CSE 105Theory of Computability Fall, 2006 Lecture 15November 9 - - PDF document

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CSE 105Theory of Computability Fall, 2006 Lecture 15November 9 Undecidability Instructor: Neil Rhodes Outline Relationship between decidable and Turing-recognizable classes Identifying a non-decidable language Some languages are not

SLIDE 1

CSE 105—Theory of Computability

Fall, 2006 Lecture 15—November 9 Undecidability Instructor: Neil Rhodes Outline

Relationship between decidable and Turing-recognizable classes Identifying a non-decidable language Some languages are not Turing-recognizable

Non-constructive proof Constructive proof

SLIDE 2

Hierarchy of languages

Regular Context-Free Decidable Turing-recognizable

r e c u r s i v e l y e n u m e r a b l e ( r . e , )

recursive

Turing-unrecognizable

Complement of a Turing-decidable language is Turing-decidable

If a language, L, is Turing-decidable (recursive), its complement, LC, is:

4 Regular Context-Free Decidable Turing-recognizable

recursively enumerable (r.e,)

r e c u r s i v e

SLIDE 3

Definition of co-language class

If L is in language class, its complement is in co-language class

If L is regular LC is ______________ If L is context-free, LC is _______________ If L is deterministic context-free, LC is _______________

Complement of non-decidable Turing-recognizable language is not Turing-recognizable

L is decidable iff L and LC are both Turing-recognizable

SLIDE 4

Turing-recognizable co-Turing-recognizable

Outline

Relationship between decidable and Turing-recognizable classes Identifying a non-decidable language Some languages are not Turing-recognizable

Non-constructive proof Constructive proof

SLIDE 5

Barber paradox

The only barber in town shaves everyone (and only those) who don’t shave themselves.

Who shaves the barber?

Diagonalization

The set of real numbers is uncountable (no bijection with the natural numbers)

SLIDE 6

The acceptance problem is undecidable

Does Turing Machine M accept input w? ATM={<M,w>|M is a TM and M accepts w}

Explicit diagonalization in proof that the acceptance problem is undecidable

Encode Turing machines as integers

Any integer describing an invalid Turing machine we’ll interpret as a TM that

halts and rejects

TMi = Turing Machine described by integer i

Make table

Columns are ith binary string Rows are jth Turing machine Each entry (i, j) is:

– 1 if TMj accepts input i – 0 if TMj doesn’t accept input i

LD = opposite of (i, i) for each i

Diagonalization language not

recognized by any TM

1 2 3 4 5 6 7 8 TM1 TM2 TM3 TM4 TM5 TM6 TM7 TM8

SLIDE 7

Outline

Relationship between decidable and Turing-recognizable classes Identifying a non-decidable language Some languages are not Turing-recognizable

Non-constructive proof Constructive proof

There are many languages which have no TM (not recursively enumerable)

How large is set of TMs? How large is set of languages? Bijection possible?

SLIDE 8

Outline

Relationship between decidable and Turing-recognizable classes Identifying a non-decidable language Some languages are not Turing-recognizable

Non-constructive proof Constructive proof

ATMC is not Turing-recognizable

Assume ATMC is Turing-recognizable ATM is Turing-recognizable Since both ATM and ATMC are Turing-recognizable, ______________________________________ But, we’ve already shown that ____________________________ So, ATMC is not Turing-recognizable