More Undecidable Problems - Automata and Complexity Theory - Lecture Slides, Slides of Theory of Automata

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1

More Undecidable Problems

Rice’s Theorem

Post’s Correspondence Problem

Some Real Problems

2

Properties of Languages

Any set of languages

is a

property

of

languages.

Example: The infiniteness property is the set of infinite languages.

4

Trivial Properties

There are two (

trivial ) properties P for

which L

P

is decidable.

The

always-false property, which contains

no RE languages.

The

always-true property, which contains

every RE language.

Rice’s Theorem: For every otherproperty P, L

P

is undecidable.

5

Plan for Proof of Rice’s Theorem

Lemma needed: recursive languagesare closed under complementation.

We need the technique known asreduction, where an algorithmconverts instances of one problem toinstances of another.

Then, we can prove the theorem.

7

Proof – Concluded

M’ simulates M.

But if M enters an accepting state, M’ halts without accepting.

If M halts without accepting, M’ instead has a move taking it to state f.

In state f, M’ halts.

8

Reductions

A

reduction

from language L to

language L’ is an algorithm (TM thatalways halts) that takes a string w andconverts it to a string x, with theproperty that:

x is in L’ if and only if w is in L.

10

Reductions – (2)

If we reduce L to L’, and L’ is decidable, then the algorithm for L’ + thealgorithm of the reduction shows that Lis also decidable.

Used in the contrapositive: If we know L is not decidable, then L’ cannot bedecidable.

11

Reductions – Aside

This form of reduction is not the most general.

Example: We “reduced” L

d

to L

u

, but in

doing so we had to complementanswers.

More in NP-completeness discussion on Karp vs. Cook reductions.

13

The Reduction

Our reduction algorithm must take M andw and produce a TM M’.

L(M’) has property P if and only if Maccepts w.

M’ has two tapes, used for:

Simulates another TM M

L

on the input to M’.

Simulates M on w. 

Note: neither M, M

L

, nor w is input to M’.

14

The Reduction – (2)

Assume that

does not have property P.



If it does, consider the complement of P,which would also be decidable by the lemma.

Let L be any language with property P, and let M

L

be a TM that accepts L.

M’ is constructed to work as follows (next slide).

16

Action of M’ if M Accepts w

Simulate M

on input w

x

Simulate M

L

on input x

On accept

Acceptiff x isin M

L

17

Design of M’ – (2)

Suppose M accepts w.

Then M’ simulates M

L

and therefore

accepts x if and only if x is in L.

That is, L(M’) = L, L(M’) has property P, and M’ is in L

P

19

Action of M’ if M Does not

Accept w

Simulate M

on input w

x

Never accepts, sonothing else happensand x is not accepted

20

Design of M’ – Conclusion

Thus, the algorithm that converts M and w to M’ is a reduction of L

u

to L

P

Thus, L

P

is undecidable.