meljun cortes automata theory 20

8/13/2019 MELJUN CORTES Automata Theory 20

1/21


2/21

Why do we care about undecidability?

decidableundecidable

... pretty much everything else

L= {0n1n: n> 0}


3/21

Decidable problems

L

= {0

n

1

n

:n

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

if nis odd reject

for i:= 0to n/2:

if xi 0or xn-i 1rejectotherwise accept

IsP

a valid java program?

Can you get from pointA

to point Bin 100steps?

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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5/21

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 of kcolors to the verticesso that no edge has both endpoints of same color

CSC 3230 CSC 2110

CSC 3160CSC 3130

...


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7/21

Shades of decidability

L= {0n

1n

: n> 0}

Sure, we can writea computer program...

...but what happens when we runit?


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How fast?

L

= {0

n

1

n

:n

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

if nis odd reject

for i:= 0to n/2:

if xi 0or xn-i 1rejectotherwise 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?




1week

a few milliseconds


10/21

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

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 of kcolors to the verticesso that no edge has both endpoints of same color

CSC 3230 CSC 2110

CSC 3160CSC 3130

100vertices, 10colors

10100assignmentsCan we do better?


13/21

Some history

first electronic computer

UNIVAC

1950s

integrated circuits


14/21

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


16/21

Fast algorithms

shortest

paths1956

matchings

1965and a few more


17/21

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.


18/21

Proving theorems

We know some true statements are not provable

But lets forget about those.

Can you get a computer to 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 color Gwith 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 harder thanfinding 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 asone another!

theorem-proving

Richard Karp

Q: So are they all easy or all hard?

A: We dont know, but we suspect

all hard

(1972)

meljun cortes automata theory 20

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