Reducibility 1 Problem is reduced to problem If

Reducibility 1

Problem is reduced to problem If we can solve problem then 2

Problem If If is reduced to problem is decidable then is undecidable then is decidable is undecidable 3

Example: the halting problem is reduced to the state-entry problem 4

The state-entry problem Inputs: • Turing Machine • State • String Question: Does on input enter state ? 5

Theorem: The state-entry problem is undecidable Proof: Reduce the halting problem to the state-entry problem 6

Suppose we have a Decider for the state-entry algorithm: state-entry problem decider YES enters NO doesn’t enter 7

We want to build a decider for the halting problem: Halting problem decider YES halts on NO doesn’t halt on 8

We want to reduce the halting problem to the state-entry problem: Halting problem decider YES State-entry problem NO NO decider 9

We need to convert one problem instance to the other problem instance Halting problem decider Convert Inputs ? YES State-entry problem NO NO decider 10

Convert to : • Add new state • From any halting state of halting states add transitions to Single halt state 11

halts on input if and only if halts on state on input 12

Halting problem decider Generate YES State-entry problem NO NO decider 13

We reduced the halting problem to the state-entry problem Since the halting problem is undecidable, the state-entry problem is undecidable END OF PROOF 14

Another example: the halting problem is reduced to the blank-tape halting problem 15

The blank-tape halting problem Input: Turing Machine Question: Does halt when started with a blank tape? 16

Theorem: The blank-tape halting problem is undecidable Proof: Reduce the halting problem to the blank-tape halting problem 17

Suppose we have a decider for the blank-tape halting problem: YES blank-tape halting problem NO decider halts on blank tape doesn’t halt on blank tape 18

We want to build a decider for the halting problem: halting problem decider YES halts on NO doesn’t halt on 19

We want to reduce the halting problem to the blank-tape halting problem: Halting problem decider YES Blank-tape halting problem NO decider YES NO 20

We need to convert one problem instance to the other problem instance Halting problem decider Convert Inputs ? YES Blank-tape halting problem NO decider YES NO 21

Construct a new machine • When started on blank tape, writes • Then continues execution like step 1 if blank tape step 2 execute then write with input 22

halts on input string if and only if halts when started with blank tape 23

Halting problem decider Generate YES blank-tape halting problem NO decider YES NO 24

We reduced the halting problem to the blank-tape halting problem Since the halting problem is undecidable, the blank-tape halting problem is undecidable END OF PROOF 25

Summary of Undecidable Problems Halting Problem: Does machine halt on input ? Membership problem: Does machine accept string ? 26

Blank-tape halting problem: Does machine on blank tape? halt when starting State-entry Problem: Does machine on input ? enter state 27

Uncomputable Functions 28

Uncomputable Functions Domain Values region A function is uncomputable if it cannot be computed for all of its domain 29

An uncomputable function: maximum number of moves until any Turing machine with states halts when started with the blank tape 30

Theorem: Function Proof: is uncomputable Assume for contradiction that is computable Then the blank-tape halting problem is decidable 31

Decider for blank-tape halting problem: Input: machine 1. Count states of : 2. Compute 3. Simulate for steps starting with empty tape If halts then return YES otherwise return NO 32

Therefore, the blank-tape halting problem is decidable However, the blank-tape halting problem is undecidable Contradiction!!! 33

Therefore, function in uncomputable END OF PROOF 34

Undecidable Problems for Recursively Enumerable Languages 35

Take a recursively enumerable language Decision problems: • is empty? • is finite? • contains two different strings of the same length? All these problems are undecidable 36

Theorem: For any recursively enumerable language it is undecidable to determine whether is empty Proof: We will reduce the membership problem to this problem 37

Let be the TM with Suppose we have a decider for the empty language problem: empty language problem decider YES empty NO not empty 38

We will build the decider for the membership problem: membership problem decider YES NO accepts rejects 39

We want to reduce the membership problem to the empty language problem: Membership problem decider empty language problem decider YES NO NO YES 40

We need to convert one problem instance to the other problem instance Membership problem decider Convert inputs ? empty language problem decider YES NO NO YES 41

Construct machine : On arbitrary input string executes the same as with When compare enters a final state, with Accept only if 42

if and only if is not empty 43

Membership problem decider construct empty language problem decider YES NO NO YES END OF PROOF 44