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### Heuristic Search

CMSC 100Tuesday, November 4, 2008Prof. Marie desJardins

Summary of Topics

- What is heuristic search?
- Examples of search problems
- Search methods
- Uninformed search
- Informed search
- Local search
- Game trees

Building Goal-Based Intelligent Agents

To build a goal-based agent we need to answer the following questions:

- What is the goal to be achieved?
- What are the actions?
- What relevant information is necessary to encode in order to describe the state of the world, describe the available transitions, and solve the problem?

Initial

state

Goal

state

Actions

Representing States

- What information is necessary to encode about the world to sufficiently describe all relevant aspects to solving the goal?
- That is, what knowledge needs to be represented in a state description to adequately describe the current state or situation of the world?
- The size of a problem is usually described in terms of the number of states that are possible.
- Tic-Tac-Toe has about 39 states.
- Checkers has about 1040 states.
- Rubik’s Cube has about 1019 states.
- Chess has about 10120 states in a typical game.

Real-world Search Problems

- Route finding
- Touring (traveling salesman)
- Logistics
- VLSI layout
- Robot navigation
- Learning

8-Puzzle

Given an initial configuration of 8 numbered tiles on a 3 x 3 board, move the tiles into a desired goal configuration of the tiles.

8-Puzzle Encoding

- State: 3 x 3 array configuration of the tiles on the board.
- 4 Operators: Move Blank Square Left, Right, Up or Down.
- This is a more efficient encoding of the operators than one in which each of four possible moves for each of the 8 distinct tiles is used.
- Initial State: A particular configuration of the board.
- Goal: A particular configuration of the board.
- What does the state space look like?

Missionaries and Cannibals

There are 3 missionaries, 3 cannibals, and 1 boat that can carry up to two people on one side of a river.

- Goal: Move all the missionaries and cannibals across the river.
- Constraint: Missionaries can never be outnumbered by cannibals on either side of river; otherwise, the missionaries are killed.
- State: Configuration of missionaries and cannibals and boat on each side of river.
- Operators: Move boat containing some set of occupants across the river (in either direction) to the other side.
- What’s the solution??

Missionaries and Cannibals Solution

Near sideFar side

0 Initial setup: MMMCCC B -

1 Two cannibals cross over: MMMC B CC

2 One comes back: MMMCC B C

3 Two cannibals go over again: MMM B CCC

4 One comes back: MMMC B CC

5 Two missionaries cross: MC B MMCC

6 A missionary & cannibal return: MMCC B MC

7 Two missionaries cross again: CC B MMMC

8 A cannibal returns: CCC B MMM

9 Two cannibals cross: C B MMMCC

10 One returns: CC B MMMC

11 And brings over the third: - B MMMCCC

Water Jug Problem

Given a full 5-gallon jug and an empty 2-gallon jug, the goal is to fill the 2-gallon jug with exactly one gallon of water.

- Possible actions:
- Empty the 5-gallon jug (pour contents down the drain)
- Empty the 2-gallon jug
- Pour the contents of the 2-gallon jug into the 5-gallon jug (only if there is enough room)
- Fill the 2-gallon jug from the 5-gallon jug
- Case 1: at least 2 gallons in the 5-gallon jug
- Case 2: less than 2 gallons in the 5-gallon jug
- What are the states?
- What are the state transitions?
- What does the state space look like?

Water Jug Problem

Given a full 5-gallon jug and an empty 2-gallon jug, the goal is to fill the 2-gallon jug with exactly one gallon of water.

- State = (x,y), where x is the # of gallons of water in the 5-gallon jug and y is the # of gallons in the 2-gallon jug
- Initial State = (5,0)
- Goal State = (*,1), where * means any amount

Operator table

Water Jug State Space

5, 0

5, 2

5, 1

4, 0

4, 2

4, 1

3, 0

3, 2

3, 1

2, 0

2, 2

2, 1

1, 0

1, 2

1, 1

0, 0

0, 2

0, 1

Empty5

Empty2

2to5

5to2

5to2part

The 8-Queens Problem

- Place eight queens on a chessboard such that no queen attacks any other
- What are the states and operators?
- What does the state space look like?

Solution Cost

- A solution is a sequence of operators that is associated with a path in a state space from a start node to a goal node.
- The cost of a solution is the sum of the arc costs on the solution path.
- If all arcs have the same (unit) cost, then the solution cost is just the length of the solution (number of steps / state transitions)

Evaluating search strategies

- Completeness
- Guarantees finding a solution whenever one exists
- Time complexity
- How long (worst or average case) does it take to find a solution? Usually measured in terms of the number of nodes expanded
- Space complexity
- How much space is used by the algorithm? Usually measured in terms of the maximum size of the “nodes” list during the search
- Optimality/Admissibility
- If a solution is found, is it guaranteed to be an optimal one? That is, is it the one with minimum cost?

Types of Search Methods

- Uninformed search strategies
- Also known as “blind search,” uninformed search strategies use no information about the likely “direction” of the goal node(s)
- Variations on “generate and test” or “trial and error” approach
- Uninformed search methods: breadth-first, depth-first, uniform-cost
- Informed search strategies
- Also known as “heuristic search,” informed search strategies use information about the domain to (try to) (usually) head in the general direction of the goal node(s)
- Informed search methods: greedy search, A, A*
- Local search strategies
- Pick a starting solution (that might not be very good) and incrementally try to improve it
- Local search methods: hill-climbing, genetic algorithms
- Game trees
- Search strategies for situations where you have an opponent who gets to make some of the moves
- Try to pick moves that will let you win most of the time by “looking ahead” to see what your opponent might do

Depth-First (DFS)

- Enqueue nodes on nodes in LIFO (last-in, first-out) order. That is, nodes used as a stack data structure to order nodes.
- May not terminate without a “depth bound,” i.e., cutting off search below a fixed depth D ( “depth-limited search”)
- Not complete (with or without cycle detection, and with or without a cutoff depth)
- Exponential time, O(bd), but only linear space, O(bd)
- Can find long solutions quickly if lucky (and short solutions slowly if unlucky!)
- When search hits a dead end, can only back up one level at a time, even if the “problem” occurs because of a bad operator choice near the top of the tree.

Depth-First Search Solution

Expanded node Nodes list

{ S0 }

S0 { A3 B1 C8 }

A3 { D6 E10 G18 B1 C8 }

D6 { E10 G18 B1 C8 }

E10 { G18 B1 C8 }

G18 { B1 C8 }

Solution path found is S A G, cost 18

Number of nodes expanded (including goal node) = 5

Breadth-First

- Enqueue nodes on nodes in FIFO (first-in, first-out) order.
- Complete
- Optimal (i.e., admissible) if all operators have the same cost. Otherwise, not optimal but finds solution with shortest path length.
- Exponential time and space complexity, O(bd), where d is the depth of the solution and b is the branching factor (i.e., number of children) at each node.
- Will take a long time to find solutions with a large number of steps because it must look at all shorter length possibilities first.
- A complete search tree of depth d where each non-leaf node has b children, has a total of 1 + b + b2 + ... + bd = (b(d+1) - 1)/(b-1) nodes
- For a complete search tree of depth 12, where every node at depths 0, ..., 11 has 10 children and every node at depth 12 has 0 children, there are 1 + 10 + 100 + 1000 + ... + 1012 = (1013 - 1)/9 = O(1012) nodes in the complete search tree. If BFS expands 1000 nodes/sec and each node uses 100 bytes of storage, then BFS will take 35 years to run in the worst case, and it will use 111 terabytes of memory!

Breadth-First Search Solution

Expanded node Nodes list

{ S0 }

S0 { A3 B1 C8 }

A3 { B1 C8 D6 E10 G18 }

B1 { C8 D6 E10 G18 G21 }

C8 { D6 E10 G18 G21 G13 }

D6 { E10 G18 G21 G13 }

E10 { G18 G21 G13 }

G18 { G21 G13 }

Solution path found is S A G , cost 18

Number of nodes expanded (including goal node) = 7

Uniform Cost Search

- Enqueue nodes by path cost. That is, let priority = cost of the path from the start node to the current node n. Sort nodes by increasing value of cost (try low-cost nodes first)
- Called “Dijkstra’s Algorithm” in the algorithms literature; similar to “Branch and Bound Algorithm” from operations research
- Complete
- Optimal/Admissible
- Exponential time and space complexity, O(bd)

Uniform-Cost Search Solution

Expanded node Nodes list

{ S0 }

S0 { B1 A3 C8 }

B1 { A3 C8 G21 }

A3 { D6 C8 E10 G18 G21 }

D6 { C8 E10 G18 G21 }

C8 { E10 G13 G18 G21 }

E10 { G13 G18 G21 }

G13 { G18 G21 }

Solution path found is S C G, cost 13

Number of nodes expanded (including goal node) = 7

Comparing Performance

- Depth-First Search:
- Expanded nodes: S A D E G
- Solution found: S A G (cost 18)
- Breadth-First Search:
- Expanded nodes: S A B C D E G
- Solution found: S A G (cost 18)
- Uniform-Cost Search:
- Expanded nodes: S A D B C E G
- Solution found: S B G (cost 13)

This is the only uninformed search method that worries about costs.

Holy Grail Search

Expanded node Nodes list

{ S0 }

S0 { C8 A3 B1 }

C8 { G13 A3 B1 }

G13 { A3 B1 }

Solution path found is S C G, cost 13 (optimal)

Number of nodes expanded (including goal node) = 3 (as few as possible!)

If only we knew where we were headed…

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]

Informed Search: Use What You Know!

- Add domain-specific information to select the best path along which to continue searching
- Define a heuristic function, h(n), that estimates the “goodness” of a node n.
- Most often, h(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

Heuristic Functions

- All domain knowledge used in the search is encoded in the heuristic functionh.
- Heuristic search is an example of a “weak method” because of the limited way that domain-specific information is used to solve the problem.
- Examples:
- Missionaries and Cannibals: Number of people on starting river bank
- 8-puzzle: Number of tiles out of place
- 8-puzzle: Sum of distances each tile is from its goal position
- 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 dead end from which a goal cannot be reached

Example

n g(n) h(n) f(n) h*(n)

S 0 8 8 13

A 3 8 11 15

B 1 4 5 20

C 8 3 11 5

D 6

E 10

G 13 0 13 0

- g(n) is the (lowest observed) cost from the start node to n
- H(n) is the estimated cost from n to the goal node
- F(n) is the heuristic value (f(n) = g(n) + h(n), estimated total cost from start to goal through n)
- h*(n) is the (hypothetical) perfect heuristic
- Since h(n) h*(n) for all n, h is admissible
- Optimal path = S C G with cost 13

Greedy Search

a

h

b

i

c

e

g

g2

d

h=2

h=4

h=1

h=1

h=0

h=1

h=1

h=0

- Use as an evaluation function f(n) = h(n), sorting nodes by increasing values of f
- Selects node to expand believed to be closest (hence “greedy”) to a goal node (i.e., select node with smallest f value)
- Not complete
- Not admissible, as in the example. Assuming all arc costs are 1, then greedy search will find goal g, which has a solution cost of 5, while the optimal solution is the path to goal g2 with cost 3

Greedy Search

f(n) = h(n)

node expanded nodes list

{ S8 }

S { C3 B4 A8 }

C { G0 B4 A8 }

G { B4 A8 }

- Solution path found is S C G, 3 nodes expanded.
- See how fast the search is!! But it is not always optimal.

Algorithm A

- Use as an evaluation function

f(n) = g(n) + h(n)

- g(n) = minimal-cost path from the start state to state n.
- The g(n) term adds a “breadth-first” component to the evaluation function.
- Ranks nodes on search frontier by estimated cost of solution from start node through the given node to goal.
- Not complete if h(n) can equal infinity.
- Not admissible.

Algorithm A*

- Algorithm A with constraint that h(n) h*(n)
- h*(n) = true cost of the minimal cost path from n to a goal.
- h is admissible when h(n) h*(n) holds.
- Using an admissible heuristic guarantees that the first solution found will be an optimal one.
- A* is complete whenever the branching factor is finite, and every operator has a fixed positive cost
- A* is admissible

A* Search

f(n) = g(n) + h(n)

node exp. nodes list

{ S8 }

S { B5 A11 C11 }

B { A11 C11 G21 }

A { C11 G18 G21 D E }

C { G13 G18 G21 D E }

G { G18 G21 D E }

- Solution path found is S B G, 5 nodes expanded..
- Still pretty fast. And optimal, too.

Dealing with Hard Problems

- For large problems, A* often requires too much space.
- Two variations conserve memory: IDA* and SMA*
- IDA* -- iterative deepening A* -- uses successive iteration with growing limits on f:
- A* but don’t consider any node n where f(n) >10
- A* but don’t consider any node n where f(n) >20
- A* but don’t consider any node n where f(n) >30, ...
- SMA* -- Simplified Memory-Bounded A*
- uses a queue of restricted size to limit memory use.

Local Search

- Another approach to search involves starting with an initial guess at a solution and gradually improving it until it is a legal solution or the best that can be found.
- Also known as “incremental improvement” search
- Some examples:
- Hill climbing
- Genetic algorithms

Hill Climbing on a Surface of States

Height Defined by Evaluation Function

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 Example

2

8

3

1

7

4

5

2

8

4

3

4

7

6

5

2

3

1

8

1

6

5

7

7

1

3

6

4

7

6

3

1

8

5

8

6

5

1

4

3

8

4

7

6

5

2

2

2

start

h = 0

goal

h = -4

-2

-5

-5

h = -3

h = -1

-3

-4

h = -2

h = -3

-4

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

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.
- Remedies:
- Random restart
- Problem reformulation
- Some problem spaces are great for hill climbing and others are terrible.

Genetic Algorithms

- Start with k random states (the initial population)
- New states are generated by “mutating” a single state or “reproducing” (combining) two parent states (selected according to their fitness)
- Encoding used for the “genome” of an individual strongly affects the behavior of the search
- Genetic algorithms / genetic programming are a large and active area of research

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 the search time, but the algorithm 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 and optimal, but space complexity is high.
- The time complexity depends on the quality of the heuristic function.
- Hill-climbing algorithms keep only a single state in memory, but can get stuck on local optima.
- Genetic algorithms can search a large space by modeling biological evolution.

Why Study Games?

- Clear criteria for success
- Offer an opportunity to study problems involving {hostile, adversarial, competing} agents.
- Historical reasons
- Fun
- Interesting, hard problems which require minimal “initial structure”
- Games often define very large search spaces
- chess 35100 nodes in search tree, 1040 legal states

State of the Art

- How good are computer game players?
- Chess:
- Deep Blue beat Gary Kasparov in 1997
- Garry Kasparav vs. Deep Junior (Feb 2003): tie!
- Kasparov vs. X3D Fritz (November 2003): tie! http://www.cnn.com/2003/TECH/fun.games/11/19/kasparov.chess.ap/
- Checkers: Chinook (an AI program with a very large endgame database) is the world champion (checkers is “solved”!)
- Go: Computer players are decent, at best
- Bridge: “Expert-level” computer players exist (but no world champions yet!)
- Good places to learn more:
- http://www.cs.ualberta.ca/~games/
- http://www.cs.unimass.nl/icga

Chinook

- Chinook is the World Man-Machine Checkers Champion, developed by researchers at the University of Alberta.
- It earned this title by competing in human tournaments, winning the right to play for the (human) world championship, and eventually defeating the best players in the world.
- Visit http://www.cs.ualberta.ca/~chinook/ to play a version of Chinook over the Internet.
- The developers claim to have fully analyzed the game of checkers, and can provably always win if they play black
- “One Jump Ahead: Challenging Human Supremacy in Checkers” Jonathan Schaeffer, University of Alberta (496 pages, Springer. $34.95, 1998).

Typical Game Setting

- 2-person game
- Players alternate moves
- Zero-sum: one player’s loss is the other’s gain
- Perfect information: both players have access to complete information about the state of the game. No information is hidden from either player.
- No chance (e.g., using dice) involved
- Examples: Tic-Tac-Toe, Checkers, Chess, Go, Nim, Othello
- Not: Bridge, Solitaire, Backgammon, ...

How to Play a Game

- A way to play such a game is to:
- Consider all the legal moves you can make
- Compute the new position resulting from each move
- Evaluate each resulting position and determine which is best
- Make that move
- Wait for your opponent to move and repeat
- Key problems are:
- Representing the “board”
- Generating all legal next boards
- Evaluating a 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

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 had over 8000 features in its evaluation function

Game Trees

- Problem spaces for typical games are represented as trees
- Root node represents the current board configuration; player must decide the best single move to make next
- Static evaluator function rates a board position. f(board) = real number with f>0 “white” (me), f<0 for black (you)
- Arcs represent the possible legal moves for a player
- If it is my turn to move, then the root is labeled a "MAX" node; otherwise it is labeled a "MIN" node, indicating my opponent's turn.
- Each level of the tree has nodes that are all MAX or all MIN; nodes at level i are of the opposite kind from those at level i+1

Minimax Procedure

- Create start node as a MAX node with current board configuration
- Expand nodes down to some depth (a.k.a. ply) of lookahead in the game
- Apply the evaluation function at each of the leaf nodes
- “Back up” values for each of the non-leaf nodes until a value is computed for the root node
- At MIN nodes, the backed-up value is the minimum of the values associated with its children.
- At MAX nodes, the backed-up value is the maximum of the values associated with its children.
- Pick the operator associated with the child node whose backed-up value determined the value at the root

Minimax Algorithm

2

1

2

2

1

2

7

1

8

2

2

2

7

7

1

1

8

8

2

1

MAX

MIN

2

7

1

8

This is the move

selected by minimax

Static evaluator

value

Partial Game Tree for Tic-Tac-Toe

- f(n) = +1 if the position is a win for X.
- f(n) = -1 if the position is a win for O.
- f(n) = 0 if the position is a draw.

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