Undecidable problems for Recursively enumerable languages continued 1

Undecidable problems for Recursively enumerable languages continued… 1

Take a recursively enumerable language Decision problems: • is empty? • is finite? • contains two different strings of the same length? All these problems are undecidable 2

Theorem: For a recursively enumerable language it is undecidable to determine whether is finite Proof: We will reduce the halting problem to this problem 3

Let be the TM with Suppose we have a decider for the finite language problem: finite language problem decider YES finite NO not finite 4

We will build a decider for the halting problem: Halting problem decider YES halts on NO doesn’t halt on 5

We want to reduce the halting problem to the finite language problem Halting problem decider YES NO finite language problem YES NO decider 6

We need to convert one problem instance to the other problem instance Halting problem decider convert input ? YES NO finite language problem YES NO decider 7

Construct machine : On arbitrary input string Initially, simulates on input If enters a halt state, accept ( inifinite language) Otherwise, reject ( finite language) 8

halts on if and only if is infinite 9

halting problem decider YES construct NO finite language problem YES decider NO 10

Take a recursively enumerable language Decision problems: • is empty? • is finite? • contains two different strings of the same length? All these problems are undecidable 11

Theorem: For a recursively enumerable language it is undecidable to determine whether contains two different strings of same length Proof: We will reduce the halting problem to this problem 12

Let be the TM with Suppose we have the decider for the two-strings problem: Two-strings problem decider YES contains NO Doesn’t contain two equal length strings 13

We will build a decider for the halting problem: Halting problem decider YES halts on NO doesn’t halt on 14

We want to reduce the halting problem to the empty language problem Halting problem decider Two-strings problem decider YES NO NO 15

We need to convert one problem instance to the other problem instance Halting problem decider convert inputs ? Two-strings problem decider YES NO NO 16

Construct machine : On arbitrary input string Initially, simulate on input When enters a halt state, accept if or (two equal length strings Otherwise, reject ) ( ) 17

halts on if and only if accepts two equal length strings accepts and 18

Halting problem decider construct Two-strings problem decider YES NO NO 19

Rice’s Theorem 20

Definition: Non-trivial properties of recursively enumerable languages: any property possessed by some (not all) recursively enumerable languages 21

Some non-trivial properties of recursively enumerable languages: • is empty • is finite • contains two different strings of the same length 22

Rice’s Theorem: Any non-trivial property of a recursively enumerable language is undecidable 23

The Post Correspondence Problem 24

Some undecidable problems for context-free languages: • Is ? are context-free grammars • Is context-free grammar ambiguous? 25

We need a tool to prove that the previous problems for context-free languages are undecidable: The Post Correspondence Problem 26

The Post Correspondence Problem Input: Two sequences of strings 27

There is a Post Correspondence Solution if there is a sequence such that: PC-solution: Indeces may be repeated or ommited 28

Example: PC-solution: 29

Example: There is no solution Because total length of strings from is smaller than total length of strings from 30

The Modified Post Correspondence Problem Inputs: MPC-solution: 31

Example: MPC-solution: 32

We will show: 1. The MPC problem is undecidable (by reducing the membership to MPC) 2. The PC problem is undecidable (by reducing MPC to PC) 33

Theorem: The MPC problem is undecidable Proof: We will reduce the membership problem to the MPC problem 34

Membership problem Input: recursive language string Question: Undecidable 35

Membership problem Input: unrestricted grammar string Question: Undecidable 36

Suppose we have a decider for the MPC problem String Sequences MPC problem decider MPC solution? YES NO 37

We will build a decider for the membership problem Membership problem decider YES NO 38

The reduction of the membership problem to the MPC problem: Membership problem decider yes MPC problem decider no no 39

We need to convert the input instance of one problem to the other Membership problem decider convert inputs ? yes MPC problem decider no no 40

Grammar : start variable : special symbol For every variable 41

Grammar string : special symbol For every production 42

Example: Grammar : String 43

44

45

Grammar : 46

47

48

49

has an MPC-solution if and only if 50

Membership problem decider Construct yes MPC problem decider no no 51

Since the membership problem is undecidable, The MPC problem is uncedecidable END OF PROOF 52

Theorem: The PC problem is undecidable Proof: We will reduce the MPC problem to the PC problem 53

Suppose we have a decider for the PC problem String Sequences PC problem decider PC solution? YES NO 54

We will build a decider for the MPC problem String Sequences MPC problem decider MPC solution? YES NO 55

The reduction of the MPC problem to the PC problem: MPC problem decider yes no no 56

We need to convert the input instance of one problem to the other MPC problem decider convert inputs ? PC problem decider yes no no 57

: input to the MPC problem 58

We construct new sequences 59

We insert a special symbol between any two symbols 60

61

Special Cases 62

has a PC solution if and only if has an MPC solution 63

PC-solution MPC-solution 64

MPC problem decider Construct PC problem decider yes no no 65

Since the MPC problem is undecidable, The PC problem is undecidable END OF PROOF 66

Some undecidable problems for context-free languages: • Is ? are context-free grammars • Is context-free grammar ambiguous? We reduce the PC problem to these problems 67