CS 8520: Artificial Intelligence Search 2 Paula Matuszek Fall, 2008 Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are in turn based

CS 8520: Artificial Intelligence

Search 2

Paula Matuszek

Fall, 2008

Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are in turn based on Russell, aima.eecs.berkeley.edu/slides-pdf.

Paula Matuszek, CSC 8520, Fall 2008. Based in part on aima.eecs.berkeley.edu/slides-ppt and www.cs.umbc.edu/671/fall03/slides/c5-6_inf_search.ppt 2

Search strategies• A search strategy is defined by picking the order of

node expansion. (e.g., breadth-first, depth-first) • Strategies are evaluated along the following

dimensions:– completeness: does it always find a solution if one exists?– time complexity: number of nodes generated– space complexity: maximum number of nodes in memory– optimality: does it always find a least-cost solution?

• Time and space complexity are measured in terms of – b: maximum branching factor of the search tree– d: depth of the least-cost solution– m: maximum depth of the state space (may be infinite)


Uninformed search strategies• Uninformed search strategies use only the

information available in the problem definition

• Breadth-first search

• Uniform-cost search

• Depth-first search

• Depth-limited search

• Iterative deepening search


Implementation: general tree search


Breadth-first search• Expand shallowest unexpanded node

• Implementation:– fringe is a FIFO queue, i.e., new successors go

at end



• Implementation:– fringe is a FIFO queue, i.e., new successors

go at end




at end




at end


Properties of breadth-first search• Complete? Yes (if b is finite)• Time? 1+b+b2+b3+… +bd + b(bd-1) = O(bd+1)• Space? O(bd+1) (keeps every node in memory)• Optimal? Yes (if cost = 1 per step)

• Space is the bigger problem (more than time)


Uniform-cost search• Expand least-cost unexpanded node• Implementation:

– fringe = queue ordered by path cost

• Equivalent to breadth-first if step costs all equal• Complete? Yes, if step cost >= epsilon (otherwise can

loop)• Time and space? O(bceiling(C*/ epsilon)) where C* is the cost of

the optimal solution and epsilon is the smallest step cost– Can be much worse than breadth-first if many small steps not on

optimal path

• Optimal? Yes – nodes expanded in increasing order of g(n)


Uniform Cost Search


Depth-first search• Expand deepest unexpanded node• Implementation:

– fringe = LIFO queue, i.e., put successors at front



































Properties of depth-first search

• Complete? No: fails in infinite-depth spaces, spaces with loops– Modify to avoid repeated states along path

complete in finite spaces

• Time? O(bm): terrible if m is much larger than d– but if solutions are dense, may be much faster than

breadth-first

• Space? O(bm), i.e., linear space!• Optimal? No


Depth-limited search• = depth-first search with depth limit l

• Nodes at depth l have no successors

• Solves problem of infinite depth

• Incomplete

• Recursive implementation:


Iterative deepening search•Repeated Depth-Limited search, incrementing limit l until a solution is found or failure.

•Repeats earlier steps at each new level, so inefficient -- but never more than doubles cost

•No longer incomplete


Iterative deepening search l =0








Properties of iterative deepening

• Complete? Yes

• Time? (d+1)b0 + d b1 + (d-1)b2 + … + bd = O(bd)

• Space? O(bd)

• Optimal? Yes, if step cost = 1


Summary of Algorithms for Ininformed Search

Criterion Breadth-first

Uniform Cost

Depth First

Depth-Limited

Iterative Deepening

Complete?

Yes Yes No No Yes

Time? O(bd+1) O(b(ceilingC*

/epsilon)

O(bm) O(bl) O(bd)

Space? O(bd+1) O(b(ceilingC*

/epsilon)

O(bm) O(bl) O(bd)

Optimal? Yes Yes No No Yes


A Caution: Repeated States• Failure to detect repeated states can turn a linear

problem into an exponential one, or even an infinite one.– For example: 8-puzzle– Simple repeat -- empty square simply moves back and

forth– More complex repeats also possible.

• Save list of expanded states -- the closed list.• Add new state to fringe only if it's not in closed list.


Graph search


Summary: Uninformed Search• Problem formulation usually requires abstracting away

real-world details to define a state space that can feasibly be explored

• Variety of uninformed search strategies

• Iterative deepening search uses only linear space and not much more time than other uninformed algorithms: usual choice

Informed search algorithms

Slides derived in part from

www.cs.berkeley.edu/~russell/slides/chapter04a.pdf, converted to powerpoint by Min-Yen Kan, National University of Singapore,

and from www.cs.umbc.edu/671/fall03/slides/c5-6_inf_search.ppt, Marie DesJardins, University of Maryland Baltimore County.


Review: Tree search

• A search strategy is defined by picking the order of node expansion


Heuristic Search• Uninformed search is generic; choice of node to

expand is dependent on shape of tree and strategy for node expansion.

• Sometimes domain knowledge can help us make a better decision.

• For the Romania problem, eyeballing it results in looking at certain cities first because they "look closer" to where we are going.

• If that domain knowledge can be captured in a heuristic, search performance can be improved by using that heuristic.

• This gives us an informed search strategy.


So What's A Heuristic?Webster's Revised Unabridged Dictionary (1913) (web1913)

Heuristic \Heu*ris"tic\, a. [Gr. ? to discover.] Serving to discover or find out.

The Free On-line Dictionary of Computing (15Feb98) heuristic 1. <programming> A rule of thumb, simplification or

educated guess that reduces or limits the search for solutions in domains that are difficult and poorly understood. Unlike algorithms, heuristics do not guarantee feasible solutions and are often used with no theoretical guarantee. 2. <algorithm> approximation algorithm.

From WordNet (r) 1.6 heuristic adj 1: (computer science) relating to or using a heuristic

rule 2: of or relating to a general formulation that serves to guide investigation [ant: algorithmic] n : a commonsense rule (or set of rules) intended to increase the probability of solving some problem [syn: heuristic rule, heuristic program]


Heuristics• For search it has a very specific meaning:

– All domain knowledge used in the search is encoded in the heuristic function h.

• Examples:– Missionaries and Cannibals: Number of people on starting river bank– 8-puzzle: Number of tiles out of place – 8-puzzle: Sum of distances from goal – Romania: straight-line distance from city to Bucharest

• In general:– h(n) >= 0 for all nodes n – h(n) = 0 implies that n is a goal node – h(n) = infinity implies that n is a deadend from which a goal cannot be

reached

• h is some estimate of how desirable a move is, or how close it gets us to our goal


Best-first search• Order nodes on the nodes list by

increasing value of an evaluation function, f(n), that incorporates domain-specific information in some way.

• This is a generic way of referring to the class of informed methods.


Best-first search• Idea: use an evaluation function f(n) for each node

– estimate of "desirability"Expand most desirable unexpanded node

• Implementation:Order the nodes in fringe in decreasing order of desirability

• Special cases:– greedy best-first search– A* search


Romania with step costs in km


Greedy best-first search• Evaluation function f(n) = h(n) (heuristic)

• = estimate of cost from n to goal

• e.g., hSLD(n) = straight-line distance from n to Bucharest

• Greedy best-first search expands the node that appears to be closest to goal


Greedy best-first search example








Properties of greedy best-first search

• Complete? No – can get stuck in loops, e.g., Iasi Neamt Iasi Neamt

• Time? O(bm), but a good heuristic can give dramatic improvement

• Space? O(bm) -- keeps all nodes in memory

• Optimal? No• Remember: Time and space complexity are measured in

terms of – b: maximum branching factor of the search tree– d: depth of the least-cost solution– m: maximum depth of the state space (may be infinite)


A* search• Idea: avoid expanding paths that are already

expensive

• Evaluation function f(n) = g(n) + h(n)

• g(n) = cost so far to reach n

• h(n) = estimated cost from n to goal

• f(n) = estimated total cost of path through n to goal


A* search example


A* search example


A* search example


A* search example


A* search example


A* search example


Admissible heuristics• A heuristic h(n) is admissible if for every node n,

h(n) <= h*(n), where h*(n) is the true cost to reach the goal state from n.

• An admissible heuristic never overestimates the cost to reach the goal, i.e., it is optimistic.

• This means that we won't ignore a better path because we think the cost is too high. (If we underestimate it we wil learn that when we explore it.)

• Example: hSLD(n) (never overestimates the actual road distance)


Admissible heuristicsE.g., for the 8-puzzle:• h1(n) = number of misplaced tiles• h2(n) = total Manhattan distance

(i.e., no. of squares from desired location of each tile)

• h1(S) = ? • h2(S) = ?


Admissible heuristicsE.g., for the 8-puzzle:• h1(n) = number of misplaced tiles• h2(n) = total Manhattan distance(i.e., no. of squares from desired location of each tile)

• h1(S) = ? 8• h2(S) = ? 3+1+2+2+2+3+3+2 = 18


Properties of A*• If h(n) is admissible

• Complete? Yes (unless there are infinitely many nodes with f ≤ f(G) )

• Time? Exponential in [relative error in h * length of solution]

• Space? Keeps all nodes in memory

• Optimal? Yes; cannot expand f i+1 until f i is finished.


Some observations on A*• Perfect heuristic: If h(n) = h*(n) for all n, then only nodes on

the optimal solution path will be expanded. So, no extra work will be performed.

• Null heuristic: If h(n) = 0 for all n, then this is an admissible heuristic and A* acts like Uniform-Cost Search.

• Better heuristic: If h1(n) < h2(n) <= h*(n) for all non-goal nodes, then h2 is a better heuristic than h1 – If A1* uses h1, and A2* uses h2, then every node expanded by A2* is

also expanded by A1*. – In other words, A1 expands at least as many nodes as A2*. – We say that A2* is better informed than A1*, or A2* dominates A1*

• The closer h is to h*, the fewer extra nodes that will be expanded


What’s a good heuristic? How do we find one?

• If h1(n) < h2(n) <= h*(n) for all n, then both are admissible and h2 is better than (dominates) h1.

• Relaxing the problem: remove constraints to create a (much) easier problem; use the solution cost for this problem as the heuristic function

• Combining heuristics: take the max of several admissible heuristics: still have an admissible heuristic, and it’s better!

• Identify good features, then use a learning algorithm to find a heuristic function: may lose admissibility


Relaxed problems• A problem with fewer restrictions on the actions is

called a relaxed problem• The cost of an optimal solution to a relaxed

problem is an admissible heuristic for the original problem

• If the rules of the 8-puzzle are relaxed so that a tile can move anywhere, then h1(n) gives the shortest solution

• If the rules are relaxed so that a tile can move to any adjacent square, then h2(n) gives the shortest solution


Some Examples of Heuristics?• 8-puzzle?

• Mapquest driving directions?

• Minesweeper?

• Crossword puzzle?

• Making a medical diagnosis?

• ??


Local search algorithms• In many optimization problems, the path to the

goal is irrelevant; the goal state itself is the solution

• State space = set of "complete" configurations• Find configuration satisfying constraints, e.g., n-

queens

• In such cases, we can use local search algorithms• Keep a single "current" state, try to improve it


Example: n-queens• Put n queens on an n x n board with no two

queens on the same row, column, or diagonal


Hill-climbing search• If 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 search• "Like climbing Everest in thick fog with

amnesia"


Hill climbing example

2 8 3

1 4

7 6 5

2 3

1 8 4

7 6 5

1 3

8 4

7 6 5

2

3

1 8 4

7 6 5

2

1 3

8 4

7 6 5

2

start goal

-5

h = -3

h = -3

h = -2

h = -1

h = 0h = -4

-5

-4

-4-3

-2

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

2 8 3

1 6 4

7 5

1 3

8 4

7 6 5

2


Drawbacks of hill climbing• Problems:

– Local Maxima: peaks that aren’t the highest point in the space

– Plateaus: 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: – Random restart.

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


Hill-climbing search• Problem: depending on initial state, can get

stuck in local maxima


Example of a local maximum

1 2 5

7 4

8 6 3

1 2 5

7 4

8 6 3

1 2 5

7 4

8 6 3

1 2 5

7 4

8 6 3

1 2 5

7 4

8 6 3

-3

-4

-4

-4

0

start goal


Simulated annealing• Simulated 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 search• Idea: escape local maxima by allowing some "bad" moves but

gradually decrease their frequency


Properties of simulated annealing search• One can prove: If T decreases slowly enough, then

simulated annealing search will find a global optimum with probability approaching 1

• Widely used in VLSI layout, airline scheduling, etc


Local beam search• Keep track of k states rather than just one

• Start with k randomly generated states

• At each iteration, all the successors of all k states are generated

• If any one is a goal state, stop; else select the k best successors from the complete list and repeat.


Summary: Informed search• Best-first search is general search where the minimum-cost nodes

(according to some measure) are expanded first. • Greedy search uses minimal estimated cost h(n) to the goal state as

measure. This reduces search time, but is neither complete nor optimal. • A* search combines uniform-cost search and greedy search: f(n) =

g(n) + h(n). A* handles state repetitions and h(n) never overestimates. – A* is complete, optimal and optimally efficient, but its space

complexity is still bad. – The time complexity depends on the quality of the heuristic

function.

• Local Search techniques are useful when you don't care about path, only result. Examples include– Hill-climbing – Simulated annealing– Local Beam Search


Search in an Adversarial Environment

• Iterative deepening and A* useful for single-agent search problems

• What if there are TWO agents?

• Goals in conflict:– Adversarial Search

• Especially common in AI:– Goals in direct conflict– IE: GAMES.


Games vs. search problems

• "Unpredictable" opponent specifying a move for every possible opponent reply

• Time limits unlikely to find goal, must approximate

• Efficiency matters a lot

• HARD.

• In AI, typically "zero sum": one player wins exactly as much as other player loses.


Types of games Deterministic ChancePerfect Info Chess Monopoly

Checkers Backgammon Othello Tic-Tac-Toe

Imperfect Info Bridge Poker Scrabble


Tic-Tac-Toe• Tic Tac Toe is one of the classic AI examples.

Let's play some.

• A Tic Tac Toe game:– http://www.ourvirtualmall.com/tictac.htm

• Try it, at various levels of difficulty.– What kind of strategy are you using?– What kind does the computer seem to be using?– Did you win? Lose?

http://www.ourvirtualmall.com/tictac.htm


Problem Definition• Formally define a two-person game as:• Two players, called MAX and MIN.

– Alternate moves

– At end of game winner is rewarded and loser penalized.

• Game has– Initial State: board position and player to go first

– Successor Function: returns (move, state) pairs• All legal moves from the current state

• Resulting state

– Terminal Test

– Utility function for terminal states.

• Initial state plus legal moves define game tree.


Tic Tac Toe Game tree


Optimal Strategies• Optimal strategy is sequence of moves

leading to desired goal state.

• MAX's strategy is affected by MIN's play.

• So MAX needs a strategy which is the best possible payoff, assuming optimal play on MIN's part.

• Determined by looking at MINIMAX value for each node in game tree.


Minimax• Perfect play for deterministic games

• Idea: choose move to position with highest minimax value = best achievable payoff against best play

• E.g., 2-ply game:


Minimax algorithm


Properties of minimax• Complete? Yes (if tree is finite)

• Optimal? Yes (against an optimal opponent)

• Time complexity? O(bm)

• Space complexity? O(bm) (depth-first exploration)

• For chess, b ≈ 35, m ≈100 for "reasonable" games exact solution completely infeasible

• Even tic-tac-toe is much too complex to diagram here, although it's small enough to implement.


Pruning the Search• “If you have an idea that is surely bad, don't

take the time to see how truly awful it is.” -- Pat Winston

• Minimax exponential with # of moves; not feasible in real-life

• But we can PRUNE some branches.

• Alpha-Beta pruning– If it is clear that a branch can't improve on the value

we already have, stop analysis.


α-β pruning example










Properties of α-β• Pruning does not affect final result

• Good move ordering improves effectiveness of pruning

• With "perfect ordering," time complexity = O(bm/2) doubles depth of search which can be carried out for a given level

of resources.

• A simple example of the value of reasoning about which computations are relevant (a form of metareasoning)


Why is it called α-β?• α is the value of the

best (i.e., highest-value) choice found so far at any choice point along the path for max

• If v is worse than α, max will avoid it prune that branch

• Define β similarly for min


The α-β algorithm


The α-β algorithm


"Informed" Search• Alpha-Beta still not feasible for large game

spaces.

• Can we improve on performance with domain knowledge?

• Yes -- if we have a useful heuristic for evaluating game states.

• Conceptually analogous to A* for single-agent search.


Resource limitsSuppose we have 100 secs, explore 104 nodes/sec

106 nodes per move

Standard approach:• cutoff test:

e.g., depth limit (perhaps add quiescence search)

• evaluation function = estimated desirability of position


Evaluation function• Evaluation function or static evaluator is used to

evaluate the “goodness” of a game position.– Contrast with heuristic search where the evaluation function was a

non-negative estimate of the cost from the start node to a goal and passing through the given node

• The zero-sum assumption allows us to use a single evaluation function to describe the goodness of a board with respect to both players. – f(n) >> 0: position n good for me and bad for you

– f(n) << 0: position n bad for me and good for you

– f(n) near 0: position n is a neutral position

– f(n) = +infinity: win for me

– f(n) = -infinity: win for you

DesJardins: www.cs.umbc.edu/671/fall03/slides/c8-9_games.ppt


Evaluation function examples• Example of an evaluation function for Tic-Tac-Toe:

f(n) = [# of 3-lengths open for me] - [# of 3-lengths open for you]

where a 3-length is a complete row, column, or diagonal

• Alan Turing’s function for chess– f(n) = w(n)/b(n) where w(n) = sum of the point value of white’s pieces

and b(n) = sum of black’s

• Most evaluation functions are specified as a weighted sum of position features:f(n) = w1*feat1(n) + w2*feat2(n) + ... + wn*featk(n)

• Example features for chess are piece count, piece placement, squares controlled, etc.

• Deep Blue (which beat Gary Kasparov in 1997) had over 8000 features in its evaluation function

DesJardins: www.cs.umbc.edu/671/fall03/slides/c8-9_games.ppt


Cutting off searchMinimaxCutoff is identical to MinimaxValue except

1. Terminal? is replaced by Cutoff?2. Utility is replaced by Eval

Does it work in practice?For chess: bm = 106, b=35 m=4

4-ply lookahead is a hopeless chess player!– 4-ply ≈ human novice– 8-ply ≈ typical PC, human master– 12-ply ≈ Deep Blue, Kasparov


Deterministic games in practice• Checkers: Chinook ended 40-year-reign of human world champion

Marion Tinsley in 1994. Used a precomputed endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 444 billion positions.

• Chess: Deep Blue defeated human world champion Garry Kasparov in a six-game match in 1997. Deep Blue searched 200 million positions per second, used very sophisticated evaluation, and undisclosed methods for extending some lines of search up to 40 ply.

• Othello: human champions refuse to compete against computers, who are too good.

• Go: Just beginning to be good enough to play human champions. In go, b > 300, so most programs use pattern knowledge bases to suggest plausible moves, but MoGo also uses a recently developed search method, Monte Carlo Tree Search.


Summary• Games are fun to work on!

• They illustrate several important points about AI

• perfection is unattainable must approximate

• good idea to think about what to think about


Search Summary• For uninformed search, tradeoffs between time

and space complexity, with iterative deepening often the best choice.

• For non-adversarial informed search, A* usually the best choice; the better the heuristic, the better the performance.

• For adversarial search, minimax with alpha-beta pruning is optimal where feasible

• Adding an evaluation-function-based cutoff increases range of feasibility.

• The better we can capture domain knowledge in the heuristic and evaluation functions, the better we can do.

Documents

CS 8520: Artificial Intelligence Search 2 Paula Matuszek Fall, 2008 Slides based on Hwee Tou Ng, aima.eecs.berkeley.edu/slides-ppt, which are in turn based