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CS 4700:

Foundations of

Artificial Intelligence

Spring 2020

Prof. Haym Hirsh

Lecture 4

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Reminder: Technology Policy

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Reminder: Due Feb 4

• Special accommodations letter:

Scan documentation letter and email to

[email protected]

(4)

First Karma Lecture

The Principal-Agent Value Alignment Problem in Artificial Intelligence

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Textbook

So far:

“Uninformed” Search: Sections 3.1-3.5 Upcoming:

“Informed” Search: Sections 3.6.1, 4.1

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1 10 6 2 3 4 5 7 8 9 11 12 13

Depth-First Search

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1 4 3 2 5 6 7 8 9 11 12 13

Breadth-First Search

10

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DFS vs BFS

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DFS vs BFS:

Handwavy Analysis

Method DFS BFS

Complete? Yes Yes

Optimal? No Yes

Time O(bm) O(bd)

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DFS vs BFS:

Handwavy Analysis

Method DFS BFS

Complete? Yes Yes

Optimal? No Yes

Time O(bm) O(bd)

Space O(bm) O(bd)

Complete: If there’s a solution you will find it

Optimal: You will find the

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DFS vs BFS:

Handwavy Analysis

Method DFS BFS

Complete? Yes Yes

Optimal? No Yes

Time O(bm) O(bd)

Space O(bm) O(bd)

DFS can miss optimal solutions BFS won’t

Complete: If there’s a solution you will find it

Optimal: You will find the

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DFS vs BFS:

Handwavy Analysis

Method DFS BFS

Complete? Yes Yes

Optimal? No Yes

Time O(bm) O(bd)

(13)

DFS vs BFS:

Handwavy Analysis

Method DFS BFS

Complete? Yes Yes

Optimal? No Yes

Time O(bm) O(bd)

Space O(bm) O(bd)

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DFS vs BFS:

Handwavy Analysis

Method DFS BFS

Complete? Yes Yes

Optimal? No Yes

Time O(bm) O(bd)

Space O(bm) O(bd)

Loosely, number of nodes expanded Worst-case exponential DFS: in depth of search space

(15)

DFS vs BFS:

Handwavy Analysis

Method DFS BFS

Complete? Yes Yes

Optimal? No Yes

Time O(bm) O(bd)

Space O(bm) O(bd)

Loosely, number of nodes expanded Worst-case exponential DFS: in depth of search space

(16)

DFS vs BFS:

Handwavy Analysis

Method DFS BFS

Complete? Yes Yes

Optimal? No Yes

Time O(bm) O(bd)

(17)

DFS vs BFS:

Handwavy Analysis

Method DFS BFS

Complete? Yes Yes

Optimal? No Yes

Time O(bm) O(bd)

Space O(bm) O(bd)

(18)

DFS vs BFS:

Handwavy Analysis

Method DFS BFS

Complete? Yes Yes

Optimal? No Yes

Time O(bm) O(bd)

Space O(bm) O(bd)

Loosely, number of nodes in Open

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DFS vs BFS: more detail

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DFS vs BFS: more detail

• b: branching factor (finite) • d: depth of solution (finite) • Assume a tree of depth m

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DFS vs BFS: more detail

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DFS vs BFS: more detail

• b: branching factor (finite) • d: depth of solution (finite) • Assume a tree of depth m

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DFS vs BFS: more detail

• b: branching factor (finite) • d: depth of solution (finite) • Assume a tree of depth m

But search space is a graph not a tree!

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DFS vs BFS: more detail

• b: branching factor (finite) • d: depth of solution (finite) • Assume a tree of depth m

Space = # of nodes in Open – what about Closed? Because we check if a state has already been visited,

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DFS vs BFS: more detail

• b: branching factor (finite) • d: depth of solution (finite) • Assume a tree of depth m

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DFS vs BFS: more detail

• b: branching factor (finite) • d: depth of solution (finite) • Assume a tree of depth m

Space = # of nodes in Open – what about Closed? (Handwavy response, see book for more handwaving)

(27)

DFS vs BFS:

Handwavy Analysis

Method DFS BFS

Complete? Yes Yes

Optimal? No Yes

Time O(bm) O(bd)

(28)

DFS vs BFS:

Handwavy Analysis

Method DFS BFS

Complete? Yes Yes

Optimal? No Yes

Time O(bm) O(bd)

Space O(bm) O(bd)

(29)

DFS vs BFS

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DFS vs BFS

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Search Algorithm Template

Initial call: Search(initialstate,ops,{},{})

Search(s,ops,open,closed) =

If goal(s) Then return(s); Else If not(s  closed)

Then

successors  {}; add(s,closed); For each o  ops that applies to s

add apply(o,s) to successors open  add successors to open;

If not(empty(open))

s’  select(open);

open  remove(s’,open);

(32)

DFS

Initial call: DFS(initialstate,ops,{},{})

DFS(s,ops,open,closed) =

If goal(s) Then return(s); Else If not(s  closed)

Then

successors  {}; add(s,closed); For each o  ops that applies to s

add apply(o,s) to successors open  add successors to open;

If not(empty(open))

s’  newest(open);

open  remove(s’,open);

(33)

Depth-Bounded Depth-First Search

Initial call: DBDFS(initialstate,ops,{},{},bound)

DBDFS(s,ops,open,closed,bound) = If goal(s) Then return(s);

Else If not(s  closed) and bound>0

Then

successors  {}; add(s,closed); For each o  ops that applies to s

add apply(o,s) to successors open  add successors to open;

If not(empty(open))

s’  newest(open);

open  remove(s’,open);

(34)

Iterative Deepening Search

IDS(s,ops) i=0

loop {result = DBDFS(s,ops,{},{},i) if result = fail then i=i+1 else return(result)}

(35)

Depth-Bounded Depth-First Search

Initial call: DBDFS(initialstate,ops,{},{},bound)

DBDFS(s,ops,open,closed,bound) = If goal(s) Then return(s);

Else If not(s  closed) and bound>0

Then

successors  {}; add(s,closed); For each o  ops that applies to s

add apply(o,s) to successors open  add successors to open;

If not(empty(open))

s’  newest(open);

open  remove(s’,open);

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(37)
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Iterative Deepening Search

1,2

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Iterative Deepening Search

1,2

(40)

Iterative Deepening Search

1,2

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Iterative Deepening Search

1,2,6

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Iterative Deepening Search

1,2,6

(43)

Iterative Deepening Search

8

1,2,6

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Iterative Deepening Search

8 9

1,2,6

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Iterative Deepening Search

8 9 10

1,2,6

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Iterative Deepening Search

8 9 10

1,2,6

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Iterative Deepening Search

8 9 10 12

1,2,6

(48)

Iterative Deepening Search

8 9 10 12 13

1,2,6

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Iterative Deepening Search

8 9 10 12 13 14

1,2,6

(50)

Iterative Deepening Search

8 9 10 12 13 14

1,2,6

(51)

Iterative Deepening Search

8 9 10 12 13 14 16

1,2,6

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Iterative Deepening Search

8 9 10 12 13 14 16 17

1,2,6

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Iterative Deepening Search

8 9 10 12 13 14 16 17 18

1,2,6

(54)

Iterative Deepening Search

What???

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b=2 # States Visited

Level Depth-First Iterative Deepening Ratio

(56)

b=2 # States Visited

Level Depth-First Iterative Deepening Ratio

(57)

b=10 # States Visited

Level Depth-First Iterative Deepening Ratio

(58)
(59)

DFS vs BFS vs IDS

Method DFS BFS IDS Complete? Yes Yes

Optimal? No Yes

Time O(bm) O(bd) O()

(60)

DFS vs BFS vs IDS

Method DFS BFS IDS Complete? Yes Yes Yes

Optimal? No Yes Yes

Time O(bm) O(bd) O(bd)

Space O(bm) O(bd) O(bd)

(61)

Time Complexity of IDS

• Time for depth-bounded depth-first search to level i: O(bi)

(62)

Time Complexity of IDS

• Time for depth-bounded depth-first search to level i: O(bi)

• Time for IDS is the sum of this time for i = 0, …, d

(63)

Time Complexity of IDS

• Time for depth-bounded depth-first search to level i: O(bi)

• Time for IDS is the sum of this time for i = 0, …, d

(64)

Time Complexity of IDS

• Time for depth-bounded depth-first search to level i: O(bi)

• Time for IDS is the sum of this time for i = 0, …, d

෍

𝑖=0 𝑑

𝑏𝑖 = 𝑏

𝑑+1 − 1

(65)

Time Complexity of IDS

• Time for depth-bounded depth-first search to level i: O(bi)

• Time for IDS is the sum of this time for i = 0, …, d

(66)

Time Complexity of IDS

• Time for depth-bounded depth-first search to level i: O(bi)

• Time for IDS is the sum of this time for i = 0, …, d

෍ 𝑖=0 𝑑 𝑏𝑖 = 𝑏 𝑑+1 − 1 𝑏 − 1 This is O(bd)

(67)

Iterative Deepening Search

• The amount of wasted effort is a fraction of the time for the final stage of the search

• That fraction gets smaller as the branching factor gets bigger

• In exchange for that, you get linear space! (In d, not m!) Looks like BFS in the right ways

References

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