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

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

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

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

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