HEURISTIC SEARCH. Luger: Artificial Intelligence, 5 th edition. © Pearson Education Limited, 2005 Portion of the state space for tic-tac-toe

HEURISTIC SEARCH

Luger: Artificial Intelligence, 5 th edition. © Pearson Education Limited, 2005

Portion of the state space for tic-tac-toe.

Heuristically reduced state space for tic-tac-toe.


First three levels of the tic-tac-toe state space reduced by symmetry


The “most wins” heuristic applied to the first children in tic-tac-toe.


Is it possible to completely cover with non-overlapping dominos an 8x8 grid having two diagonally opposite corners removed?



Issues:1) Representation

(organization)2) Algorithm

(process)

The local maximum problem for hill-climbing with 3-level look ahead




priority queue

not necessary if states are added to open in sorted order

not necessary if using a monotone heuristic

Heuristic search of a hypothetical state space.


(assume P3 is the goal state and lower scores are preferred)

A trace of the execution of best_first_search for previous figure.


Heuristic search of a hypothetical state space with open and closed states highlighted.


Eight Puzzle…

Compare:best-first with depth factor: 26 moves to solution(7725 in CLOSED; 7498 left in OPEN)

best-first without depth factor: 48 moves to solution(272 in CLOSED; 332 in OPEN)

30 shuffle moves

Consider this screen shot of a brute-force breath-first search in process for the 15-puzzle where the optimal solution is known (by another method) to be located at depth 26.

How long will it take to find the solution?

The start state, first moves, and goal state for an example-8 puzzle.


1 2 34 5 67 8

or

Goal

Three heuristics applied to states in the 8-puzzle.



The heuristic f applied to states in the 8-puzzle.


(but should use a better heuristic…)

State space generated in heuristic search of the 8-puzzle graph.


Notice that Luger is using the “Tiles out of place” heuristic.

State space generated in heuristic search of the 8-puzzle graph.

The successive stages of open and closed that generate this graph are:


Open and closed as they appear after the 3rd iteration of heuristic search


Heuristic search with inclusion of a depth factor(heuristic here is # of tiles out of place) Nilsson

i.e., as long as h(n) is not overestimated(i.e., because that could prevent search of the optimal path)



(i.e., locally admissible)

Note: If the graph search algorithm for best-first search is used with a monotonic heuristic, a new path cannot be shorter than one already found.

Note: the best h(n) can be is the actual cost of the optimal path from node n to the goal. If h1(n) < h2(n), then h1(n) is actually underestimating the cost more than h2(n) and is, therefore, less informed.


Comparison of state space searched using heuristic search with space searched by breadth-first search. The proportion of the graph searched heuristically is shaded. The optimal search selection is in bold. Heuristic used is f(n) = g(n) + h(n) where h(n) is tiles out of place.


Number of nodes generated as a function of branching factor, B, for various lengths, L, of solution paths. The relating equation is T = (BL+1 – 1)/(B – 1), adapted from Nilsson (1980)*.

*Note: equation in text is incorrect but the one shown here is correct for root node at depth 0

Bra

nchi

ng F

acto

r =

B


Informal plot of cost of searching and cost of computing heuristic evaluation against informedness of heuristic, adapted from Nilsson (1980).


(i.e., cost to run the heuristic)

(i.e., cost to traverse the search graph)

Two-“Person” GamesTwo-“Person” Games

State space for a variant of Nim. Each state partitions the seven matches into one or more piles of different sizes. First player who cannot make a legal move, wins.


Exhaustive minimax for the game of Nim. Bold lines indicate forced win for MIN. Each node is marked with its derived value (0 or 1) under minimax.


Note: Figure is slightly different from text...

Getting ready to apply minimax to a hypothetical state space. Leaf states show heuristic values.


Applying minimax to a hypothetical state space. Leaf states show heuristic values; internal states show backed-up values.


Heuristic measuring conflict applied to states of tic-tac-toe.


Two-ply minimax applied to the opening move of tic-tac-toe, from Nilsson (1971).


Two ply minimax, and one of two possible MAX second moves, from Nilsson (1971).


Two-ply minimax applied to X’s move near the end of the game, from Nilsson (1971).


Let’s reconsider the application of minimax to the hypothetical state space shown a few slides back...


Alpha-Beta Pruning













Alpha-beta pruning applied to the previous minimax state space search. States without numbers are not evaluated.


Alpha-Beta SearchAlpha-Beta SearchAs successors of a node are given backed-up values, the bounds As successors of a node are given backed-up values, the bounds on the backed-up values can change but:on the backed-up values can change but:– Alpha values of MAX nodes can never decreaseAlpha values of MAX nodes can never decrease– Beta values of MIN nodes can never increaseBeta values of MIN nodes can never increase

Rules for discontinuing search:Rules for discontinuing search:– Discontinue search below any MIN node having a beta value ≤ to Discontinue search below any MIN node having a beta value ≤ to

alpha value of any of its MAX node ancestors. Final backed-up value alpha value of any of its MAX node ancestors. Final backed-up value of this MIN node can be set to its beta value.of this MIN node can be set to its beta value.

– Discontinue search below any MAX node having an alpha value ≥ Discontinue search below any MAX node having an alpha value ≥ the beta value of any of its MIN node ancestors. Final backed-up the beta value of any of its MIN node ancestors. Final backed-up value of this MAX node can be set to its alpha value.value of this MAX node can be set to its alpha value.

Rules for computing alpha and beta values:Rules for computing alpha and beta values:– The alpha value of a MAX node is set equal to the current largest The alpha value of a MAX node is set equal to the current largest

final backed-up value of its successors.final backed-up value of its successors.– The beta value of a MIN node is set equal to the current smallest The beta value of a MIN node is set equal to the current smallest

final backed-up value of its successors.final backed-up value of its successors.

102 8 4 5 8 3 2 7 6 2 5 1 0

Mini-Max

102 8 4 5 8 3 2 7 6 2 5 1 0

10 8 8 3 7 5 1

138

8 Max’s move

Here are the backed-up values

Min’s move

Max takes this move

Mini-Max

102 8 4 5 8 3 2 7 6 2 5 1 0

Mini-Max with Alpha-Beta Cutoffs

Mini-Max with Alpha-Beta Cutoffs

Node α-β valuesA (max) α = -, β = B (min) α = -, β = E (max) α = -, β = LE (max) α = 2, β = ME (max) α = 10, β = B (min) α = -, β = 10F (max) α = -, β = 10NF (max) α = 8, β = 10OF (max) α = 8, β = 10B (min) α = -, β = 8A (max) α = 8, β = C (min) α = 8, β = G (max) α = 8, β = PG (max) α = 8, β = QG (max) α = 8, β = C (min) α = 8, β = 8Search terminated below C because β @ C <= α @ A A (max) α = 8, β = D (min) α = 8, β = J (max) α = 8, β = VJ (max) α = 8, β = WJ (max) α = 8, β = D (min) α = 8, β = 5Search terminated below D because β @ D <= α @ A

Processing sequence for α-β assignment and visiting leaf nodes:

32 8 5 7 6 0 1 5 2 8 4 10 2

E

Max

MinB

A

F MaxG H

C

J

D

KI

Another Example of Mini-Max with Alpha-Beta Cutoffs

32 8 5 7 6 0 1 5 2 8 4 10 2

Eα=3

Max’s move

Here are the final alpha-beta (=backed-up) values(X denotes branches that do not need to be searched)

Min’s move

Max takes this move

Bβ=3

Aα=8

Fα=8

Max’s move

X

Gα=7

X

Hα=1

Cβ=1

X

Jα=8

Dβ=8

Kα=10

X

I

Another Example of Mini-Max with Alpha-Beta Cutoffs

There is no need to search below a MAX node whose value is greater than the parent node’s value (because the MIN node above wouldn’t accept anything > than its value), nor to search below a MIN node (e.g., node C) whose value is less than the parent’s value (because the MAX node above wouldn’t accept anything < than its value).

Processing sequence:

Node α or β valueE α=3B β=3F α=8No search of F’s right subtree because α @ F > β @ BG α=7No search of G’s right subtree because α @ G > β @ BA α=3H α=1C β=1No search of C’s right subtree because β @ C < α @ AJ α=8D β=8K α=10No search of K’s right subtree because α @ K > β @ DA α=8

note: α or β values are also the backed-up values...

32 8 5 7 6 0 1 5 2 8 4 10 2

Eα=3

Max’s move

Here are the final alpha-beta (=backed-up) values(X denotes branches that do not need to be searched)

Min’s move

Max takes this move

Bβ=3

Aα=8

Fα=8

Max’s move

X

Gα=7

X

Hα=1

Cβ=1

X

Jα=8

Dβ=8

Kα=10

X

I

Processing Sequence for Previous Search

Documents

HEURISTIC SEARCH. Luger: Artificial Intelligence, 5 th edition. © Pearson Education Limited, 2005 Portion of the state space for tic-tac-toe