CSNB234CSNB234ARTIFICIAL INTELLIGENCEARTIFICIAL INTELLIGENCE

Chapter 5State Space Search

Chapter 5State Space Search

Instructor: Alicia Tang Y. C.

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State Space Representation

A state space can be considered as consisting of a collection of points, each point corresponding to a state of a problem. So there will be as many number of points in the state space as the number of different possible states of the problem. Thus, a state space is the collection of all possible states, a problem can have. States can be represented as nodes of a tree. The searching of a goal node is by traversing the tree’s nodes using some well-established algorithms.

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Defining A Search ProblemDefining A Search ProblemState Space

– is described by an initial state space a set of possible actions (operators) goal state

Path– is any sequence of actions that lead from a

state to another state and finally reached (hopefully) the goal node

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Defining a Search ProblemDefining a Search Problem

Other considerations are:– Path cost: it is only relevant if there

exists more than one path leading to the goal state.

& certainly that we want the shortest path

– Goal test: it is applicable to single state problem, I.e. only one goal is found in the state space problem

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Relationship between Initial states, actions and goal(s)

InitialStates

Actions Goal(s)

ControlStrategy

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Why Why ‘‘SearchSearch’’??Search is an important (and

powerful ) aspect in AI problem solving

Search will– will help to explore alternatives in tree– will find sequence of steps in a

planning situation/caseAll goal-driven activities (such as

one used in B. C. reasoning) occur in a state space, and we call the problem a state space problem

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Classical Search DomainsClassical Search Domains

8-PuzzleWater JugBlocks WorldTravelling SalesmanMazeChessTower of Hanoietc.

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An Example of Search PreparationAn Example of Search Preparation

Initial State: The location of each of the 8-puzzle in one of the nine squares

Path cost: Each step costs 1 point, total path cost will be equal to the number of steps taken

Goal test: State matches the goal configuration (given board pattern)

Operators: blank moves [1] Left, [2] Right, [3] Up & [4] Down.

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6InitialState

GoalState

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SolutionTest Search ProcessSearch

Approaches

Blind

Heuristics

Optimisation

complete

partialCheck only

some alternatives

All possible solutions are

checked

Comparisonsstop

when all checked

Comparisons

optimal

Best among

alternatives

Onlypromisingsolutions

Stop whensolution is

goodenough

Goodenough

Generateimproved solutions

optimalStop when no improvement

is possible

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Methods of Search (I)

Popular blind search methods are Depth-first Breadth-first Bi-directional Iterative deepening Depth-limited Uniform cost

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Heuristic Search Hill-climbing Best-first Generate & test Induction Greedy search

Optimal Searching A* search

Methods of Search (II)

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Depth-first– explores search tree branch by branch

Breadth-first– examines search tree row by row

Hill-climbing– it is depth first but most promising child node is

examined first Best-first

– expands the most promising partial path of all those so far discovered

A*– Best-first and Branch and bound algorithm

To becalculated

Decision required

Methods of search at the first glance

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..…… Blind Blind……..

The entire search tree is examined in an orderly manner.

It can be classified as exhaustive or partial.

Two well-known partial methods are– breadth-first – depth-first

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DEPTH-FIRST SEARCHDEPTH-FIRST SEARCH(DFS)(DFS)

A ~ begins at the root node and works downward to successively

deeper level.

This process continues until a solution is found or backtracking is forced by reaching a dead end.

Blind ....

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DEPTH-FIRST SEARCHDEPTH-FIRST SEARCH

Delete FIRSTNODE from start of QUEUETake children of FIRSTNODEAdd to the front of QUEUEPut result in NEWQUEUE

Basic steps used

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Given a search tree of this: S A B C D E F H G Traversing order:

S A C D B E H G F

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BREADTH-FIRST SEARCH(BFS)

A ~ examines all the nodes in a search tree, beginning with the

root node.

The nodes in each level are examined completely before moving

to the next level.

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BREADTH-FIRST SEARCH

Take children of FIRSTNODEDelete FIRSTNODE from start of QUEUE

Append list of children to the end of QUEUEPut result in NEWQUEUE

The algorithm

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Given a search tree of this: S A B C D E F H G Traversing order:

S A B C D E F H G

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Heuristics Search Heuristics Search

Heuristics search is designed to reduce the amount of search for a solution.

It is based on rule-of-thumb principle.When a problem is presented, such as a

goal state is given, this approach tries to reduce the size of the search tree by pruning non promising or inferior nodes.

It can normally speed up the search process in obtaining a good enough solution.

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Benefits of HeuristicsBenefits of Heuristics

It has inherent flexibilityIt is better when an optimal/best solution is

too costly to generate– even though it can only produce “good enough”

answerSimpler for decision maker to understand

(especially the managers)– because they think in the same way as

“heuristics” does

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A heuristics search in generalA heuristics search in generalProblem: Tic-Tac-Toe

x

x

x

x x

ox x

ox

x

oxx

ox

x

ox

x

oxx

oxx

xxo

o o

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Consider the Tic-Tac-Toe gameConsider the Tic-Tac-Toe game

Each nine first moves has eight possible responses, which in turn have seven continuing moves, and so on.

A simple analysis tells that the number of states to be considered in an exhaustive search is 9 * 8 * 7 *… or 9!

However, symmetry reduction in second level leads to only 12 * 7! (much smaller than above), further reduction in search

space will further reduce the number or search required

Blind search is surely not practical…

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HILL-CLIMBING SEARCH

Delete FIRSTNODE from start of QUEUETake children of FIRSTNODE

Order children with most promising firstPlace them at the front of QUEUE

Put result in NEWQUEUE

Main steps used

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HILL-CLIMBING SEARCH

Set Queue to be the List[StartNode]If Queue is non-empty, then:

Take the first element, FirstNode, in QueueIf FirstNode is a goal node, then:

Return FirstNodeOtherwise:

Delete FirstNode fro start of QueueTake children of FirstNodeOrder children with most promising firstPlace them at the front of QueuePut result in NewQueue

Otherwise: return failure(Recursive step) Repeat the procedure with NewQueue in place of Queue, let the result ofthis call be NodeReturn Node

A complete function:

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Hill-ClimbingHill-Climbing

Hill-climbing combines depth-first search with a method for ordering the alternatives by measuring the probability of success at each decision point. In other word it tries to reach a goal by choosing those nodes which are predicted to be nearest to the goal. Quality measurements turn depth-first search into hill-climbing.

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Earlier slide says that it is quite similar to depth-first blind search.

However, paths are not selected arbitrarily, but in relationship to their proximity to the desired goal (i.e. how close to it)

let’s take a look at this tree:– notice that there are numbers attached nodes

those are potential numbers of defects in a product

It is about fin ding the shortest h(n)

Hill-ClimbingHill-Climbing

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Apply hill-climbing search to find ‘1’ defect node

A

B C D

FE G H

I J

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4

2

13

3

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7 5

1

III III

START

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Each production process I, II and III can continue for several stages.

If this is done by depth-first, it goes to I then to II and then to III .

In Hill-climbing method, nodes B, C and D are compared, and it starts in branch I (lowest, 8 here).

Since one defect was not found, backtracking is exercised to branch III. (branch III is done before II, why)?

The search goes to D and H; backtracking is then done and D, G, J path leads to the desired solution.

Path A--C--F was not visitedtherefore much time and cost are saved This is exactly the aims of ‘heuristics’!

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Hill-climbingHill-climbing

Few more words on hill-climbing method: If we imagine the goal as the top of hill, h(n) as the difference in heights between n and the top, the hill-climbing procedure corresponds to climbing the hill by always going upwards.

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Problems with hill-climbing:Problems with hill-climbing:

Foothill problem.– Go up to the top of a hill and not advancing

further. - misses the overall maximumPlateau problem.

– Problem with multidimensional problem space, i.e. no hill to climb (is flat). Gradient equals to zero. - not informative

Ridges.– Where there are steep slopes and the search

direction is not towards the way up (but side)

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Some solutions to hill-Some solutions to hill-climbing problems are:climbing problems are:

Random restart hill-climbing– where random initial states are generated

Simulated annealing– allow for bad moves as well, where the

probability of such a move decreases exponentially with its badness

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BEST-FIRST SEARCHBased on some heuristics evaluation

function– More flexible than hill-climbing

An evaluation function is used to assign a score to each candidate node– Next move is made by selecting the best value

nodeIt expands the best partial path, for that It

could lead to “shortsighted” situation

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BEST-FIRST SEARCHBest-first search has been used in such applications as

games and web crawlers In a web crawler, each web page is treated as a node,

and all the hyperlinks on the page are treated as unvisited successor nodes in the search space.

A crawler that uses best-first search generally uses an evaluation function that assigns priority to links based on how closely the contents of their parent page resemble the search query

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BEST-FIRST SEARCH

Delete FIRSTNODE from start of QUEUETake children of FIRSTNODEAppend children to QUEUE

Order result with most promising firstPut ordered result in NEWQUEUE

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Apply best-first search to find ‘3’ defect in a production

A

B C D

FE

G HI

J

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14 16

13

23

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3Traversing order (i.e. the search path) isA B D H G J

If hill climbing is used, the solution path will beA B E I D H G J (it is longer, less ‘intelligent’)

[A]

[B D C]

[D C E I]

[H G C E I]

[G C E I]

[J C E I]

[A]

[B D C]

[E I D C]

[I D C]

[D C]

[H G C]

[G C]

[J C]

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Optimal SearchOptimal Search

Will produce optimal (best) answerBased on some optimisation functionMathematical functions are used

– for improvements – for “optimisation”

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BRANCH AND BOUND ALGORITHM

One way to find optimal paths with less work is to use branch-and-bound.

It always keeps track of all partial paths contending for further consideration.

The shortest one is extended one level, creating as many new partial paths.

Next, these new paths are considered, along with the remaining old ones, again, the shortest is extended.

The process is repeated until the goal is reached along some path.

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Branch-and-BoundBranch-and-BoundKeys to remember:

– To turn likely to certain, you have to extend all partial paths until they are as long or longer than the complete path. The reason is that the last step in reaching the goal may be long enough to make the supposed solution longer than one or more partial paths.

– It might be that only a tiny step would extend one of the partial paths to the solution node.

– To be sure that this is not so, instead of terminating when a path is found, you terminate when the shortest partial path is longer than the shortest complete path

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An Example for B-N-BAn Example for B-N-B

S

D

A

B

E

F

G1515

The length of the completepath from S to G, S-D-E-F-G is 15.

Similarly, the length of the partial path S-D-A-B also 15and any additional movement along a branch will make it

longer than 15.Accordingly, there is no need to try S-D-A-B any further. Because it will be longerthan the complete path

already known.Only other paths emerging from S and from S-D-E have to be considered, as they may provide a shorter path.

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To conduct a branch-and-bound search:

Form a one-element queue consisting of zero length path (only root node)– Until the first path in the queue terminated at

the goal node or the queue is empty,– remove the first path from the queue; create new paths by

extending the first path to all the neighbours of the terminal node

– reject all paths with loops– add the reaming new paths, if any, to the queue– sort the entire queue by path length with least-cost paths in

front

– If the goal is found, announce success; otherwise announce failure.

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Branch-and-bound Method The search tree: S Start Node 4 1 A B 1 2 2 3 C D E F 4 3 2 1 H I J G Goal Node

Consider this search tree:

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Search path to goal node:

[S(0)] [B(1) A(4)] [E(3) A(4) F(4)] [A(4) F(4) I(6) H(7)] [F(4) C(5) I(6) D(6) H(7)] [G(5) C(5) I(6) D(6) J(6) H(7)] Goal node reached and stopped.

Solution

Use your tie breaker solution

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A* SearchAll A* algorithms are admissible

– A search algorithm is admissible if it always produces an optimal solution

The A* search is branch-and-bound, with an estimate of remaining cost combined with dynamic programming principle

Theorem of A* algorithm– For each node n, let h*(n) denote the cost of an

optimal path from n to a goal node– The algorithm uses a heuristic function h that

satisfies h(n) <= h*(n) for all n in the state space, is admissible

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A* SEARCH

Delete FIRSTNODE from start of QUEUEAppend children of FIRSTNODEOrder resulting list according to

cost-so-far + underestimate of remaining costPut result in NEWQUEUE

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A* SEARCH

Set Queue to be the List[StartNode]If Queue is non-empty, then:

Take the first element, FirstNode, in QueueIf FirstNode is a goal node, then:

Return FirstNodeOtherwise:

Delete FirstNode fro start of QueueAppend children of FirstNodeOrder resulting list according to

cost-so-far + underestimate of remaining cost

Put result in NewQueueOtherwise: return failure(Recursive step) Repeat the procedure with NewQueue in place of Queue, let the result ofthis call be NodeReturn Node

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To conduct a A* search: – Form a one-element queue consisting of zero length

path (only root node)– Until the first path in the queue terminated at the goal

node or the queue is empty,– remove the first path from the queue; create new paths by

extending the first path to all the neighbours of the terminal node

– reject all paths with loops– if two or more paths reach a common node, delete all

those paths except the one that reaches the common node with the minimum cost

– sort the entire queue by SUM of the path length and a lower-bound estimate of the cost remaining, with least-cost paths in front

– If the goal is found, announce success; otherwise announce failure.

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Function definition in A*Function definition in A*

Consider the evaluation function f(n) = g(n) + h(n)

where n is any state in the search

g(n) is the cost from the start state

h(n) is the heuristic estimate of the cost

going from n state to goal node

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Using Using A*A* in 8-puzzle problem in 8-puzzle problem

Figure 1 shows a start state and the first set of moves

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2 8 3

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2 8 3

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2 8 3

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START

First level ofThe tree search

Which one will the A* takes?& Why? On what basis it is chosen?

A1 A2 A3

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1 2 3

8 47 5

To reach this GOAL state

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Many levels of search and many solutions are possible to reach the goal state

What is the GOAL we wanted to achieve? i.e. what is the question?

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Using Using A*A* in 8-puzzle problem in 8-puzzle problemFigure 2 : Three heuristics applied to states in the 8-puzzle

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2 8 3

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1 2 3

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Recall the goal state5 6 0

3 4 0

5 6 0

Tiles out ofplace

Sum of distances out of place

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2 * the no.of direct tilesreversals

These are potentialCriteria for the A* formula

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Using Using A*A* in 8-puzzle problem in 8-puzzle problemFigure 3 : The heuristic f applied to states in the 8-puzzle

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1 6 47 5

2 8 3

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2 8 3

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1 2 3

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GOAL

2 8 3

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g(n) = 0

g(n)=1

6 4 6f(n)f(n) = g(n) + h(n)where g(n) actual distance from n (cost from start state) h(n) the no. of tiles out of place

6

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Exercise #1Exercise #1

List 4(four) criteria that are normally used to evaluate search

methods

Exercise #2Exercise #2

List 2(two) limitations of heuristics search methods

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Answer to Exercise #1Answer to Exercise #1Completeness

– will the solution be eventually found? if there is one at all?

Space complexity– how much memory will it need or is necessary?

Time complexity– how long will it take to complete the search?

Optimality– will the search method find the highest quality of

solution path when there are several.

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Supplementary SlidesSupplementary Slides

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Properties of Search Algorithms

Breadth-first Depth-first A* Greedy

Complete Yes No Yes No Space Exponential Linear Exponential Linear

Time Exponential Exponential Exponential Exponential

Optimal Yes No Yes No

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Bi-directional SearchBi-directional Search It searches both forward and backward in the

same state space (run simultaneously).It stops when the two moves meet in the

middle.

Start Goal

A schematic view of ~ that when a branch from the start node meets aBranch of the goal node, it stops

Blind Search..

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Iterative Deepening (Korf 1987)

As depth-first search gets quickly into a deep level in the search space. Depth-first search can get lost deep in the search tree/graph. It may also stuck in infinite path that does not lead to a goal.

A compromise is to use a depth bound on depth-first :– the depth bound forces failure on a search

path once it gets below certain level

Blind Search..

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Iterative Deepening (Korf 1987)

The hard part of Depth-limit search is to determine what limit is “good” limit.– Iterative deepening tries all possible depth

limits and pick up the best one!Thus:

– It is optimal and complete.– It uses modest memory.

Blind Search..

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Depth-Limited search

Similar to DFS, except that it avoids the pitfalls of DFS by imposing a cut off on the maximum depth of a given path

Drawback: if a chosen limit is too small, the scheme is not

“complete” optimality = no

Blind Search..

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Uniform Cost SearchUniform Cost SearchWe learnt that BFS finds the

shallowest path leading to the goal state

uniform cost search modifies BFS by expanding only the lowest cost node (n)

the cost of a path should remain low but not decreasing hence the term “uniform”Note that if all step costs are equal, this is identical to BFS

Blind Search..

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Comparison of Comparison of 6 Blind search methods6 Blind search methods

Breadth Depth Depth Bi-directional Iterative Uniform first first -limited cost

Time bd bm bl bd/2 bd bd

Space bd bm bl bd/2 bd bd

Complete yes no yes, l d yes yes yesOptimal yes no no yes yes yes

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Induction

Induction means to generalise from a smaller version of the same problem

Two essential features– problem must be modelled in terms of the

associated data– induced result must be tested against real

examples

Heuristics …

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Generate & Test The basic idea is to generate possible

solutions and devise a test to determine if the solutions are indeed good, i.e. acceptable.

Steps: add a specification criterion try to open a path that satisfies the specification determine whether the path is plausible, prune it

if it is not plausible move to the next path check whether all specifications have been

mentioned. If not, add the next specification and reiterate the steps by returning to first step

Heuristics …

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Greedy SearchGreedy Search

There is a heuristic function, h(n), which serves to hold values of prediction of path-cost left to the goal

Greedy search is to minimise the estimated cost to reach the goal

Therefore, the node (state) closest to the goal will be expanded first

Heuristics …

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Greedy SearchGreedy Search

It resembles DFS in the way that it follows a single path all the way to the goal, if there is no dead end

Thus, it is incomplete and not optimal

With good quality heuristic function, however, the space an time complexity can be reduced

Heuristics …