[PDF] [PDF] BBM402-Lecture 9: Reducibility

[PDF] Lecture Notes: The Halting Problem; Reductions - Computer

20 mar 2012 · The Halting Problem; Reductions COMS W3261 Columbia halts in an accepting state iff its input is in the language Definition 2 A language 

[PDF] 1 Reductions

Reductions A reduction is a way of converting one problem into another problem such that a solution to the second problem can be used to solve the first problem Proof We will reduce Atm to HALT Mapping Reductions On input w The Halting Problem On input x then accept x = {〈M1,M2〉 L(M1) = L(M2)} is not r e

[PDF] co-RE and Reducibility

Mapping Reductions ○ A tool The halting problem is the following problem: Given a Proof: By contradiction; assume HALT ∈ R Then there must be some

[PDF] Theory of Computer Science - Halting Problem and Reductions

10 mai 2017 · undecidability results from the undecidability of the special halting problem a new problem to an already known problem x ∈ A if and only if f (x) ∈ B Then we say that A can be reduced to B (in symbols: A ≤ B), and f is called reduction from A to B

[PDF] BBM402-Lecture 9: Reducibility

The Halting Problem Proposition The language HALT = {〈M,w〉 M halts on input w} is undecidable Proof We will reduce Atm to HALT Based on a machine  

[PDF] Reductions

The Halting problem HALTTM = {M,w M is a DTM and M halts on w} The reduction machine outputs a DTM that loops whenever M reaches the rejecting state

[PDF] 11 Reductions We mentioned above the idea of solving problems by

Question: Show, by reduction from Halting, that the Uniform Halting problem is unde- cidable Answer: It is a theorem that if Q can be reduced by a many–one ( or 

[PDF] Reducibility The Halting Problem for TMs - Computer Science Kent

ML that such TMaisM M E w string input on halts that TMaisM wM HALT TM A reduction is a way of converting one problem into another problem in such a

[PDF] Other undecidable problems Examples of undecidable problems

Languages and Automata Undecidability, problem reduction, and Rice's Theorem (12 2) Other undecidable problems • Once we have shown that the halting

[PDF] Chapter 7 Uncomputability

The problem of deciding if a Turing machine stops for at least one input word (the existential halting problem) is undecidable One proceeds by reduction from the 

[PDF] halton till

[PDF] ham cooking temperature chart

[PDF] ham cooking time calculator

[PDF] ham radio codes 10 codes

[PDF] ham radio programming software for mac

[PDF] ham roasting times

[PDF] hamiltonian of coupled harmonic oscillators

[PDF] hamiltonian path

[PDF] hamiltonian path and circuit

[PDF] hamlet act 1

[PDF] hamlet act 2

[PDF] hamlet passage

[PDF] hamlet pdf with footnotes

[PDF] hamlet website

[PDF] hamlet with explanatory notes

Lecturer: Lale Özkahya

Undecidability

ReductionsInformal Overview

Definition and PropertiesReductions

A reduction is a w ayof converting one p robleminto another problem such that a solution to the second problem can be used to solve the first problem. We say the first problem reduces to the second problem.Informal Examples:Measuring the a reaof rectangle reduces to measuring the length of the sides; Solving a system of linear equations reduces to inverting a matrixThe problemLdreduces to the problemAtmas follows: "To see ifw2Ldcheck ifhw,wi 2Atm."Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesReductions

A reduction is a w ayof converting one p robleminto another problem such that a solution to the second problem can be used to solve the first problem. We say the first problem reduces to the second problem.Informal Examples:Measuring the a reaof rectangle reduces to measuring the length of the sides; Solving a system of linear equations reduces to inverting a matrixThe problemLdreduces to the problemAtmas follows: "To see ifw2Ldcheck ifhw,wi 2Atm."Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesReductions

A reduction is a w ayof converting one p robleminto another problem such that a solution to the second problem can be used to solve the first problem. We say the first problem reduces to the second problem.Informal Examples:Measuring the a reaof rectangle reduces to measuring the length of the sides; Solving a system of linear equations reduces to inverting a matrixThe problemLdreduces to the problemAtmas follows: "To see ifw2Ldcheck ifhw,wi 2Atm."Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesReductions

A reduction is a w ayof converting one p robleminto another problem such that a solution to the second problem can be used to solve the first problem. We say the first problem reduces to the second problem.Informal Examples:Measuring the a reaof rectangle reduces to measuring the length of the sides; Solving a system of linear equations reduces to inverting a matrixThe problemLdreduces to the problemAtmas follows: "To see ifw2Ldcheck ifhw,wi 2Atm."Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesUndecidability using Reductions

Proposition

Suppose L

1reduces to L2and L1is undecidable. Then L2is

undecidable.Proof Sketch. Suppose for contradictionL2is decidable. Then there is aMthat always halts and decidesL2. Then the following algorithm decides L

1On inputw, apply reduction to transformwinto an inputw?

for problem 2RunMonw?, and use its answer.Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesUndecidability using Reductions

Proposition

Suppose L

1reduces to L2and L1is undecidable. Then L2is

undecidable.Proof Sketch. Suppose for contradictionL2is decidable. Then there is aMthat always halts and decidesL2. Then the following algorithm decides L

1On inputw, apply reduction to transformwinto an inputw?

for problem 2RunMonw?, and use its answer.Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesUndecidability using Reductions

Proposition

Suppose L

1reduces to L2and L1is undecidable. Then L2is

undecidable.Proof Sketch. Suppose for contradictionL2is decidable. Then there is aMthat always halts and decidesL2. Then the following algorithm decides L

1On inputw, apply reduction to transformwinto an inputw?

for problem 2RunMonw?, and use its answer.Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesSchematic View

Algorithm for Problem 1

ReductionfAlgorithm for

Problem 2wf(w)yes

no

Reductions schematically

Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesSchematic View

Algorithm for Problem 1

ReductionfAlgorithm for

Problem 2wf(w)yes

no

Reductions schematically

Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesSchematic View

Algorithm for Problem 1

ReductionfAlgorithm for

Problem 2wf(w)yes

no

Reductions schematically

Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesSchematic View

Algorithm for Problem 1

ReductionfAlgorithm for

Problem 2wf(w)yes

no

Reductions schematically

Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesThe Halting Problem

Proposition

The language HALT=fhM,wi jM halts on input wgis

undecidable.Proof. We will reduceAtmto HALT. Based on a machineM, let us consider a new machinef(M) as follows:On inputx

RunMonx

IfMaccepts then halt and accept

IfMrejects then go into an infinite loopObserve thatf(M) halts on inputwif and only ifMaccepts w!Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesThe Halting Problem

Proposition

The language HALT=fhM,wi jM halts on input wgis

undecidable.Proof. We will reduceAtmto HALT. Based on a machineM, let us consider a new machinef(M) as follows:On inputx

RunMonx

IfMaccepts then halt and accept

IfMrejects then go into an infinite loopObserve thatf(M) halts on inputwif and only ifMaccepts w!Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesThe Halting Problem

Proposition

The language HALT=fhM,wi jM halts on input wgis

undecidable.Proof. We will reduceAtmto HALT. Based on a machineM, let us consider a new machinef(M) as follows:On inputx

RunMonx

IfMaccepts then halt and accept

IfMrejects then go into an infinite loopObserve thatf(M) halts on inputwif and only ifMaccepts w!Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesThe Halting Problem

Proposition

The language HALT=fhM,wi jM halts on input wgis

undecidable.Proof. We will reduceAtmto HALT. Based on a machineM, let us consider a new machinef(M) as follows:On inputx

RunMonx

IfMaccepts then halt and accept

IfMrejects then go into an infinite loopObserve thatf(M) halts on inputwif and only ifMaccepts w!Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesThe Halting Problem

Completing the proofProof (contd).

Suppose HALT is decidable. Then there is a Turing machineH that always halts andL(H) = HALT.Consider the following programT

On inputhM,wi

Construct programf(M)

RunHonhf(M),wi

Accept ifHaccepts and reject ifHrejectsTdecidesAtm.But,Atmis undecidable, which gives us the contradiction.?Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesThe Halting Problem

Completing the proofProof (contd).

Suppose HALT is decidable. Then there is a Turing machineH that always halts andL(H) = HALT.Consider the following programT

On inputhM,wi

Construct programf(M)

RunHonhf(M),wi

Accept ifHaccepts and reject ifHrejectsTdecidesAtm.But,Atmis undecidable, which gives us the contradiction.?Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesThe Halting Problem

Completing the proofProof (contd).

Suppose HALT is decidable. Then there is a Turing machineH that always halts andL(H) = HALT.Consider the following programT

On inputhM,wi

Construct programf(M)

RunHonhf(M),wi

Accept ifHaccepts and reject ifHrejectsTdecidesAtm.But,Atmis undecidable, which gives us the contradiction.?Agha-ViswanathanCS373

Undecidability

ReductionsInformal Overview

Definition and PropertiesThe Halting Problem

quotesdbs_dbs17.pdfusesText_23