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"""Search (Chapters 3-4)
The way to use this code is to subclass Problem to create a class of problems,
then create problem instances and solve them with calls to the various search
functions."""
from utils import (
is_in, argmin, argmax, argmax_random_tie, probability, weighted_sampler,
memoize, print_table, open_data, PriorityQueue, name,
distance, vector_add
)
from collections import defaultdict, deque
import math
import random
import sys
import bisect
from operator import itemgetter
infinity = float('inf')
# ______________________________________________________________________________
class Problem(object):
"""The abstract class for a formal problem. You should subclass
this and implement the methods actions and result, and possibly
__init__, goal_test, and path_cost. Then you will create instances
of your subclass and solve them with the various search functions."""
def __init__(self, initial, goal=None):
"""The constructor specifies the initial state, and possibly a goal
state, if there is a unique goal. Your subclass's constructor can add
other arguments."""
self.initial = initial
self.goal = goal
def actions(self, state):
"""Return the actions that can be executed in the given
state. The result would typically be a list, but if there are
many actions, consider yielding them one at a time in an
iterator, rather than building them all at once."""
raise NotImplementedError
def result(self, state, action):
"""Return the state that results from executing the given
action in the given state. The action must be one of
self.actions(state)."""
raise NotImplementedError
def goal_test(self, state):
"""Return True if the state is a goal. The default method compares the
state to self.goal or checks for state in self.goal if it is a
list, as specified in the constructor. Override this method if
checking against a single self.goal is not enough."""
if isinstance(self.goal, list):
return is_in(state, self.goal)
else:
return state == self.goal
def path_cost(self, c, state1, action, state2):
"""Return the cost of a solution path that arrives at state2 from
state1 via action, assuming cost c to get up to state1. If the problem
is such that the path doesn't matter, this function will only look at
state2. If the path does matter, it will consider c and maybe state1
and action. The default method costs 1 for every step in the path."""
return c + 1
def value(self, state):
"""For optimization problems, each state has a value. Hill-climbing
and related algorithms try to maximize this value."""
raise NotImplementedError
# ______________________________________________________________________________
class Node:
"""A node in a search tree. Contains a pointer to the parent (the node
that this is a successor of) and to the actual state for this node. Note
that if a state is arrived at by two paths, then there are two nodes with
the same state. Also includes the action that got us to this state, and
the total path_cost (also known as g) to reach the node. Other functions
may add an f and h value; see best_first_graph_search and astar_search for
an explanation of how the f and h values are handled. You will not need to
subclass this class."""
def __init__(self, state, parent=None, action=None, path_cost=0):
"""Create a search tree Node, derived from a parent by an action."""
self.state = state
self.parent = parent
self.action = action
self.path_cost = path_cost
self.depth = 0
if parent:
self.depth = parent.depth + 1
def __repr__(self):
return "<Node {}>".format(self.state)
def __lt__(self, node):
return self.state < node.state
def expand(self, problem):
"""List the nodes reachable in one step from this node."""
return [self.child_node(problem, action)
for action in problem.actions(self.state)]
def child_node(self, problem, action):
"""[Figure 3.10]"""
next_state = problem.result(self.state, action)
next_node = Node(next_state, self, action,
problem.path_cost(self.path_cost, self.state,
action, next_state))
return next_node
def solution(self):
"""Return the sequence of actions to go from the root to this node."""
return [node.action for node in self.path()[1:]]
def path(self):
"""Return a list of nodes forming the path from the root to this node."""
node, path_back = self, []
while node:
path_back.append(node)
node = node.parent
return list(reversed(path_back))
# We want for a queue of nodes in breadth_first_graph_search or
# astar_search to have no duplicated states, so we treat nodes
# with the same state as equal. [Problem: this may not be what you
# want in other contexts.]
def __eq__(self, other):
return isinstance(other, Node) and self.state == other.state
def __hash__(self):
return hash(self.state)
# ______________________________________________________________________________
class SimpleProblemSolvingAgentProgram:
"""Abstract framework for a problem-solving agent. [Figure 3.1]"""
def __init__(self, initial_state=None):
"""State is an abstract representation of the state
of the world, and seq is the list of actions required
to get to a particular state from the initial state(root)."""
self.state = initial_state
self.seq = []
def __call__(self, percept):
"""[Figure 3.1] Formulate a goal and problem, then
search for a sequence of actions to solve it."""
self.state = self.update_state(self.state, percept)
if not self.seq:
goal = self.formulate_goal(self.state)
problem = self.formulate_problem(self.state, goal)
self.seq = self.search(problem)
if not self.seq:
return None
return self.seq.pop(0)
def update_state(self, state, percept):
raise NotImplementedError
def formulate_goal(self, state):
raise NotImplementedError
def formulate_problem(self, state, goal):
raise NotImplementedError
def search(self, problem):
raise NotImplementedError
# ______________________________________________________________________________
# Uninformed Search algorithms
def breadth_first_tree_search(problem):
"""Search the shallowest nodes in the search tree first.
Search through the successors of a problem to find a goal.
The argument frontier should be an empty queue.
Repeats infinitely in case of loops. [Figure 3.7]"""
frontier = deque([Node(problem.initial)]) # FIFO queue
while frontier:
node = frontier.popleft()
if problem.goal_test(node.state):
return node
frontier.extend(node.expand(problem))
return None
def depth_first_tree_search(problem):
"""Search the deepest nodes in the search tree first.
Search through the successors of a problem to find a goal.
The argument frontier should be an empty queue.
Repeats infinitely in case of loops. [Figure 3.7]"""
frontier = [Node(problem.initial)] # Stack
while frontier:
node = frontier.pop()
if problem.goal_test(node.state):
return node
frontier.extend(node.expand(problem))
return None
def depth_first_graph_search(problem):
"""Search the deepest nodes in the search tree first.
Search through the successors of a problem to find a goal.
The argument frontier should be an empty queue.
Does not get trapped by loops.
If two paths reach a state, only use the first one. [Figure 3.7]"""
frontier = [(Node(problem.initial))] # Stack
explored = set()
while frontier:
node = frontier.pop()
if problem.goal_test(node.state):
return node
explored.add(node.state)
frontier.extend(child for child in node.expand(problem)
if child.state not in explored and
child not in frontier)
return None
def breadth_first_graph_search(problem):
"""[Figure 3.11]
Note that this function can be implemented in a
single line as below:
return graph_search(problem, FIFOQueue())
"""
node = Node(problem.initial)
if problem.goal_test(node.state):
return node
frontier = deque([node])
explored = set()
while frontier:
node = frontier.popleft()
explored.add(node.state)
for child in node.expand(problem):
if child.state not in explored and child not in frontier:
if problem.goal_test(child.state):
return child
frontier.append(child)
return None
def best_first_graph_search(problem, f):
"""Search the nodes with the lowest f scores first.
You specify the function f(node) that you want to minimize; for example,
if f is a heuristic estimate to the goal, then we have greedy best
first search; if f is node.depth then we have breadth-first search.
There is a subtlety: the line "f = memoize(f, 'f')" means that the f
values will be cached on the nodes as they are computed. So after doing
a best first search you can examine the f values of the path returned."""
f = memoize(f, 'f')
node = Node(problem.initial)
frontier = PriorityQueue('min', f)
frontier.append(node)
explored = set()
while frontier:
node = frontier.pop()
if problem.goal_test(node.state):
return node
explored.add(node.state)
for child in node.expand(problem):
if child.state not in explored and child not in frontier:
frontier.append(child)
elif child in frontier:
if f(child) < frontier[child]:
del frontier[child]
frontier.append(child)
return None
def uniform_cost_search(problem):
"""[Figure 3.14]"""
return best_first_graph_search(problem, lambda node: node.path_cost)
def depth_limited_search(problem, limit=50):
"""[Figure 3.17]"""
def recursive_dls(node, problem, limit):
if problem.goal_test(node.state):
return node
elif limit == 0:
return 'cutoff'
else:
cutoff_occurred = False
for child in node.expand(problem):
result = recursive_dls(child, problem, limit - 1)
if result == 'cutoff':
cutoff_occurred = True
elif result is not None:
return result
return 'cutoff' if cutoff_occurred else None
# Body of depth_limited_search:
return recursive_dls(Node(problem.initial), problem, limit)
def iterative_deepening_search(problem):
"""[Figure 3.18]"""
for depth in range(sys.maxsize):
result = depth_limited_search(problem, depth)
if result != 'cutoff':
return result
# ______________________________________________________________________________
# Bidirectional Search
# Pseudocode from https://webdocs.cs.ualberta.ca/%7Eholte/Publications/MM-AAAI2016.pdf
def bidirectional_search(problem):
e = problem.find_min_edge()
gF, gB = {problem.initial : 0}, {problem.goal : 0}
openF, openB = [problem.initial], [problem.goal]
closedF, closedB = [], []
U = infinity
def extend(U, open_dir, open_other, g_dir, g_other, closed_dir):
"""Extend search in given direction"""
n = find_key(C, open_dir, g_dir)
open_dir.remove(n)
closed_dir.append(n)
for c in problem.actions(n):
if c in open_dir or c in closed_dir:
if g_dir[c] <= problem.path_cost(g_dir[n], n, None, c):
continue
open_dir.remove(c)
g_dir[c] = problem.path_cost(g_dir[n], n, None, c)
open_dir.append(c)
if c in open_other:
U = min(U, g_dir[c] + g_other[c])
return U, open_dir, closed_dir, g_dir
def find_min(open_dir, g):
"""Finds minimum priority, g and f values in open_dir"""
m, m_f = infinity, infinity
for n in open_dir:
f = g[n] + problem.h(n)
pr = max(f, 2*g[n])
m = min(m, pr)
m_f = min(m_f, f)
return m, m_f, min(g.values())
def find_key(pr_min, open_dir, g):
"""Finds key in open_dir with value equal to pr_min
and minimum g value."""
m = infinity
state = -1
for n in open_dir:
pr = max(g[n] + problem.h(n), 2*g[n])
if pr == pr_min:
if g[n] < m:
m = g[n]
state = n
return state
while openF and openB:
pr_min_f, f_min_f, g_min_f = find_min(openF, gF)
pr_min_b, f_min_b, g_min_b = find_min(openB, gB)
C = min(pr_min_f, pr_min_b)
if U <= max(C, f_min_f, f_min_b, g_min_f + g_min_b + e):
return U
if C == pr_min_f:
# Extend forward
U, openF, closedF, gF = extend(U, openF, openB, gF, gB, closedF)
else:
# Extend backward
U, openB, closedB, gB = extend(U, openB, openF, gB, gF, closedB)
return infinity
# ______________________________________________________________________________
# Informed (Heuristic) Search
greedy_best_first_graph_search = best_first_graph_search
# Greedy best-first search is accomplished by specifying f(n) = h(n).
def astar_search(problem, h=None):
"""A* search is best-first graph search with f(n) = g(n)+h(n).
You need to specify the h function when you call astar_search, or
else in your Problem subclass."""
h = memoize(h or problem.h, 'h')
return best_first_graph_search(problem, lambda n: n.path_cost + h(n))
# ______________________________________________________________________________
# A* heuristics
class EightPuzzle(Problem):
""" The problem of sliding tiles numbered from 1 to 8 on a 3x3 board,
where one of the squares is a blank. A state is represented as a tuple of length 9,
where element at index i represents the tile number at index i (0 if it's an empty square) """
def __init__(self, initial, goal=(1, 2, 3, 4, 5, 6, 7, 8, 0)):
""" Define goal state and initialize a problem """
self.goal = goal
Problem.__init__(self, initial, goal)
def find_blank_square(self, state):
"""Return the index of the blank square in a given state"""
return state.index(0)
def actions(self, state):
""" Return the actions that can be executed in the given state.
The result would be a list, since there are only four possible actions
in any given state of the environment """
possible_actions = ['UP', 'DOWN', 'LEFT', 'RIGHT']
index_blank_square = self.find_blank_square(state)
if index_blank_square % 3 == 0:
possible_actions.remove('LEFT')
if index_blank_square < 3:
possible_actions.remove('UP')
if index_blank_square % 3 == 2:
possible_actions.remove('RIGHT')
if index_blank_square > 5:
possible_actions.remove('DOWN')
return possible_actions
def result(self, state, action):
""" Given state and action, return a new state that is the result of the action.
Action is assumed to be a valid action in the state """
# blank is the index of the blank square
blank = self.find_blank_square(state)
new_state = list(state)
delta = {'UP':-3, 'DOWN':3, 'LEFT':-1, 'RIGHT':1}
neighbor = blank + delta[action]
new_state[blank], new_state[neighbor] = new_state[neighbor], new_state[blank]
return tuple(new_state)
def goal_test(self, state):
""" Given a state, return True if state is a goal state or False, otherwise """
return state == self.goal
def check_solvability(self, state):
""" Checks if the given state is solvable """
inversion = 0
for i in range(len(state)):
for j in range(i+1, len(state)):
if (state[i] > state[j]) and state[i] != 0 and state[j]!= 0:
inversion += 1
return inversion % 2 == 0
def h(self, node):
""" Return the heuristic value for a given state. Default heuristic function used is
h(n) = number of misplaced tiles """
return sum(s != g for (s, g) in zip(node.state, self.goal))
# ______________________________________________________________________________
class PlanRoute(Problem):
""" The problem of moving the Hybrid Wumpus Agent from one place to other """
def __init__(self, initial, goal, allowed, dimrow):
""" Define goal state and initialize a problem """
self.dimrow = dimrow
self.goal = goal
self.allowed = allowed
Problem.__init__(self, initial, goal)
def actions(self, state):
""" Return the actions that can be executed in the given state.
The result would be a list, since there are only three possible actions
in any given state of the environment """
possible_actions = ['Forward', 'TurnLeft', 'TurnRight']
x, y = state.get_location()
orientation = state.get_orientation()
# Prevent Bumps
if x == 1 and orientation == 'LEFT':
if 'Forward' in possible_actions:
possible_actions.remove('Forward')
if y == 1 and orientation == 'DOWN':
if 'Forward' in possible_actions:
possible_actions.remove('Forward')
if x == self.dimrow and orientation == 'RIGHT':
if 'Forward' in possible_actions:
possible_actions.remove('Forward')
if y == self.dimrow and orientation == 'UP':
if 'Forward' in possible_actions:
possible_actions.remove('Forward')
return possible_actions
def result(self, state, action):
""" Given state and action, return a new state that is the result of the action.
Action is assumed to be a valid action in the state """
x, y = state.get_location()
proposed_loc = list()
# Move Forward
if action == 'Forward':
if state.get_orientation() == 'UP':
proposed_loc = [x, y + 1]
elif state.get_orientation() == 'DOWN':
proposed_loc = [x, y - 1]
elif state.get_orientation() == 'LEFT':
proposed_loc = [x - 1, y]
elif state.get_orientation() == 'RIGHT':
proposed_loc = [x + 1, y]
else:
raise Exception('InvalidOrientation')
# Rotate counter-clockwise
elif action == 'TurnLeft':
if state.get_orientation() == 'UP':
state.set_orientation('LEFT')
elif state.get_orientation() == 'DOWN':
state.set_orientation('RIGHT')
elif state.get_orientation() == 'LEFT':
state.set_orientation('DOWN')
elif state.get_orientation() == 'RIGHT':
state.set_orientation('UP')
else:
raise Exception('InvalidOrientation')
# Rotate clockwise
elif action == 'TurnRight':
if state.get_orientation() == 'UP':
state.set_orientation('RIGHT')
elif state.get_orientation() == 'DOWN':
state.set_orientation('LEFT')
elif state.get_orientation() == 'LEFT':
state.set_orientation('UP')
elif state.get_orientation() == 'RIGHT':
state.set_orientation('DOWN')
else:
raise Exception('InvalidOrientation')
if proposed_loc in self.allowed:
state.set_location(proposed_loc[0], [proposed_loc[1]])
return state
def goal_test(self, state):
""" Given a state, return True if state is a goal state or False, otherwise """
return state.get_location() == tuple(self.goal)
def h(self, node):
""" Return the heuristic value for a given state."""
# Manhattan Heuristic Function
x1, y1 = node.state.get_location()
x2, y2 = self.goal
return abs(x2 - x1) + abs(y2 - y1)
# ______________________________________________________________________________
# Other search algorithms
def recursive_best_first_search(problem, h=None):
"""[Figure 3.26]"""
h = memoize(h or problem.h, 'h')
def RBFS(problem, node, flimit):
if problem.goal_test(node.state):
return node, 0 # (The second value is immaterial)
successors = node.expand(problem)
if len(successors) == 0:
return None, infinity
for s in successors:
s.f = max(s.path_cost + h(s), node.f)
while True:
# Order by lowest f value
successors.sort(key=lambda x: x.f)
best = successors[0]
if best.f > flimit:
return None, best.f
if len(successors) > 1:
alternative = successors[1].f
else:
alternative = infinity
result, best.f = RBFS(problem, best, min(flimit, alternative))
if result is not None:
return result, best.f
node = Node(problem.initial)
node.f = h(node)
result, bestf = RBFS(problem, node, infinity)
return result
def hill_climbing(problem):
"""From the initial node, keep choosing the neighbor with highest value,
stopping when no neighbor is better. [Figure 4.2]"""
current = Node(problem.initial)
while True:
neighbors = current.expand(problem)
if not neighbors:
break
neighbor = argmax_random_tie(neighbors,
key=lambda node: problem.value(node.state))
if problem.value(neighbor.state) <= problem.value(current.state):
break
current = neighbor
return current.state
def exp_schedule(k=20, lam=0.005, limit=100):
"""One possible schedule function for simulated annealing"""
return lambda t: (k * math.exp(-lam * t) if t < limit else 0)
def simulated_annealing(problem, schedule=exp_schedule()):
"""[Figure 4.5] CAUTION: This differs from the pseudocode as it
returns a state instead of a Node."""
current = Node(problem.initial)
for t in range(sys.maxsize):
T = schedule(t)
if T == 0:
return current.state
neighbors = current.expand(problem)
if not neighbors:
return current.state
next_choice = random.choice(neighbors)
delta_e = problem.value(next_choice.state) - problem.value(current.state)
if delta_e > 0 or probability(math.exp(delta_e / T)):
current = next_choice
def simulated_annealing_full(problem, schedule=exp_schedule()):
""" This version returns all the states encountered in reaching
the goal state."""
states = []
current = Node(problem.initial)
for t in range(sys.maxsize):
states.append(current.state)
T = schedule(t)
if T == 0:
return states
neighbors = current.expand(problem)
if not neighbors:
return current.state
next_choice = random.choice(neighbors)
delta_e = problem.value(next_choice.state) - problem.value(current.state)
if delta_e > 0 or probability(math.exp(delta_e / T)):
current = next_choice
def and_or_graph_search(problem):
"""[Figure 4.11]Used when the environment is nondeterministic and completely observable.
Contains OR nodes where the agent is free to choose any action.
After every action there is an AND node which contains all possible states
the agent may reach due to stochastic nature of environment.
The agent must be able to handle all possible states of the AND node (as it
may end up in any of them).
Returns a conditional plan to reach goal state,
or failure if the former is not possible."""
# functions used by and_or_search
def or_search(state, problem, path):
"""returns a plan as a list of actions"""
if problem.goal_test(state):
return []
if state in path:
return None
for action in problem.actions(state):
plan = and_search(problem.result(state, action),
problem, path + [state, ])
if plan is not None:
return [action, plan]
def and_search(states, problem, path):
"""Returns plan in form of dictionary where we take action plan[s] if we reach state s."""
plan = {}
for s in states:
plan[s] = or_search(s, problem, path)
if plan[s] is None:
return None
return plan
# body of and or search
return or_search(problem.initial, problem, [])
# Pre-defined actions for PeakFindingProblem
directions4 = { 'W':(-1, 0), 'N':(0, 1), 'E':(1, 0), 'S':(0, -1) }
directions8 = dict(directions4)
directions8.update({'NW':(-1, 1), 'NE':(1, 1), 'SE':(1, -1), 'SW':(-1, -1) })
class PeakFindingProblem(Problem):
"""Problem of finding the highest peak in a limited grid"""
def __init__(self, initial, grid, defined_actions=directions4):
"""The grid is a 2 dimensional array/list whose state is specified by tuple of indices"""
Problem.__init__(self, initial)
self.grid = grid
self.defined_actions = defined_actions
self.n = len(grid)
assert self.n > 0
self.m = len(grid[0])
assert self.m > 0
def actions(self, state):
"""Returns the list of actions which are allowed to be taken from the given state"""
allowed_actions = []
for action in self.defined_actions:
next_state = vector_add(state, self.defined_actions[action])
if next_state[0] >= 0 and next_state[1] >= 0 and next_state[0] <= self.n - 1 and next_state[1] <= self.m - 1:
allowed_actions.append(action)
return allowed_actions
def result(self, state, action):
"""Moves in the direction specified by action"""
return vector_add(state, self.defined_actions[action])
def value(self, state):
"""Value of a state is the value it is the index to"""
x, y = state
assert 0 <= x < self.n
assert 0 <= y < self.m
return self.grid[x][y]
class OnlineDFSAgent:
"""[Figure 4.21] The abstract class for an OnlineDFSAgent. Override
update_state method to convert percept to state. While initializing
the subclass a problem needs to be provided which is an instance of
a subclass of the Problem class."""
def __init__(self, problem):
self.problem = problem
self.s = None
self.a = None
self.untried = dict()
self.unbacktracked = dict()
self.result = {}
def __call__(self, percept):
s1 = self.update_state(percept)
if self.problem.goal_test(s1):
self.a = None
else:
if s1 not in self.untried.keys():
self.untried[s1] = self.problem.actions(s1)
if self.s is not None:
if s1 != self.result[(self.s, self.a)]:
self.result[(self.s, self.a)] = s1
self.unbacktracked[s1].insert(0, self.s)
if len(self.untried[s1]) == 0:
if len(self.unbacktracked[s1]) == 0:
self.a = None
else:
# else a <- an action b such that result[s', b] = POP(unbacktracked[s'])
unbacktracked_pop = self.unbacktracked.pop(s1)
for (s, b) in self.result.keys():
if self.result[(s, b)] == unbacktracked_pop:
self.a = b
break
else:
self.a = self.untried.pop(s1)
self.s = s1
return self.a
def update_state(self, percept):
"""To be overridden in most cases. The default case
assumes the percept to be of type state."""
return percept
# ______________________________________________________________________________
class OnlineSearchProblem(Problem):
"""
A problem which is solved by an agent executing
actions, rather than by just computation.
Carried in a deterministic and a fully observable environment."""
def __init__(self, initial, goal, graph):
self.initial = initial
self.goal = goal
self.graph = graph
def actions(self, state):
return self.graph.graph_dict[state].keys()
def output(self, state, action):
return self.graph.graph_dict[state][action]
def h(self, state):
"""Returns least possible cost to reach a goal for the given state."""
return self.graph.least_costs[state]
def c(self, s, a, s1):
"""Returns a cost estimate for an agent to move from state 's' to state 's1'."""
return 1
def update_state(self, percept):
raise NotImplementedError
def goal_test(self, state):
if state == self.goal:
return True
return False
class LRTAStarAgent:
""" [Figure 4.24]
Abstract class for LRTA*-Agent. A problem needs to be
provided which is an instance of a subclass of Problem Class.
Takes a OnlineSearchProblem [Figure 4.23] as a problem.
"""
def __init__(self, problem):
self.problem = problem
# self.result = {} # no need as we are using problem.result
self.H = {}
self.s = None
self.a = None
def __call__(self, s1): # as of now s1 is a state rather than a percept
if self.problem.goal_test(s1):
self.a = None
return self.a
else:
if s1 not in self.H:
self.H[s1] = self.problem.h(s1)
if self.s is not None:
# self.result[(self.s, self.a)] = s1 # no need as we are using problem.output
# minimum cost for action b in problem.actions(s)
self.H[self.s] = min(self.LRTA_cost(self.s, b, self.problem.output(self.s, b),
self.H) for b in self.problem.actions(self.s))
# an action b in problem.actions(s1) that minimizes costs
self.a = argmin(self.problem.actions(s1),
key=lambda b: self.LRTA_cost(s1, b, self.problem.output(s1, b), self.H))
self.s = s1
return self.a
def LRTA_cost(self, s, a, s1, H):
"""Returns cost to move from state 's' to state 's1' plus
estimated cost to get to goal from s1."""
print(s, a, s1)
if s1 is None:
return self.problem.h(s)
else:
# sometimes we need to get H[s1] which we haven't yet added to H
# to replace this try, except: we can initialize H with values from problem.h
try:
return self.problem.c(s, a, s1) + self.H[s1]
except:
return self.problem.c(s, a, s1) + self.problem.h(s1)
# ______________________________________________________________________________
# Genetic Algorithm
def genetic_search(problem, fitness_fn, ngen=1000, pmut=0.1, n=20):
"""Call genetic_algorithm on the appropriate parts of a problem.
This requires the problem to have states that can mate and mutate,
plus a value method that scores states."""
# NOTE: This is not tested and might not work.
# TODO: Use this function to make Problems work with genetic_algorithm.
s = problem.initial_state
states = [problem.result(s, a) for a in problem.actions(s)]
random.shuffle(states)
return genetic_algorithm(states[:n], problem.value, ngen, pmut)
def genetic_algorithm(population, fitness_fn, gene_pool=[0, 1], f_thres=None, ngen=1000, pmut=0.1):
"""[Figure 4.8]"""
for i in range(ngen):
population = [mutate(recombine(*select(2, population, fitness_fn)), gene_pool, pmut)
for i in range(len(population))]
fittest_individual = fitness_threshold(fitness_fn, f_thres, population)
if fittest_individual:
return fittest_individual
return argmax(population, key=fitness_fn)
def fitness_threshold(fitness_fn, f_thres, population):
if not f_thres:
return None
fittest_individual = argmax(population, key=fitness_fn)
if fitness_fn(fittest_individual) >= f_thres:
return fittest_individual
return None
def init_population(pop_number, gene_pool, state_length):
"""Initializes population for genetic algorithm
pop_number : Number of individuals in population
gene_pool : List of possible values for individuals
state_length: The length of each individual"""
g = len(gene_pool)
population = []
for i in range(pop_number):
new_individual = [gene_pool[random.randrange(0, g)] for j in range(state_length)]
population.append(new_individual)
return population
def select(r, population, fitness_fn):
fitnesses = map(fitness_fn, population)
sampler = weighted_sampler(population, fitnesses)
return [sampler() for i in range(r)]
def recombine(x, y):
n = len(x)
c = random.randrange(0, n)
return x[:c] + y[c:]
def recombine_uniform(x, y):
n = len(x)
result = [0] * n
indexes = random.sample(range(n), n)
for i in range(n):
ix = indexes[i]
result[ix] = x[ix] if i < n / 2 else y[ix]
return ''.join(str(r) for r in result)
def mutate(x, gene_pool, pmut):
if random.uniform(0, 1) >= pmut:
return x
n = len(x)
g = len(gene_pool)
c = random.randrange(0, n)
r = random.randrange(0, g)
new_gene = gene_pool[r]
return x[:c] + [new_gene] + x[c+1:]
# _____________________________________________________________________________
# The remainder of this file implements examples for the search algorithms.
# ______________________________________________________________________________
# Graphs and Graph Problems
class Graph:
"""A graph connects nodes (vertices) by edges (links). Each edge can also
have a length associated with it. The constructor call is something like:
g = Graph({'A': {'B': 1, 'C': 2})
this makes a graph with 3 nodes, A, B, and C, with an edge of length 1 from
A to B, and an edge of length 2 from A to C. You can also do:
g = Graph({'A': {'B': 1, 'C': 2}, directed=False)
This makes an undirected graph, so inverse links are also added. The graph
stays undirected; if you add more links with g.connect('B', 'C', 3), then
inverse link is also added. You can use g.nodes() to get a list of nodes,
g.get('A') to get a dict of links out of A, and g.get('A', 'B') to get the
length of the link from A to B. 'Lengths' can actually be any object at
all, and nodes can be any hashable object."""
def __init__(self, graph_dict=None, directed=True):
self.graph_dict = graph_dict or {}
self.directed = directed
if not directed:
self.make_undirected()
def make_undirected(self):
"""Make a digraph into an undirected graph by adding symmetric edges."""
for a in list(self.graph_dict.keys()):
for (b, dist) in self.graph_dict[a].items():
self.connect1(b, a, dist)
def connect(self, A, B, distance=1):
"""Add a link from A and B of given distance, and also add the inverse
link if the graph is undirected."""
self.connect1(A, B, distance)
if not self.directed:
self.connect1(B, A, distance)
def connect1(self, A, B, distance):
"""Add a link from A to B of given distance, in one direction only."""
self.graph_dict.setdefault(A, {})[B] = distance
def get(self, a, b=None):
"""Return a link distance or a dict of {node: distance} entries.
.get(a,b) returns the distance or None;
.get(a) returns a dict of {node: distance} entries, possibly {}."""
links = self.graph_dict.setdefault(a, {})
if b is None:
return links
else:
return links.get(b)
def nodes(self):
"""Return a list of nodes in the graph."""
s1 = set([k for k in self.graph_dict.keys()])
s2 = set([k2 for v in self.graph_dict.values() for k2, v2 in v.items()])
nodes = s1.union(s2)
return list(nodes)
def UndirectedGraph(graph_dict=None):
"""Build a Graph where every edge (including future ones) goes both ways."""
return Graph(graph_dict = graph_dict, directed=False)
def RandomGraph(nodes=list(range(10)), min_links=2, width=400, height=300,
curvature=lambda: random.uniform(1.1, 1.5)):
"""Construct a random graph, with the specified nodes, and random links.
The nodes are laid out randomly on a (width x height) rectangle.
Then each node is connected to the min_links nearest neighbors.
Because inverse links are added, some nodes will have more connections.
The distance between nodes is the hypotenuse times curvature(),
where curvature() defaults to a random number between 1.1 and 1.5."""
g = UndirectedGraph()
g.locations = {}
# Build the cities
for node in nodes:
g.locations[node] = (random.randrange(width), random.randrange(height))
# Build roads from each city to at least min_links nearest neighbors.
for i in range(min_links):
for node in nodes:
if len(g.get(node)) < min_links:
here = g.locations[node]
def distance_to_node(n):
if n is node or g.get(node, n):
return infinity
return distance(g.locations[n], here)
neighbor = argmin(nodes, key=distance_to_node)
d = distance(g.locations[neighbor], here) * curvature()
g.connect(node, neighbor, int(d))
return g
""" [Figure 3.2]
Simplified road map of Romania
"""
romania_map = UndirectedGraph(dict(
Arad=dict(Zerind=75, Sibiu=140, Timisoara=118),
Bucharest=dict(Urziceni=85, Pitesti=101, Giurgiu=90, Fagaras=211),
Craiova=dict(Drobeta=120, Rimnicu=146, Pitesti=138),
Drobeta=dict(Mehadia=75),
Eforie=dict(Hirsova=86),
Fagaras=dict(Sibiu=99),
Hirsova=dict(Urziceni=98),
Iasi=dict(Vaslui=92, Neamt=87),
Lugoj=dict(Timisoara=111, Mehadia=70),
Oradea=dict(Zerind=71, Sibiu=151),
Pitesti=dict(Rimnicu=97),
Rimnicu=dict(Sibiu=80),
Urziceni=dict(Vaslui=142)))
romania_map.locations = dict(
Arad=(91, 492), Bucharest=(400, 327), Craiova=(253, 288),
Drobeta=(165, 299), Eforie=(562, 293), Fagaras=(305, 449),
Giurgiu=(375, 270), Hirsova=(534, 350), Iasi=(473, 506),
Lugoj=(165, 379), Mehadia=(168, 339), Neamt=(406, 537),
Oradea=(131, 571), Pitesti=(320, 368), Rimnicu=(233, 410),
Sibiu=(207, 457), Timisoara=(94, 410), Urziceni=(456, 350),
Vaslui=(509, 444), Zerind=(108, 531))
""" [Figure 4.9]
Eight possible states of the vacumm world
Each state is represented as
* "State of the left room" "State of the right room" "Room in which the agent
is present"
1 - DDL Dirty Dirty Left
2 - DDR Dirty Dirty Right
3 - DCL Dirty Clean Left
4 - DCR Dirty Clean Right
5 - CDL Clean Dirty Left
6 - CDR Clean Dirty Right
7 - CCL Clean Clean Left
8 - CCR Clean Clean Right
"""
vacuum_world = Graph(dict(
State_1=dict(Suck=['State_7', 'State_5'], Right=['State_2']),
State_2=dict(Suck=['State_8', 'State_4'], Left=['State_2']),
State_3=dict(Suck=['State_7'], Right=['State_4']),
State_4=dict(Suck=['State_4', 'State_2'], Left=['State_3']),
State_5=dict(Suck=['State_5', 'State_1'], Right=['State_6']),
State_6=dict(Suck=['State_8'], Left=['State_5']),
State_7=dict(Suck=['State_7', 'State_3'], Right=['State_8']),
State_8=dict(Suck=['State_8', 'State_6'], Left=['State_7'])
))
""" [Figure 4.23]
One-dimensional state space Graph
"""
one_dim_state_space = Graph(dict(
State_1=dict(Right='State_2'),
State_2=dict(Right='State_3', Left='State_1'),
State_3=dict(Right='State_4', Left='State_2'),
State_4=dict(Right='State_5', Left='State_3'),
State_5=dict(Right='State_6', Left='State_4'),
State_6=dict(Left='State_5')
))
one_dim_state_space.least_costs = dict(
State_1=8,
State_2=9,
State_3=2,
State_4=2,
State_5=4,
State_6=3)
""" [Figure 6.1]
Principal states and territories of Australia
"""
australia_map = UndirectedGraph(dict(
T=dict(),
SA=dict(WA=1, NT=1, Q=1, NSW=1, V=1),
NT=dict(WA=1, Q=1),
NSW=dict(Q=1, V=1)))
australia_map.locations = dict(WA=(120, 24), NT=(135, 20), SA=(135, 30),
Q=(145, 20), NSW=(145, 32), T=(145, 42),
V=(145, 37))
class GraphProblem(Problem):
"""The problem of searching a graph from one node to another."""
def __init__(self, initial, goal, graph):
Problem.__init__(self, initial, goal)
self.graph = graph
def actions(self, A):
"""The actions at a graph node are just its neighbors."""
return list(self.graph.get(A).keys())
def result(self, state, action):
"""The result of going to a neighbor is just that neighbor."""
return action
def path_cost(self, cost_so_far, A, action, B):
return cost_so_far + (self.graph.get(A, B) or infinity)
def find_min_edge(self):
"""Find minimum value of edges."""
m = infinity
for d in self.graph.graph_dict.values():
local_min = min(d.values())
m = min(m, local_min)
return m
def h(self, node):
"""h function is straight-line distance from a node's state to goal."""
locs = getattr(self.graph, 'locations', None)
if locs:
if type(node) is str:
return int(distance(locs[node], locs[self.goal]))
return int(distance(locs[node.state], locs[self.goal]))
else:
return infinity
class GraphProblemStochastic(GraphProblem):
"""
A version of GraphProblem where an action can lead to
nondeterministic output i.e. multiple possible states.
Define the graph as dict(A = dict(Action = [[<Result 1>, <Result 2>, ...], <cost>], ...), ...)
A the dictionary format is different, make sure the graph is created as a directed graph.
"""
def result(self, state, action):
return self.graph.get(state, action)
def path_cost(self):
raise NotImplementedError
# ______________________________________________________________________________
class NQueensProblem(Problem):
"""The problem of placing N queens on an NxN board with none attacking
each other. A state is represented as an N-element array, where
a value of r in the c-th entry means there is a queen at column c,
row r, and a value of -1 means that the c-th column has not been
filled in yet. We fill in columns left to right.
>>> depth_first_tree_search(NQueensProblem(8))
<Node (7, 3, 0, 2, 5, 1, 6, 4)>
"""
def __init__(self, N):
self.N = N
self.initial = tuple([-1] * N)
Problem.__init__(self, self.initial)
def actions(self, state):
"""In the leftmost empty column, try all non-conflicting rows."""
if state[-1] is not -1:
return [] # All columns filled; no successors
else:
col = state.index(-1)
return [row for row in range(self.N)
if not self.conflicted(state, row, col)]
def result(self, state, row):
"""Place the next queen at the given row."""
col = state.index(-1)
new = list(state[:])
new[col] = row
return tuple(new)
def conflicted(self, state, row, col):
"""Would placing a queen at (row, col) conflict with anything?"""
return any(self.conflict(row, col, state[c], c)
for c in range(col))
def conflict(self, row1, col1, row2, col2):
"""Would putting two queens in (row1, col1) and (row2, col2) conflict?"""
return (row1 == row2 or # same row
col1 == col2 or # same column
row1 - col1 == row2 - col2 or # same \ diagonal
row1 + col1 == row2 + col2) # same / diagonal
def goal_test(self, state):
"""Check if all columns filled, no conflicts."""
if state[-1] is -1:
return False
return not any(self.conflicted(state, state[col], col)
for col in range(len(state)))
def h(self, node):
"""Return number of conflicting queens for a given node"""
num_conflicts = 0
for (r1, c1) in enumerate(node.state):
for (r2, c2) in enumerate(node.state):
if (r1, c1) != (r2, c2):
num_conflicts += self.conflict(r1, c1, r2, c2)
return num_conflicts
# ______________________________________________________________________________
# Inverse Boggle: Search for a high-scoring Boggle board. A good domain for
# iterative-repair and related search techniques, as suggested by Justin Boyan.
ALPHABET = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'
cubes16 = ['FORIXB', 'MOQABJ', 'GURILW', 'SETUPL',
'CMPDAE', 'ACITAO', 'SLCRAE', 'ROMASH',
'NODESW', 'HEFIYE', 'ONUDTK', 'TEVIGN',
'ANEDVZ', 'PINESH', 'ABILYT', 'GKYLEU']
def random_boggle(n=4):
"""Return a random Boggle board of size n x n.
We represent a board as a linear list of letters."""
cubes = [cubes16[i % 16] for i in range(n * n)]
random.shuffle(cubes)
return list(map(random.choice, cubes))
# The best 5x5 board found by Boyan, with our word list this board scores
# 2274 words, for a score of 9837
boyan_best = list('RSTCSDEIAEGNLRPEATESMSSID')
def print_boggle(board):
"""Print the board in a 2-d array."""
n2 = len(board)
n = exact_sqrt(n2)
for i in range(n2):
if i % n == 0 and i > 0:
print()
if board[i] == 'Q':
print('Qu', end=' ')
else:
print(str(board[i]) + ' ', end=' ')
print()
def boggle_neighbors(n2, cache={}):
"""Return a list of lists, where the i-th element is the list of indexes
for the neighbors of square i."""
if cache.get(n2):
return cache.get(n2)
n = exact_sqrt(n2)
neighbors = [None] * n2
for i in range(n2):
neighbors[i] = []
on_top = i < n
on_bottom = i >= n2 - n
on_left = i % n == 0
on_right = (i+1) % n == 0
if not on_top:
neighbors[i].append(i - n)
if not on_left:
neighbors[i].append(i - n - 1)
if not on_right:
neighbors[i].append(i - n + 1)
if not on_bottom:
neighbors[i].append(i + n)
if not on_left:
neighbors[i].append(i + n - 1)
if not on_right:
neighbors[i].append(i + n + 1)
if not on_left:
neighbors[i].append(i - 1)
if not on_right:
neighbors[i].append(i + 1)
cache[n2] = neighbors
return neighbors
def exact_sqrt(n2):
"""If n2 is a perfect square, return its square root, else raise error."""
n = int(math.sqrt(n2))
assert n * n == n2
return n
# _____________________________________________________________________________
class Wordlist:
"""This class holds a list of words. You can use (word in wordlist)
to check if a word is in the list, or wordlist.lookup(prefix)
to see if prefix starts any of the words in the list."""
def __init__(self, file, min_len=3):
lines = file.read().upper().split()
self.words = [word for word in lines if len(word) >= min_len]
self.words.sort()
self.bounds = {}
for c in ALPHABET:
c2 = chr(ord(c) + 1)
self.bounds[c] = (bisect.bisect(self.words, c),
bisect.bisect(self.words, c2))
def lookup(self, prefix, lo=0, hi=None):
"""See if prefix is in dictionary, as a full word or as a prefix.
Return two values: the first is the lowest i such that
words[i].startswith(prefix), or is None; the second is
True iff prefix itself is in the Wordlist."""
words = self.words
if hi is None:
hi = len(words)
i = bisect.bisect_left(words, prefix, lo, hi)
if i < len(words) and words[i].startswith(prefix):
return i, (words[i] == prefix)
else:
return None, False
def __contains__(self, word):
return self.lookup(word)[1]
def __len__(self):
return len(self.words)
# _____________________________________________________________________________
class BoggleFinder:
"""A class that allows you to find all the words in a Boggle board."""
wordlist = None # A class variable, holding a wordlist
def __init__(self, board=None):
if BoggleFinder.wordlist is None:
BoggleFinder.wordlist = Wordlist(open_data("EN-text/wordlist.txt"))
self.found = {}
if board:
self.set_board(board)
def set_board(self, board=None):
"""Set the board, and find all the words in it."""
if board is None:
board = random_boggle()
self.board = board
self.neighbors = boggle_neighbors(len(board))
self.found = {}
for i in range(len(board)):
lo, hi = self.wordlist.bounds[board[i]]
self.find(lo, hi, i, [], '')
return self
def find(self, lo, hi, i, visited, prefix):
"""Looking in square i, find the words that continue the prefix,
considering the entries in self.wordlist.words[lo:hi], and not
revisiting the squares in visited."""
if i in visited:
return
wordpos, is_word = self.wordlist.lookup(prefix, lo, hi)
if wordpos is not None:
if is_word:
self.found[prefix] = True
visited.append(i)
c = self.board[i]
if c == 'Q':
c = 'QU'
prefix += c
for j in self.neighbors[i]:
self.find(wordpos, hi, j, visited, prefix)
visited.pop()
def words(self):
"""The words found."""
return list(self.found.keys())
scores = [0, 0, 0, 0, 1, 2, 3, 5] + [11] * 100
def score(self):
"""The total score for the words found, according to the rules."""
return sum([self.scores[len(w)] for w in self.words()])
def __len__(self):
"""The number of words found."""
return len(self.found)
# _____________________________________________________________________________
def boggle_hill_climbing(board=None, ntimes=100, verbose=True):
"""Solve inverse Boggle by hill-climbing: find a high-scoring board by
starting with a random one and changing it."""
finder = BoggleFinder()
if board is None:
board = random_boggle()
best = len(finder.set_board(board))
for _ in range(ntimes):
i, oldc = mutate_boggle(board)
new = len(finder.set_board(board))
if new > best:
best = new
if verbose:
print(best, _, board)
else:
board[i] = oldc # Change back
if verbose:
print_boggle(board)
return board, best
def mutate_boggle(board):
i = random.randrange(len(board))
oldc = board[i]
# random.choice(boyan_best)
board[i] = random.choice(random.choice(cubes16))
return i, oldc
# ______________________________________________________________________________
# Code to compare searchers on various problems.
class InstrumentedProblem(Problem):
"""Delegates to a problem, and keeps statistics."""
def __init__(self, problem):
self.problem = problem
self.succs = self.goal_tests = self.states = 0
self.found = None
def actions(self, state):
self.succs += 1
return self.problem.actions(state)
def result(self, state, action):
self.states += 1
return self.problem.result(state, action)
def goal_test(self, state):
self.goal_tests += 1
result = self.problem.goal_test(state)
if result:
self.found = state
return result
def path_cost(self, c, state1, action, state2):
return self.problem.path_cost(c, state1, action, state2)
def value(self, state):
return self.problem.value(state)
def __getattr__(self, attr):
return getattr(self.problem, attr)
def __repr__(self):
return '<{:4d}/{:4d}/{:4d}/{}>'.format(self.succs, self.goal_tests,
self.states, str(self.found)[:4])
def compare_searchers(problems, header,
searchers=[breadth_first_tree_search,
breadth_first_graph_search,
depth_first_graph_search,
iterative_deepening_search,
depth_limited_search,
recursive_best_first_search]):
def do(searcher, problem):
p = InstrumentedProblem(problem)
searcher(p)
return p
table = [[name(s)] + [do(s, p) for p in problems] for s in searchers]
print_table(table, header)
def compare_graph_searchers():
"""Prints a table of search results."""
compare_searchers(problems=[GraphProblem('Arad', 'Bucharest', romania_map),
GraphProblem('Oradea', 'Neamt', romania_map),
GraphProblem('Q', 'WA', australia_map)],
header=['Searcher', 'romania_map(Arad, Bucharest)',
'romania_map(Oradea, Neamt)', 'australia_map'])
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