Solutions to Problem Set 4
Feb 23 2007 If. A is finite
CS 373: Theory of Computation
decidable (or simply decidable) if there exists a TM M which decides L. • Every finite language is decidable: For example by a TM that has all the strings in ...
BBM401-Lecture 7: Decidable Languages and the Halting Problem
Every finite language is decidable: For e.g. by a TM that has all the Proof. If Atm is recognizable
A note on algebras of languages
study some immunity properties: for instance we prove that for every coinfinite decidable language L there exists a decidable language L′ such that L ⊆ L′ L′
CSE 135: Introduction to Theory of Computation Decidability and
▷ Every finite language is decidable. Page 10. Decidable and Recognizable Languages. Recall: Definition. A Turing machine M is said to recognize a language L
CSE 6321 - Solutions to Problem Set 1
Show that the collection of decidable languages is closed under the following operations. 1. complementation. Solution: Proof. Let L be a decidable language and
Adriana Palacio - University of California San Diego Instructor
Jul 29 2004 Since all finite languages are regular
Practice Problems for Final Exam: Solutions CS 341: Foundations of
In each part below if you need to prove that the given language L is decidable
Lecture 33: Reductions and Undecidability SD & Turing Enumerable
• Can only happen if L is finite. • But all finite languages are decidable. • Fixes proof. • But not decidable whether L is finite!! Undecidable Problems. The
Decidability
If any generate w accept
CS 373: Theory of Computation
Every finite language is decidable: For example by a TM that has all the Proposition 2. If L and L are recognizable
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23 févr. 2007 these machines only recognize regular languages). ... A is finite it is decidable because all finite languages are decidable (just hardwire ...
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In each part below if you need to prove that the given language L is decidable
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Every finite language is decidable: For e.g. by a TM that has Proposition. If L and L are recognizable
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Show that the collection of decidable languages is closed under the following operations. 1. complementation. Solution: Proof. Let L be a decidable language and
Equations over finite sets of words and equivalence problems in
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Lecturer: Lale Özkahya
High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable Language ComplementationDecidable and Recognizable LanguagesRecall: Definition
A Turing machineMis said torecognize a language LifL=L(M).A Turing machineMis said todecide a langua geLifL=L(M)
andMhalts on every input.Lis said to beT uring-recognizable(o rsimply recognizable) if there exists a TMMwhich recognizesL.Lis said to beT uring-decidable(or simply decidable) if there exists a TMMwhich decidesL.Every finite language is decidable: For e.g., by a TM that has
all the strings in the language "hard-coded" into itWe just saw some example algorithms all of which terminate
in a finite number of steps, and output yes or no (accept or reject).i.e., They decide the corresponding languages.Agha-ViswanathanCS373
High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable Language ComplementationDecidable and Recognizable LanguagesRecall: Definition
A Turing machineMis said torecognize a language LifL=L(M).A Turing machineMis said todecide a langua geLifL=L(M)
andMhalts on every input.Lis said to beT uring-recognizable(o rsimply recognizable) if there exists a TMMwhich recognizesL.Lis said to beT uring-decidable(or simply decidable) if there exists a TMMwhich decidesL.Every finite language is decidable: For e.g., by a TM that has
all the strings in the language "hard-coded" into itWe just saw some example algorithms all of which terminate
in a finite number of steps, and output yes or no (accept or reject).i.e., They decide the corresponding languages.Agha-ViswanathanCS373
High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable Language ComplementationDecidable and Recognizable LanguagesRecall: Definition
A Turing machineMis said torecognize a language LifL=L(M).A Turing machineMis said todecide a langua geLifL=L(M)
andMhalts on every input.Lis said to beT uring-recognizable(o rsimply recognizable) if there exists a TMMwhich recognizesL.Lis said to beT uring-decidable(or simply decidable) if there exists a TMMwhich decidesL.Every finite language is decidable: For e.g., by a TM that has
all the strings in the language "hard-coded" into itWe just saw some example algorithms all of which terminate
in a finite number of steps, and output yes or no (accept or reject).i.e., They decide the corresponding languages.Agha-ViswanathanCS373
High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable Language ComplementationDecidable and Recognizable LanguagesRecall: Definition
A Turing machineMis said torecognize a language LifL=L(M).A Turing machineMis said todecide a langua geLifL=L(M)
andMhalts on every input.Lis said to beT uring-recognizable(o rsimply recognizable) if there exists a TMMwhich recognizesL.Lis said to beT uring-decidable(or simply decidable) if there exists a TMMwhich decidesL.Every finite language is decidable: For e.g., by a TM that has
all the strings in the language "hard-coded" into itWe just saw some example algorithms all of which terminate
in a finite number of steps, and output yes or no (accept or reject).i.e., They decide the corresponding languages.Agha-ViswanathanCS373
High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable Language ComplementationDecidable and Recognizable LanguagesRecall: Definition
A Turing machineMis said torecognize a language LifL=L(M).A Turing machineMis said todecide a langua geLifL=L(M)
andMhalts on every input.Lis said to beT uring-recognizable(o rsimply recognizable) if there exists a TMMwhich recognizesL.Lis said to beT uring-decidable(or simply decidable) if there exists a TMMwhich decidesL.Every finite language is decidable: For e.g., by a TM that has
all the strings in the language "hard-coded" into itWe just saw some example algorithms all of which terminate
in a finite number of steps, and output yes or no (accept or reject).i.e., They decide the corresponding languages.Agha-ViswanathanCS373
High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable Language ComplementationDecidable and Recognizable LanguagesRecall: Definition
A Turing machineMis said torecognize a language LifL=L(M).A Turing machineMis said todecide a langua geLifL=L(M)
andMhalts on every input.Lis said to beT uring-recognizable(o rsimply recognizable) if there exists a TMMwhich recognizesL.Lis said to beT uring-decidable(or simply decidable) if there exists a TMMwhich decidesL.Every finite language is decidable: For e.g., by a TM that has
all the strings in the language "hard-coded" into itWe just saw some example algorithms all of which terminate
in a finite number of steps, and output yes or no (accept or reject).i.e., They decide the corresponding languages.Agha-ViswanathanCS373
High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable Language ComplementationDecidable and Recognizable LanguagesRecall: Definition
A Turing machineMis said torecognize a language LifL=L(M).A Turing machineMis said todecide a langua geLifL=L(M)
andMhalts on every input.Lis said to beT uring-recognizable(o rsimply recognizable) if there exists a TMMwhich recognizesL.Lis said to beT uring-decidable(or simply decidable) if there exists a TMMwhich decidesL.Every finite language is decidable: For e.g., by a TM that has
all the strings in the language "hard-coded" into itWe just saw some example algorithms all of which terminate
in a finite number of steps, and output yes or no (accept or reject).i.e., They decide the corresponding languages.Agha-ViswanathanCS373
High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable Language ComplementationDecidable and Recognizable LanguagesRecall: Definition
A Turing machineMis said torecognize a language LifL=L(M).A Turing machineMis said todecide a langua geLifL=L(M)
andMhalts on every input.Lis said to beT uring-recognizable(o rsimply recognizable) if there exists a TMMwhich recognizesL.Lis said to beT uring-decidable(or simply decidable) if there exists a TMMwhich decidesL.Every finite language is decidable: For e.g., by a TM that has
all the strings in the language "hard-coded" into itWe just saw some example algorithms all of which terminate
in a finite number of steps, and output yes or no (accept or reject).i.e., They decide the corresponding languages.Agha-ViswanathanCS373
High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable Language ComplementationDecidable and Recognizable Languages But not all languages a redecidable !In the next class we will see an example:A tm={?M,w? |Mis a TM andMacceptsw}is undecidableHoweverAtmisT uring-recognizable!Proposition There are languages which are recognizable, but not decidableAgha-ViswanathanCS373
High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable Language ComplementationDecidable and Recognizable Languages But not all languages a redecidable !In the next class we will see an example:A tm={?M,w? |Mis a TM andMacceptsw}is undecidableHoweverAtmisT uring-recognizable!Proposition There are languages which are recognizable, but not decidableAgha-ViswanathanCS373
High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable Language ComplementationDecidable and Recognizable Languages But not all languages a redecidable !In the next class we will see an example:A tm={?M,w? |Mis a TM andMacceptsw}is undecidableHoweverAtmisT uring-recognizable!Proposition There are languages which are recognizable, but not decidableAgha-ViswanathanCS373
High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable Language ComplementationDecidable and Recognizable Languages But not all languages a redecidable !In the next class we will see an example:A tm={?M,w? |Mis a TM andMacceptsw}is undecidableHoweverAtmisT uring-recognizable!Proposition There are languages which are recognizable, but not decidableAgha-ViswanathanCS373
High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable LanguageComplementationRecognizingAtm
ProgramUforrec ognizingAtm:
On input?M,w?
simulateMonw if simulatedMacceptsw, then accept else reject (by moving toqrej)U(the Universal TM) accepts?M,w?iffMacceptsw. i.e., L(U) =AtmButUdoes notdecide Atm: IfMrejectswby not halting,U rejects?M,w?by not halting.Indeed (as we shall see) no TM decidesAtm.Agha-ViswanathanCS373High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable LanguageComplementationRecognizingAtm
ProgramUforrec ognizingAtm:
On input?M,w?
simulateMonw if simulatedMacceptsw, then accept else reject (by moving toqrej)U(the Universal TM) accepts?M,w?iffMacceptsw. i.e., L(U) =AtmButUdoes notdecide Atm: IfMrejectswby not halting,U rejects?M,w?by not halting.Indeed (as we shall see) no TM decidesAtm.Agha-ViswanathanCS373High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable LanguageComplementationRecognizingAtm
ProgramUforrec ognizingAtm:
On input?M,w?
simulateMonw if simulatedMacceptsw, then accept else reject (by moving toqrej)U(the Universal TM) accepts?M,w?iffMacceptsw. i.e., L(U) =AtmButUdoes notdecide Atm: IfMrejectswby not halting,U rejects?M,w?by not halting.Indeed (as we shall see) no TM decidesAtm.Agha-ViswanathanCS373High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable LanguageComplementationRecognizingAtm
ProgramUforrec ognizingAtm:
On input?M,w?
simulateMonw if simulatedMacceptsw, then accept else reject (by moving toqrej)U(the Universal TM) accepts?M,w?iffMacceptsw. i.e., L(U) =AtmButUdoes notdecide Atm: IfMrejectswby not halting,U rejects?M,w?by not halting.Indeed (as we shall see) no TM decidesAtm.Agha-ViswanathanCS373High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable LanguageComplementationRecognizingAtm
ProgramUforrec ognizingAtm:
On input?M,w?
simulateMonw if simulatedMacceptsw, then accept else reject (by moving toqrej)U(the Universal TM) accepts?M,w?iffMacceptsw. i.e., L(U) =AtmButUdoes notdecide Atm: IfMrejectswby not halting,U rejects?M,w?by not halting.Indeed (as we shall see) no TM decidesAtm.Agha-ViswanathanCS373High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable LanguageComplementationDeciding vs. Recognizing
Proposition
If L andL are recognizable, then L is decidable
Proof.
ProgramPfordec idingL, given programsPLandPL
for recognizingLandL:On inputx, simulatePLandPL on inputx.Whetherx?Lor x??L, one ofPLandPL will halt in finite number of steps.Which one to simulate first?Either could go on forever.On inputx, simulatein pa rallelPLandPL
on inputxuntil eitherPLorPL acceptsIfPLaccepts, acceptxand halt. IfPL accepts, rejectxand halt.··→Agha-ViswanathanCS373High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable LanguageComplementationDeciding vs. Recognizing
Proposition
If L andL are recognizable, then L is decidable
Proof.
ProgramPfordec idingL, given programsPLandPL
for recognizingLandL:On inputx, simulatePLandPL on inputx.Whetherx?Lor x??L, one ofPLandPL will halt in finite number of steps.Which one to simulate first?Either could go on forever.On inputx, simulatein pa rallelPLandPL
on inputxuntil eitherPLorPL acceptsIfPLaccepts, acceptxand halt. IfPL accepts, rejectxand halt.··→Agha-ViswanathanCS373High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable LanguageComplementationDeciding vs. Recognizing
Proposition
If L andL are recognizable, then L is decidable
Proof.
ProgramPfordec idingL, given programsPLandPL
for recognizingLandL:On inputx, simulatePLandPL on inputx.Whetherx?Lor x??L, one ofPLandPL will halt in finite number of steps.Which one to simulate first?Either could go on forever.On inputx, simulatein pa rallelPLandPL
on inputxuntil eitherPLorPL acceptsIfPLaccepts, acceptxand halt. IfPL accepts, rejectxand halt.··→Agha-ViswanathanCS373High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable LanguageComplementationDeciding vs. Recognizing
Proposition
If L andL are recognizable, then L is decidable
Proof.
ProgramPfordec idingL, given programsPLandPL
for recognizingLandL:On inputx, simulatePLandPL on inputx.Whetherx?Lor x??L, one ofPLandPL will halt in finite number of steps.Which one to simulate first?Either could go on forever.On inputx, simulatein pa rallelPLandPL
on inputxuntil eitherPLorPL acceptsIfPLaccepts, acceptxand halt. IfPL accepts, rejectxand halt.··→Agha-ViswanathanCS373High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable LanguageComplementationDeciding vs. Recognizing
Proposition
If L andL are recognizable, then L is decidable
Proof.
ProgramPfordec idingL, given programsPLandPL
for recognizingLandL:On inputx, simulatePLandPL on inputx.Whetherx?Lor x??L, one ofPLandPL will halt in finite number of steps.Which one to simulate first?Either could go on forever.On inputx, simulatein pa rallelPLandPL
on inputxuntil eitherPLorPL acceptsIfPLaccepts, acceptxand halt. IfPL accepts, rejectxand halt.··→Agha-ViswanathanCS373High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable LanguageComplementationDeciding vs. Recognizing
Proposition
If L andL are recognizable, then L is decidable
Proof.
ProgramPfordec idingL, given programsPLandPL
for recognizingLandL:On inputx, simulatePLandPL on inputx.Whetherx?Lor x??L, one ofPLandPL will halt in finite number of steps.Which one to simulate first?Either could go on forever.On inputx, simulatein pa rallelPLandPL
on inputxuntil eitherPLorPL acceptsIfPLaccepts, acceptxand halt. IfPL accepts, rejectxand halt.··→Agha-ViswanathanCS373High-Level Descriptions of Computation
Deciding vs. Recognizing
Recursive EnumerationAn Undecidable but Recognizable LanguageComplementationDeciding vs. Recognizing
Proof (contd).
In more detail,Pworks as follows:
On input x
fori= 1,2,3,... simulatePLon inputxforisteps simulatePL on inputxforisteps if either simulation accepts, break ifPLaccepted, acceptx(and halt)quotesdbs_dbs22.pdfusesText_28[PDF] all fonts family css
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