02 search problems

Search ProblemsSearch Problems

Explorating AlternativesExplorating Alternatives

R&N: Chap. 3, Sect. 3.1–2 + 3.6R&N: Chap. 3, Sect. 3.1–2 + 3.6

Example: 8-PuzzleExample: 8-Puzzle

8 1 2 3

Initial state Goal state

Search is about the exploration of alternatives

Exploratory search is an old idea: Exploratory search is an old idea: The Labyrinth and the Ariadne ThreadThe Labyrinth and the Ariadne Thread

According to Greek mythology, Theseus came to Crete to slay the Minotaur, a monster who lived in a Labyrinth. Ariadne gave Theseus a ball of yarn which he unwound as he entered the Labyrinth. After killing the Minotaur, Theseus traced the thread back to the entrance of the Labyrinth, rejoined Ariadne, and successfully escaped Crete

Since the dawn of civilization, puzzles and Since the dawn of civilization, puzzles and games that require the exploration of games that require the exploration of alternative paths have fascinated mankind alternative paths have fascinated mankind and have been considered a challenge for and have been considered a challenge for human intelligencehuman intelligence

ChessChess originated in Persia and India originated in Persia and India about 4000 years ago about 4000 years ago

CheckersCheckers appeared as early as 1600 B.C appeared as early as 1600 B.C in Egyptian paintingsin Egyptian paintings

GoGo originated in China over 3000 years originated in China over 3000 years agoago

15-Puzzle15-PuzzleIntroduced in 1878 by Sam Loyd, who dubbed Introduced in 1878 by Sam Loyd, who dubbed himself “America’s greatest puzzle-expert”himself “America’s greatest puzzle-expert”

5 6 7 8

15-Puzzle15-PuzzleSam Loyd offered Sam Loyd offered $1,000$1,000 of his own money to of his own money to the first person who would solve the following the first person who would solve the following problem:problem:

5 6 7 8

But no one ever won the prize !!But no one ever won the prize !!

8-Puzzle: State Space8-Puzzle: State Space

8-Puzzle: Successor Function8-Puzzle: Successor Function

Stating a Problem as Stating a Problem as a Search Problema Search Problem

State space S Successor function: x ∈ S → SUCCESSORS(x) ∈ 2S

Arc cost Initial state s0

Goal test: x∈S → GOAL?(x) =T or F

State GraphState Graph It is defined as follows:It is defined as follows:

• Each state is represented by a distinct Each state is represented by a distinct nodenode

• An arc connects a node s to a node s’ if An arc connects a node s to a node s’ if s’ s’ ∈∈ SUCCESSORSSUCCESSORS(s) (s)

The state graph may contain more than The state graph may contain more than one connected componentone connected component

Solution to the Search ProblemSolution to the Search Problem A A solutionsolution is a path connecting the initial is a path connecting the initial

node to a goal node (any one)node to a goal node (any one)

Solution to the Search ProblemSolution to the Search Problem A A solutionsolution is a path connecting the initial is a path connecting the initial

to a goal node (any one)to a goal node (any one) The The costcost of a path is the sum of the of a path is the sum of the

edge costs along this pathedge costs along this path An An optimaloptimal solution is a solution path of solution is a solution path of

minimum costminimum cost There might be no solution !There might be no solution !

How big is the state space of How big is the state space of the (nthe (n22-1)-puzzle?-1)-puzzle?

8-puzzle 8-puzzle 9! = 362,880 states 9! = 362,880 states 15-puzzle 15-puzzle 16! ~ 1.3 x 10 16! ~ 1.3 x 101212 states states 24-puzzle 24-puzzle 25! ~ 10 25! ~ 102525 states states

But But only halfonly half of these states are of these states are reachable from any given statereachable from any given state

Let the goal be:Let the goal be:

Let nLet nii be the number of tiles j < i that appear after tile i be the number of tiles j < i that appear after tile i(from left to right and top to bottom)(from left to right and top to bottom)

N = nN = n22 + n + n33 + + …… + n + n15 15 + row number of empty tile+ row number of empty tile

Permutation InversionsPermutation Inversions

121511

1395 6 7 8

121511

1395 10 7 8

4321 n2 = 0 n3 = 0 n4 = 0n5 = 0 n6 = 0 n7 = 1n8 = 1 n9 = 1 n10 = 4n11 = 0 n12 = 0 n13 = 0n14 = 0 n15 = 0

N = 7 + 4

Proposition: (N mod 2) is invariant under Proposition: (N mod 2) is invariant under any legal move of the empty tileany legal move of the empty tile

Proof:Proof:• Any horizontal move of the empty tile Any horizontal move of the empty tile

leaves N unchangedleaves N unchanged• A vertical move of the empty tile changes A vertical move of the empty tile changes

N by an even incrementN by an even increment

121511

1395 6 7

s’ = N(s’) = N(s) + 3 + 1

Proposition: (N mod 2) is invariant under Proposition: (N mod 2) is invariant under any legal move of the empty tileany legal move of the empty tile

For a goal state g to be reachable For a goal state g to be reachable from a state s, a necessary condition is from a state s, a necessary condition is that N(g) and N(s) have the same paritythat N(g) and N(s) have the same parity

It can be shown that this is also a It can be shown that this is also a sufficient conditionsufficient condition

The state graph consists of two The state graph consists of two connected components of equal sizeconnected components of equal size

So, the second state is So, the second state is not reachable from the not reachable from the first, and Sam Loyd took first, and Sam Loyd took no risk with his money ...no risk with his money ...

15-PuzzleSam Loyd offered $1,000 of his own money to the first person who would solve the following problem:

5 6 7 8

N = 4 N = 5

What is the Actual State Space?What is the Actual State Space?• The set of all states? The set of all states?

[e.g., a set of 16! states for the 15-puzzle][e.g., a set of 16! states for the 15-puzzle]• The set of all states from which a given goal The set of all states from which a given goal

state is reachable? state is reachable? [e.g., a set of 16!/2 states for the 15-puzzle][e.g., a set of 16!/2 states for the 15-puzzle]

• The set of all states reachable from a given The set of all states reachable from a given initial state?initial state?

In general, the answer is In general, the answer is a)a)

What is the Actual State Space?What is the Actual State Space?• The set of all states? The set of all states?

[e.g., a set of 16! states for the 15-puzzle][e.g., a set of 16! states for the 15-puzzle]• The set of all states from which a given goal The set of all states from which a given goal

state is reachable? state is reachable? [e.g., a set of 16!/2 states for the 15-puzzle][e.g., a set of 16!/2 states for the 15-puzzle]

• The set of all states reachable from a given The set of all states reachable from a given initial state?initial state?

In general, the answer is a)In general, the answer is a)But a fast test determining whether a state is reachable from another is very useful, as search-based problem solvers are often very inefficient when a problem has no solution

Stating a Problem as Stating a Problem as a Search Problema Search Problem

State space S Successor function: x ∈ S → SUCCESSORS(x) ∈ 2S

Arc cost Initial state s0

Goal test: x∈S → GOAL?(x) =T or F A solution is a path joining the initial to a goal node

Searching the State SpaceSearching the State Space Often it is not Often it is not

feasible to build feasible to build a complete a complete representation representation of the state of the state graphgraph

8-puzzle 362,880 states

15-puzzle 1.3 x 1012 states

24-puzzle 1025 states

100 millions states/sec

0.036 sec

< 4 hours

> 109 years

8-, 15-, 24-Puzzles8-, 15-, 24-Puzzles

Searching the State SpaceSearching the State Space Often it is not Often it is not

feasible to build feasible to build a complete a complete representation representation of the state of the state graphgraph

A problem solver A problem solver must construct a must construct a solution by solution by exploring a small exploring a small portion of the portion of the graph graph

Searching the State SpaceSearching the State Space

Search tree

Simple Problem-Solving-Agent Simple Problem-Solving-Agent AlgorithmAlgorithm

ss00 sense/read initial state sense/read initial state GOAL? GOAL? select/read goal test select/read goal test Succ Succ select/read successor function select/read successor function solution solution searchsearch(s(s00, GOAL?, Succ) , GOAL?, Succ) perform(solution)perform(solution)

State SpaceState Space Each state is an Each state is an abstractabstract representation representation

of a collection of possible worlds sharing of a collection of possible worlds sharing some crucial properties and differing on some crucial properties and differing on non-important details onlynon-important details onlyE.g.: In assembly planning, a state does not E.g.: In assembly planning, a state does not define exactly the absolute position of each partdefine exactly the absolute position of each part

The state space is The state space is discretediscrete. It may be . It may be finite, or infinitefinite, or infinite

Successor FunctionSuccessor Function It implicitly represents all the actions It implicitly represents all the actions

that are feasible in each statethat are feasible in each state

Successor FunctionSuccessor Function It implicitly represents all the actions It implicitly represents all the actions

that are feasible in each statethat are feasible in each state Only the results of the actions (the Only the results of the actions (the

successor states) and their costs are successor states) and their costs are returned by the functionreturned by the function

The successor function is a “black box”: The successor function is a “black box”: its content is unknownits content is unknownE.g., inE.g., in assembly planning, the function does assembly planning, the function does not say if it only allows two sub-assemblies to not say if it only allows two sub-assemblies to be merged or if it makes assumptions about be merged or if it makes assumptions about subassembly stabilitysubassembly stability

Path CostPath Cost An arc cost is a positive number An arc cost is a positive number

measuring the “cost” of performing the measuring the “cost” of performing the action corresponding to the arc, e.g.:action corresponding to the arc, e.g.:• 1 in the 8-puzzle example1 in the 8-puzzle example• expected time to merge two sub-assembliesexpected time to merge two sub-assemblies

We will assume that for any given We will assume that for any given problem the cost c of an arc always problem the cost c of an arc always verifies: verifies: c c ≥ ≥ εε >> 0 0, where , where εε is a constant is a constant

Path CostPath Cost An arc cost is a positive number An arc cost is a positive number

measuring the “cost” of performing the measuring the “cost” of performing the action corresponding to the arc, e.g.:action corresponding to the arc, e.g.:• 1 in the 8-puzzle example1 in the 8-puzzle example• expected time to merge two sub-assembliesexpected time to merge two sub-assemblies

We will assume that for any given We will assume that for any given problem the cost c of an arc always problem the cost c of an arc always verifies: verifies: c c ≥ ≥ εε >> 0 0, where , where εε is a constant is a constant[This condition guarantees that, if path becomes [This condition guarantees that, if path becomes arbitrarily long, its cost also becomes arbitrarily large]arbitrarily long, its cost also becomes arbitrarily large]

Why is this needed?

It may be explicitly described:It may be explicitly described:

or partially described:or partially described:

or defined by a condition, or defined by a condition, e.g., the sum of every row, of every column, e.g., the sum of every row, of every column, and of every diagonals equals 30and of every diagonals equals 30

Goal StateGoal State1 2 34 5 67 8

384 10 9 7

122115

a aa a

aa (“a” stands for “any”)

Other examplesOther examples

8-Queens Problem8-Queens ProblemPlace 8 queens in a chessboard so that no two queens are in the same row, column, or diagonal.

A solution Not a solution

Formulation #1Formulation #1 States:States: all arrangements of 0, all arrangements of 0,

1, 2, ..., or 8 queens on the 1, 2, ..., or 8 queens on the boardboard

Initial state:Initial state: 0 queen on the 0 queen on the boardboard

Successor function:Successor function: each of each of the successors is obtained by the successors is obtained by adding one queen in an empty adding one queen in an empty squaresquare

Arc cost:Arc cost: irrelevantirrelevant Goal test:Goal test: 8 queens are on the 8 queens are on the

board, with no two of them board, with no two of them attacking each otherattacking each other

64x63x...x53 ~ 3x1014 states

Formulation #2Formulation #2 States:States: all arrangements of k = all arrangements of k =

0, 1, 2, ..., or 8 queens in the k 0, 1, 2, ..., or 8 queens in the k leftmost columns with no two leftmost columns with no two queens attacking each otherqueens attacking each other

Initial state:Initial state: 0 queen on the 0 queen on the boardboard

Successor function:Successor function: each each successor is obtained by adding successor is obtained by adding one queen in any square that is one queen in any square that is not attacked by any queen not attacked by any queen already in the board, in the already in the board, in the leftmost empty column leftmost empty column

Arc cost:Arc cost: irrelevantirrelevant Goal test:Goal test: 8 queens are on the 8 queens are on the

boardboard 2,057 states

n-Queens Problemn-Queens Problem A solution is a A solution is a goal nodegoal node, not a path to this , not a path to this

node (typical of design problem)node (typical of design problem) Number of states in state space:Number of states in state space:

• 8-queens 8-queens 2,057 2,057• 100-queens 100-queens 10 105252

But techniques exist to solve n-queens But techniques exist to solve n-queens problems efficiently for large values of nproblems efficiently for large values of nThey exploit the fact that there are many They exploit the fact that there are many solutions well distributed in the state spacesolutions well distributed in the state space

Path PlanningPath Planning

What is the state space?

Formulation #1Formulation #1

Cost of one horizontal/vertical step = 1Cost of one diagonal step = √2

Optimal SolutionOptimal Solution

This path is the shortest in the discretized state space, but not in the original continuous space

Formulation #2Formulation #2sweep-line

StatesStates

Successor FunctionSuccessor Function

Solution PathSolution Path

A path-smoothing post-processing step is usually needed to shorten the path further

Cost of one step: length of segment

Visibility graph

Solution PathSolution Path

The shortest path in this state space is also the shortest in the original continuous space

Assembly (Sequence) PlanningAssembly (Sequence) Planning

Possible FormulationPossible Formulation States:States: All decompositions of the assembly All decompositions of the assembly

into subassemblies (subsets of parts in their into subassemblies (subsets of parts in their relative placements in the assembly)relative placements in the assembly)

Initial state:Initial state: All subassemblies are made of a All subassemblies are made of a single partsingle part

Goal state:Goal state: Un-decomposed assembly Un-decomposed assembly Successor function:Successor function: Each successor of a state Each successor of a state

is obtained by merging two subassemblies (the is obtained by merging two subassemblies (the successor function must check if the merging successor function must check if the merging is feasible: collision, stability, grasping, ...)is feasible: collision, stability, grasping, ...)

Arc cost:Arc cost: 1 or time to carry the merging 1 or time to carry the merging

A Portion of State SpaceA Portion of State Space

But the formulation rules out But the formulation rules out “non-monotonic” assemblies“non-monotonic” assemblies

XThis “subassembly” is not allowed in the definition ofthe state space: the 2 partsare not in their relative placements in the assembly

Allowing any grouping of parts as a valid subassembly would make the state space much bigger and more difficult to search

Assumptions in Basic SearchAssumptions in Basic Search

The world is The world is staticstatic The world is The world is discretizablediscretizable The world is The world is observableobservable The actions are The actions are deterministicdeterministic

But many of these assumptions can be removed, and search still remains an important problem-solving tool

Vacuum Cleaner ProblemVacuum Cleaner Problem A vacuum robot lives in a two-room environmentA vacuum robot lives in a two-room environment

States:States: The robot is in one of the two rooms, The robot is in one of the two rooms, and each room may or may not contain dirt and each room may or may not contain dirt 8 states 8 states

Successor function:Successor function: the successors of a state the successors of a state correspond to trying 3 actions: Right, Left, correspond to trying 3 actions: Right, Left, Suck. Suck.

Initial state:Initial state: UnknownUnknown (not observable) (not observable) Goal state:Goal state: No dust in the rooms No dust in the rooms

Re-Formulation with “Belief States”Re-Formulation with “Belief States” Belief states:Belief states: sets of states sets of states 2 288 =256 =256

belief statesbelief states Initial belief state:Initial belief state: set of 8 states set of 8 states Successor function:Successor function: the successors of a the successors of a

belief state correspond to trying Right, belief state correspond to trying Right, Left, Suck on all states in the belief Left, Suck on all states in the belief state. state.

Goal belief state:Goal belief state: any set of states with any set of states with none having dust in any of the roomsnone having dust in any of the rooms

Left RightSuck

Search and AISearch and AI Search methods are Search methods are ubiquitousubiquitous in AI systems. in AI systems.

They often are the backbones of both core They often are the backbones of both core and peripheral modules and peripheral modules

An An autonomous robotautonomous robot uses search methods: uses search methods:• to decide which actions to take and which sensing to decide which actions to take and which sensing

operations to perform, operations to perform, • to quickly anticipate and prevent collision, to quickly anticipate and prevent collision, • to plan trajectories,to plan trajectories,• to interpret large numerical datasets provided by to interpret large numerical datasets provided by

sensors into compact symbolic representations, sensors into compact symbolic representations, • to diagnose why something did not happen as to diagnose why something did not happen as

expected, expected, • etc...etc...

ApplicationsApplicationsSearch plays a key role in many applications, e.g.:Search plays a key role in many applications, e.g.:

Route finding: airline travel, networksRoute finding: airline travel, networks Package/mail distributionPackage/mail distribution Pipe routingPipe routing Comparison and classification of protein foldsComparison and classification of protein folds Pharmaceutical drug designPharmaceutical drug design Design of protein-like moleculesDesign of protein-like molecules Video gamesVideo games

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