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CS 385 Fall 2006Chapter 3

Structures and Strategies for State Space Search

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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.

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Famous example: Konigsburg bridge problem

Is there a walk that crosses each bridge exactly once?

Can we represent this in a better way?

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Graph of the Königsberg bridge system:

Predicate calculus representation: connect(i1,i2, b1) ....

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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.

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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 boardsS: 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

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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.

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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:

1 2 4 5 3 6

2 3 1 3 7 6

1 3 6

4 5 6 7 How do we pick the "best" path?

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Function backtrack algorithm (general, no examination order specified)

CS: current stateSL: states in current pathNSL states awaiting evaluationDE: dead ends

typo in book, p should be ≠

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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 successelse 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 deepeninguse 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 r u

s → r

t → r v t 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

open closed

[t s] []

[s r] [t]

[r u] [s t]

[u p] [r s t]

[p] [u r s t]

p

q r u

v t s

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Is p true?

DFS

open closed

[t s] []

[r s] [t]

[p u] [r t]

p

q r u

v 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

b

c

a b → d

a c → e

b d → f

f → g

a e → h

Graph?

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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

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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 consistsof 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.

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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: