1

Reducibility

2

Problem is reduced to problemA B

If we can solve problem thenwe can solve problem

BA

B

A

3

If is undecidable then is undecidable

If is decidable then is decidableB A

A B

Problem is reduced to problemA B

4

Example: the halting problem

is reduced to

the state-entry problem

5

The state-entry problem

Inputs: M•Turing Machine

•State q

Question: Does M

•Stringw

enter state q

on input ?w

6

Theorem:

The state-entry problem is undecidable

Proof: Reduce the halting problem to

the state-entry problem

7

Suppose we have an algorithm (Turing Machine)

that solves the state-entry problem

We will construct an algorithmthat solves the halting problem

8

Algorithm for state-entry problem

M

w

q

YES

NO

entersM q

doesn’t enterM q

Assume we have the state-entry algorithm:

9

Algorithm for Halting problem

M

w

YES

NO

halts onM w

doesn’t halt onM w

We want to design the halting algorithm:

10

Modify input machine :M

•Add new state q

•From any halting state add transitions to q

M q

halting statesSinglehalt state

M

11

M halts

M halts on state q

if and only if

12

machine and string

Algorithm for halting problem:

Inputs: M w

1. Construct machine with state M q

2. Run algorithm for state-entry problem with inputs: M wq, ,

13

Generate

M M

w

M qw

State-entry

algorithm

Halting problem algorithm

YES

NO

YES

NO

14

Since the halting problem is undecidable,it must be that the state-entry problemis also undecidable

END OF PROOF

We reduced the halting problemto the state-entry problem

15

Another example:

the halting problem

is reduced to

the blank-tape halting problem

16

The blank-tape halting problem

Input: MTuring Machine

Question: Does M halt when started with

a blank tape?

17

Theorem:

Proof:Reduce the halting problem to the

blank-tape halting problem

The blank-tape halting problem is undecidable

18

Suppose we have an algorithmfor the blank-tape halting problem

We will construct an algorithmfor the halting problem

19

Algorithm for blank-tape halting problem

M

YES

NO

halts onblanks tape

M

doesn’t halton blank tape

M

Assume we have the blank-tape halting algorithm:

20

Algorithm for halting problem

M

w

YES

NO

halts onM w

doesn’t halt onM w

We want to design the halting algorithm:

21

wMConstruct a new machine

• On blank tape writes w

• Then continues execution like M

wM

Mthen write w

step 1 step2

if blank tape executewith input w

22

M halts on input string

wM halts when started with blank tape

if and only if

w

23

Algorithm for halting problem:

1. Construct wM

2. Run algorithm for blank-tape halting problem with input

, ,

wM

machine and stringInputs: M w

24

Generate

wMM

w

blank-tape halting algorithm

Halting problem algorithm

YES

NOwM

YES

NO

25

Since the halting problem is undecidable,the blank-tape halting problem is also undecidable

END OF PROOF

We reduced the halting problemto the blank-tape halting problem

26

Does machine halt on input ?

Summary of Undecidable Problems

Halting Problem:

M w

Membership problem:

Is a string member of a

recursively enumerable language ? Lw

Does machine accept string ?M w

In other words:

27

Does machine enter state on input ?

Does machine halt when startingon blank tape?

Blank-tape halting problem:

M

State-entry Problem:

Mw

q

28

Uncomputable Functions

29

Uncomputable Functions

A function is uncomputable if it cannotbe computed for all of its domain

Domain Valuesregion

f

30

An uncomputable function:

)(nfmaximum number of moves untilany Turing machine with stateshalts when started with the blank tape

n

31

Theorem:Function is uncomputable)(nf

Proof:

Then the blank-tape halting problem is decidable

Assume for contradiction that is computable)(nf

32

Algorithm for blank-tape halting problem:

Input: machineM

1. Count states of : M m

2. Compute )(mf

3. Simulate for steps starting with empty tape

M )(mf

If halts then return YES otherwise return NO

M

33

Therefore, the blank-tape haltingproblem is decidable

However, the blank-tape haltingproblem is undecidable

Contradiction!!!

34

Therefore, function in uncomputable)(nf

END OF PROOF

35

Rice’s Theorem

36

Non-trivial properties of recursively enumerable languages:

any property possessed by some (not all)recursively enumerable languages

Definition:

37

Some non-trivial properties of recursively enumerable languages:

• is emptyL

L• is finite

L• contains two different strings of the same length

38

Rice’s Theorem:

Any non-trivial property of a recursively enumerable languageis undecidable

39

We will prove some non-trivial propertieswithout using Rice’s theorem

40

Theorem:

For any recursively enumerable language Lit is undecidable to determine whether is empty L

Proof:

We will reduce the membership problemto this problem

41

Algorithm for empty languageproblem

M

YES

NO

Assume we have the empty language algorithm:

Let be the machine that accepts M L

)(ML

)(ML

empty

not empty

LML )(

42

Algorithm for membershipproblem

M

w

YES

NO

acceptsM w

rejectsM w

We will design the membership algorithm:

43

First construct machine : wM

When enters a final state, compare original input string with w

M

Accept if original input is the same with w

44

Lw

)( wML is not empty

if and only if

}{)( wML w

45

Algorithm for membership problem:

Inputs: machine and string M w

1. Construct wM

2. Determine if is empty )( wML

YES: then )(MLw

NO: then )(MLw

46

construct

wM

Check if)( wML

is empty

YES

NO

M

w

NO

YES

Membership algorithm

wM

END OF PROOF