Solving problems by searching

Artificial Intelligence
Solving problems by searching
Fall 2008

professor: Luigi Ceccaroni

Problem solving
• We want:
– To automatically solve a problem

• We need:
– A representation of the problem
– Algorithms that use some strategy to solve
the problem defined in that representation

Problem representation
• General:
– State space: a problem is divided into a set
of resolution steps from the initial state to the
goal state
– Reduction to sub-problems: a problem is
arranged into a hierarchy of sub-problems
• Specific:
– Game resolution
– Constraints satisfaction

States
• A problem is defined by its elements and their
relations.
• In each instant of the resolution of a problem,
those elements have specific descriptors (How
to select them?) and relations.
• A state is a representation of those elements in
a given moment.
• Two special states are defined:
– Initial state (starting point)
– Final state (goal state)

State modification:
successor function
• A successor function is needed to move
between different states.
• A successor function is a description of
possible actions, a set of operators. It is a
transformation function on a state
representation, which convert it into another
state.
• The successor function defines a relation of
accessibility among states.
• Representation of the successor function:
– Conditions of applicability
– Transformation function

State space
• The state space is the set of all states
reachable from the initial state.
• It forms a graph (or map) in which the nodes are
states and the arcs between nodes are actions.
• A path in the state space is a sequence of
states connected by a sequence of actions.
• The solution of the problem is part of the map
formed by the state space.

Problem solution
• A solution in the state space is a path from the
initial state to a goal state or, sometimes, just a
goal state.
• Path/solution cost: function that assigns a
numeric cost to each path, the cost of applying
the operators to the states
• Solution quality is measured by the path cost
function, and an optimal solution has the
lowest path cost among all solutions.
• Solutions: any, an optimal one, all. Cost is
important depending on the problem and the
type of solution sought.

Problem description
• Components:
– State space (explicitly or implicitly defined)
– Initial state
– Goal state (or the conditions it has to fulfill)
– Available actions (operators to change state)
– Restrictions (e.g., cost)
– Elements of the domain which are relevant to the
problem (e.g., incomplete knowledge of the starting
point)
– Type of solution:
• Sequence of operators or goal state
• Any, an optimal one (cost definition needed), all

Example: 8-puzzle

1 2 3

4 5 6

7 8

Example: 8-puzzle
• State space: configuration of the eight
tiles on the board
• Initial state: any configuration
• Goal state: tiles in a specific order
• Operators or actions: “blank moves”
– Condition: the move is within the board
– Transformation: blank moves Left, Right, Up,
or Down
• Solution: optimal sequence of operators

Example: n queens
(n = 4, n = 8)

Example: n queens
(n = 4, n = 8)
• State space: configurations from 0 to n queens
on the board with only one queen per row and
column
• Initial state: configuration without queens on the
board
• Goal state: configuration with n queens such that
no queen attacks any other
• Operators or actions: place a queen on the
board
 Condition: the new queen is not attacked by any
other already placed
 Transformation: place a new queen in a particular
square of the board
• Solution: one solution (cost is not considered)

Structure of the state space
• Data structures:
– Trees: only one path to a given node
– Graphs: several paths to a given node
• Operators: directed arcs between nodes
• The search process explores the state
space.
• In the worst case all possible paths
between the initial state and the goal state
are explored.

Search as goal satisfaction
• Satisfying a goal
– Agent knows what the goal is
– Agent cannot evaluate intermediate solutions
(uninformed)
– The environment is:
• Static
• Observable
• Deterministic

Example: holiday in Romania
• On holiday in Romania; currently in Arad
• Flight leaves tomorrow from Bucharest at
13:00
• Let’s configure this to be an AI problem

Romania
• What’s the problem?
– Accomplish a goal
• Reach Bucharest by 13:00

• So this is a goal-based problem

Romania
• What’s an example of a non-goal-based
problem?
– Live long and prosper
– Maximize the happiness of your trip to
Romania
– Don’t get hurt too much

Romania
• What qualifies as a solution?
– You can/cannot reach Bucharest by 13:00
– The actions one takes to travel from Arad to
Bucharest along the shortest (in time) path

Romania
• What additional information does one
need?
– A map

More concrete problem
definition
Which cities could you be
A state space in?
Which city do you start
An initial state from?
Which city do you aim to
A goal state reach?
When in city foo, the
A function defining state following cities can be
transitions reached
A function defining the How long does it take to
“cost” of a state travel through a city
sequence?
sequence

More concrete problem
definition
A state space Choose a representation

Choose an element from the
An initial state representation
Create goal_function(state) such
A goal state that TRUE is returned upon reaching
goal
successor_function(statei) =
A function defining state
{<actiona, statea>, <actionb, stateb>,
transitions …}

A function defining the “cost” of a
cost (sequence) = number
state sequence

Important notes about this
example
– Static environment (available states,
successor function, and cost functions don’t
change)
– Observable (the agent knows where it is)
– Discrete (the actions are discrete)
– Deterministic (successor function is always
the same)

Tree search algorithms
• Basic idea:
– Simulated
exploration of
state space by
generating
successors of
already explored
states (AKA
expanding
states) Sweep out from start (breadth)

Tree search algorithms
• Basic idea:
– Simulated
exploration of
state space by
generating
successors of
already explored
states (AKA
expanding Go East, young man! (depth)
states)

Implementation: general search
algorithm
Algorithm General Search
Open_states.insert (Initial_state)
Current= Open_states.first()
while not is_final?(Current) and not Open_states.empty?() do
Open_states.delete_first()
Closed_states.insert(Current)
Successors= generate_successors(Current)
Successors= process_repeated(Successors, Closed_states,
Open_states)
Open_states.insert(Successors)
Current= Open_states.first()
eWhile
eAlgorithm

Example: Arad  Bucharest
Algorithm General Search
Open_states.insert (Initial_state)

Arad

• Current= Open_states.first()

Arad

Open_states)

Arad

Zerind (75) Timisoara (118) Sibiu (140)

• Current= Open_states.first()

Arad


Open_states)

Arad


Dradea (151) Faragas (99) Rimnicu Vilcea (80)

Implementation: states vs.
nodes
• State
– (Representation of) a physical configuration
• Node
– Data structure constituting part of a search
tree
• Includes parent, children, depth, path cost g(x)
• States do not have parents, children,
depth, or path cost!

Search strategies
• A strategy is defined by picking the order of node
expansion
• Strategies are evaluated along the following
dimensions:
– Completeness – does it always find a solution if
one exists?
– Time complexity – number of nodes
generated/expanded
– Space complexity – maximum nodes in memory
– Optimality – does it always find a least-cost
solution?

Search strategies
• Time and space complexity are measured in
terms of:
– b – maximum branching factor of the search tree
(may be infinite)
– d – depth of the least-cost solution
– m – maximum depth of the state space (may be
infinite)

Uninformed Search Strategies
• Uninformed strategies use only the
information available in the problem
definition
– Breadth-first search
– Uniform-cost search
– Depth-first search
– Depth-limited search
– Iterative deepening search

Nodes
• Open nodes:
– Generated, but not yet explored
– Explored, but not yet expanded

• Closed nodes:
– Explored and expanded

35

Breadth-first search
• Expand shallowest unexpanded node
• Implementation:
– A FIFO queue, i.e., new successors go at end

Space cost of BFS

• Because you must be able to generate the path upon
finding the goal state, all visited nodes must be stored
• O (bd+1)

Properties of breadth-first
search
• Complete?
– Yes (if b (max branch factor) is finite)
• Time?
– 1 + b + b2 + … + bd + b(bd-1) = O(bd+1), i.e., exponential in d
• Space?
– O(bd+1) (keeps every node in memory)
• Optimal?
– Only if cost = 1 per step, otherwise not optimal in general
• Space is the big problem; it can easily generate nodes at
10 MB/s, so 24 hrs = 860GB!

Depth-first search
• Expand deepest unexpanded node
• Implementation:
– A LIFO queue, i.e., a stack

Depth-first search
• Complete?
– No: fails in infinite-depth spaces, spaces with loops.
– Can be modified to avoid repeated states along path 
complete in finite spaces
• Time?
– O(bm): terrible if m is much larger than d, but if solutions are
dense, may be much faster than breadth-first
• Space?
– O(bm), i.e., linear space!
• Optimal?
– No

Depth-limited search
• It is depth-first search with an imposed
limit on the depth of exploration, to
guarantee that the algorithm ends.

41

Treatment of repeated states
• Breadth-first:
– If the repeated state is in the structure of closed or open
nodes, the actual path has equal or greater depth than the
repeated state and can be forgotten.

42

Treatment of repeated states
• Depth-first:
– If the repeated state is in the structure of closed nodes, the
actual path is kept if its depth is less than the repeated state.
– If the repeated state is in the structure of open nodes, the
actual path has always greater depth than the repeated state
and can be forgotten.

43

Iterative deepening search
• The algorithm consists of iterative, depth-first
searches, with a maximum depth that increases at
each iteration. Maximum depth at the beginning is 1.
• Behavior similar to BFS, but without the spatial
complexity.
• Only the actual path is kept in memory; nodes are
regenerated at each iteration.
• DFS problems related to infinite branches are
avoided.
• To guarantee that the algorithm ends if there is no
solution, a general maximum depth of exploration can
be defined. 45

Summary
– All uninformed searching techniques are more alike
than different.
– Breadth-first has space issues, and possibly optimality
issues.
– Depth-first has time and optimality issues, and possibly
completeness issues.
– Depth-limited search has optimality and completeness
issues.
– Iterative deepening is the best uninformed search we
have explored.

Uninformed vs. informed
• Blind (or uninformed) search algorithms:
– Solution cost is not taken into account.

• Heuristic (or informed) search algorithms:
– A solution cost estimation is used to guide the
search.
– The optimal solution, or even a solution, are
not guaranteed.

48

Solving problems by searching

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