Instructor: Eyal AmirGrad TAs: Wen Pu, Yonatan Bisk

Undergrad TAs: Sam Johnson, Nikhil Johri

CS 440 / ECE 448Introduction to Artificial Intelligence

Spring 2010Lecture #4

When to Use Search Techniques?

The search space is small, andThere is no other available techniques, orIt is not worth the effort to develop a more efficient technique

The search space is large, andThere is no other available techniques, andThere exist “good” heuristics

Models, |=, math proofs

Alpha |= BetaAlpha, Beta – propositional formulasM |= Alpha “M models Alpha” means “Alpha evaluated to TRUE in model M”

Math. Proofs: exampleA � B |= A(another one later in today’s class)

Search AlgorithmsBlind search – BFS, DFS, ID, uniform cost

no notion concept of the “right direction”can only recognize goal once it’s achieved

Heuristic search – we have rough idea of how good various states are, and use this knowledge to guide our search

Types of heuristic search

Best FirstA* is a special caseBFS is a special case

ID A*ID is a special case

Hill climbingSimulated Annealing

A* Example: 8-Puzzle

0+4

1+5

1+5

1+3

3+3

3+4

3+4

3+2 4+1

5+2

5+0

2+3

2+4

2+3

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

ID A*: 8-Puzzle Example

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

4

4

6

Cutoff=4

6

ID A*: 8-Puzzle Examplef(N) = g(N) + h(N) with h(N) = number of misplaced tiles

4

4

6

Cutoff=4

6

5

ID A*: 8-Puzzle Examplef(N) = g(N) + h(N) with h(N) = number of misplaced tiles

4

4

6

Cutoff=4

6

5

5

ID A*: 8-Puzzle Examplef(N) = g(N) + h(N) with h(N) = number of misplaced tiles

4

4

6

Cutoff=4

6

5

56

ID A*: 8-Puzzle Examplef(N) = g(N) + h(N) with h(N) = number of misplaced tiles

4

6

Cutoff=5

ID A*: 8-Puzzle Examplef(N) = g(N) + h(N) with h(N) = number of misplaced tiles

4

4

6

Cutoff=5

6

ID A*: 8-Puzzle Examplef(N) = g(N) + h(N) with h(N) = number of misplaced tiles

4

4

6

Cutoff=5

6

5

ID A*: 8-Puzzle Examplef(N) = g(N) + h(N) with h(N) = number of misplaced tiles

4

4

6

Cutoff=5

6

5

7

ID A*: 8-Puzzle Examplef(N) = g(N) + h(N) with h(N) = number of misplaced tiles

4

4

6

Cutoff=5

6

5

7

5

ID A*: 8-Puzzle Examplef(N) = g(N) + h(N) with h(N) = number of misplaced tiles

4

4

6

Cutoff=5

6

5

7

5 5

ID A*: 8-Puzzle Examplef(N) = g(N) + h(N) with h(N) = number of misplaced tiles

4

4

6

Cutoff=5

6

5

7

5 5

ID A*: 8-Puzzle Examplef(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Hill climbing example 2 8 31 6 47 5

2 8 31 47 6 5

2 31 8 47 6 5

1 38 4

7 6 5

2

31 8 47 6 5

2

1 38 47 6 5

2start

goal

-5

h = -3

h = -3

h = -2

h = -1

h = 0

h = -4

-5

-4

-4

-3

-2

f(n) = -(number of tiles out of place)

Best-first search

Idea: use an evaluation function f(n) for each node nExpand unexpanded node n with min f(n)Implementation: FRINGE is queue sorted by decreasing order of desirability

Greedy searchA* search

Greedy Search

h(n) – a ‘heuristic’ function estimating the distance to the goal

Greedy Best First: expand argmin_n h(n)thus, f(v) = h(v)

Informed Search

Add domain-specific information to select the best path along which to continue searchingh(n) = estimated cost (or distance) of minimal cost path from n to a goal state. The heuristic function is an estimate, based on domain-specific information that is computable from the current state description, of how close we are to a goal

Robot Navigation

Robot Navigation

0 211

58 7

7

3

4

7

6

7

6 3 2

8

6

45

23 3

36 5 24 43 5

54 6

5

6

4

5

f(N) = h(N), with h(N) = Manhattan distance to the goal

Robot Navigation

0 211

58 7

7

3

4

7

6

7

6 3 2

8

6

45

23 3

36 5 24 43 5

54 6

5

6

4

5

f(N) = h(N), with h(N) = Manhattan distance to the goal

7

0What happened???

Greedy Search

f(N) = h(N) � greedy best-firstIs it complete?

If we eliminate endless loops, yesIs it optimal?

More informed search

Our goal is not to minimize the distance from the current head of our path to the goal, we want to minimize the overall length of the path to the goal!Let g(N) be the cost of the bestpath found so far between the initial node and Nf(N) = g(N) + h(N)

Robot Navigation

f(N) = g(N)+h(N), with h(N) = Manhattan distance to goal

0 211

58 7

7

3

4

7

6

7

6 3 2

8

6

45

23 3

36 5 24 43 5

54 6

5

6

4

57+0

6+1

6+1

8+1

7+0

7+2

6+1

7+2

6+1

8+1

7+2

8+3

7+2

6+3

6+3

5+4

5+4

4+5

4+5

3+6

3+6

2+7

8+3

7+4

7+4

6+5

5+6

6+3

5+6

2+7

3+8

4+7

5+6

4+7

3+8

4+7

3+8

3+8

2+9

2+9

3+10

2+9

3+8

2+9

1+10

1+10

0+11

Can we Prove Anything?

If the state space is finite and we avoid repeated states, the search is complete, but in general is not optimalProof: ?If the state space is finite and we do not avoid repeated states, the search is in general not complete

Admissible heuristic

Let h*(N) be the true cost of the optimal path from N to a goal nodeHeuristic h(N) is admissible if: 0 � h(N) � h*(N)

An admissible heuristic is always optimistic

A* Search

Evaluation function:f(N) = g(N) + h(N)where:

g(N) is the cost of the best path found so far to Nh(N) is an admissible heuristic

Then, best-first search with this evaluation function is called A* search

Important AI algorithm developed by Fikes and Nilsson in early 70s. Originally used in Shakey robot.

Completeness & Optimality of A*

Claim 1: If there is a path from the initial to a goal node, A* (with no removal of repeated states) terminates by finding the best path, hence is:

completeoptimal

requirements:0 < � � c(N,N’) - c(N,N’) – cost of going from N to N’

Completeness of A*

Theorem: If there is a finite path from the initial state to a goal node, A* will find it.

Proof of Completeness

Intuition (not math. Proof):

Let g be the cost of a best path to a goal nodeNo path in search tree can get longer than g/�, before the goal node is expanded

Optimality of A*

Theorem: If h(n) is admissable, then A* is optimal (finds an optimal path).

Proof of Optimality

G1

N

G2

f(G1) = g(G1)

f(N) = g(N) + h(N) � g(N) + h*(N)

f(G1) � g(N) + h(N) � g(N) + h*(N)

Cost of best path to a goal thru N

Example of Evaluation Function

f(N) = (sum of distances of each tile to its goal)+ 3 x (sum of score functions for each tile)

where score function for a non-central tile is 2 if it is not followed by the correct tile in clockwise order and 0 otherwise

1 2 34 5 67 8

123

45

67

8

N goal

f(N) = 2 + 3 + 0 + 1 + 3 + 0 + 3 + 13x(2 + 2 + 2 + 2 + 2 + 0 + 2)= 49

Heuristic Function

Function h(N) that estimates the cost of the cheapest path from node N to goal node.Example: 8-puzzle

1 2 34 5 67 8

123

45

67

8

N goal

h(N) = number of misplaced tiles= 6

Heuristic Function

Function h(N) that estimate the cost of the cheapest path from node N to goal node.Example: 8-puzzle

1 2 34 5 67 8

123

45

67

8

N goal

h(N) = sum of the distances of every tile to its goal position= 2 + 3 + 0 + 1 + 3 + 0 + 3 + 1= 13

8-Puzzle

4

5

5

3

3

4

3 4

4

2 1

2

0

3

4

3

f(N) = h(N) = number of misplaced tiles

8-Puzzle

0+4

1+5

1+5

1+3

3+3

3+4

3+4

3+2 4+1

5+2

5+0

2+3

2+4

2+3

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

8-Puzzle

5

6

6

4

4

2 1

2

0

5

5

3

f(N) = h(N) = � distances of tiles to goal

8-Puzzle

1 2 3

4 5 6

7 8

12

3

4

5

67

8

N goalh1(N) = number of misplaced tiles = 6 is

admissibleh2(N) = sum of distances of each tile to goal = 13is admissibleh3(N) = (sum of distances of each tile to goal)+ 3 x (sum of score functions for each tile) = 49is not admissible

8-Puzzle

0+4

1+5

1+5

1+3

3+3

3+4

3+4

3+2 4+1

5+2

5+0

2+3

2+4

2+3

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Robot navigation

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

f(N) = g(N) + h(N), with h(N) = straight-line distance from N to goal

About Repeated States

N N1S S

1f(N)=g(N)+h(N)

f(N1)=g(N1)+h(N1)

g(N1) < g(N)h(N) < h(N1)f(N) < f(N1)

N2f(N2)=g(N2)+h(N)

g(N2) < g(N)

Consistent Heuristic

The admissible heuristic h is consistent (or satisfies the monotone restriction) if for every node N and every successor N’ of N:

h(N) � c(N,N’) + h(N’)

(triangle inequality)

N

N’

h(N)

h(N’)

c(N,N’)

8-Puzzle

1 2 3

4 5 6

7 8

12

3

4

5

67

8

N goal

h1(N) = number of misplaced tilesh2(N) = sum of distances of each tile to goal

are both consistent

Robot navigation

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

h(N) = straight-line distance to the goal is consistent

Claims

If h is consistent, then the function f alongany path is non-decreasing:

f(N) = g(N) + h(N)f(N’) = g(N) +c(N,N’) + h(N’)

N

N’

h(N)

h(N’)

c(N,N’)

Claims

If h is consistent, then the function f alongany path is non-decreasing:

f(N) = g(N) + h(N)f(N’) = g(N) +c(N,N’) + h(N’)h(N) � c(N,N’) + h(N’)f(N) � f(N’)

N

N’

h(N)

h(N’)

c(N,N’)

Claims

If h is consistent, then the function f alongany path is non-decreasing:

f(N) = g(N) + h(N)f(N’) = g(N) +c(N,N’) + h(N’)h(N) � c(N,N’) + h(N’)f(N) � f(N’) If h is consistent, then whenever A* expands a node it has already found an optimal path to the state associated with this node

N

N’

h(N)

h(N’)

c(N,N’)

Avoiding Repeated States in A*

If the heuristic h is consistent, then:Let CLOSED be the list of states associated with expanded nodesWhen a new node N is generated:

If its state is in CLOSED, then discard NIf it has the same state as another node in the fringe, then discard the node with the largest f

Complexity of Consistent A*

Let s be the size of the state space Let r be the maximal number of states that can be attained in one step from any stateAssume that the time needed to test if a state is in CLOSED is O(1)The time complexity of A* is O(s r log s)

Heuristic Accuracy

h(N) = 0 for all nodes is admissible and consistent. Hence, breadth-first is a special case of A* !!!Let h1 and h2 be two admissible and consistent heuristics such that for all nodes N: h1(N) � h2(N). Then, every node expanded by A* using h2 is also expanded by A* using h1.h2 is more informed than h1

Iterative Deepening A* (IDA*)

Use f(N) = g(N) + h(N) with admissible and consistent hEach iteration is depth-first with cutoff on the value of f of expanded nodes

8-Puzzle

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

6

8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

6

5

8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

6

5

5

4

8-Puzzle

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=4

6

5

56

8-Puzzle

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

6

8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

6

5

8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

6

5

7

8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

6

5

7

5

8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

6

5

7

5 5

8-Puzzle

4

4

6

f(N) = g(N) + h(N) with h(N) = number of misplaced tiles

Cutoff=5

6

5

7

5 5

About Heuristics

Heuristics are intended to orient the search along promising pathsThe time spent computing heuristics must be recovered by a better searchAfter all, a heuristic function could consist of solving the problem; then it would perfectly guide the searchDeciding which node to expand is sometimes called meta-reasoningHeuristics may not always look like numbers and may involve large amount of knowledge

What’s the Issue?

Search is an iterative local procedureGood heuristics should provide some global look-ahead (at low computational cost)

Another approach…

for optimization problemsrather than constructing an optimal solution from scratch, start with a suboptimal solution and iteratively improve it

Local Search AlgorithmsHill-climbing or Gradient descentPotential FieldsSimulated AnnealingGenetic Algorithms, others…

Hill-climbing searchIf there exists a successor s for the current state n such that

h(s) < h(n)h(s) <= h(t) for all the successors t of n,

then move from n to s. Otherwise, halt at n. Looks one step ahead to determine if any successor is better than the current state; if there is, move to the best successor. Similar to Greedy search in that it uses h, but does not allow backtracking or jumping to an alternative path since it doesn’t “remember” where it has been.Not complete since the search will terminate at "local minima," "plateaus," and "ridges."

Hill climbing on a surface of states

Height Defined by Evaluation Function

Hill climbing

Steepest descent (~ greedy best-first with no search) � may get stuck into local minimum

Robot Navigation

f(N) = h(N) = straight distance to the goal

Local-minimum problem

Examples of problems with HC

applet

Drawbacks of hill climbing

Problems:Local Maxima: peaks that aren’t the highest point in the spacePlateaus: the space has a broad flat region that gives the search algorithm no direction (random walk)Ridges: flat like a plateau, but with dropoffs to the sides; steps to the North, East, South and West may go down, but a step to the NW may go up.

Remedy: Introduce randomness

Random restart.

Some problem spaces are great for hill climbing and others are terrible.

What’s the Issue?

Search is an iterative local procedureGood heuristics should provide some global look-ahead (at low computational cost)

Hill climbing example 2 8 31 6 47 5

2 8 31 47 6 5

2 31 8 47 6 5

1 38 4

7 6 5

2

31 8 47 6 5

2

1 38 47 6 5

2start

goal

-5

h = -3

h = -3

h = -2

h = -1

h = 0

h = -4

-5

-4

-4

-3

-2

f(n) = -(number of tiles out of place)

Example of a local maximum

1 2 57 4

8 6 3

1 2 57 4

8 6 3

1 2 57 4

8 6 3

1 2 57 4

8 6 3

1 2 57 4

8 6 3

-3

-4

-4

-4

0

start

goal

Potential FieldsIdea: modify the heuristic functionGoal is gravity well, drawing the robot toward itObstacles have repelling fields, pushing the robot away from themThis causes robot to “slide” around obstaclesPotential field defined as sum of attractor field which get higher as you get closer to the goal and the indivual obstacle repelling field (often fixed radius that increases exponentially closer to the obstacle)

Does it always work?

No.But, it often works very well in practiceAdvantage #1: can search a very large search space without maintaining fringe of possiblitiesScales well to high dimensions, where no other methods workExample: motion planningAdvantage #2: local method. Can be done online

Example: RoboSoccerAll robots have same field: attracted to the ball

Repulsive potential to other playersKicking field: attractive potential to the ball and local repulsive potential if clase to the ball, but not facing the direction of the opponent’s goal. Result is tangent, player goes around the ball.Single team: kicking field + repulsive field to avoid hitting other players + player position fields (paraboilic if outside your area of the field, 0 inside). Player nearest to the ball has the largest attractive coefficient, avoids all players crowding the ball.Two teams: identical potential fields.

Simulated annealingSimulated annealing (SA) exploits an analogy between the way in which a metal cools and freezes into a minimum-energy crystalline structure (the annealing process) and the search for a minimum [or maximum] in a more general system. SA can avoid becoming trapped at local minima.SA uses a random search that accepts changes that increase objective function f, as well as some that decrease it.SA uses a control parameter T, which by analogy with the original application is known as the system “temperature.”T starts out high and gradually decreases toward 0.

Simulated annealing (cont.)

A “bad” move from A to B is accepted with a probability

(f(B)-f(A)/T)

eThe higher the temperature, the more likely it is that a bad move can be made.As T tends to zero, this probability tends to zero, and SA becomes more like hill climbingIf T is lowered slowly enough, SA is complete and admissible.

The simulated annealing algorithm

Summary: Local Search Algorithms

Steepest descent (~ greedy best-first with no search) � may get stuck into local minimumBetter Heuristics: Potential FieldsSimulated annealingGenetic algorithms

When to Use Search Techniques?

The search space is small, andThere is no other available techniques, orIt is not worth the effort to develop a more efficient technique

The search space is large, andThere is no other available techniques, andThere exist “good” heuristics

Summary

Heuristic functionBest-first searchAdmissible heuristic and A*A* is complete and optimalConsistent heuristic and repeated statesHeuristic accuracyIDA*

Modified Search Algorithm

1. INSERT(initial-node,FRINGE)2. Repeat:

If FRINGE is empty then return failuren � REMOVE(FRINGE)s � STATE(n)If GOAL?(s) then return path or goal stateFor every state s’ in SUCCESSORS(s)

Create a node n’INSERT(n’,FRINGE)