MELJUN CORTES -- AUTOMATA THEORY LECTURE - 20

7/31/2019 MELJUN CORTES -- AUTOMATA THEORY LECTURE - 20

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CSC 3130: Automata theory and formal languages

Andrej Bogdanov

http://www.cse.cuhk.edu.hk/~andrejb/csc3130

Efficient computation


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Why do we care about undecidability?

decidableundecidable

... pretty much everything else

L= {0n1n: n> 0}


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Decidable problems

L

= {0

n

1

n

:n

> 0}On input x:n:= |x|

ifnis odd reject

fori:= 0 to n/2:

ifxi 0 orxn-i 1 rejectotherwise accept

IsP

a valid java program?

Can you get from pointA

to point B in 100 steps?

Option 1: Try all derivations

Option 2: CYK algorithm

Option 3: LR(1) algorithm

8

6 10 14

14

3

3

9

8 4

25

5

9A

B

For all paths out of pointA

of length at most 100:Try this pathIf path reaches B, accept

Otherwise, reject


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Scheduling

Can you schedule final examsso that there are no conflicts?

Say we have nexams, k slots(maybe n= 200, k = 10)

Exams vertices

Slots colors

Conflicts edges

CSC 3230 CSC 2110

CSC 3160CSC 3130

...

Task: Assign one ofk colors to the vertices

so that no edge has both endpoints of same color


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Scheduling algorithm

For every possible assignment

of colors to vertices:

Try this assignment

If every edge has endpoints

of different colors, accept

If all colorings failed, reject

Task: Assign one ofk colors to the verticesso that no edge has both endpoints of same color

CSC 3230 CSC 2110

CSC 3160CSC 3130

...


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Matching

Can you pair up people sothat every pair is happy?

People vertices

Happy together edgesPairing matching

For every possible pairing:

If each pair is happy, acceptIf all pairings failed, reject


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Shades of decidability

L= {0n

1n

: n> 0}

Sure, we can write a computer program...

...but what happens when we run it?


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How fast?

L

= {0

n

1

n

:n

> 0}On input x:n:= |x|

ifnis odd reject

fori:= 0 to n/2:

ifxi 0 orxn-i 1 rejectotherwise accept

Is Pa valid java program?






8

6 10 14

14

3

3

9

8 4

25

5

9

A

B


of length at most 100:Try this pathIf path reaches B, accept

Otherwise, reject


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How fast?

the lifetime of the universe

P= 100 line java programIs Pa valid java program?




1 week

a few milliseconds


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Finding paths



8

610 14

14

3

3

9

8 4

25

5

9

A

B


of length at most 1000:

Try this pathIf path reaches B, accept

Otherwise, reject

Is there a better wayto do this?

Yes! Dijkstras algorithm

Edsger Dijkstra(1930-2002)


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Matching

For every possible pairing:If each pair is happy, accept

If all pairings failed, reject

Can we do better?

Yes! Edmonds algorithm

Jack Edmonds


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Scheduling algorithm

Forevery possible assignment

of colors to vertices:

Try this assignment

If every edge has endpoints

of different colors, accept

If all colorings failed, reject

Task: Assign one ofk colors to the verticesso that no edge has both endpoints of same color

CSC 3230 CSC 2110

CSC 3160CSC 3130

100 vertices, 10 colors

10100 assignmentsCan we do better?


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Some history

first electronic computer

UNIVAC

1950s

integrated circuits


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The need for optimization

routing problems

how to route calls throughthe phone network?

packing problems

how many trucks do youneed to fit all the boxes?


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The need for optimization

scheduling problems

how do you schedule jobsto complete them

as soon as possible?

constraint satisfaction

can you satisfy a systemof local constraints?

30 min

40 min

45 min

50 min

50 min

0 min


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Fast algorithms

shortest

paths1956

matchings

1965and a few more


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Proving theorems

Kurt Gdel

J. von Neumann

Princeton, 20 March 1956

Dear Mr. von Neumann:

With the greatest sorrow I have learned of your

illness. []

Since you now, as I hear, are feeling stronger, I

would like to allow myself to write you about a

mathematical problem: [] This would haveconsequences of the greatest importance. It

would obviously mean that in spite of the

undecidability of the Entscheidungsproblem,the mental work of a mathematician concerning

Yes-or-No questions could be completely

replaced by a machine.


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Proving theorems

We know some true statements are not provable

But lets forget about those.

Can you get a computerto find the

proof reasonably quickly?

Better than try all possible proofs?

52 years later, we still dont know!


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Optimization versus proofs

If we can find proofs quickly, then we can alsooptimize quickly!

Question: Can we colorGwith so there are no

conflicts?

Proof: We can, set

v1 v2

v3 v4

v1 v2 v3 v4

(v1,R), (v2,Y), (v3,R), (v4,Y)


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The Cook-Levin Theorem

However,

Constraint satisfaction is no harderthanfinding proofs quickly

It is no easier, either

Stephen Cook Leonid Levin

Cook 1971, Levin 1973


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Equivalence of optimization problems

constraint

satisfaction

scheduling packing

coveringrouting

are all as hard as one another!

theorem-proving

Richard Karp

Q: So are they all easy orall hard?

A: We dont know, but we suspect

all hard

(1972)

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MELJUN CORTES -- AUTOMATA THEORY LECTURE - 20