Intro to Theory of Computation LECTURE 18 Last

  • Slides: 15
Download presentation
Intro to Theory of Computation LECTURE 18 Last time • ATM is unrecognizable •

Intro to Theory of Computation LECTURE 18 Last time • ATM is unrecognizable • Reductions Today • Reductions • Mapping reductions Homework 7 due Homework 8 out Sofya Raskhodnikova 3/17/2016

Problems in language theory ADFA ATM decidable undecidable EDFA ECFG ETM decidable EQDFA EQCFG

Problems in language theory ADFA ATM decidable undecidable EDFA ECFG ETM decidable EQDFA EQCFG decidable 3/17/2016 ACFG ? undecidable EQTM ? L 18. 2

Using reductions to prove undecidability We want to prove that language L is undecidable.

Using reductions to prove undecidability We want to prove that language L is undecidable. Idea: Use a proof by contradiction. 1. Suppose to the contrary that L is decidable. 2. Use a decider for L as a subroutine to construct a decider for ATM. 3. But ATM is undecidable. Contradiction! 3/17/2016 L 18. 3

I-clicker question (frequency: AC) To prove that ETM is undecidable A. we assumed ETM

I-clicker question (frequency: AC) To prove that ETM is undecidable A. we assumed ETM had a decider and used it to construct a decider for ATM B. we assumed ATM had a decider and used it to construct a decider for ETM C. we constructed a TM S that on input <M, w> decides whether M accepts w, assuming the existence of a TM R that decides on input <M’> whether the language of <M’> is empty D. There is more than one correct answer. E. None of the above. 3/17/2016 L 18. 4

Prove that CFLTM is undecidable Proof: Suppose to the contrary that CFLTM is decidable,

Prove that CFLTM is undecidable Proof: Suppose to the contrary that CFLTM is decidable, and let R be a TM that decides it. We construct TM S that decides ATM. 3/17/2016 it rejects L 18. 5

Prove that EQTM is undecidable Proof: Suppose to the contrary that EQTM is decidable,

Prove that EQTM is undecidable Proof: Suppose to the contrary that EQTM is decidable, and let R be a TM that decides it. We construct TM S that decides ATM. 3/17/2016 it accepts L 18. 6

Proof 2 that EQTM is undecidable Proof: Suppose to the contrary that EQTM is

Proof 2 that EQTM is undecidable Proof: Suppose to the contrary that EQTM is decidable, and let R be a TM that decides it. We construct TM S that decides ETM. What do we change? 3/17/2016 it accepts L 18. 7

Problems in language theory ADFA ATM decidable undecidable EDFA ECFG ETM decidable undecidable EQDFA

Problems in language theory ADFA ATM decidable undecidable EDFA ECFG ETM decidable undecidable EQDFA EQCFG EQTM decidable 3/17/2016 ACFG ? undecidable L 18. 8

Proving undecidability and unrecognizability Mapping Reductions 3/17/2016 L 18. 9

Proving undecidability and unrecognizability Mapping Reductions 3/17/2016 L 18. 9

Computable functions 3/17/2016 L 18. 10

Computable functions 3/17/2016 L 18. 10

Mapping reductions A B 3/17/2016 L 18. 11

Mapping reductions A B 3/17/2016 L 18. 11

Using mapping reductions to prove decidability 3/17/2016 L 18. 12

Using mapping reductions to prove decidability 3/17/2016 L 18. 12

Using mapping reductions to prove undecidability 3/17/2016 L 18. 13

Using mapping reductions to prove undecidability 3/17/2016 L 18. 13

Using mapping reductions to prove recognizability 3/17/2016 L 18. 14

Using mapping reductions to prove recognizability 3/17/2016 L 18. 14

Using mapping reductions to prove unrecognizability 3/17/2016 L 18. 15

Using mapping reductions to prove unrecognizability 3/17/2016 L 18. 15