CS 385 Fall 2006
Chapter 3
Structures and Strategies for State Space
Search
1
Where are we?
Predicate calculus:
– a way to describe objects and relationships
Inference rules:
– a way to infer new knowledge, defining a state space that is
searched to find a solution to a problem
Strategy
– generate all possible elements of the state space and see if
your answer is there
– works for tic-tac-toe
– doesn't work for chess
Goal: intelligent ways to search a state space
Tool: state space graphs.
2
Famous example: Konigsburg bridge problem
Is there a walk that crosses each bridge exactly once?
3
Can we represent this in a better way?
Graph of the Königsberg bridge system:
Predicate calculus representation:
connect(i1,i2, b1)
....
4
Euler's Proof
A node is odd or even depending on the number of arcs
leading into it
Odd degree nodes can only be at the beginning of the path
A walk must contain 0 or 2 odd nodes (why?)
Is there a path?
Does predicate calculus suffice for this argument?
No, no notion of odd/even
We need graph theory to do more with this.
5
What are N, A, S, and GD for tic-tac-toe?
N could be all possible configurations of 0s and 1s or just reachable ones.
A: allowable moves between boards
S: empty board, GD winning boards
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The 8 Puzzle
N = 3 x 3 configurations of tiles 1-8 and 1 blank
Start state:
Goal state:
Arcs: move the blank up/down/left/right
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Figure 3.6: State space of the 8-puzzle generated by “move blank” operations.
Is this like tic-tac toe?
No, you might go around in circles 8
Traveling Salesperson
N= cities
A = paths between cities with weights (mileage, time, cost)
S = home
GD = home
Path: visit each exactly once
Goal: minimize something
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Figure 3.8: Search of the traveling salesperson problem. Each arc is marked
with the total weight of all paths from the start node (A) to its
endpoint.
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Traveling Salesperson
How many potential paths?
Can we solve this for 50 states?
Of great interest to people who like algorithms because it
gets large so fast.
Exhaustive search out of the question
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Figure 3.9: The traveling salesperson problem with the nearest neighbor path in bold.
Note that this path (A, E, D, B, C, A), at a cost of 550, is not the shortest.
The comparatively high cost of arc (C, A) defeated the heuristic.
12
Data- versus Goal-Driven for Finding a
Route from Aurora to LA
Where you start. E.g. routes from Aurora to LA
Data driven (forward chaining)
From Aurora one can get to x, y, z...
From x one can get to ..
From y one can get to ...
Keep checking to see if LA ends up in one of the destinations
Goal driven (backward chaining)
One can get to LA from x, y, z...
One can get to x from ..
One can get to y from
Keep checking to see if Aurora ends up in one of the sources
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Better examples
Lineage: am I related to Reverend Thomas Carter?
Which is better, data or goal driven?
Could I prune extraneous paths?
Is this the same as "Am I related to Thomas Jefferson"?
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Medical Diagnosis
"Do I have strep throat" vs. "What disease do I have"
strep symptoms vs. lots of symptoms
take a culture
vs. run a spectrum of tests
Which is data-driven/forward chaining?
Which is goal-driven/backward chaining?
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Backtracking
A technique for systematically trying all paths through a
state space.
Begin at start and pursue a path until goal or dead end
If dead end, backtrack to most recent node with
unexamined siblings
E.g. is 6 in this tree?
1

Possible trials:
2

4
1 2 4 5 3 6

3


5
6
1 3 7 6
1 3 6

7
How do we pick the "best" path?
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Function backtrack algorithm (general, no examination order specified)
typo in book, p should be ≠
CS: current state
SL: states in current path
NSL states awaiting evaluation
DE: dead ends
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A trace of backtrack on the graph of figure 3.12
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A trace of backtrack on the graph of figure 3.12
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A trace of backtrack on the graph of figure 3.12
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A trace of backtrack on the graph of figure 3.12
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Observations
No order is specified for adding nodes to NSL
(opportunity for intelligence)
SL gives us the path to the current solution
(hence to the goal at the end)
When C is the current state, F is not added to NSL
(because it is in DE)
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How used on a maze?
ifgoal return success
else
try north
try east
try south
try west
backtrack
Track:
• where you are
• where you can go
from each state
• where you came from
• visited states
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Function breadth_first search algorithm
X is CS in backtrack, closed = DE + SL
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breadth_first_search on Figure 3.13
open
closed
[A]
[]
[B C D]
[A]
[C D E F]
[B A]
[D E F G H]
[C B A]
[E F G H I J]
[D C B A]
[F G H I J K L]
[E D C B A]
[G H I J K L M]
[F E D C B A]
[H I J K L M N]
[G F E D B C A]
[I J K L M N O P]
[F T L S K E B A]
[J K L M N O P Q][M F T L S K E B A]
[K L M N O P Q R]
[J M F T L S K E B A]
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Observations
Cleaner
But no path to start state
Solution: associate the parent with each node. E.g.
[B, C, D]
→ [(B,A), (C,A), (D,A)]
[C, D, E, F] → [(C,A), (D,A), (B,A), (F, B)]
Is the first algorithm ever better?
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depth_first_search on Figure 3.13
Put new nodes at the beginning
of the open list.
open
closed
[A]
[]
[B C D]
[A]
[E F C D]
[B A]
[K L F C D]
[E B A]
[S L F C D]
[K E B A]
[L F C D][S K E B A]
[T F C D]
[L S K E B A]
[F C D]
[T L S K E B A]
[M C D]
[F T L S K E B A]
[C D]
[M F T L S K E B A]
[G H D]
[C M F T L S K E B A]
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Comparison
BFS finds the shortest path
DFS
gets quickly into a deep search space
good if you know the solution is "far away"
wrong path: inefficient
DFS with iterative deepening
use a depth bound
retreat at the bound
no luck: increase the bound
Later: use knowledge about the problem to order nodes on
the open list
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Using State Space to Represent
Predicate Calculus
node: state of the problem
arc: inference
search: to decide if an assertion is implied by others
q→p
p
r→p

v→q

q
s→r
r

t→r
v

t
u


s
s→u
s, t
Determining truth: path from boxed nodes to proposition
Data-driven: start with boxed; goal-driven: start with goal.
DFS or BFS?
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Is p true?
BFS
p
open
closed
[t s]
[]
[s r]
[t]
[r u]
[s t]
[u p]
[r s t]
[p]
[u r s t]

q
r

v

u
 
t

s
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Is p true?
DFS
p
open
closed
[t s]
[]
[r s]
[t]
[p u]
[r t]

q
r

v

u
 
t

s
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And/Or Graphs
in the graph above t  s →r:
r
t
s
To express t  s →r, connect incoming arcs
r
t
s
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Example 3.3.2
a
Graph?
b
c
a b→d
a c→e
b d→f
f→g
a e→h
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Example 3.3.2
a
b
c
a b→d
a c→e
b d→f
f→g
a e→h
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Search for h:
Goal-directed:
h: try to prove a and e
a is true
e is true if c and a
a is true
c is true
← e is true
← h is true
Data-directed:
a, b, c are true
a and b → d
a and c → e
a and e → h
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Symbolic Integration
MACSYMA
Symbolic algebra, including integration
http://integrals.wolfram.com/index.jsp
How do you think this works?
Decomposition using and/or graphs
∫f + g decomposes to ∫f and ∫g
More complicated expressions decompose into possible
transformations
Graph is generated on the fly
Goal-directed
How is it searched? BFS? DFS?
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Figure 3.24:
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And/Or for Financial Advisor?
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Figure 3.26: And/or graph
for the financial advisor
39
Five rules for a simple subset of English grammar (rewrite rules):
Does 1 look like AND?
Do 2 and 3 this look like OR?
Construct the graph
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Figure 3.27: And/or graph for the grammar of Example 3.3.6.
How do we use this?
A sentence is well-formed if it consists
of terminal symbols and there is a series
of substitutions that reduce it to the
sentence symbol
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Figure 3.28: Parse tree for “The dog bites
the man.” Note that this is a subtree Figure 3.26.
42
Parse "the dog bites the man"
7: art ↔ the
gives
art dog bites the man
7: art ↔ the
gives
art dog bites art man
8: n ↔ man
gives
art dog bites art n*
3: np ↔ art n
gives
art dog bites np**
9: n ↔ dog
gives
art n bites np
3: np↔ art n
gives
np bites np
11: v ↔ bites
gives
np v bites
5: vp ↔ v np
gives
np vp
1: sentence ↔ np vp ...
* Why isn't dog rewritten first?
** Why didn't 11 fire instead:
v ↔ bites
gives
art n v the man
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Parse "dog the man"
7: art ↔ the
gives
dog art man
8: n ↔ man
gives
dog art man
9: n ↔ dog
gives
n art n
There are no rules that rewrite this
How does the C++ compiler work?
Can you write a grammer for C++?
program ↔ declarations body
...
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Figure 3.29:
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