"""CSP (Constraint Satisfaction Problems) problems and solvers. (Chapter 5).""" from __future__ import generators from utils import * import search import types class CSP(search.Problem): """This class describes finite-domain Constraint Satisfaction Problems. A CSP is specified by the following three inputs: vars 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: succ() Return a list of (action, state) pairs goal_test(a) 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, vars, domains, neighbors, constraints): "Construct a CSP problem. If vars is empty, it becomes domains.keys()." vars = vars or domains.keys() update(self, vars=vars, domains=domains, neighbors=neighbors, constraints=constraints, initial={}, curr_domains=None, pruned=None, nassigns=0) def assign(self, var, val, assignment): """Add {var: val} to assignment; Discard the old value if any. Do bookkeeping for curr_domains and nassigns.""" self.nassigns += 1 assignment[var] = val if self.curr_domains: if self.fc: self.forward_check(var, val, assignment) if self.mac: AC3(self, [(Xk, var) for Xk in self.neighbors[var]]) def unassign(self, var, assignment): """Remove {var: val} from assignment; that is backtrack. DO NOT call this if you are changing a variable to a new value; just call assign for that.""" if var in assignment: # Reset the curr_domain to be the full original domain if self.curr_domains: self.curr_domains[var] = self.domains[var][:] del assignment[var] 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): val2 = assignment.get(var2, None) return val2 != None and not self.constraints(var, val, var2, val2) return count_if(conflict, self.neighbors[var]) def forward_check(self, var, val, assignment): "Do forward checking (current domain reduction) for this assignment." if self.curr_domains: # Restore prunings from previous value of var for (B, b) in self.pruned[var]: self.curr_domains[B].append(b) self.pruned[var] = [] # Prune any other B=b assignement that conflict with var=val for B in self.neighbors[var]: if B not in assignment: for b in self.curr_domains[B][:]: if not self.constraints(var, val, B, b): self.curr_domains[B].remove(b) self.pruned[var].append((B, b)) 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 'CSP:', self, 'with assignment:', assignment ## These methods are for the tree and graph search interface: def succ(self, assignment): "Return a list of (action, state) pairs." if len(assignment) == len(self.vars): return [] else: var = find_if(lambda v: v not in assignment, self.vars) result = [] for val in self.domains[var]: if self.nconflicts(self, var, val, assignment) == 0: a = assignment.copy; a[var] = val result.append(((var, val), a)) return result def goal_test(self, assignment): "The goal is to assign all vars, with all constraints satisfied." return (len(assignment) == len(self.vars) and every(lambda var: self.nconflicts(var, assignment[var], assignment) == 0, self.vars)) ## This is for min_conflicts search def conflicted_vars(self, current): "Return a list of variables in current assignment that are in conflict" return [var for var in self.vars if self.nconflicts(var, current[var], current) > 0] #______________________________________________________________________________ # CSP Backtracking Search def backtracking_search(csp, mcv=False, lcv=False, fc=False, mac=False): """Set up to do recursive backtracking search. Allow the following options: mcv - If true, use Most Constrained Variable Heuristic lcv - If true, use Least Constraining Value Heuristic fc - If true, use Forward Checking mac - If true, use Maintaining Arc Consistency. [Fig. 5.3] >>> backtracking_search(australia) {'WA': 'B', 'Q': 'B', 'T': 'B', 'V': 'B', 'SA': 'G', 'NT': 'R', 'NSW': 'R'} """ if fc or mac: csp.curr_domains, csp.pruned = {}, {} for v in csp.vars: csp.curr_domains[v] = csp.domains[v][:] csp.pruned[v] = [] update(csp, mcv=mcv, lcv=lcv, fc=fc, mac=mac) return recursive_backtracking({}, csp) def recursive_backtracking(assignment, csp): """Search for a consistent assignment for the csp. Each recursive call chooses a variable, and considers values for it.""" if len(assignment) == len(csp.vars): return assignment var = select_unassigned_variable(assignment, csp) for val in order_domain_values(var, assignment, csp): if csp.fc or csp.nconflicts(var, val, assignment) == 0: csp.assign(var, val, assignment) result = recursive_backtracking(assignment, csp) if result is not None: return result csp.unassign(var, assignment) return None def select_unassigned_variable(assignment, csp): "Select the variable to work on next. Find" if csp.mcv: # Most Constrained Variable unassigned = [v for v in csp.vars if v not in assignment] return argmin_random_tie(unassigned, lambda var: -num_legal_values(csp, var, assignment)) else: # First unassigned variable for v in csp.vars: if v not in assignment: return v def order_domain_values(var, assignment, csp): "Decide what order to consider the domain variables." if csp.curr_domains: domain = csp.curr_domains[var] else: domain = csp.domains[var][:] if csp.lcv: # If LCV is specified, consider values with fewer conflicts first key = lambda val: csp.nconflicts(var, val, assignment) domain.sort(lambda(x,y): cmp(key(x), key(y))) while domain: yield domain.pop() def num_legal_values(csp, var, assignment): if csp.curr_domains: return len(csp.curr_domains[var]) else: return count_if(lambda val: csp.nconflicts(var, val, assignment) == 0, csp.domains[var]) #______________________________________________________________________________ # Constraint Propagation with AC-3 def AC3(csp, queue=None): """[Fig. 5.7]""" if queue == None: queue = [(Xi, Xk) for Xi in csp.vars for Xk in csp.neighbors[Xi]] while queue: (Xi, Xj) = queue.pop() if remove_inconsistent_values(csp, Xi, Xj): for Xk in csp.neighbors[Xi]: queue.append((Xk, Xi)) def remove_inconsistent_values(csp, Xi, Xj): "Return true if we remove a value." removed = False for x in csp.curr_domains[Xi][:]: # If Xi=x conflicts with Xj=y for every possible y, eliminate Xi=x if every(lambda y: not csp.constraints(Xi, x, Xj, y), csp.curr_domains[Xj]): csp.curr_domains[Xi].remove(x) removed = True return removed #______________________________________________________________________________ # Min-conflicts hillclimbing search for CSPs def min_conflicts(csp, max_steps=1000000): """Solve a CSP by stochastic hillclimbing on the number of conflicts.""" # Generate a complete assignement for all vars (probably with conflicts) current = {}; csp.current = current for var in csp.vars: val = min_conflicts_value(csp, var, current) csp.assign(var, val, current) # Now repeapedly 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 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], lambda val: csp.nconflicts(var, val, current)) #______________________________________________________________________________ # Map-Coloring Problems 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 vars 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: %r}' % self.value def different_values_constraint(A, a, B, b): "A constraint saying two neighboring variables must differ in value." return a != b 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(neighbors.keys(), UniversalDict(colors), neighbors, different_values_constraint) def parse_neighbors(neighbors, vars=[]): """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']} """ dict = DefaultDict([]) for var in vars: dict[var] = [] specs = [spec.split(':') for spec in neighbors.split(';')] for (A, Aneighbors) in specs: A = A.strip(); dict.setdefault(A, []) for B in Aneighbors.split(): dict[A].append(B) dict[B].append(A) return dict australia = MapColoringCSP(list('RGB'), 'SA: WA NT Q NSW V; NT: WA Q; NSW: Q V; T: ') usa = 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; 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; IN: KY; MS: TN AL; AL: TN GA FL; MI: OH; 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 CA NJ; NJ: DE; DE: MD; MD: DC; VT: NH MA; MA: NH RI CT; CT: RI; ME: NH; HI: ; AK: """) #______________________________________________________________________________ # n-Queens Problem 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) class NQueensCSP(CSP): """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 vars, then variable selection will also be O(1). >>> len(backtracking_search(NQueensCSP(8))) 8 >>> len(min_conflicts(NQueensCSP(8))) 8 """ def __init__(self, n): """Initialize data structures for n Queens.""" CSP.__init__(self, range(n), UniversalDict(range(n)), UniversalDict(range(n)), queen_constraint) update(self, rows=[0]*n, ups=[0]*(2*n - 1), downs=[0]*(2*n - 1)) 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.vars) c = self.rows[val] + self.downs[var+val] + self.ups[var-val+n-1] if assignment.get(var, None) == val: c -= 3 return c def assign(self, var, val, assignment): "Assign var, and keep track of conflicts." oldval = assignment.get(var, None) if val != oldval: if oldval is not None: # Remove old val if there was one self.record_conflict(assignment, var, oldval, -1) self.record_conflict(assignment, var, val, +1) CSP.assign(self, var, val, assignment) 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) def record_conflict(self, assignment, var, val, delta): "Record conflicts caused by addition or deletion of a Queen." n = len(self.vars) self.rows[val] += delta self.downs[var + val] += delta self.ups[var - val + n - 1] += delta def display(self, assignment): "Print the queens and the nconflicts values (for debugging)." n = len(self.vars) 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, print ' ', for var in range(n): if assignment.get(var,'') == val: ch ='*' else: ch = ' ' print str(self.nconflicts(var, val, assignment))+ch, print #______________________________________________________________________________ # The Zebra Puzzle 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 Ukranian Japanese'.split() Smokes = 'Kools Chesterfields Winston LuckyStrike Parliaments'.split() vars = Colors + Pets + Drinks + Countries + Smokes domains = {} for var in vars: domains[var] = 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; Ukranian: Tea; Japanese: Parliaments; Kools: Horse; Coffee: Green; Green: Ivory""", vars) 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 == 'Ukranian' 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 'error' return CSP(vars, domains, neighbors, zebra_constraint) def solve_zebra(algorithm=min_conflicts, **args): z = Zebra() ans = algorithm(z, **args) for h in range(1, 6): print 'House', h, for (var, val) in ans.items(): if val == h: print var, print return ans['Zebra'], ans['Water'], z.nassigns, ans,