1. Intro to Theory of Computation
LECTURE 16
Last time
Undecidable/unrecognizable languages (ATM is undecidable)
Diagonalization
Today
ATM is unrecognizable
Reductions CS
464
Sofya Raskhodnikova
5/16/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

2. Sofya Raskhodnikova; based on slides by Nick Hopper
Classes of languages
recognizable
ATM
decidable
ATM
CFL
{0 𝑛 1 𝑛 0 𝑛 ∣𝑛≥0}
{0 𝑛 1 𝑛 ∣𝑛≥0}
regular 1*
5/16/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

3. Sofya Raskhodnikova; based on slides by Nick Hopper
Theorem. Language L is decidable iff
L and L are Turing-recognizable
Corollary is not Turing-recognizable.
ATM
5/16/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

4. Sofya Raskhodnikova; based on slides by Nick Hopper
Prove that the following language are Turing-recognizable
ATM = { 𝑴, 𝒘 ∣ 𝑴 is a TM that accepts string 𝒘 }
HALTTM = { 𝑴, 𝒘 ∣ 𝑴 is a TM that halts on string 𝒘 }
5/16/2019
Sofya Raskhodnikova; based on slides by Nick Hopper

5. Sofya Raskhodnikova; based on slides by Nick Hopper
Prove that the following language are undecidable via reduction from ATM
HALTTM= { 𝑴, 𝒘 ∣ 𝑴 is a TM that halts on string 𝒘 }
ETM = { 𝑴 ∣ 𝑴 is a TM and 𝑳 𝑴 =∅}
CFLTM = { 𝑴 ∣ 𝑴 is a TM and 𝑳 𝑴 is context-free}
5/16/2019
Sofya Raskhodnikova; based on slides by Nick Hopper