04 problem solving in ai - search algorithms

8/8/2019 04 Problem Solving in AI - Search Algorithms

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Problem Solving in AI: Search Algorithms

Most AI problem solving techniques areimplemented as a search for a solution



The nature of problem solving

Human problem solving involves searching for a solution

The solution is a structured object built, i.e. synthesised,

along the way

Three generic problem types are

State space transformation

Manipulate a current situation in an allowable way starting from

some initial position until a satisfactory end position is reached Transformation is achieved by bringing an operator to bear on the

current state

There is usually a choice of operators at each point



State space transformation

Uses the notion that each intermediate problemposition is a 'state of the world

The set of all possible positions is the state space

There are three ingredients in the statement of sucha problem:1. The set of initial states

2. The set of operators for moving from one state to another

3. The set of goal states or conditions that a state must satisfy to be agoal state

The problem is solved by finding a path, i.e. asequence from an initial state to a goal state



Search spaces, graphs and trees

A problem is represented as a graph with the nodes characterisingsituations and arcs (edges) indicating the permitted transformations

This is usually implicit since it is too vast to generate wholly

Arcs are usually directed (giving direction of permitted travel)

If there is a directed arc from a to b, then a is a predecessor or parent of b and b is a successor of a

Additionally they may be labelled (say with costs of traversal)

The average number of successors to a node is the branching factor

and is a key indicator in considering the computational complexity of thesearch



Search spaces, graphs and trees ctd A directed acyclic graph (DAG) is one containing no loops or cycles

An important type of DAG is a tree

This has a special node called the root

For each node (other than the root) there is one parent, i.e. there is a singlearc directed from the parent to the nod.

A node having no successor is called a leaf

The maximum distances in terms of arcs from the root to a leaf is the depth of the tree

As the search progresses an explicit graph (or more usually a tree) is

built - this is the search graph or search tree It can be used to keep track of the development of the solution

The 'solution' may require finding a path with particular properties(e.g. least cost)



A simple problem

Start: aGoal: f, j

Successors:

a p b, c b p d, ec p f, g d p h

e p i, j f p k



Search tree

b

g

a

d e

ih k

c

f

j

Goals at depth 2 and 3

Average branching factor < 2



A basic search algorithm

Specifying a search algorithm involves giving a recipe for

which successor nodes to try And, optionally, for recording what has already been tried

Examining the successors of a node is called expanding thenode

1. Set L, the open list, to be the list of start nodes

2. If L is empty, fail. Otherwise pick a node, n, from L

3. If n is a goal node, stop and return it together with the path from

the initial node to n

4. Otherwise remove n from L and add all of n's successors to L

labelling each with its path from the initial node Return to step 2.



Types of search

This general procedure does not specif y how a node is chosenfor expansion,

i.e. in what order such nodes are chosen This specification is all important leading to different actual

algorithms

Search can be blind or heuristic

A blind search method follows a fixed prescription regardless of the

particular characteristics of the problem This is unrealistic except for toy problems

A heuristic search method uses knowledge, in the form of heuristicsappropriate to the problem in hand, to guide search

thus narrowing the space actually covered



Blind search

Depth-first (with or without a depth bound)

Dive deep into the search tree

Nodes for expansion are selected from the front of L and successors to a node are also added to the front of L

Use of depth bound, M, prevents excessive pursuit of a goal tosubstantial depths when there may be a shallow goal on anotherbranch

Br eadth-first Move acr oss th e s ear ch tr ee th en down

Nodes for expa nsi on a r e s elect ed f r om th e f r ont of L and successors to a node are also added to the end of L



Example of depth-first search

b

g

a

d e

ih k

c

f

j

Nodes are visited in the order : a, b, d, h, e, i, j.

Solution path is : a, b, e, j



Example of breadth-first search

b

g

a

d e

ih k

c

f

j

Nodes are visited in the order : a, b, c, d, e, f

Solution path is : a, c, f



Complexit y of algorithms

Search algorithms have properties of

time complexit y

i.e. how long they take to complete the search memory (or space) complexit y,

i. e. how much storage space they need for intermediate results

Calculations are made in general or average terms as theact ual req uirements will differ from problem to problem

Complexities are specified in terms of parameters of theproblem, e.g. branching factor, using o notation (meaningorder of) o(x2) means complexit y varies as x2



Complexities for depth and breadth-first

Both have same time complexit y

exponential in depth

It is impossible to improve on this as a general result for any blind search algorithm

Memory req uirement for breadth-first is much worse again, exponential in depth

because alternative paths are stored

M e t o T im e M e m o r y

D e p t h i r s t~ ( )

2

d

d bo b

( 1) 1 ( )d b o d b

B r e a d t h i r s t (1 1 / ) ~ ( )2

d

d b

b o b 1 ( )d d b o b



Other blind search algorithmsDepth-limited

A variation on depth-first

A maximum depth parameter is specified for the search.

This acts a glass floor

when the max depth is reached, backtracking occurs and the searchmoves across instead of continuing down

Iterative deepening

A seq uence of depth-limited searches beginning with max depth =1 If the goal is not found the search is repeated with max depth =2 and

so on ...

Has the same complexit y as depth-first yet finds shallow nodes first like breadth-first



Avoiding loops in the search If a node has more than one parent then expansion may

lead to re-visiting of nodes alread y considered

making a loop or repetition in the path

Here it is usef ul to maintain another list to store the nodeswhich have been visited alread y

This is called the closed list

After a node has been expanded, it is placed in the closed list

A successor node is added to L (the open list) provided:

it is not in L alread y and it is not in the closed list



Sample problem:

missionaries and cannibals

T h ree missionaries and th ree cannibals are on one side of a river

Using a boat wh ich can h old at most tw o people th ey must all get t o

th e oth er side If th e missionaries are ever ou t numbered on one side of th e river

th ey w ill be eat en by cannibals

A st at e can be represent ed as:(Left, Boat-left, Right)

Left represents the people on the left bank,

Boat represents those on the boat and

Right those on the right

Each of Left, Boat and Right is a pair (M,C) where M is the number of missionaries and C the number of cannibals

Initial state: ( (3,3), (0,0)-left, (0,0) )

Goal state: ( (0,0), (0,0)-right, (3,3) )



Sample problem:

water containers

Given bot h a seven and a five litre container, initially empty, t h e goal is tofind a sequence of actions wh ich leaves four litres of liquid in t h e seven litrecontainer

T h ere are t h ree kinds of actions wh ich can alter t h e state of t h e containers:

i. A container can be filled

ii. A container can be emptied

iii. Liquid can be poured from one container into t h e ot h er, until t h e first isempty or t h e second is full

A state can be represented as:

(C1, C2) where C1 and C2 are the current volumes in the first (seven litre) and second (five litre) containers

Initial state is is (0, 0)

Goal state is (4, _) , i.e the volume of the second container is irrelevant



Heuristic search

A heuristic, h, is an inspired guess or rule-of thumb about where to go next in the search:

It is often a measure of distance from (or, closeness to) thegoal - called an evaluation f unction

It uses knowledge of the particular problem

Nodes not yet visited are evaluated using the

measure and the one with the best score is chosen Heuristics offer no guarantees

They are used because they tend to give betterresults



Types of heuristic search Hill climbing

From the current node choose (if possible) a successor node with the best evaluation and which improves upon that of the node itself

This is a local choice of best leading to a depth-first approach

B est-first Choose the best node globally to expand next from the open list

Depth and breadth first are special cases

Heuristic search will only improve on blind search if theheuristic contains real information about how near thegoal is. This will enable it to:

find a short, direct path to the goal and

find this path with little meandering along the way



Example of hill-climbing search

b

g

a

d e

ih k

c

f

j

Nodes are visited in the order : a, b, e, j

Solution path is : a, b, e, j

h =1.6

h = 0.7

h=0.8

h =1.8 h =0.9 h =2.7

h =0h =3.7h =4.9 h =6.2

h=0



Example of best first search

b

g

a

d e

ih k

c

f

j

Nodes are visited in the order : a, b, c, f

Solution path is : a, c, f

h=1.6

h=0.7

h=0.8

h=1.8 h=0.9 h=2.7

h=0h=3.7h=4.9 h=6.2

h=0



The A* algorithm

This is a version of best-first which applies to a specialclass of evaluation f unction: f(n)=g(n) + h'(n)

g(n) is the 'cost' incurred in reaching n Costs are attached to arcs of the graph

If a solution with shortest path length is req uired, arc cost = 1

g(n) is the act ual cost in reaching n

There may be a path (not found) to n with lower cost

Thus g(n) over-estimates the minimum cost

h'(n) is a heuristic that estimates the cost that will beincurred in reaching the goal from n



The A* algorithm ctd If the estimate h'(n) always under-estimates the true cost,

then h' is said to be admissible

In this case A* will always find a least cost path to thegoal assuming such a path exists

Ideally h'(n) should be a tight estimate Should be close to the true cost without going over

The tighter the estimate the more direct or focused the search willbe

there will be less meandering on the way

The h'(n) = true cost (as determined from arc costs, thensearches progresses directly to the goal



Example of A* search

b

g

a

d e

ih k

c

f

j

h=1.6

h=0.7

h=0.8

h=1.8 h=0.9 h=2.7

h=0h=3.7h=4.9 h=6.2

h=0

Cost per arc = 1

h values are admissible, e.g. at b, act ual cost of reaching goal (j) is 1+1=2 but h

is only 0.7. At b, f (b) =g (b) + h (b) = 1 + 0.7 =1.8



Sample problem:the 8-puzzle Req uires the eight numbered tiles in a 3*3 grid to be rearranged

(an example is shown below)

An individ ual move consists of a horizontal or vertical shift of a tileinto the space

Can generalise problem to n*n-1

2 1 6

4 8

7 5 3

1 2 3

8 4

7 6 5



Heuristic for the 8-puzzle

A good heuristic is provided by the total, over numbered tiles, of the Manhatten distance (md) of a tile from its final

positionmd = sum of horizontal and vertical distance to correct position

E.g. in diagram, tile number 6 has to travel distances of 1to the left and 2 dow n : md = 1+2

Can be improved by adding a seq uence misalignment factor A tile in the centre scores 1

A tile on a non-central sq uare scores 0 if it is follow ed the clockw isedirection by the correct tile in the goal configuration; else its score is 2

04 problem solving in ai - search algorithms

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