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restructure sat.py: clean up and more general interface

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Maximilian Keßler 2023-03-13 17:21:07 +01:00
parent ab2cda2207
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sat.py
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@ -1,121 +1,270 @@
from pysmt.shortcuts import Symbol, Bool, Not, Implies, Iff, And, Or, AtMostOne, ExactlyOne, get_model, get_atoms, get_formula_size, get_unsat_core from pysmt.shortcuts import Symbol, Bool, Not, Implies, Iff, And, Or, AtMostOne, ExactlyOne, get_model, get_atoms, get_formula_size, get_unsat_core
from pysmt.rewritings import conjunctive_partition from pysmt.rewritings import conjunctive_partition
import json import json
from typing import List
MAX_MOVES = 63 from compress import DeckCard, Action, ActionType, link
NUM_STRIKES = 3
COLORS = 'rygbp'
NUM_CARDS = 50
deck_str = 'p5 p3 b4 r5 y4 y4 y5 r4 b2 y2 y3 g5 g2 g3 g4 p4 r3 b2 b3 b3 p4 b1 p2 b1 b1 p2 p1 p1 g1 r4 g1 r1 r3 r1 g1 r1 p1 b4 p3 g2 g3 g4 b5 y1 y1 y1 r2 r2 y2 y3' STANDARD_HAND_SIZE = {2: 5, 3: 5, 4: 4, 5: 4, 6: 3}
NUM_STRIKES_TO_LOSE = 3
# instance independent names of variables
# literals to model game as sat instance to check for feasibility
# variants 'throw it in a hole not handled', 'clue starved' and 'up or down' currently not handled
class Literals():
# num_suits is total number of suits, i.e. also counts the dark suits
# default distribution among all suits is assumed
def __init__(self, num_players, num_suits, num_dark_suits=0):
assert ( 2 <= num_players <= 6 )
## some game parameters
self.num_players = num_players
self.num_suits = num_suits
self.num_dark_suits = num_dark_suits
self.hand_size = STANDARD_HAND_SIZE[num_players]
self.num_strikes = NUM_STRIKES_TO_LOSE
self.deck_size = 10 * num_suits - 5 * num_dark_suits
self.distributed_cards = self.num_players * self.hand_size
self.draw_pile_size = self.deck_size - self.distributed_cards
## maximum number of moves in any game that can achieve max score
# each suit gives 15 moves, as we can play and discard 5 cards each and give 5 clues. dark suits only give 5 moves, since no discards are added
# number of cards that remain in players hands after end of game. they cost 2 turns each, since we cannot discard them and also have one clue less
# 8 clues at beginning, one further clue for each suit but one (the clue of the last 5 is never useful since it is gained in the extra-round)
# subtract a further move for a second 5-clue that can't be used in 5 or 6-player games, since the extraround starts too soon
self.max_moves = 15 * num_suits - 10 * num_dark_suits \
- 2 * num_players * (self.hand_size - 1) \
+ 8 + (num_suits - 1) \
+ (-1 if num_players >= 5 else 0)
###
# note that we generate 'literals' always one out of boundary and set them to explicit truth values. This makes sat formulation easier but has no actual overhead in solving it
# move are numbered starting with 0
# clues[m][i] == "after move m we have at least i clues" # clues[m][i] == "after move m we have at least i clues"
clues = {-1: {i: Bool(i < 9) for i in range(0, 10)}, **{m: {0: Bool(True), 9: Bool(False), **{i: Symbol('m{}c{}'.format(m, i)) for i in range(1, 9)}} for m in range(MAX_MOVES)}} self.clues = {
-1: { i: Bool(i < 9) for i in range(0, 10) } # we have 8 clues after turn -1
, **{
m: {
0: Bool(True), # always at least 0 clues
**{ i: Symbol('m{}c{}'.format(m, i)) for i in range(1, 9) },
9: Bool(False) # never 9 or more clues. This will implicitly forbid discarding at 8 clues later
}
for m in range(self.max_moves)
}
}
# strikes[m][i] == "after move m we have at least i strikes" # strikes[m][i] == "after move m we have at least i strikes"
strikes = {-1: {i: Bool(i == 0) for i in range(0,NUM_STRIKES+1)}, **{m: {0: Bool(True), NUM_STRIKES: Bool(False), **{s: Symbol('m{}s{}'.format(m,s)) for s in range(1,NUM_STRIKES)}} for m in range(MAX_MOVES)} } self.strikes = {
-1: {i: Bool(i == 0) for i in range(0, self.num_strikes + 1)} # no strikes when we start
, **{
m: {
0: Bool(True),
**{ s: Symbol('m{}s{}'.format(m,s)) for s in range(1, self.num_strikes) },
self.num_strikes: Bool(False) # never so many clues that we lose. Implicitly forbids striking out
}
for m in range(self.max_moves)
}
}
# extraturn[m] = "turn m is a move part of the extra round or a dummy turn" # extraturn[m] = "turn m is a move part of the extra round or a dummy turn"
extraround = {-1: Bool(False), **{m: Bool(False) if m <= 29 else Symbol('m{}e'.format(m)) for m in range(0, MAX_MOVES)}} self.extraround = {
-1: Bool(False)
, **{
m: Bool(False) if m < self.draw_pile_size else Symbol('m{}e'.format(m)) # it takes at least as many turns as cards in the draw pile to start the extra round
for m in range(0, self.max_moves)
}
}
# dummyturn[m] = "turn m is a dummy nurn and not actually part of the game" # dummyturn[m] = "turn m is a dummy nurn and not actually part of the game"
dummyturn = {-1: Bool(False), **{m: Bool(False) if m <= 34 else Symbol('m{}dt'.format(m)) for m in range(0, MAX_MOVES)}} self.dummyturn = {
# strike[m] = "at move m we get a strike" -1: Bool(False)
strike = {-1: Bool(False), **{m: Symbol('m{}s+'.format(m)) for m in range(MAX_MOVES)}} , **{
m: Bool(False) if m < self.draw_pile_size + self.num_players else Symbol('m{}dt'.format(m))
for m in range(0, self.max_moves)
}
}
# draw[m][i] == "at move m we play/discard deck[i]" # draw[m][i] == "at move m we play/discard deck[i]"
discard = {m: {i: Symbol('m{}-{}'.format(m, i)) for i in range(NUM_CARDS)} for m in range(MAX_MOVES)} self.discard = {
# progress[m][c, k] == "after move m we have played in color c until k" m: {i: Symbol('m{}-{}'.format(m, i)) for i in range(self.deck_size)}
progress = {-1: {(c, k): Bool(k == 0) for c in COLORS for k in range(6)}, **{m: {**{(c, 0): Bool(True) for c in COLORS}, **{(c, k): Symbol('m{}:{}{}'.format(m, c, k)) for c in COLORS for k in range(1, 6)}} for m in range(MAX_MOVES)}} for m in range(self.max_moves)
}
# draw[m][i] == "at move m we draw deck card i"
self.draw = {
-1: { i: Bool(i == self.distributed_cards - 1) for i in range(self.distributed_cards - 1, self.deck_size) }
, **{
m: {
self.distributed_cards - 1: Bool(False),
**{i: Symbol('m{}+{}'.format(m, i)) for i in range(self.distributed_cards, self.deck_size)}
}
for m in range(self.max_moves)
}
}
# strike[m] = "at move m we get a strike"
self.strike = {
-1: Bool(False)
, **{
m: Symbol('m{}s+'.format(m))
for m in range(self.max_moves)
}
}
# progress[m][card = (suitIndex, rank)] == "after move m we have played in suitIndex up to rank"
self.progress = {
-1: {(s, r): Bool(r == 0) for s in range(0, self.num_suits) for r in range(0, 6)} # at start, have only played rank zero
, **{
m: {
**{(s, 0): Bool(True) for s in range(0, self.num_suits)},
**{(s, r): Symbol('m{}:{}{}'.format(m, s, r)) for s in range(0, self.num_suits) for r in range(1, 6)}
}
for m in range(self.max_moves)
}
}
## Utility variables
# discard_any[m] == "at move m we play/discard a card" # discard_any[m] == "at move m we play/discard a card"
discard_any = {m: Symbol('m{}d'.format(m)) for m in range(MAX_MOVES)} self.discard_any = { m: Symbol('m{}d'.format(m)) for m in range(self.max_moves) }
# draw_any[m] == "at move m we draw a card" # draw_any[m] == "at move m we draw a card"
draw_any = {m: Symbol('m{}D'.format(m)) for m in range(MAX_MOVES)} self.draw_any = {m: Symbol('m{}D'.format(m)) for m in range(self.max_moves)}
# play[m] == "at move m we play a card" # play[m] == "at move m we play a card"
play = {m: Symbol('m{}p'.format(m)) for m in range(MAX_MOVES)} self.play = {m: Symbol('m{}p'.format(m)) for m in range(self.max_moves)}
# play5[m] == "at move m we play a 5" # play5[m] == "at move m we play a 5"
play5 = {m: Symbol('m{}p5'.format(m)) for m in range(MAX_MOVES)} self.play5 = {m: Symbol('m{}p5'.format(m)) for m in range(self.max_moves)}
# incr_clues[m] == "at move m we obtain a clue" # incr_clues[m] == "at move m we obtain a clue"
incr_clues = {m: Symbol('m{}c+'.format(m)) for m in range(MAX_MOVES)} self.incr_clues = {m: Symbol('m{}c+'.format(m)) for m in range(self.max_moves)}
### this is dependent on the number of players
# draw[m][i] == "at move m we draw deck[i]"
hand_size = {2: 5, 3: 5, 4: 4, 5: 4, 6: 3}
last_hand_card = {2: 9, 3: 14, 4: 15, 5: 19, 6: 17}
max_moves = {2: -1, 3: 63, 4: 63, 5: 56, 6: 62}
draw = {p: {-1: {i: Bool(i == last_hand_card[p]) for i in range(last_hand_card[p], NUM_CARDS)}, **{m: {last_hand_card[p]: Bool(False), **{i: Symbol('m{}+{}'.format(m, i)) for i in range(last_hand_card[p] + 1, NUM_CARDS)}} for m in range(max_moves[p])}} for p in range(2,7)}
def solve(deck_str, num_players=5):
deck = [(s[0], int(s[1])) for s in deck_str.split(' ')] def solve(deck: List[DeckCard], num_players=5):
hand_size = globals()['hand_size'][num_players]
last_hand_card = globals()['last_hand_card'][num_players] num_suits = max(map(lambda card: card.suitIndex, deck)) + 1
max_moves = globals()['max_moves'][num_players] num_dark_suits = (len(deck) - 10 * num_suits) // (-5)
draw = globals()['draw'][num_players]
ls = Literals(num_players, num_suits, num_dark_suits)
valid_move = lambda m: And( valid_move = lambda m: And(
Implies(dummyturn[m], Not(discard_any[m])), # in dummy turns, nothing can be discarded
Implies(ls.dummyturn[m], Not(ls.discard_any[m])),
# definition of discard_any # definition of discard_any
Iff(discard_any[m], Or(discard[m][i] for i in range(NUM_CARDS))), Iff(ls.discard_any[m], Or(ls.discard[m][i] for i in range(ls.deck_size))),
# definition of draw_any # definition of draw_any
Iff(draw_any[m], Or(draw[m][i] for i in range(last_hand_card + 1, NUM_CARDS))), Iff(ls.draw_any[m], Or(ls.draw[m][i] for i in range(ls.distributed_cards, ls.deck_size))),
# draw implies discard (and converse true before last 5 moves)
Implies(draw_any[m], discard_any[m]), # ls.draw implies ls.discard (and converse true before the ls.extraround)
Implies(discard_any[m], Or(extraround[m], draw_any[m])), Implies(ls.draw_any[m], ls.discard_any[m]),
# play requires discard Implies(ls.discard_any[m], Or(ls.extraround[m], ls.draw_any[m])),
Implies(play[m], discard_any[m]),
# definition of play5 # ls.play requires ls.discard
Iff(play5[m], And(play[m], Or(discard[m][i] for i in range(NUM_CARDS) if deck[i][1] == 5))), Implies(ls.play[m], ls.discard_any[m]),
# definition of incr_clues
Iff(incr_clues[m], And(discard_any[m], Implies(play[m], And(play5[m], Not(clues[m-1][8]))))), # definition of ls.play5
#Iff(incr_clues[m], And(discard_any[m], Implies(play[m], play5[m]))), Iff(ls.play5[m], And(ls.play[m], Or(ls.discard[m][i] for i in range(ls.deck_size) if deck[i].rank == 5))),
# change of clues
*[Iff(clues[m][i], Or(clues[m-1][i+1], And(clues[m-1][i], Or(discard_any[m], dummyturn[m])), And(clues[m-1][i-1], incr_clues[m]))) for i in range(1, 9)], # definition of ls.incr_clues
## more than 8 clues not allowed, discarding produces a strike Iff(ls.incr_clues[m], And(ls.discard_any[m], Implies(ls.play[m], And(ls.play5[m], Not(ls.clues[m-1][8]))))),
Iff(strike[m], And(discard_any[m], Not(play[m]), clues[m-1][8])),
# change of ls.clues
*[Iff(ls.clues[m][i], Or(ls.clues[m-1][i+1], And(ls.clues[m-1][i], Or(ls.discard_any[m], ls.dummyturn[m])), And(ls.clues[m-1][i-1], ls.incr_clues[m]))) for i in range(1, 9)],
## more than 8 clues not allowed, ls.discarding produces a strike
# Note that this means that we will never strike while not at 8 clues.
# It's easy to see that if there is any solution to the instance, then there is also one where we only strike at 8 clues
# (or not at all) -> Just strike later if neccessary
# So, we decrease the solution space with this formulation, but do not change whether it's empty or not
Iff(ls.strike[m], And(ls.discard_any[m], Not(ls.play[m]), ls.clues[m-1][8])),
# change of strikes # change of strikes
*[Iff(strikes[m][i], Or(strikes[m-1][i], And(strikes[m-1][i-1], strike[m]))) for i in range(1, NUM_STRIKES+1)], *[Iff(ls.strikes[m][i], Or(ls.strikes[m-1][i], And(ls.strikes[m-1][i-1], ls.strike[m]))) for i in range(1, ls.num_strikes + 1)],
# less than 0 clues not allowed # less than 0 clues not allowed
Implies(Not(discard_any[m]), Or(clues[m-1][1], dummyturn[m])), Implies(Not(ls.discard_any[m]), Or(ls.clues[m-1][1], ls.dummyturn[m])),
# we can only draw card i if the last drawn card was i-1
*[Implies(draw[m][i], Or(And(draw[m0][i-1], *[Not(draw_any[m1]) for m1 in range(m0+1, m)]) for m0 in range(max(-1, m-9), m))) for i in range(last_hand_card + 1, NUM_CARDS)], # we can only draw card i if the last ls.drawn card was i-1
#*[Implies(draw[m][i], Not(draw[m0][i])) for m0 in range(m) for i in range(20, 50)], *[Implies(ls.draw[m][i], Or(And(ls.draw[m0][i-1], *[Not(ls.draw_any[m1]) for m1 in range(m0+1, m)]) for m0 in range(max(-1, m-9), m))) for i in range(ls.distributed_cards, ls.deck_size)],
#*[Implies(draw[m][i], Or(draw[m0][i-1] for m0 in range(max(-1, m-9), m))) for i in range(20, 50)],
# we can only draw at most one card (NOTE: redundant, FIXME: avoid quadratic formula) # we can only draw at most one card (NOTE: redundant, FIXME: avoid quadratic formula)
AtMostOne(draw[m][i] for i in range(last_hand_card + 1, NUM_CARDS)), AtMostOne(ls.draw[m][i] for i in range(ls.distributed_cards, ls.deck_size)),
#*[Not(And(draw[m][i], draw[m][j])) for i in range(20, 50) for j in range(20, i)],
# we can only discard a card if we drew it earlier... # we can only discard a card if we drew it earlier...
*[Implies(discard[m][i], Or(draw[m0][i] for m0 in range(m-num_players, -1, -num_players))) for i in range(last_hand_card + 1, NUM_CARDS)], *[Implies(ls.discard[m][i], Or(ls.draw[m0][i] for m0 in range(m-ls.num_players, -1, -ls.num_players))) for i in range(ls.distributed_cards, ls.deck_size)],
# ...or if it was part of the initial hand # ...or if it was part of the initial hand
*[Not(discard[m][i]) for i in range(last_hand_card + 1) if i//hand_size != m % num_players], *[Not(ls.discard[m][i]) for i in range(0, ls.distributed_cards) if i // ls.hand_size != m % ls.num_players],
# we can only discard a card if we did not discard it yet # we can only discard a card if we did not discard it yet
*[Implies(discard[m][i], And(Not(discard[m0][i]) for m0 in range(m-num_players, -1, -num_players))) for i in range(NUM_CARDS)], *[Implies(ls.discard[m][i], And(Not(ls.discard[m0][i]) for m0 in range(m-ls.num_players, -1, -ls.num_players))) for i in range(ls.deck_size)],
# we can only discard at most one card (FIXME: avoid quadratic formula) # we can only discard at most one card (FIXME: avoid quadratic formula)
AtMostOne(discard[m][i] for i in range(NUM_CARDS)), AtMostOne(ls.discard[m][i] for i in range(ls.deck_size)),
#*[Not(And(discard[m][i], discard[m][j])) for i in range(NUM_CARDS) for j in range(i)],
# we can only play a card if it matches the progress # we can only play a card if it matches the progress
*[Implies(And(discard[m][i], play[m]), And(Not(progress[m-1][deck[i]]), progress[m-1][deck[i][0], deck[i][1]-1])) for i in range(NUM_CARDS)], *[Implies(
And(ls.discard[m][i], ls.play[m]),
And(
Not(ls.progress[m-1][deck[i].suitIndex, deck[i].rank]),
ls.progress[m-1][deck[i].suitIndex, deck[i].rank-1 ]
)
)
for i in range(ls.deck_size)
],
# change of progress # change of progress
*[Iff(progress[m][c, k], Or(progress[m-1][c, k], And(play[m], Or(discard[m][i] for i in range(NUM_CARDS) if deck[i] == (c, k))))) for c in COLORS for k in range(1, 6)], *[
Iff(
ls.progress[m][s, r],
Or(
ls.progress[m-1][s, r],
And(ls.play[m], Or(ls.discard[m][i]
for i in range(0, ls.deck_size)
if deck[i] == DeckCard(s, r) ))
)
)
for s in range(0, ls.num_suits)
for r in range(1, 6)
],
# extra round bool # extra round bool
Iff(extraround[m], Or(extraround[m-1], draw[m-1][NUM_CARDS-1])), Iff(ls.extraround[m], Or(ls.extraround[m-1], ls.draw[m-1][ls.deck_size-1])),
# dummy turn bool # dummy turn bool
*[Iff(dummyturn[m], Or(dummyturn[m-1], draw[m-1 - num_players][NUM_CARDS-1])) for i in range(0,1) if m >= num_players] *[Iff(ls.dummyturn[m], Or(ls.dummyturn[m-1], ls.draw[m-1 - ls.num_players][ls.deck_size-1])) for i in range(0,1) if m >= ls.num_players]
) )
win = And( win = And(
# maximum progress at each color # maximum progress at each color
*[progress[max_moves-1][c, 5] for c in COLORS], *[ls.progress[ls.max_moves-1][s, 5] for s in range(0, ls.num_suits)],
# played every color/value combination (NOTE: redundant)
*[Or(And(discard[m][i], play[m]) for m in range(max_moves) for i in range(50) if deck[i] == (c, k)) for c in COLORS for k in range(1, 6)] # played every color/value combination (NOTE: redundant, but makes solving faster)
*[
Or(
And(ls.discard[m][i], ls.play[m])
for m in range(ls.max_moves)
for i in range(ls.deck_size)
if deck[i] == DeckCard(s, r)
)
for s in range(0, ls.num_suits)
for r in range(1, 6)
]
) )
constraints = And(*[valid_move(m) for m in range(max_moves)], win) constraints = And(*[valid_move(m) for m in range(ls.max_moves)], win)
# print('Solving instance with {} variables, {} nodes'.format(len(get_atoms(constraints)), get_formula_size(constraints))) # print('Solving instance with {} variables, {} nodes'.format(len(get_atoms(constraints)), get_formula_size(constraints)))
model = get_model(constraints) model = get_model(constraints)
if model: if model:
# print_model(model, deck) # print_model(model, deck)
solution = toJSON(model, deck, num_players) solution = toJSON(model, deck, ls)
return True, solution return True, solution
else: else:
print('unsatisfiable') print('unsatisfiable')
@ -140,44 +289,35 @@ def print_model(model, deck, num_players):
flags = ['discard_any', 'draw_any', 'play', 'play5', 'incr_clues', 'strike', 'extraround', 'dummyturn'] flags = ['discard_any', 'draw_any', 'play', 'play5', 'incr_clues', 'strike', 'extraround', 'dummyturn']
print(', '.join(f for f in flags if model.get_py_value(globals()[f][m]))) print(', '.join(f for f in flags if model.get_py_value(globals()[f][m])))
def toJSON(model, deck, num_players): def toJSON(model, deck: List[DeckCard], ls: Literals) -> dict:
deck_json = [{"suitIndex": COLORS.index(s), "rank": r} for (s,r) in deck] players = ["Alice", "Bob", "Cathy", "Donald", "Emily"][:ls.num_players]
players = ["Alice", "Bob", "Cathy", "Donald", "Emily"][:num_players] # we keep track of the hands to output some dummy clues
hands = [deck[hand_size[num_players]*p:hand_size[num_players]*(p+1)] for p in range(0,num_players)] hands = [deck[ls.hand_size * p : ls.hand_size *(p+1)] for p in range(0, ls.num_players)]
draw = globals()['draw'][num_players]
actions = [] actions = []
for m in range(max_moves[num_players]): for m in range(ls.max_moves):
if model.get_py_value(dummyturn[m]): if model.get_py_value(ls.dummyturn[m]):
break break
if model.get_py_value(discard_any[m]): if model.get_py_value(ls.discard_any[m]):
discarded = next(i for i in range(0,NUM_CARDS) if model.get_py_value(discard[m][i])) discarded = next(i for i in range(0,ls.deck_size) if model.get_py_value(ls.discard[m][i]))
icard = hands[m % num_players].index(deck[discarded]) icard = hands[m % ls.num_players].index(deck[discarded])
for i in range(icard, hand_size[num_players] - 1): for i in range(icard, ls.hand_size - 1):
hands[m % num_players][i] = hands[m % num_players][i + 1] hands[m % ls.num_players][i] = hands[m % ls.num_players][i + 1]
if model.get_py_value(draw_any[m]): if model.get_py_value(ls.draw_any[m]):
hands[m % num_players][hand_size[num_players] - 1] = next(deck[i] for i in range(last_hand_card[num_players] + 1, NUM_CARDS) if model.get_py_value(draw[m][i])) hands[m % ls.num_players][ls.hand_size - 1] = next(deck[i] for i in range(ls.distributed_cards, ls.deck_size) if model.get_py_value(ls.draw[m][i]))
if model.get_py_value(play[m]) or model.get_py_value(strike[m]): if model.get_py_value(ls.play[m]) or model.get_py_value(ls.strike[m]):
actions.append({ actions.append( Action( ActionType.Play, target=discarded) )
"type": 0,
"target": discarded
})
else: else:
actions.append({ actions.append( Action( ActionType.Discard, target=discarded) )
"type": 1,
"target": discarded
})
else: else:
actions.append({ actions.append( Action(
"type": 3, ActionType.RankClue,
"target": (m + 1) % num_players, target=(m+1) % ls.num_players, # clue next player
"value": hands[(m+1) % num_players][0][1] value=hands[(m+1) % ls.num_players][0].rank # clue rank of rightmost card
}) )
actions.append({ )
"type": 4, actions.append( Action (ActionType.EndGame, target=0, value=1))
"value": 1
})
game = { game = {
"deck": deck_json, "deck": deck,
"players": players, "players": players,
"actions": actions, "actions": actions,
"first_player": 0, "first_player": 0,
@ -185,9 +325,16 @@ def toJSON(model, deck, num_players):
"variant": "No Variant", "variant": "No Variant",
} }
} }
return json.dumps(game) return game
if __name__ == "__main__": if __name__ == "__main__":
solvable, sol = solve(deck_str) COLORS = 'rygbp'
deck_str = 'p5 p3 b4 r5 y4 y4 y5 r4 b2 y2 y3 g5 g2 g3 g4 p4 r3 b2 b3 b3 p4 b1 p2 b1 b1 p2 p1 p1 g1 r4 g1 r1 r3 r1 g1 r1 p1 b4 p3 g2 g3 g4 b5 y1 y1 y1 r2 r2 y2 y3'
deck = [DeckCard(COLORS.index(c[0]), int(c[1])) for c in deck_str.split(" ")]
print(deck)
solvable, sol = solve(deck, num_players=5)
if solvable: if solvable:
print(sol) print(sol)
print(link(sol))