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Search Problems(Where reasoning consists of
exploring alternatives)
R&N: Chap. 3, Sect. 3.1²2 + 3.6
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Declarative knowledgecreates alternatives: Which pieces of
knowledge to use? How to use them?
Search is a aboutexploring alternatives.
It is a major approachto exploit knowledge
Search
Knowledgerep.Planning
Reasoning
Learning
Agent
RoboticsPerception
Naturallanguage
... ExpertSystems
Constraintsatisfaction
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Example: 8-Puzzle
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3 45 6
7
8 1 2 3
4 5 67 8
Initial state Goal state
State: Any arrangement of 8 numbered tiles and an empty tile on a 3x3 board
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8-Puzzle: Successor Function
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3 4
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3 4
5 6
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Search is about the
exploration of alternatives
SUCC(state) subset of states
The successor function is knowledgeabout the 8-puzzle game, but it doesnot tell us which outcome to use, nor to
which state of the board to apply it.
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Across history, puzzles and gamesrequiring the exploration of alternatives
have been considered a challenge forhuman intelligence:
Chess originated in Persia and India
about 4000 years ago Checkers appear in 3600-year-old
Egyptian paintings
Go originated in China over 3000 yearsago
So, it·s not surprising that AI uses games
to design and test algorithms
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(n2-1)-puzzle
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3 4
5 6
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8
12
15
11
14
10
13
9
5 6 7 8
4321
....
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15-Puzzle
Introduced (?) in 1878 by Sam Loyd, who dubbedhimself ´America·s greatest puzzle-expertµ
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15-Puzzle
Sam Loyd offered $1,000 of his own money tothe first person who would solve the followingproblem:
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15
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5 6 7 8
4321
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5 6 7 8
4321
?
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But no one ever won the prize !!
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Stating a Problem as
a Search Problem
State space S
Successor function:x Sp SUCCESSORS(x) 2S
Initial state s0
Goal test:
x
Sp
GOAL?(x) =T or F Arc cost
S
1 3 2
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State Graph
Each state isrepresented by adistinct node
An arc (or edge)connects a node sto a node s· if
s·
SUCCESSORS(s) The state graph may
contain more than oneconnected component
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Solution to the Search Problem
A solution is a path connecting the initialnode to a goal node (any one)
I
G
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Solution to the Search Problem
A solution is a path connecting the initialnode to a goal node (any one)
The cost of a path is the sum of the arc
costs along this path An optimal solution is a solution path of
minimum cost
There might beno solution !
I
G
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How big is the state space of
the (n2
-1)-puzzle? 8-puzzle ?? states
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How big is the state space of
the (n2
-1)-puzzle? 8-puzzle 9! = 362,880 states 15-puzzle 16! ~ 2.09 x 1013 states 24-puzzle 25! ~ 1025 states
But only half of these states are
reachable from any given state(but you may not know that in advance)
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Wlg, let the goal be:
A tile j appears after a tile i if either j appears on the same row asi to the right of i, or on another row below the row of i.
For every i = 1, 2, ..., 15, let ni be the number of tiles j < i thatappear after tile i (permutation inversions)
N = n2 + n3 + ~ + n15 + row number of empty tile
Permutation Inversions
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5 6 7 8
4321
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5 10 7 8
4321 n2 = 0 n3 = 0 n4 = 0n5 = 0 n6 = 0 n7 = 1n8 = 1 n9 = 1 n10 = 4n11 = 0 n12 = 0n13 = 0
n14 = 0n15 = 0
N = 7 + 4
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Proposition: (N mod 2) is invariant under
any legal move of the empty tile Proof:
Any horizontal move of the empty tileleaves N unchanged
A vertical move of the empty tile changesN by an even increment (s 1 s 1 s 1 s 1)
1215
11
14
10
13
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5 6 7
8
4321
s =
1215
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13
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5 6 7
8
4321
s· = N(s·) = N(s) + 3 + 1
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Proposition: (N mod 2) is invariant under
any legal move of the empty tile For a goal state g to be reachable
from a state s, a necessary condition is
that N(g) and N(s) have the same parity It can be shown that this is also a
sufficient condition
The state graph consists of twoconnected components of equal size
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So, the second state is
not reachable from thefirst, and Sam Loyd tookno risk with his money ...
15-Puzzle
Sam Loyd offered $1,000 of his own money to
the first person who would solve the followingproblem:
12
14
11
15
10
13
9
5 6 7 8
4321
12
14
11
15
10
13
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5 6 7 8
4321
12
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5 6 7 8
4321
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5 6 7 8
4321
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5 6 7 8
4321
N = 4 N = 5
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What is the Actual State Space?
a) The set of all states?[e.g., a set of 16! states for the 15-puzzle]
b) The set of all states reachable from a giveninitial state?[e.g., a set of 16!/2 states for the 15-puzzle]
In general, the answer is a)[because one does not know in advance which statesare reachable]
But a fast test determining whether a state is reachablefrom another is very useful, as search techniques areoften inefficient when a problem has no solution
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Searching the State Space
It is often notfeasible (or tooexpensive) tobuild a complete
representationof the stategraph
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8-puzzle 362,880 states
15-puzzle 2.09 x 1013 states
24-puzzle 1025 states
100 millions states/sec
0.036 sec
~ 55 hours
> 109 years
8-, 15-, 24-Puzzles
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Searching the State Space
Often it is notfeasible (or tooexpensive) tobuild a complete
representationof the stategraph
A problem solvermust construct a
solution byexploring a smallportion of thegraph
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Searching the State Space
Search tree
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Searching the State Space
Search tree
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Searching the State Space
Search tree
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Searching the State Space
Search tree
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Searching the State Space
Search tree
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Searching the State Space
Search tree
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Simple Problem-Solving-Agent
Algorithm
1. I sense/read initial state2. GOAL? select/read goal test3. Succ select/read successor function4. solution search(I, GOAL?, Succ)
5. perform(solution)
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State Space
Each state is an abstract representationof a collection of possible worlds sharingsome crucial properties and differing onnon-important details onlyE.g.: In assembly planning, a state does notdefine exactly the absolute position of each part
The state space is discrete. It may be
finite, or infinite
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Successor Function
It implicitly represents all the actionsthat are feasible in each state
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Successor Function
It implicitly represents all the actionsthat are feasible in each state
Only the results of the actions (the
successor states) and their costs arereturned by the function
The successor function is a ´black boxµ:
its content is unknownE.g., in assembly planning, the successorfunction may be quite complex (collision,stability, grasping, ...)
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Path Cost
An arc cost is a positive numbermeasuring the ´costµ of performing theaction corresponding to the arc, e.g.:
1 in the 8-puzzle example expected time to merge two sub-assemblies
We will assume that for any given
problem the cost c of an arc alwaysverifies: c u " 0, where is a constant
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Path Cost
An arc cost is a positive numbermeasuring the ´costµ of performing theaction corresponding to the arc, e.g.:
1 in the 8-puzzle example expected time to merge two sub-assemblies
We will assume that for any given
problem the cost c of an arc alwaysverifies: c u " 0, where is a constant[This condition guarantees that, if path becomesarbitrarily long, its cost also becomes arbitrarily large]
Why is this needed?
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It may be explicitly described:
or partially described:
or defined by a condition,e.g., the sum of every row, of every column,
and of every diagonal equals 30
Goal State
1 2 34 5 6
7 8
11
14
5
13
6
3
8
4 10 9 7
122115
1
58
a a
a aaa (´aµ stands for ´anyµother than 1, 5, and 8)
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O
ther examples
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8-Queens Problem
Place 8 queens in a chessboard so that no twoqueens are in the same row, column, or diagonal.
A solution Not a solution
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Formulation #1
States: all arrangements of 0,1, 2, ..., 8 queens on the board
Initial state: 0 queens on theboard
Successor function: each ofthe successors is obtained byadding one queen in an emptysquare
Arc cost: irrelevant
Goal test: 8 queens are on theboard, with no queens attackingeach other
~ 64x63x...x57 ~ 3x1014 states
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Formulation #2
States: all arrangements of k =0, 1, 2, ..., 8 queens in the kleftmost columns with no twoqueens attacking each other
Initial state: 0 queens on the
board Successor function: each
successor is obtained by addingone queen in any square that isnot attacked by any queenalready in the board, in theleftmost empty column
Arc cost: irrelevant Goal test: 8 queens are on the
board 2,057 states
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n-Queens Problem
A solution is a goal node, not a path to thisnode (typical of design problem)
Number of states in state space:
8-queens
2,057 100-queens 1052
But techniques exist to solve n-queensproblems efficiently for large values of n
They exploit the fact that there are manysolutions well distributed in the state space
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Path Planning
What is the state space?
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Formulation #1
Cost of one horizontal/vertical step = 1Cost of one diagonal step = 2
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Optimal Solution
This path is the shortest in the discretized statespace, but not in the original continuous space
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Formulation #2sweep-line
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Formulation #2
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States
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Successor Function
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Solution Path
A path-smoothing post-processing step isusually needed to shorten the path further
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Formulation #3
Cost of one step: length of segment
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Formulation #3
Cost of one step: length of segment
Visibility graph
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Solution Path
The shortest path in this state space is also theshortest in the original continuous space
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Assembly (Sequence) Planning
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Possible Formulation
States: All decompositions of the assemblyinto subassemblies (subsets of parts in theirrelative placements in the assembly)
Initial state: All subassemblies are made of asingle part
Goal state: Un-decomposed assembly Successor function: Each successor of a state
is obtained by merging two subassemblies (thesuccessor function must check if the mergingis feasible: collision, stability, grasping, ...)
Arc cost: 1 or time to carry the merging
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A Portion of State Space
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But the formulation rules out
´non-monotonicµ assemblies
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But the formulation rules out
´non-monotonicµ assemblies
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But the formulation rules out
´non-monotonicµ assemblies
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But the formulation rules out
´non-monotonicµ assemblies
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But the formulation rules out
´non-monotonicµ assemblies
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But the formulation rules out
´non-monotonicµ assembliesThis ´subassemblyµ is notallowed in the definition ofthe state space: the 2 parts
are not in their relativeplacements in the assembly
Allowing any grouping of parts
as a valid subassembly wouldmake the state space muchbigger and more difficultto search
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Assumptions in Basic Search
The world is static
The world is discretizable
The world is observable
The actions are deterministic
But many of these assumptions can beremoved, and search still remains animportant problem-solving tool
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Search and AI
Search methods are ubiquitous in AI systems.They often are the backbones of both coreand peripheral modules
An autonomous robot uses search methods: to decide which actions to take and which sensing
operations to perform, to quickly anticipate collision, to plan trajectories, to interpret large numerical datasets provided by
sensors into compact symbolic representations, to diagnose why something did not happen as
expected, etc...
Many searches may occur concurrently and
sequentially
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Applications
Search plays a key role in many applications, e.g.:
Route finding: airline travel, networks
Package/mail distribution Pipe routing, VLSI routing
Comparison and classification of protein folds
Pharmaceutical drug design
Design of protein-like molecules Video games