""
n = len(self.variables)
self.rows[val] += delta
self.downs[var + val] += delta
self.ups[var - val + n - 1] += delta
[docs]
def display(self, assignment):
"""Print the queens and the nconflicts values (for debugging)."""
n = len(self.variables)
for val in range(n):
for var in range(n):
if assignment.get(var, '') == val:
ch = 'Q'
elif (var + val) % 2 == 0:
ch = '.'
else:
ch = '-'
print(ch, end=' ')
print(' ', end=' ')
for var in range(n):
if assignment.get(var, '') == val:
ch = '*'
else:
ch = ' '
print(str(self.nconflicts(var, val, assignment)) + ch, end=' ')
print()
# ______________________________________________________________________________
# Sudoku
[docs]
def flatten(seqs):
"""Concatenate a sequence of lists into a single flat list."""
return sum(seqs, [])
easy1 = '..3.2.6..9..3.5..1..18.64....81.29..7.......8..67.82....26.95..8..2.3..9..5.1.3..'
harder1 = '4173698.5.3..........7......2.....6.....8.4......1.......6.3.7.5..2.....1.4......'
_R3 = list(range(3))
_CELL = itertools.count().__next__
_BGRID = [[[[_CELL() for x in _R3] for y in _R3] for bx in _R3] for by in _R3]
_BOXES = flatten([list(map(flatten, brow)) for brow in _BGRID])
_ROWS = flatten([list(map(flatten, zip(*brow))) for brow in _BGRID])
_COLS = list(zip(*_ROWS))
_NEIGHBORS = {v: set() for v in flatten(_ROWS)}
for unit in map(set, _BOXES + _ROWS + _COLS):
for v in unit:
_NEIGHBORS[v].update(unit - {v})
[docs]
class Sudoku(CSP):
"""
A Sudoku problem.
The box grid is a 3x3 array of boxes, each a 3x3 array of cells.
Each cell holds a digit in 1..9. In each box, all digits are
different; the same for each row and column as a 9x9 grid.
>>> e = Sudoku(easy1)
>>> e.display(e.infer_assignment())
. . 3 | . 2 . | 6 . .
9 . . | 3 . 5 | . . 1
. . 1 | 8 . 6 | 4 . .
------+-------+------
. . 8 | 1 . 2 | 9 . .
7 . . | . . . | . . 8
. . 6 | 7 . 8 | 2 . .
------+-------+------
. . 2 | 6 . 9 | 5 . .
8 . . | 2 . 3 | . . 9
. . 5 | . 1 . | 3 . .
>>> AC3(e) # doctest: +ELLIPSIS
(True, ...)
>>> e.display(e.infer_assignment())
4 8 3 | 9 2 1 | 6 5 7
9 6 7 | 3 4 5 | 8 2 1
2 5 1 | 8 7 6 | 4 9 3
------+-------+------
5 4 8 | 1 3 2 | 9 7 6
7 2 9 | 5 6 4 | 1 3 8
1 3 6 | 7 9 8 | 2 4 5
------+-------+------
3 7 2 | 6 8 9 | 5 1 4
8 1 4 | 2 5 3 | 7 6 9
6 9 5 | 4 1 7 | 3 8 2
>>> h = Sudoku(harder1)
>>> backtracking_search(h, select_unassigned_variable=mrv, inference=forward_checking) is not None
True
"""
R3 = _R3
Cell = _CELL
bgrid = _BGRID
boxes = _BOXES
rows = _ROWS
cols = _COLS
neighbors = _NEIGHBORS
def __init__(self, grid):
"""Build a Sudoku problem from a string representing the grid:
the digits 1-9 denote a filled cell, '.' or '0' an empty one;
other characters are ignored."""
squares = iter(re.findall(r'\d|\.', grid))
domains = {var: [ch] if ch in '123456789' else '123456789'
for var, ch in zip(flatten(self.rows), squares)}
for _ in squares:
raise ValueError("Not a Sudoku grid", grid) # Too many squares
CSP.__init__(self, None, domains, self.neighbors, different_values_constraint)
[docs]
def display(self, assignment):
"""Print the Sudoku grid for the given assignment as a 9x9 board."""
def show_box(box): return [' '.join(map(show_cell, row)) for row in box]
def show_cell(cell): return str(assignment.get(cell, '.'))
def abut(lines1, lines2): return list(
map(' | '.join, list(zip(lines1, lines2))))
print('\n------+-------+------\n'.join(
'\n'.join(reduce(
abut, map(show_box, brow))) for brow in self.bgrid))
# ______________________________________________________________________________
# The Zebra Puzzle
[docs]
def Zebra():
"""Return an instance of the Zebra Puzzle."""
Colors = 'Red Yellow Blue Green Ivory'.split()
Pets = 'Dog Fox Snails Horse Zebra'.split()
Drinks = 'OJ Tea Coffee Milk Water'.split()
Countries = 'Englishman Spaniard Norwegian Ukrainian Japanese'.split()
Smokes = 'Kools Chesterfields Winston LuckyStrike Parliaments'.split()
variables = Colors + Pets + Drinks + Countries + Smokes
domains = {}
for var in variables:
domains[var] = list(range(1, 6))
domains['Norwegian'] = [1]
domains['Milk'] = [3]
neighbors = parse_neighbors("""Englishman: Red;
Spaniard: Dog; Kools: Yellow; Chesterfields: Fox;
Norwegian: Blue; Winston: Snails; LuckyStrike: OJ;
Ukrainian: Tea; Japanese: Parliaments; Kools: Horse;
Coffee: Green; Green: Ivory""")
for type in [Colors, Pets, Drinks, Countries, Smokes]:
for A in type:
for B in type:
if A != B:
if B not in neighbors[A]:
neighbors[A].append(B)
if A not in neighbors[B]:
neighbors[B].append(A)
def zebra_constraint(A, a, B, b, recurse=0):
same = (a == b)
next_to = abs(a - b) == 1
if A == 'Englishman' and B == 'Red':
return same
if A == 'Spaniard' and B == 'Dog':
return same
if A == 'Chesterfields' and B == 'Fox':
return next_to
if A == 'Norwegian' and B == 'Blue':
return next_to
if A == 'Kools' and B == 'Yellow':
return same
if A == 'Winston' and B == 'Snails':
return same
if A == 'LuckyStrike' and B == 'OJ':
return same
if A == 'Ukrainian' and B == 'Tea':
return same
if A == 'Japanese' and B == 'Parliaments':
return same
if A == 'Kools' and B == 'Horse':
return next_to
if A == 'Coffee' and B == 'Green':
return same
if A == 'Green' and B == 'Ivory':
return a - 1 == b
if recurse == 0:
return zebra_constraint(B, b, A, a, 1)
if ((A in Colors and B in Colors) or
(A in Pets and B in Pets) or
(A in Drinks and B in Drinks) or
(A in Countries and B in Countries) or
(A in Smokes and B in Smokes)):
return not same
raise Exception('error')
return CSP(variables, domains, neighbors, zebra_constraint)
[docs]
def solve_zebra(algorithm=min_conflicts, **args):
"""Solve the Zebra Puzzle with the given CSP algorithm, print which person is in
each house, and return a tuple (zebra_owner_house, water_drinker_house, nassigns,
full_assignment)."""
z = Zebra()
ans = algorithm(z, **args)
for h in range(1, 6):
print('House', h, end=' ')
for (var, val) in ans.items():
if val == h:
print(var, end=' ')
print()
return ans['Zebra'], ans['Water'], z.nassigns, ans
# ______________________________________________________________________________
# n-ary Constraint Satisfaction Problem
[docs]
class NaryCSP:
"""
A nary-CSP consists of:
domains : a dictionary that maps each variable to its domain
constraints : a list of constraints
variables : a set of variables
var_to_const: a variable to set of constraints dictionary
"""
def __init__(self, domains, constraints):
"""Domains is a variable:domain dictionary
constraints is a list of constraints
"""
self.variables = set(domains)
self.domains = domains
self.constraints = constraints
self.var_to_const = {var: set() for var in self.variables}
for con in constraints:
for var in con.scope:
self.var_to_const[var].add(con)
def __str__(self):
"""String representation of CSP"""
return str(self.domains)
[docs]
def display(self, assignment=None):
"""More detailed string representation of CSP"""
if assignment is None:
assignment = {}
print(assignment)
[docs]
def consistent(self, assignment):
"""assignment is a variable:value dictionary
returns True if all of the constraints that can be evaluated
evaluate to True given assignment.
"""
return all(con.holds(assignment)
for con in self.constraints
if all(v in assignment for v in con.scope))
[docs]
class Constraint:
"""
A Constraint consists of:
scope : a tuple of variables
condition: a function that can applied to a tuple of values
for the variables.
"""
def __init__(self, scope, condition):
self.scope = scope
self.condition = condition
def __repr__(self):
return self.condition.__name__ + str(self.scope)
[docs]
def holds(self, assignment):
"""Returns the value of Constraint con evaluated in assignment.
precondition: all variables are assigned in assignment
"""
return self.condition(*tuple(assignment[v] for v in self.scope))
[docs]
def all_diff_constraint(*values):
"""Returns True if all values are different, False otherwise"""
return len(values) is len(set(values))
[docs]
def is_word_constraint(words):
"""Returns True if the letters concatenated form a word in words, False otherwise"""
def isw(*letters):
return "".join(letters) in words
return isw
[docs]
def meet_at_constraint(p1, p2):
"""Returns a function that is True when the words meet at the positions (p1, p2), False otherwise"""
def meets(w1, w2):
return w1[p1] == w2[p2]
meets.__name__ = "meet_at(" + str(p1) + ',' + str(p2) + ')'
return meets
[docs]
def adjacent_constraint(x, y):
"""Returns True if x and y are adjacent numbers, False otherwise"""
return abs(x - y) == 1
[docs]
def sum_constraint(n):
"""Returns a function that is True when the the sum of all values is n, False otherwise"""
def sumv(*values):
return sum(values) is n
sumv.__name__ = str(n) + "==sum"
return sumv
[docs]
def is_constraint(val):
"""Returns a function that is True when x is equal to val, False otherwise"""
def isv(x):
return val == x
isv.__name__ = str(val) + "=="
return isv
[docs]
def ne_constraint(val):
"""Returns a function that is True when x is not equal to val, False otherwise"""
def nev(x):
return val != x
nev.__name__ = str(val) + "!="
return nev
[docs]
def no_heuristic(to_do):
"""Return the to_do arc set unchanged (no ordering heuristic for GAC)."""
return to_do
[docs]
def sat_up(to_do):
"""Order the to_do arc set so that (variable, constraint) pairs whose constraint
has the smallest scope are processed first (a SortedSet keyed by 1 / scope size)."""
return SortedSet(to_do, key=lambda t: 1 / len([var for var in t[1].scope]))
[docs]
class ACSolver:
"""Solves a CSP with arc consistency and domain splitting"""
def __init__(self, csp):
"""a CSP solver that uses arc consistency
* csp is the CSP to be solved
"""
self.csp = csp
[docs]
def GAC(self, orig_domains=None, to_do=None, arc_heuristic=sat_up):
"""
Makes this CSP arc-consistent using Generalized Arc Consistency
orig_domains: is the original domains
to_do : is a set of (variable,constraint) pairs
returns the reduced domains (an arc-consistent variable:domain dictionary)
"""
if orig_domains is None:
orig_domains = self.csp.domains
if to_do is None:
to_do = {(var, const) for const in self.csp.constraints for var in const.scope}
else:
to_do = to_do.copy()
domains = orig_domains.copy()
to_do = arc_heuristic(to_do)
checks = 0
while to_do:
var, const = to_do.pop()
other_vars = [ov for ov in const.scope if ov != var]
new_domain = set()
if len(other_vars) == 0:
for val in domains[var]:
if const.holds({var: val}):
new_domain.add(val)
checks += 1
# new_domain = {val for val in domains[var]
# if const.holds({var: val})}
elif len(other_vars) == 1:
other = other_vars[0]
for val in domains[var]:
for other_val in domains[other]:
checks += 1
if const.holds({var: val, other: other_val}):
new_domain.add(val)
break
# new_domain = {val for val in domains[var]
# if any(const.holds({var: val, other: other_val})
# for other_val in domains[other])}
else: # general case
for val in domains[var]:
holds, checks = self.any_holds(domains, const, {var: val}, other_vars, checks=checks)
if holds:
new_domain.add(val)
# new_domain = {val for val in domains[var]
# if self.any_holds(domains, const, {var: val}, other_vars)}
if new_domain != domains[var]:
domains[var] = new_domain
if not new_domain:
return False, domains, checks
add_to_do = self.new_to_do(var, const).difference(to_do)
to_do |= add_to_do
return True, domains, checks
[docs]
def new_to_do(self, var, const):
"""
Returns new elements to be added to to_do after assigning
variable var in constraint const.
"""
return {(nvar, nconst) for nconst in self.csp.var_to_const[var]
if nconst != const
for nvar in nconst.scope
if nvar != var}
[docs]
def any_holds(self, domains, const, env, other_vars, ind=0, checks=0):
"""
Returns True if Constraint const holds for an assignment
that extends env with the variables in other_vars[ind:]
env is a dictionary
Warning: this has side effects and changes the elements of env
"""
if ind == len(other_vars):
return const.holds(env), checks + 1
else:
var = other_vars[ind]
for val in domains[var]:
# env = dict_union(env, {var:val}) # no side effects
env[var] = val
holds, checks = self.any_holds(domains, const, env, other_vars, ind + 1, checks)
if holds:
return True, checks
return False, checks
[docs]
def domain_splitting(self, domains=None, to_do=None, arc_heuristic=sat_up):
"""
Return a solution to the current CSP or False if there are no solutions
to_do is the list of arcs to check
"""
if domains is None:
domains = self.csp.domains
consistency, new_domains, _ = self.GAC(domains, to_do, arc_heuristic)
if not consistency:
return False
elif all(len(new_domains[var]) == 1 for var in domains):
return {var: first(new_domains[var]) for var in domains}
else:
var = first(x for x in self.csp.variables if len(new_domains[x]) > 1)
if var:
dom1, dom2 = partition_domain(new_domains[var])
new_doms1 = extend(new_domains, var, dom1)
new_doms2 = extend(new_domains, var, dom2)
to_do = self.new_to_do(var, None)
return self.domain_splitting(new_doms1, to_do, arc_heuristic) or \
self.domain_splitting(new_doms2, to_do, arc_heuristic)
[docs]
def partition_domain(dom):
"""Partitions domain dom into two"""
split = len(dom) // 2
dom1 = set(list(dom)[:split])
dom2 = dom - dom1
return dom1, dom2
[docs]
class ACSearchSolver(search.Problem):
"""A search problem with arc consistency and domain splitting
A node is a CSP"""
def __init__(self, csp, arc_heuristic=sat_up):
self.cons = ACSolver(csp)
consistency, self.domains, _ = self.cons.GAC(arc_heuristic=arc_heuristic)
if not consistency:
raise Exception('CSP is inconsistent')
self.heuristic = arc_heuristic
super().__init__(self.domains)
[docs]
def goal_test(self, node):
"""Node is a goal if all domains have 1 element"""
return all(len(node[var]) == 1 for var in node)
[docs]
def actions(self, state):
"""Return the successor states obtained by splitting some still-multi-valued
variable's domain in two and enforcing GAC on each half; only arc-consistent
results are kept."""
var = first(x for x in state if len(state[x]) > 1)
neighs = []
if var:
dom1, dom2 = partition_domain(state[var])
to_do = self.cons.new_to_do(var, None)
for dom in [dom1, dom2]:
new_domains = extend(state, var, dom)
consistency, cons_doms, _ = self.cons.GAC(new_domains, to_do, self.heuristic)
if consistency:
neighs.append(cons_doms)
return neighs
[docs]
def result(self, state, action):
"""Return the successor state, which is just the chosen reduced domains."""
return action
[docs]
def ac_solver(csp, arc_heuristic=sat_up):
"""Arc consistency (domain splitting interface)"""
return ACSolver(csp).domain_splitting(arc_heuristic=arc_heuristic)
[docs]
def ac_search_solver(csp, arc_heuristic=sat_up):
"""Arc consistency (search interface)"""
from search import depth_first_tree_search
solution = None
try:
solution = depth_first_tree_search(ACSearchSolver(csp, arc_heuristic=arc_heuristic)).state
except:
return solution
if solution:
return {var: first(solution[var]) for var in solution}
# ______________________________________________________________________________
# Crossword Problem
csp_crossword = NaryCSP({'one_across': {'ant', 'big', 'bus', 'car', 'has'},
'one_down': {'book', 'buys', 'hold', 'lane', 'year'},
'two_down': {'ginger', 'search', 'symbol', 'syntax'},
'three_across': {'book', 'buys', 'hold', 'land', 'year'},
'four_across': {'ant', 'big', 'bus', 'car', 'has'}},
[Constraint(('one_across', 'one_down'), meet_at_constraint(0, 0)),
Constraint(('one_across', 'two_down'), meet_at_constraint(2, 0)),
Constraint(('three_across', 'two_down'), meet_at_constraint(2, 2)),
Constraint(('three_across', 'one_down'), meet_at_constraint(0, 2)),
Constraint(('four_across', 'two_down'), meet_at_constraint(0, 4))])
crossword1 = [['_', '_', '_', '*', '*'],
['_', '*', '_', '*', '*'],
['_', '_', '_', '_', '*'],
['_', '*', '_', '*', '*'],
['*', '*', '_', '_', '_'],
['*', '*', '_', '*', '*']]
words1 = {'ant', 'big', 'bus', 'car', 'has', 'book', 'buys', 'hold',
'lane', 'year', 'ginger', 'search', 'symbol', 'syntax'}
[docs]
class Crossword(NaryCSP):
"""An n-ary CSP for a crossword puzzle. The puzzle is a grid where '_' marks a
writable cell and '*' a blocked cell; each maximal run of two or more cells (across
or down) becomes a variable sequence constrained to spell one of the given words."""
def __init__(self, puzzle, words):
domains = {}
constraints = []
for i, line in enumerate(puzzle):
scope = []
for j, element in enumerate(line):
if element == '_':
var = "p" + str(j) + str(i)
domains[var] = list(string.ascii_lowercase)
scope.append(var)
else:
if len(scope) > 1:
constraints.append(Constraint(tuple(scope), is_word_constraint(words)))
scope.clear()
if len(scope) > 1:
constraints.append(Constraint(tuple(scope), is_word_constraint(words)))
puzzle_t = list(map(list, zip(*puzzle)))
for i, line in enumerate(puzzle_t):
scope = []
for j, element in enumerate(line):
if element == '_':
scope.append("p" + str(i) + str(j))
else:
if len(scope) > 1:
constraints.append(Constraint(tuple(scope), is_word_constraint(words)))
scope.clear()
if len(scope) > 1:
constraints.append(Constraint(tuple(scope), is_word_constraint(words)))
super().__init__(domains, constraints)
self.puzzle = puzzle
[docs]
def display(self, assignment=None):
"""Print the crossword grid, filling in letters from assignment where known."""
for i, line in enumerate(self.puzzle):
puzzle = ""
for j, element in enumerate(line):
if element == '*':
puzzle += "[*] "
else:
var = "p" + str(j) + str(i)
if assignment is not None:
if isinstance(assignment[var], set) and len(assignment[var]) == 1:
puzzle += "[" + str(first(assignment[var])).upper() + "] "
elif isinstance(assignment[var], str):
puzzle += "[" + str(assignment[var]).upper() + "] "
else:
puzzle += "[_] "
else:
puzzle += "[_] "
print(puzzle)
# ______________________________________________________________________________
# Kakuro Problem
# difficulty 0
kakuro1 = [['*', '*', '*', [6, ''], [3, '']],
['*', [4, ''], [3, 3], '_', '_'],
[['', 10], '_', '_', '_', '_'],
[['', 3], '_', '_', '*', '*']]
# difficulty 0
kakuro2 = [
['*', [10, ''], [13, ''], '*'],
[['', 3], '_', '_', [13, '']],
[['', 12], '_', '_', '_'],
[['', 21], '_', '_', '_']]
# difficulty 1
kakuro3 = [
['*', [17, ''], [28, ''], '*', [42, ''], [22, '']],
[['', 9], '_', '_', [31, 14], '_', '_'],
[['', 20], '_', '_', '_', '_', '_'],
['*', ['', 30], '_', '_', '_', '_'],
['*', [22, 24], '_', '_', '_', '*'],
[['', 25], '_', '_', '_', '_', [11, '']],
[['', 20], '_', '_', '_', '_', '_'],
[['', 14], '_', '_', ['', 17], '_', '_']]
# difficulty 2
kakuro4 = [
['*', '*', '*', '*', '*', [4, ''], [24, ''], [11, ''], '*', '*', '*', [11, ''], [17, ''], '*', '*'],
['*', '*', '*', [17, ''], [11, 12], '_', '_', '_', '*', '*', [24, 10], '_', '_', [11, ''], '*'],
['*', [4, ''], [16, 26], '_', '_', '_', '_', '_', '*', ['', 20], '_', '_', '_', '_', [16, '']],
[['', 20], '_', '_', '_', '_', [24, 13], '_', '_', [16, ''], ['', 12], '_', '_', [23, 10], '_', '_'],
[['', 10], '_', '_', [24, 12], '_', '_', [16, 5], '_', '_', [16, 30], '_', '_', '_', '_', '_'],
['*', '*', [3, 26], '_', '_', '_', '_', ['', 12], '_', '_', [4, ''], [16, 14], '_', '_', '*'],
['*', ['', 8], '_', '_', ['', 15], '_', '_', [34, 26], '_', '_', '_', '_', '_', '*', '*'],
['*', ['', 11], '_', '_', [3, ''], [17, ''], ['', 14], '_', '_', ['', 8], '_', '_', [7, ''], [17, ''], '*'],
['*', '*', '*', [23, 10], '_', '_', [3, 9], '_', '_', [4, ''], [23, ''], ['', 13], '_', '_', '*'],
['*', '*', [10, 26], '_', '_', '_', '_', '_', ['', 7], '_', '_', [30, 9], '_', '_', '*'],
['*', [17, 11], '_', '_', [11, ''], [24, 8], '_', '_', [11, 21], '_', '_', '_', '_', [16, ''], [17, '']],
[['', 29], '_', '_', '_', '_', '_', ['', 7], '_', '_', [23, 14], '_', '_', [3, 17], '_', '_'],
[['', 10], '_', '_', [3, 10], '_', '_', '*', ['', 8], '_', '_', [4, 25], '_', '_', '_', '_'],
['*', ['', 16], '_', '_', '_', '_', '*', ['', 23], '_', '_', '_', '_', '_', '*', '*'],
['*', '*', ['', 6], '_', '_', '*', '*', ['', 15], '_', '_', '_', '*', '*', '*', '*']]
[docs]
class Kakuro(NaryCSP):
"""An n-ary CSP for a Kakuro puzzle. Each '_' cell is a variable with domain 1..9;
every horizontal or vertical run of cells must contain distinct digits that sum to
the clue given in the adjacent '[down, right]' header cell."""
def __init__(self, puzzle):
variables = []
for i, line in enumerate(puzzle):
# print line
for j, element in enumerate(line):
if element == '_':
var1 = str(i)
if len(var1) == 1:
var1 = "0" + var1
var2 = str(j)
if len(var2) == 1:
var2 = "0" + var2
variables.append("X" + var1 + var2)
domains = {}
for var in variables:
domains[var] = set(range(1, 10))
constraints = []
for i, line in enumerate(puzzle):
for j, element in enumerate(line):
if element != '_' and element != '*':
# down - column
if element[0] != '':
x = []
for k in range(i + 1, len(puzzle)):
if puzzle[k][j] != '_':
break
var1 = str(k)
if len(var1) == 1:
var1 = "0" + var1
var2 = str(j)
if len(var2) == 1:
var2 = "0" + var2
x.append("X" + var1 + var2)
constraints.append(Constraint(x, sum_constraint(element[0])))
constraints.append(Constraint(x, all_diff_constraint))
# right - line
if element[1] != '':
x = []
for k in range(j + 1, len(puzzle[i])):
if puzzle[i][k] != '_':
break
var1 = str(i)
if len(var1) == 1:
var1 = "0" + var1
var2 = str(k)
if len(var2) == 1:
var2 = "0" + var2
x.append("X" + var1 + var2)
constraints.append(Constraint(x, sum_constraint(element[1])))
constraints.append(Constraint(x, all_diff_constraint))
super().__init__(domains, constraints)
self.puzzle = puzzle
[docs]
def display(self, assignment=None):
"""Print the Kakuro grid, showing clue cells and filling in digits from
assignment where known."""
for i, line in enumerate(self.puzzle):
puzzle = ""
for j, element in enumerate(line):
if element == '*':
puzzle += "[*]\t"
elif element == '_':
var1 = str(i)
if len(var1) == 1:
var1 = "0" + var1
var2 = str(j)
if len(var2) == 1:
var2 = "0" + var2
var = "X" + var1 + var2
if assignment is not None:
if isinstance(assignment[var], set) and len(assignment[var]) == 1:
puzzle += "[" + str(first(assignment[var])) + "]\t"
elif isinstance(assignment[var], int):
puzzle += "[" + str(assignment[var]) + "]\t"
else:
puzzle += "[_]\t"
else:
puzzle += "[_]\t"
else:
puzzle += str(element[0]) + "\\" + str(element[1]) + "\t"
print(puzzle)
# ______________________________________________________________________________
# Cryptarithmetic Problem
# [Figure 6.2]
# T W O + T W O = F O U R
two_two_four = NaryCSP({'T': set(range(1, 10)), 'F': set(range(1, 10)),
'W': set(range(0, 10)), 'O': set(range(0, 10)), 'U': set(range(0, 10)), 'R': set(range(0, 10)),
'C1': set(range(0, 2)), 'C2': set(range(0, 2)), 'C3': set(range(0, 2))},
[Constraint(('T', 'F', 'W', 'O', 'U', 'R'), all_diff_constraint),
Constraint(('O', 'R', 'C1'), lambda o, r, c1: o + o == r + 10 * c1),
Constraint(('W', 'U', 'C1', 'C2'), lambda w, u, c1, c2: c1 + w + w == u + 10 * c2),
Constraint(('T', 'O', 'C2', 'C3'), lambda t, o, c2, c3: c2 + t + t == o + 10 * c3),
Constraint(('F', 'C3'), eq)])
# S E N D + M O R E = M O N E Y
send_more_money = NaryCSP({'S': set(range(1, 10)), 'M': set(range(1, 10)),
'E': set(range(0, 10)), 'N': set(range(0, 10)), 'D': set(range(0, 10)),
'O': set(range(0, 10)), 'R': set(range(0, 10)), 'Y': set(range(0, 10)),
'C1': set(range(0, 2)), 'C2': set(range(0, 2)), 'C3': set(range(0, 2)),
'C4': set(range(0, 2))},
[Constraint(('S', 'E', 'N', 'D', 'M', 'O', 'R', 'Y'), all_diff_constraint),
Constraint(('D', 'E', 'Y', 'C1'), lambda d, e, y, c1: d + e == y + 10 * c1),
Constraint(('N', 'R', 'E', 'C1', 'C2'), lambda n, r, e, c1, c2: c1 + n + r == e + 10 * c2),
Constraint(('E', 'O', 'N', 'C2', 'C3'), lambda e, o, n, c2, c3: c2 + e + o == n + 10 * c3),
Constraint(('S', 'M', 'O', 'C3', 'C4'), lambda s, m, o, c3, c4: c3 + s + m == o + 10 * c4),
Constraint(('M', 'C4'), eq)])