1. Problem Spaces & Search
Dan Weld
CSE 573
A handful of GENERAL SEARCH TECHNIQUES lie at the heart of practically all work in AI
We will encounter the SAME PRINCIPLES again and again in this course, whether we are talking about
GAMES
LOGICAL REASONING
MACHINE LEARNING
These are principles for SEARCHING THROUGH A SPACE OF POSSIBLE SOLUTIONS to a problems.

2. Weak Methods “In the knowledge lies the power…” [Feigenbaum]
But what if you have very little knowledge????
© Daniel S. Weld

3. Generate & Test As weak as it gets… Works on semi-decidable problems!
© Daniel S. Weld

4. + Mass Spectrometry C12O4N2H20 Relative abundance m/z value
© Daniel S. Weld

5. Improving G&T Use knowledge of problem domain
Add ‘smarts’ to the generator
© Daniel S. Weld

6. Problem Space / State Space
Search thru a
Problem Space / State Space
Input:
Set of states
Operators [and costs]
Start state
Goal state [test]
Output:
Path: start  a state satisfying goal test
[May require shortest path]
[Sometimes just need state passing test]
State Space Search is basically a GRAPH SEARCH PROBLEM
STATES are whatever you like!
OPERATORS are a compact DESCRIPTION of possible STATE TRANSITIONS – the EDGES of the GRAPH
May have different COSTS or all be UNIT cost
OUTPUT may be specified either as ANY PATH (REACHABILITY), or a SHORTEST PATH
© Daniel S. Weld

7. Example: Route Planning
Input:
Set of states
Operators [and costs]
Start state
Goal state (test)
Output:
States = cities
Start state = starting city
Goal state test = is state the destination city?
Operators = move to an adjacent city; cost = distance
Output: a shortest path from start state to goal state
SUPPOSE WE REVERSE START AND GOAL STATES?
© Daniel S. Weld

8. Example: N Queens Input: Set of states Operators [and costs]
Start state
Goal state (test)
Output
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9. Ex: Blocks World Input: Set of states Operators [and costs]
Start state
Goal state (test)
Partially specified plans
Plan modification operators
The null plan (no actions)
State = configuration of blocks: ON(A,B), ON(B, TABLE), ON(C, TABLE)
Start state = starting configuration
Goal state test = does state satisfy some test, e.g. “ON(A,B)”?
Operators: MOVE x FROM y TO z
or PICKUP(x,y) and PUTDOWN(x,y)
Output: Sequence of actions that transform start state into goal.
HOW HARD if don’t care about number of actions used? POLYNOMIAL
HOW HARD to find smallest number of actions? NP-COMPLETE
Suppose we want to search BACKWARDS from GOAL?
Simple if goal COMPLETELY specified
NEED to have a DIFFERENT notion of STATE if goals are PARTIALLY SPECIFIED
A plan which provably achieves
The desired world configuration
© Daniel S. Weld

10. Multiple Problem Spaces
Real World
States of the world (e.g. block configurations)
Actions (take one world-state to another)
Robot’s Head
Problem Space 1
PS states =
models of world states
Operators =
models of actions
Problem Space 2
PS states =
partially spec. plan
Operators =
plan modificat’n ops
State = configuration of blocks: ON(A,B), ON(B, TABLE), ON(C, TABLE)
Start state = starting configuration
Goal state test = does state satisfy some test, e.g. “ON(A,B)”?
Operators: MOVE x FROM y TO z
or PICKUP(x,y) and PUTDOWN(x,y)
Output: Sequence of actions that transform start state into goal.
HOW HARD if don’t care about number of actions used? POLYNOMIAL
HOW HARD to find smallest number of actions? NP-COMPLETE
Suppose we want to search BACKWARDS from GOAL?
Simple if goal COMPLETELY specified
NEED to have a DIFFERENT notion of STATE if goals are PARTIALLY SPECIFIED
© Daniel S. Weld

11. Classifying Search GUESSING (“Tree Search”) SIMPLIFYING (“Inference”)
Guess how to extend a partial solution to a problem.
Generates a tree of (partial) solutions.
The leafs of the tree are either “failures” or represent complete solutions
SIMPLIFYING (“Inference”)
Infer new, stronger constraints by combining one or more constraints (without any “guessing”)
Example: X+2Y = 3
X+Y =1
therefore Y = 2
WANDERING (“Markov chain”)
Perform a (biased) random walk through the space of (partial or total) solutions
HERE ARE THE THREE BASIC PRINCIPLES
Guessing – guess how to EXTEND A PARTIAL SOLUTION one step at a time
Simplifying – make inferences about how the PROBLEM CONSTRAINTS interact
X + 2Y = 3
X + Y = 1
Y = 2
3. Wandering – RANDOMLY step through the space of POSSIBILITIES
© Daniel S. Weld

12. Search Strategies Blind Search Informed Search Constraint Satisfaction
Adversary Search
Depth first search
Breadth first search
Iterative deepening search
Iterative broadening search
© Daniel S. Weld

13. Depth First Search Maintain stack of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
Not for infinite spaces a b
O(b^d) e g h c d f
O(bd)
© Daniel S. Weld
d=max depth of tree

14. Breadth First Search Maintain queue of nodes to visit Evaluation
Complete?
Time Complexity?
Space Complexity?
Yes (if branching factor is finite) a
O(bd+1) b e g h c d f
O(bd+1)
© Daniel S. Weld

15. Memory a Limitation? Suppose: 2 GHz CPU 4 GB main memory
100 instructions / expansion
5 bytes / node
200,000 expansions / sec
Memory filled in 400 sec … < 7 min
© Daniel S. Weld

16. Iterative Deepening Search
DFS with limit; incrementally grow limit
Evaluation
Complete?
Time Complexity?
Space Complexity? a
Yes b e
O(b^d) g h c d f
O(d)
© Daniel S. Weld

17. Cost of Iterative Deepeningb
ratio ID to DFS 2 3 5
1.5 10
1.2 25
1.08
100
1.02
© Daniel S. Weld

18. Speed BFS Nodes Time Iter. Deep. Nodes Time 8 Puzzle 2x2x2 Rubik’s
Assuming 10M nodes/sec & sufficient memory
BFS
Nodes Time
Iter. Deep.
Nodes Time
8 Puzzle
2x2x2 Rubik’s
15 Puzzle
3x3x3 Rubik’s
24 Puzzle
sec
sec
days
k yrs
B yrs
sec
sec
k yrs
k yrs
yrs
1Mx 8x
Why the difference? Rubik has higher branch factor puzzle has greater depth
# of duplicates
© Daniel S. Weld
Adapted from Richard Korf presentation

19. When to Use Iterative Deepening
N Queens? Q Q Q Q
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20. Search Space with Uniform Structure
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21. Search Space with Clustered Structure
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22. Iterative Broadening Search
What if know solutions lay at depth N?
No sense in doing iterative deepening
Increment sum of sibling indicies a d b e h f c g i
© Daniel S. Weld

23. Forwards vs. Backwards
start
end
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24. vs. Bidirectional
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25. Problem All these methods are slow (blind) Solution
 add guidance (“heurstic estimate”)
 “informed search”
… coming next …
© Daniel S. Weld

26. Problem Space / State Space
Recap: Search thru a
Problem Space / State Space
Input:
Set of states
Operators [and costs]
Start state
Goal state [test]
Output:
Path: start  a state satisfying goal test
[May require shortest path]
State Space Search is basically a GRAPH SEARCH PROBLEM
STATES are whatever you like!
OPERATORS are a compact DESCRIPTION of possible STATE TRANSITIONS – the EDGES of the GRAPH
May have different COSTS or all be UNIT cost
OUTPUT may be specified either as ANY PATH (REACHABILITY), or a SHORTEST PATH
© Daniel S. Weld

27.  Symbolic Integration E.g. x2ex dx = ex(x2-2x+2) + C States?
Formulae
Operators?
Integration by parts
Integration by substitution …
© Daniel S. Weld

28. Concept Learning Labeled Training Examples
<p1,blond,32,mc,ok>
<p2,red,47,visa,ok>
<p3,blond,23,cash,ter>
<p4,…
Output: f: <pn…>  {ok, ter}
Input:
Set of states
Operators [and costs]
Start state
Goal state (test)
Output:
STATE: partial assignment of letters to numbers
GOAL state: TOTAL assignment that is a correct sum
OPERATOR: assign a number to a letter
DON’T CARE about PATH LENGTH. In fact, ALL solution paths exactly 8 LETTERS long!
© Daniel S. Weld

29. Search Strategies Blind Search Informed Search Constraint Satisfaction
Adversary Search
Best-first search
A* search
Iterative deepening A* search
Local search
© Daniel S. Weld

30. Best-first Search Generalization of breadth first search
Priority queue of nodes to be explored
Cost function f(n) applied to each node
Add initial state to priority queue
While queue not empty
Node = head(queue)
If goal?(node) then return node
Add children of node to queue
© Daniel S. Weld

31. Old Friends Breadth first = best first With f(n) = depth(n)
Dijkstra’s Algorithm = best first
With f(n) = g(n), i.e. the sum of edge costs from start to n
Space bound (stores all generated nodes)
© Daniel S. Weld

32. { { A* Search Hart, Nilsson & Rafael 1968
Best first search with f(n) = g(n) + h(n)
Where g(n) = sum of costs from start to n
h(n) = estimate of lowest cost path n  goal
h(goal) = 0
If h(n) is admissible (and monotonic) then A* is optimal {
Underestimates cost
of any solution which
can reached from node {
f values increase from
node to descendants
(triangle inequality)
© Daniel S. Weld

33. Optimality of A*
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34. Optimality Continued
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35. A* Summary
Pros
Cons
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36. Iterative-Deepening A*
Like iterative-deepening depth-first, but...
Depth bound modified to be an f-limit
Start with f-limit = h(start)
Prune any node if f(node) > f-limit
Next f-limit = min-cost of any node pruned
FL=21 e
FL=15 d a f b c
© Daniel S. Weld

37. IDA* Analysis Complete & Optimal (ala A*)
Space usage  depth of solution
Each iteration is DFS - no priority queue!
# nodes expanded relative to A*
Depends on # unique values of heuristic function
In 8 puzzle: few values  close to # A* expands
In traveling salesman: each f value is unique
 1+2+…+n = O(n2) where n=nodes A* expands
if n is too big for main memory, n2 is too long to wait!
Generates duplicate nodes in cyclic graphs
© Daniel S. Weld