Monotonicity

Admissible Search:

“That finds the shortest path to the Goal”

Monotonicity:

local admissibility is called MONOTONICITY

This property ensures consistently minimal path to each state they encounter in the search.

It takes accumulative effect into consideration

(for a distance problem)

Along any path from the root,

the cost never decreases

(If this is true then the Heuristic

is Monotonic in nature.)

f(n) = g(n) + h(n) = 3 + 4 = 7

f’ (n) = g' (n’) + h' (n’) = 4 + 2 = 6

366

390

526417

n

n’

Non-Monotonic Monotonic

If f(n`) < f(n) (Non - Monotonic)

Then

f(n`) = max (f(n), g(n`) + h(n`))

“We take the cost of parent Node”.

Pathmax. This when only heuristic cost is taken

Another representation = h(n) - h(n`) <= cost (n, n`)

Informedness.

For two A “heuristic h1 and h2 if h1(n) <= h2(n) for all states ‘n’ in the search space, heuristic h2 is said to be “more informed” than h1.

Both h1 and h2 can give OPTIMAL path but h2 examines very few states in the process.

Monotonic Heuristics are Admissible

States = S1, S2, …, Sg S1 = Start

Sg = goal

h(s1) - h(s2) <= cost (s1, s2)

h(s2) - h(s3) <= cost (s2, s3)

h(g-1) - h(g) <= cost (Sg-1, Sg)

ADD h(s1) - h(g) <= cost (S1, Sg)

h(n) = 0 Uninformed search

Example

Breadth - First search

A* is more informed then Breadth - first search

Adversary Search (Games)

AIM: The aim is to move in such a way as to ‘stop’ the opponent from making a good / winning move.

Game playing can use Tree - Search.

The tree or game - tree alternates between two players.

Things to Remember:

1. Every move is vital

2. The opponent could win at the next move or subsequent moves.

3. Keep track of the safest moves

4. The opponent is well - informed

5. How the opponent is likely to response to your moves.

Two move win

Player 1 = P1

Player 2 = P2

Safest move for P1 is always A C

Safest move for P2 is always A D (if allowed 1st move)

A

BC

D

E F G H I J

P1 moves

P2 moves

P1 P2 P1 P1 P2 P2 wins

Minimax Procedure for Games

Assumption: Opponent has same knowledge of state space and make a consistent effort to WIN.

MIN: Label for the opponent trying to minimize other player’s (MAX) score.

MAX: Player trying to win (maximise advantage)

Both MAX and MIN are equally informed

Rules

1. Label level’s MAX and MIN

2. Assign values to leaf nodes:

0 if MIN wins

1 if MAX wins

3. Propagate values up the graph.

If parent is MAX, assign it

Max-value of its children

If parent is MIN, assign it

min-value of its children

Minimaxing to fixed to play depth(Complex games)

Strategy: n - move look ahead

- Suppose you start in the middle of the game.

- One cannot assign WIN/LOOSE values at that stage

- In this case some heuristics evaluation is applied

- values are then projected back to supply indicate WINNING/LOOSING trend.

Summary

• Assign heuristic values to leaves of n-level graph

• Propagate value to root node

• This value indicates best state that can be reached in ‘n’ moves from this start - state or node

MAXIMIZE for MAX Parents

MINIMIZE for MIN parents

Example:

TIC - TAC - TOE

M(n) = Total of my possible

winning lines

O(n) = Trial of Opponents E(n) = M(n) - O(n)

winning lines

X

X

X

O

O

O

Horizon Effect

• Heuristics applied with limited ahead may lead to a bad situation and the person may leave the game.

• Same depth in search can be added to partially reduce this affect.

Alpha - Beta Procedures

• Minimax procedure pursues all branches in the space. Same of them could have been ignored or pruned.

• To improve efficiency pruning is applied to two person games

Simple Idea

if A > 5 or B < 0 If A >5 and B <0

If the first condition If the first condition

A > 5 succeeds then failed then evaluating

B < 0 may not be evaluated. B < 0 is unnecessary.

- MAX can score maximum of -0.2 when moves a - c - e.

MAX has a batter option to move to ‘b’

a

b = 0.4

d = 0.6

f = -0.5 g = -0.2

e

c

-0.2

-0.2

MAX

MIN

MAX

MIN

- MAX node neglects values <= a (atleast it can score) at MIN nodes below it.

- MIN node neglects values >= b (almost it can score) at MAX nodes below it

A

B =10

C

G=0 H

MAX

MIN

C node can score ATMOST 0 nothing above 0 (beta)

A node can score ATLEAST 10 nothing less than 10 (alpha)

Complexity Reduction

Complexity Cost: “Can be estimated roughly through measuring the size of open and closed list.”

(A) Beam Search: “In beam search only the ‘n’ most pronishing state are best for future consideration”

- The procedure may miss the solution by pruning it too early.

“Bound applied to the open list.”

(B) More Informed ness

- Apply more informed heuristics to reduce the complexity.

- This may increase the computational cost in computing the heuristic