Solving Problems by Searching

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Artificial Intelligence, Spring, 2009

Solving Problems bySearching

Instructor: Tsung-Che [email protected]

Department of Computer Science and Information EngineeringNational Taiwan Normal University

2Artificial Intelligence, Spring, 2009

Outline

Problem-solving AgentsExample ProblemsSearching for SolutionsUninformed Search StrategiesAvoiding Repeated StatesSearching with Partial InformationSummary

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Problem-solving Agents

The problem-solving agent is a kind of goal-based agent.

They decide what to do by findingsequences of actions that lead to desirablestates.



Intelligent agents are supposed tomaximize their performance measure.

Achieving this is sometimes simplified byadopting a goal and aim at satisfying it.

The agent’s task is to find out an actionsequence to a set of world states in whichthe goal is satisfied.The process of looking for such a sequence is

called search.

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Example: The agent is driving to Bucharest fromArad.

Artificial Intelligence: A Modern Approach, 2nd ed., Figure 3.2



Goal formulation,based on the current situation and theperformance measure, is the first step in problemsolving.

Problem formulationis to decide what actions and states to consider,given a goal.

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If the agent has no knowledge, it can justchoose a random road.

If a map is given, it has the information about the states it might get

into and the actions it can take




“An agent with several immediate options ofunknown value can decide what to do by

first examining different possiblesequences of actions that lead to states ofknown value, and then choosing the bestsequence.”

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Agent design


Formulate|

Search|

Execute



Assumptions on the environment in theprevious designStatic (formulation, searching, & execution)Observable (initial state)Discrete (enumeration of actions)Deterministic

We are dealing with a very easyenvironment.

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Problem Formulation

A problem can be defined formally by fourcomponents:

Initial state e.g. In(Arad)

Possible actionsGoal test Path cost



Problem Formulation

Possible actionsThe most common formulation uses a successor

function. e.g. From the state In(Arad), the successor

function returns

{<Go(Sibui), In(SIbui)>, <Go(Timisoara),In(Timisoara)>, <Go(Zerind), In(Zerind)>}

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Problem Formulation

Possible actionsThe most common formulation uses a successor

function: a set of <action, successor> pairs.The initial state and successor function

implicitly define the state space of theproblem –the set of all states reachable from the initialstate.

The state space forms a graph in which thenodes are states and arcs are actions.

A path in the state space is a sequence ofstates connected by a sequence of actions.


Problem Formulation

Goal testSometimes there is an explicit set of possible

goal states.Sometimes the goal is specified by an abstract

property.

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Problem Formulation

A solution to a problem is a path from theinitial state to a goal state.

An optimal solution has the lowest pathcost among all solutions.

Artificial Intelligence: A ModernApproach, 2nd ed., Figure 3.2


Problem Formulation

During formulating a problem,considerations that are irrelevant to theproblem (including both states and actions) can befiltered out –abstraction.

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Problem Formulation

The abstraction should be valid, so we can expand any abstract solution

into a solution in the more detailed world; useful, so the actions can be carried out

without further search or planning.

Were it not for the ability to constructuseful abstractions, intelligent agentswould be completely swamped by the realworld.


Example Problems

Toy problemsVacuum world 8-puzzle 8-queen

Real-world problems Route-findingTouringVLSI layout Robot navigationScheduling

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Example Problems

Vacuum worldStates: 2 location x {clean, dirty}2 = 8 states Initial state: any state can be.Successor function:



Example Problems

Vacuum worldGoal test: This checks whether all squares are

clean. Path cost: Each step costs 1.

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Example Problems

8-puzzleStates: locations of eight tiles and the blank Initial state: any state can be.Successor function: left, right, up, downGoal test: any state can be. Path cost: Each step costs 1



Example Problems

8-puzzle

Possible?

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Example Problems

8-puzzleThe states are divided into two disjoint sets.Why?

Sliding the blank along a row doesn’t change the sum. Sliding the blank between rows doesn’t change the

sum%2.

http://www.8puzzle.com/8_puzzle_algorithm.html


Example Problems

8-puzzle It belongs to the family of sliding-block puzzles,

known to be NP-complete.

Number of states 8-puzzle: 9!/2 = 181,400 …very easy 15-puzzle: 1.3 trillion states …still easy 24-puzzle: 1025 states …quite difficult

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Example Problems

8-queens (Basic formulation)States: any arrangement of 0 to 8 queens Initial state: no queens on the boardSuccessor function: add a queen to any empty

squareGoal test: 8 queens on the

board, none attacked Path cost: 0


Note. The formulation here is incremental, wewill see the complete formation later.


Example Problems

8-queens (Smart formulation)

States:arrangement of n queens, one per column in theleftmost n columns, with no queen attackinganother

Successor function:Add a queen to any square in the leftmostempty column such that it is not attacked byany other queen.

This formulation reducesthe state space from31014 to just 2,057.

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Searching for Solutions

Searching through the state spacegenerates a search tree (maybe a searchgraph). It is important to distinguish between the state

space and the search tree.

state space: states + actionssearch tree: nodes + actions



Artificial Intelligence: A Modern Approach,2nd ed., Figure 3.2 & 3.6(a)

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Artificial Intelligence: A Modern Approach, 2nd ed., Figure 3.6(b) & 3.6(c)

expanded

generated



A node is assumed to have five components:State Parent-nodeAction: action applied to the parent to generate the node

Path-cost: the cost from the initial state to the node

Depth: the number of steps along the path


Note. States and node aredifferent. Different nodes cancontain the same world state.

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The fringe is the collection of nodes thathave been generated but not yet expanded. Every element of the fringe is a leaf node.

The search strategy selects the next nodeto be expanded from the fringe.



General tree-search algorithm


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Measuring performance CompletenessOptimalityTime complexity

usu. in terms of the number of nodes generated

Space complexity maximum number of nodes stored in the memory

b, branching factord, depth of the shallowest goal nodem, maximum length of any path


Uninformed Search Strategies

Uninformed search (blind search) can onlygenerate successors and distinguish a goalstate from a non-goal state.

Informed search (heuristic search) knowswhether one non-goal state is morepromising than another.

All search strategies are distinguished bythe order in which nodes are expanded.

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Breadth-First Search

BFS can be implemented by calling TREE-SEARCH with an empty fringe that is afirst-in-first-out (FIFO) queue.



(1)

(2)

(3)

(4)


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It is complete, provided the branchingfactor b is finite.

The shallowest goal node is not necessarilyoptimal. It is optimal when all step costs are equal.



Complexity analysisAssume branching factor is b, the solution is at

depth d. In the worst case, we would expand all but the

last node at level d.

Total number of nodes generated isb + b2 + b3 + … + bd + (bd+1 –b) = O(bd+1). Time complexity

Every node that is generated must remain in memory,because it is either part of the fringe or is an ancestorof a fringe node. Space complexity

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1000 P3,523 years101514

10 P35 years101312

101 T129 days101110

1 T31 hours1098

10 G19 minutes1076

106 M11 seconds111,1004

1 megabyte.11 seconds11002

MemoryTimeNodesDepth


Assume that (1) b = 10; (2) 10,000 nodes can be generated per second; (3) a noderequires 1000 bytes of storage.



Lessons we learnedThe time requirements are a major factor.The memory requirements are a bigger program

for BFS than is the execution time.

(In general, exponential-complexity searchproblems cannot be solved by uninformedmethods for any but the smallest instances.)

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Uniform-Cost Search

It expands the node with the lowest pathcost (not step cost!), instead of theshallowest node.

It is complete and optimal with any stepcost function

provided the cost of every step is at leasta small positive constant .


Uniform-Cost Search

Exercise: BFS & UCS

Artificial Intelligence: A ModernApproach, 2nd ed., Figure 3.2

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Uniform-Cost Search

Worst-case time and space complexity:

O(b1+C*/), where

C* is the cost of the optimal solution, and every action costs at least .

0.1

0.1

0.1

0.1

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goalIt often explores large trees ofsmall steps before exploring pathsinvolving large and perhaps usefulsteps.


Depth-First Search

It always expands the deepest node in thefringe.

After reaching the deepest level, it backsup the next shallowest node that still hasunexplored successors.

It can be implemented by TREE-SEARCHwith a last-in-first-out (LIFO) queue.

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Depth-First Search



Depth-First Search

It has very modest memory requirementssince it only stores a single path from the root to the leaf node,

and the remaining unexpanded sibling nodes.

Once a node is expanded, it can be removedafter all its descendants are fully explored.

Given branching factor b and maximumdepth m, the memory requirement is bm+1nodes.

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Depth-First Search

A variant: Backtracking search If each partially expanded node can remember

which successor to generate next, only onesuccessor is generated at a time. Memory requirement O(bm) O(m)


Depth-First Search

A variant: Backtracking search If we can undo the action when we go back, we

can keep only one state rather than O(m) states.

These techniques are critical to success forsolving problems with large state descriptionssuch as robotic assembly.

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Depth-First Search

It is neither complete nor optimal. It can make a wrong choice and get stuck going

down a very long (even infinite) path.

In the worst case, it will generate all ofthe O(bm) nodes in the search tree.Note that m can be much larger than d.


Depth-Limited Search

The problem of unbounded tree can bealleviated supplying DFS with a depth limitl.

Unfortunately, it introduces an additionalsource of incompleteness if l < d.

It is nonoptimal if l > d.Time complexity: O(bl)

Space complexity: O(bl)

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Sometimes depth limits can be set basedon knowledge of the problem.

What value willyou set for l ?





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Iterative-Deepening DFS

It combines the benefits of DFS and BFS. Its memory requirement are modest. (DFS) It is complete when the branching factor is

finite. (BFS) It is optimal when the path cost is a non-

decreasing function of the depth. (BFS)


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It is not as costly as you may imagine.

N(IDS) = (d)b + (d –1)b2 + … + (1)bd

N(BFS) = b + b2 + b3 + … + bd + (bd+1 –b)

If b = 10 and d = 5,N(IDS) = 123,450 and N(BFS) = 1,111,100.

In general, IDS is the preferred uninformedsearch method when there is a large search spaceand the depth of the solution is unknown.

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Bidirectional Search

Motivation: bd/2 + bd/2 is much less than bd.It is implemented by having one or both of

the searches check each node beforeexpansion to see if it is in the fringe of theother searchtree.




Whether the algorithm is complete/optimaldepends on the search strategies in bothsearches.

Time complexity: O(bd/2) Checking node for membership in the other

search tree can be done in constant time.

Space complexity: O(bd/2)At least one of the search tree must be kept in

memory for membership checking.

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In fact, searching backward is not easy.How can we find the predecessors of a state?

Are actions reversible? Is there only one goal state?

e.g. “clean world”in vacuum world,“checkmate”in chess.


Comparison of UninformedSearch Strategies


* *

*: optimal if step costs are all identical

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Avoiding Repeated States

For some problems, the state space is atree, and there is only one path to eachstate. e.g. the smart formulation of the 8-queens

problem

However, for some problems, we need to becareful about the possibility of expandingstates that have been expanded. e.g. the basic formulation of the 8-queens

problem



For the problems with reversible actions,repeated states are unavoidable. e.g. route-finding problems and sliding-blocks

puzzles

Repeated states can cause a solvableproblem to become unsolvable.

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An extreme example:a state space of size d+1 becomes a tree with 2d

leaves.




A more realistic exampleAbout 2d2 distinct states but 4d leaves within d

steps

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For DFS, the only nodes in memory arethose on the path from the root to thecurrent node. Checking those nodes to the current node can

detect the looping paths.However, it cannot avoid the exponential

proliferation of nonlooping paths.There is a fundamental tradeoff between space

and time. “Algorithms that forget their history are doomed to

repeat it.”



If an algorithm remembers every statethat it visited, then it can be viewed asexploring the state-space graph directly.


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On problems with many repeated states,GRAPH-SEARCH is much more efficientthan TREE-SEARCH. Its worst-case time and space complexity are

proportional to the size of the state space.[This may be much smaller than O(bd).]

Because GRAPH-SEARCH keeps every node inmemory, some searches are infeasible due tomemory limitations.



The optimality of GRAPH-SEARCH It needs to check whether a newly discovered

path to a node is better than the original one. If so, it might need to revise the depths and

path costs of that node’s descendants.

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Searching with PartialInformationWhat if the knowledge of the states or

actions is incomplete? (Not fully observable or notdeterministic)

Different types of incompleteness lead tothree distinct problem types:Sensorless problems Contingency problems Exploration problems (Sec. 4.5 & Chap. 6)


Searching with PartialInformationSensorless problems (Example: the vacuum world)

The agent has no sensors at all.It could be in one of several possible initialstates.

It knows all the effects of its actions.Each action might lead to one of severalpossible successor states.

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Searching with PartialInformation

Left

Suck

Right Suck


Searching with PartialInformationThe agent can reach state 8 with the

action sequence [Left, Suck, Right, Suck]without knowing the initial state.

“When the world is not fully observable,the agent must reason about sets of states(belief state), rather than a single state.”

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Searching with PartialInformationSearching in the space of belief states

An action is applied to a belief state by unioningthe results of applying the action to eachphysical state in the belief state.

A path now connects belief states.A solution now is a path leading to a belief state,

all of whose members are goal states.

In general, there are 2S belief states given Sphysical states. (But not all of them arereachable.)


Searching with PartialInformationThe analysis is essentially the same if the

environment is not deterministic.We just need to add the various outcomes

of the action to the successor belief state.

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Searching with PartialInformationContingency problems

The environment is only partially observableand/or nondeterministic, but the agent canobtain new information from sensors afteracting.

For example, in the vacuum world, the Suckaction sometimes deposits dirt when there is nodirt already.


Searching with PartialInformationExample

Percept [L, Dirty]

Suck Right Then what?

No fixed action sequence guarantees a solution to this problem.

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Searching with PartialInformationMany problems in the real world are

contingency problems, because exactprediction is impossible.

They lead to a different agent design, inwhich the agent can act before finding aguaranteed plan.This type of interleaving of search and

execution is also useful for exploration andgame playing.


Summary

In the deterministic, observable, andstatic environments, the agent canconstruct sequences of actions to achieveits goal –search.

A problem consists of the initial state, a set of actions, a goal test function, and a path cost function

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Summary

Common blind search strategies

Bidirectional search can be efficient, but notalways applicable and may require too much space.


Summary

GRAPH-SEARCH is efficient when thestate space is a graph rather than a tree.But it also suffers from the memoryrequirements.

When the environment is partiallyobservable, the agent can search in thespace of belief states.Sometimes a single solution can be constructed.Sometimes a contingency plan is needed.

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Classwork/Homework 2

Missionaries and Cannibals Problemhttp://www.learn4good.com/games/puzzle/boat.htm



Draw a diagram of the complete statespace.

Implement and solve the problem optimallyusing an appropriate search algorithm.The optimality means the use of the smallest

number of moves.Show the solution step by step on the monitor. Report how many nodes are generated.

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Everyone should submit his/her own results.

Deadline: 2009/4/13 23:59 Title:

AI-HW2-your student ID(9 digits)

Submit a .rar (or .zip) file including your entire program (a single file is encouraged for the

convenience of testing) a document describing your search algorithm

No re-submission is accepted.

Solving Problems by Searching

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