The module provides a simple abstract class `SearchProblem[S,A]`

to be specialized for a specific problem.

The concrete class `SearchSolver`

is used to configure, query, and run a solver for a given problem.

For an example, see the `puzzle.nit`

program in the `examples`

subdirectory.

ai :: SearchNode

A node in the search-tree visited by a`SearchSolver`

.
ai :: SearchProblem

Abstract search problem over immutable states (`S`

) and actions (`A`

).
ai :: SearchSolver

A running solver for a given problem, to configure and control.ai $ SearchNode

A node in the search-tree visited by a`SearchSolver`

.
`S`

) and actions (`A`

).
core :: union_find

union–find algorithm using an efficient disjoint-set data structure```
# Basic framework for search problems and solver.
#
# The module provides a simple abstract class `SearchProblem[S,A]` to be specialized for a specific problem.
#
# The concrete class `SearchSolver` is used to configure, query, and run a solver for a given problem.
#
# For an example, see the `puzzle.nit` program in the `examples` subdirectory.
module search
import realtime
import trees
# Abstract search problem over immutable states (`S`) and actions (`A`).
#
# This class serves to model problems of planing and path-finding.
# A state, `S`, is a point in the search problem and fully models a given state of the world.
# An action, `A`, is an available mean of transition between two states.
#
# This abstract class is generic made to work with any kind of states and actions.
# Therefore, specific subclasses must be developed to implement the required services:
#
# * `initial_state`
# * `actions`
# * `apply_action`
# * `is_goal`
#
# Note that the implemented methods should not temper with the parameters since it is expected
# that they remain unmodified.
#
#
# # Basic search
#
# These tree method are enough to trigger a basic search.
#
# The methods `breadth_first` and `depth_first` return pre-configured solvers for, respectively,
# a breadth-first search, a depth-first search.
#
#
# # Advanced search
#
# The `cost` method can be implemented to represent the cost of a single transition.
# By default, the cost is 1.
#
# The `heuristic` method can be implemented to represent a lower-bound estimation of the remaining
# cost to reach a goal state.
# By default, the heuristic is 0.
#
# If one of these (or both) methods are implemented, the `astar` method will return a pre-configured
# solver for a A* search.
#
# More configuration and optimization on the search can to be done in the `SearchSolver` class.
interface SearchProblem[S: Object, A]
# The starting state for the problem
fun initial_state: S is abstract
# The available applicable actions for a given state.
# While there is a potential large number of distinct states and actions, there should be only
# a small number of possible action from a specific state (a small, or at least finite, branching factor).
fun actions(state: S): nullable SequenceRead[A] is abstract
# The new state when applying a given action
#
# The returned state can be:
# * a completely new state,
# * an existing state,
# * a new state but equals to an existing state
# in this case, ensure that the `==` and `hash` method
# are correctly implemented.
#
# Remember, this method should not modify its parameters.
#
# REQUIRE: `actions(state).has(action)`
fun apply_action(state:S, action:A): S is abstract
# Is the state a goal state?
# Once a goal state is found, the solver is automatically stopped.
# A problem can have 0, one or more goals if it makes sense
# but solver must be used accordingly.
# Default: no goals
fun is_goal(state:S): Bool do return false
# The cost for `action` from `old_state` to `new_state`
# REQUIRE: `apply_action(old_state, action) == new_state`
# Default: `1`.
# Note that having a 0 or negative value can make some search
# algorithms fail, or not terminate.
fun cost(state:S, action:A, state2: S): Float do return 1.0
# An heuristic of the estimated `cost` going from `state` to a goal state.
#
# Is is expected that the heuristic is *admissible*, it means its is an
# optimistic estimation and never an over-estimate, thus is cannot be
# higher than the lowest possible remaining cost.
# See `SearchSolver::do_revisit` for details.
#
# Default: `0`.
fun heuristic(state: S): Float do return 0.0
# return a new breadth-first solver
fun breadth_first: SearchSolver[S, A]
do
var todo = (new Array[SearchNode[S, A]]).as_fifo
var sol = new SearchSolver[S, A](self, todo)
return sol
end
# return a new depth-first solver
fun depth_first: SearchSolver[S, A]
do
var todo = (new List[SearchNode[S, A]]).as_lifo
var sol = new SearchSolver[S, A](self, todo)
return sol
end
# return a new best-first solver
#
# notes:
# * if `heuristic` is not defined, this is basically a Dijkstra search.
# * if `cost` in not defined either, this is basically a breadth-first search.
fun astar: SearchSolver[S, A]
do
var cpt = new NodeComparator[S, A]
var todo = new MinHeap[SearchNode[S, A]](cpt)
var sol = new SearchSolver[S, A](self, todo)
return sol
end
# Create the initial node in the search-tree.
# Used internally by the solvers but made public for those who want to replay a plan.
fun initial_node: SearchNode[S, A]
do
var res = new SearchNode[S,A](self, initial_state, null, null, 0.0, 0)
res.compute_heuristic
return res
end
# Negligible quantity for float comparisons
# Because of float imprecision, two really near float values should be considered equals.
# However, the specific epsilon value could be specific to the problem.
#
# The epsilon value is used for cost comparisons.
#
# Default: 1E-9
fun epsilon: Float do return 0.000000001
# Run and evaluate solvers with multiple configuration.
# This method can be used to evaluate which configuration to choose by default for a given problem.
#
# `steps` is the maximum number of steps a giver configuration can run.
fun run_configs(steps: Int)
do
var c = 0
loop
if astar.run_config(steps, c, "A*") then break
c += 1
end
end
# Various Map implementations of memory.
# In order to try more configurations with `run_config`, this method
# is called to provide alternative implementation.
#
# For instance, a subclass can redefine this method and extends the result with an additional `RBTreeMap`.
# Note: because the true nature of the sates `S` is left to the user, some
# specific Map implementation could be more efficient than a HashMop.
#
# Default: A `HashMap`
fun make_memory: Array[Map[S, SearchNode[S, A]]]
do
var res = new Array[Map[S, SearchNode[S, A]]]
res.add new HashMap[S, SearchNode[S, A]]
return res
end
end
# A running solver for a given problem, to configure and control.
#
# For a given problem, a lot of variation of search algorithms can be made.
# Thus this class permit the user to control the parameters of the search algorithm.
#
# Note that this class is not meant to be specialized, and usually not instantiated directly.
#
#
# # Basic run and result.
#
# 1. Instantiate it with the method `breadth_first`, `depth_first`, or `astar` from `SearchProblem`.
# 2. Apply the method `run`, that will search and return a solution.
# 3. Retrieve information from the solution.
#
# ~~~~nitish
# var p: SearchProblem = new MyProblem
# var res = p.astar.run
# if res != null then print "Found plan with {res.depth} actions, that cost {res.cost}: {res.plan.join(", ")}"
# ~~~~
#
#
# # Step-by-step runs and multiple runs
#
# The `run_steps` method (see also `steps`, and `steps_limit`) can be used to run only a maximum number of steps.
# This method can be used as a *co-routine* and run them periodically for a small amount of time.
#
# `run` and `run_steps` return the next solution.
# A subsequent call to `run` returns the following solution and so on.
#
# When there is no more solutions available, `is_running` become false.
#
#
# # Search-trees
#
# Internally, solvers use a search-tree where nodes are labeled with states, and edges are labeled with actions.
# See `SearchNode` for details.
#
# The `run` method return a `SearchNode` that can be used to retrieve a lot of useful information,
# like the full `path` or the `plan`.
#
#
# # Configuration
#
# The solver provide some *knobs* to control how the search-tree is visited.
#
# * `memorize` (see also `memorize_late`)
# * `do_revisit` (see also `revisits`)
# * `depth_limit` (see also `iterative_deepening` and `depth_limit_reached`)
class SearchSolver[S: Object, A]
# The problem currently solved
var problem: SearchProblem[S, A]
# The currently open nodes to process.
# They are the open nodes.
#
# It is the nature of the queue that control how the solver works.
# However, the methods `SearchProblem::breadth_first`, `SearchProblem::depth_first`,
# and `SearchProblem::astar` takes care of its correct initialization.
private var todo: Queue[SearchNode[S, A]]
# Is the solver still running?
# A running solver has not yet exhausted all the possible solutions.
var is_running: Bool = true
# Does the solver need to memorize visited states?
# When activated, there is an increased memory consumption since
# all visited states must be kept in memory,
# However, there is real a gain, since visited nodes are not
# revisited (unless needed by `do_revisit`)
#
# Default: `true`
#
# Note: If the graph of states has circuits, then a memory-less search may not terminate.
var memorize: Bool = true is writable
# Use memory only on visited (closed) state.
# Less memory operations, but two big drawbacks:
# * duplicated nodes can fill the `todo` queue (and the memory)
# * duplicated nodes require more invocation of `SearchProblem::heuristic`
#
# Note: if `memorize` is false, then this has no effect.
#
# Default: `false`
var memorize_late: Bool = false is writable
# Storage of nodes when `memorize` is activated
# Each state is associated with a node.
# This permit:
# * to avoid to revisit visited nodes (see `do_revisit`)
# * to avoid to reinvoke `heuristic` on known states (unless `memorize_late` is set)
private var memory: Map[S, SearchNode[S, A]] = new HashMap[S, SearchNode[S, A]]
# Total number of time an already memorized node is seen again.
# If `memorize_late` is set, then only visited nodes are counted.
# Otherwise, nodes in the todo-list are also considered.
var memorized = 0
# Revisit states when a better path to them is found.
# Such revisits generally triggers more revisits because they yield
# better path to their neighbors.
#
# If `false`, visited states are never revisited.
#
# With astar and an admissible heuristic, no visited node should be revisited.
# If the heuristic is not admissible, one may consider set this to `true`.
#
# Obviously, if `memorize` is false, then the value has no specific effect
# since all states are considered unvisited.
#
# Default: `false`.
#
# See also `revisits` and `SearchNode::revisits`.
var do_revisit: Bool = false is writable
# Total number of states (potentially) revisited.
#
# It is the number of time that a better path to a visited state is found.
# With astar and a really admissible heuristic, this number should stay 0.
# So check this value if you are not sure of the heuristic.
#
# Note that states are effectively revisited if `do_revisit` is activated.
var revisits = 0
# The solution found by the last `run`.
#
# ensure `solution != null implies problem.is_goal(solution.state)`
var solution: nullable SearchNode[S,A] = null
# Nearest solution found (up to date).
# The nearest solution is the one with the smallest heuristic value.
# The cost is not considered.
var nearest_solution: nullable SearchNode[S,A] = null
# Limit in the depth search.
#
# States found above this limit are not considered.
#
# Use 0 for no limit.
# Default: 0
# See also: `iterative_deepening`
var depth_limit: Int = 0 is writable
# How much time a `depth_limit` was reached?
#
# This can be used to query if some solutions may have been
# ignored because of a `depth_limit`.
#
# This is also used automatically if `iterative_deepening` is activated.
var depth_limit_reached: Int = 0
# Increase amount for an iterative deepening search.
# It =0, then the iterative deepening search is disabled.
# If >0, then `depth_limit` is automatically increased when the todo
# queue is empty but the `depth_limit` was reached in the previous iteration.
# Default: 0
var iterative_deepening: Int = 0
# The total steps executed since the beginning
# A step is the visit of a node in the `todo`-list
var steps: Int = 0
# The total number of nodes created
var nodes = 0
# Limit in the number of steps for a `run`.
#
# One can modify this value then `run` or just call `run_steps`.
#
# Use 0 for no limit.
# Default: 0
var steps_limit: Int = 0 is writable
# Total number of neighbors considered.
var neighbors = 0
# The average number of neighbors by nodes.
fun branching_factor: Float do return neighbors.to_f / steps.to_f
# Update `steps_limit` then just run some additional steps
# Return the best solution so far (if any)
fun run_steps(steps: Int): nullable SearchNode[S,A]
do
assert steps > 0
self.steps_limit = self.steps + steps
return run
end
# Reset the search from the initial state.
# Is used at the beginning and with `iterative_deepening`.
private fun start
do
assert todo.is_empty
depth_limit_reached = 0
var initial_node = problem.initial_node
if memorize and not memorize_late then memory[initial_node.state] = initial_node
initial_node.id = nodes
nodes += 1
todo.add initial_node
end
# Run the solver and return the next solution (if any)
# Return null is one of these is true:
# * `steps_limit` is reached
# * the `todo` queue is empty (eg. no reachable solution)
fun run: nullable SearchNode[S,A]
do
if steps == 0 then start
var nearest = nearest_solution
loop
# Enough work
if steps_limit > 0 and steps >= steps_limit then break
#print "todo={todo.length}"
#print " {todo.join("\n ")}"
# Next node, please
if todo.is_empty then
# iterative depth search?
if depth_limit <= 0 or iterative_deepening <= 0 or depth_limit_reached == 0 then
is_running = false
break
end
depth_limit += iterative_deepening
start
end
var node = todo.take
# Skip `old` stuff
# Because `Queue` has no remove :(
if node.drop then continue
var state = node.state
if memorize and memorize_late then
# Is the state already visited?
var old = memory.get_or_null(state)
if old != null then
memorized += 1
if old.cost - node.cost < problem.epsilon then continue
revisits += 1
if not do_revisit then
old.revisits += 1
continue
end
node.revisits = old.revisits + 1
end
memory[state] = node
end
steps += 1
assert node.steps == 0
node.steps = steps
self.node = node
# Keep trace to the nearest
if nearest == null or node.heuristic < nearest.heuristic then
nearest = node
nearest_solution = node
end
#print "try {node}"
#print "try {node}; todo={todo.length}"
# Won?
if problem.is_goal(state) then
solution = node
return node
end
# Ignore successors states if the depth limit is reached
if depth_limit > 0 and node.depth >= depth_limit then
depth_limit_reached += 1
continue
end
# Now, expand!
var actions = problem.actions(state)
if actions == null then continue
for action in actions do
neighbors += 1
# Fast track if no memory or late memory
if not memorize or memorize_late then
var new_node = node.apply_action(action)
new_node.id = nodes
nodes += 1
todo.add(new_node)
continue
end
# Get the state and the cost. Do not create the node yet.
var new_state = problem.apply_action(state, action)
var new_cost = node.cost + problem.cost(state, action, new_state)
# So check if the state was already seen
var old = memory.get_or_null(new_state)
if old != null then
memorized += 1
# If not better, then skip
if old.cost - new_cost < problem.epsilon then continue
# If visited and do not revisit, then skip
if old.steps > 0 and not do_revisit then
old.revisits += 1
revisits += 1
continue
end
# Even if `==`, reuse the same state object so
# * it may helps the GC
# * user-cached things in the previous state can be reused
new_state = old.state
end
# Finally, create the node
var new_node = new SearchNode[S, A](problem, new_state, node, action, new_cost, node.depth+1)
new_node.id = nodes
nodes += 1
if old == null then
# Compute heuristic and cost
new_node.compute_heuristic
else
# Reuse heuristic and update the cost
var h = old.heuristic
new_node.heuristic = h
new_node.score = new_cost + h
# Is `old` a visited node?
if old.steps == 0 then
# Old is still in the todo list, so drop it
old.drop = true
else
# Old was visited, so revisit it
new_node.revisits = old.revisits + 1
revisits += 1
#print "found {old.cost}>{new_cost}:{old.cost>new_cost} d={old.cost-new_cost}\n\t{old}\nthat is worse than\n\t{new_node}"
end
end
memory[new_state] = new_node
todo.add(new_node)
end
end
return null
end
# The last visited node.
# Unless when debugging, the last visited node is not really meaningful.
var node: nullable SearchNode[S, A] = null
redef fun to_s
do
var res ="steps={steps} nodes={nodes} todo={todo.length}"
if neighbors > 0 then res += " n={neighbors} (bf={branching_factor})"
if revisits > 0 then res += " re={revisits}"
if memorized > 0 then res += " mem={memorized}"
var n = solution
if n != null then
res += " last={n}"
else
n = nearest_solution
if n != null then res += " near={n}"
end
return res
end
# Run the configuration number `i`, for `steps` number of steps.
# The message `msg` suffixed by the name of the configuration is printed followed by the result
#
# This method is used by `SearchProblem::run_configs`
fun run_config(steps: Int, i: Int, msg: String): Bool
do
do
if i == 0 then
msg += " -mem"
memorize = false
break
end
i -= 1
var mems = problem.make_memory
memory = mems[i % mems.length]
msg += " {memory.class_name}"
i = i / mems.length
if i % 2 == 0 then
msg += " +mem"
memorize = true
memorize_late = false
else
msg += " +mem_late"
memorize = true
memorize_late = true
end
i = i / 2
if i % 2 == 0 then
msg += " +revisit"
do_revisit = true
else
msg += " -revisit"
do_revisit = false
end
i = i / 2
if i >= 1 then return true
end
print msg
var t = new Clock
run_steps(steps)
print "\t{self}"
var l = t.lapse
print "\ttime={l}"
return false
end
end
# Used to compare nodes with their score.
# Smaller is score, smaller is the node.
private class NodeComparator[S: Object, A]
super Comparator
redef type COMPARED: SearchNode[S, A]
redef fun compare(a,b) do return a.score <=> b.score
end
# A node in the search-tree visited by a `SearchSolver`.
# In search-trees, nodes are labeled with states (`S`), and edges by actions (`A`).
#
# The root node is labeled by the initial state of the problem.
#
# This class is exposed to allow queries on the solution provided by the solver.
class SearchNode[S: Object, A]
# A flag that indicate that `self` is virtually removed from the todo-list.
# `self` was added to the todo-list but that a better node for the
# same state was found latter.
private var drop = false
# The associated problem
var problem: SearchProblem[S, A]
# The state associated to `self`.
# The state labels the node `self`.
var state: S
# Is `self` a root node of the search-tree?
# ensure: `result` == `parent == null` and `result`== `action == null`
fun is_root: Bool do return parent == null
# The previous node in the search-tree (if not root).
var parent: nullable SearchNode[S, A]
# The action used to go from `parent` to `self` (if not root)
# The action labels the edge from `parent` to `self`.
var action: nullable A
# The past cost (g) from the root node to `self`.
var cost: Float
# The heuristic from self to the goal (according to `problem.heuristic(state)`
# It is the future cost (h)
var heuristic: Float is noinit
# The sum of `cost` and `heuristic`
# It is the f function.
var score: Float is noinit
# Update `heuristic` and `score` according to `problem`.
private fun compute_heuristic
do
var h = problem.heuristic(state)
heuristic = h
score = cost + h
end
# The depth of `self` in the search tree
# It is the number of parents to the root node.
var depth: Int
# The number of steps needed by the solver to process `self`
# It is just a useless generation number, but could be used to evaluate
# the behavior of search algorithms.
var steps: Int = 0
# The rank of creation of nodes by the solver.
# It is just a useless generation number, but could be used to evaluate
# the behavior of search algorithms.
var id: Int = 0
# The number of (potential) revisits of `node`.
# This information can be used to debug search algorithms.
# And to detect when heuristics are not admissible.
#
# See `SearchSolver::revisits` and `SearchSolver::do_revisit`
# for details.
var revisits: Int = 0
# Create a new child node for the next state, according to `problem`.
# Used internally by the solvers but made public for those who want to replay a plan.
#
# ensure `result.parent == self`
# ensure `result.action == action`
fun apply_action(action: A): SearchNode[S, A]
do
var new_state = problem.apply_action(state, action)
var new_cost = problem.cost(state, action, new_state)
var new_node = new SearchNode[S, A](problem, new_state, self, action, cost + new_cost, depth+1)
new_node.compute_heuristic
return new_node
end
# Build the sequence of nodes from the initial node to `self`
#
# ensure `result.first.is_root and result.last == self`
fun path: Sequence[SearchNode[S, A]]
do
var res = new List[SearchNode[S, A]]
res.add(self)
var node = parent
while node != null do
res.unshift(node)
node = node.parent
end
return res
end
# Build a sequence of actions from the initial state to `self`
# See `path` for a more detailed plan.
fun plan: Sequence[A]
do
var res = new List[A]
var node: nullable SearchNode[S, A] = self
while node != null do
var a = node.action
if a != null then res.unshift(a)
node = node.parent
end
return res
end
# Just print a detailed path on the screen
fun dump
do
print "result:{state}"
for n in path do
var a = n.action
if a != null then print " + {a}"
print " {n.steps}: {n.state} ({n.cost}$)"
end
end
redef fun to_s do return "#{steps}/{id} d={depth} f={cost+heuristic} g={cost} h={heuristic}: {state}"
#redef fun to_s do return "#{steps} f={(cost+heuristic).to_i} g={cost.to_i} h={heuristic.to_i}"
end
```

lib/ai/search.nit:11,1--741,3