1 Reductions
Then L2 is undecidable. Proof Sketch. Suppose for contradiction L2 is decidable. Then there is a M that always halts and decides L2. Then the
CS21 Decidability and Tractability Outline Definition of reduction
4 févr. 2022 decidable/undecidable. • Rice's Theorem. • Post Correspondence Problem. February 4 2022. CS21 Lecture 14. 3. Definition of reduction.
Decidability: Reduction Proofs • Basic technique for proving a
Decidability: Reduction Proofs (2). • Steps of a reduction proof. Given language L2 where want to show that L2 /? D (or SD).
Decidability
Turing reduction. E.g. ETM: Create an algorithm for a known undecidable problem using the problem to be reduced to as a subroutine.
Calculabilité Cours 3 : réductions
Une réduction many-one Turing du langage A au langage B est une gage L. Une réduction montre que si le langage B est décidable alors il en est de même.
Lecture 33: Reductions and Undecidability Undecidable Problems
for sixth cannot exist . Reductions. • A reduction R from L1 to L2 is one or more. Turing machines such that:.
CS21 Decidability and Tractability Outline So far… RE and co-RE
2 févr. 2022 Theorem: a language L is decidable if and ... if L is decidable then complement of L is ... undecidable by reduction from ATM.
On the Decidability of Membership in Matrix-exponential Semigroups
3 déc. 2020 The decidability proof is by reduction to a version of integer programming that has transcendental constants. We give a decision procedure ...
How to Understand and Create Mapping Reductions
A mapping reduction A ?m B (or A ?P B) is an algorithm (respectively To prove decidability: If A ?m B and B is decidable
THE DECIDABILITY OF HINDLEYS AXIOMS FOR STRONG
In his paper [3] Hindley shows that strong reduction in combinatory logic (see axiom schemes and that the property of being an axiom is decidable.
CS21 Decidability and Tractability - California Institute of
Example reduction •Preceding reduction proved: Theorem: A TM is undecidable Proof (recap): –suppose A TMis decidable –we showed how to use A TMto decide HALT –conclude HALT is decidable Contradiction
Council tax/ Rate relief - Carers UK
Proving that a problem is undecidable by a reduction from the halting problem Define reduction Describe at a high level how we can use reduction to prove that a decision problem is undecidable Prove that a decision problem is undecidable by using a reduction from the halting problem CS 245 Logic and Computation Fall 2019 3 / 13
Busch Complexity Lectures: Decidability and Reductions
Decidability and Reductions 2 Recall that: A language is decidable if there is a Turing machine (decider) that accepts the language and for reductionf 18 L 1
Lecture Notes: The Halting Problem; Reductions
the language while for decidability we require that the TM halt on every input Now recall our two most important results: Theorem 1 There exist (uncountably many!) languages which are not Turing-recognizable Proof (intuitive) There are as many strings as natural numbers because every ( nite) string
Searches related to reduction decidability filetype:pdf
A reduction from 3SAT will demonstrate that all problems in NP reduce to an instance of the 3-coloring problem The reduction has several parts: 1) Construct a palette graph which is a 3-clique; this will require 3 colors to color in correctly We can designate the three colors as “True” “False” and “Dummy” as we
What is a disability reduction?
- A disability reduction will mean that the council tax bill is reduced to the amount payable for a home in the valuation band below yours. If you are in the lowest band already (Band A) you get a reduction of one sixth of the bill.
What is a reduction of attributes?
- § 1.108-7 Reduction of attributes. (a) In general. (1) If a taxpayer excludes discharge of indebtedness income ( COD income) from gross income under section 108 (a) (1) (A), (B), or (C), then the amount excluded shall be applied to reduce the following tax attributes of the taxpayer in the following order: (i) Net operating losses.
What is reduction?
- Reduction: ?technique of setting a displaced fracture to a proper alignment. ? may be done non-operating or operatively so called closed and open reduction respectively. 31 37. 1.
What is diimide reduction?
- Diimide reduction saturated the double and triple bonds while keeping the sulfinyl group intact. Reduction of a unsaturated organic compound to an alkane by diimide (H2N2), which itself is oxidized to N2. The reaction is syn-stereospecific and chemoselective. Copyright © 2020 Elsevier Limited. Reaxys is a trademark of Elsevier Limited.
How to Understand and Create Mapping Reductions
What is a mapping reduction?
A mapping reductionAmB(orAPB) is an algorithm (respectively, polytime algorithm) that can transform any instance of decision problemAinto an instance of decision problem B, in such a way that theanswer correspondenceproperty holds. Answer correspondence property: The answer (yes/no) to anyAproblem instance must be the same as the answer to the correspondingBproblem instance to which the reduction transforms it. 1 Thus the following two-step algorithm can be used to solve anyAproblem instance:1. Transform the givenAproblem instance to a correspondingBproblem instance.
2. Call a solver forBproblem instances and return whatever answer (yes/no) it gives.
How should one make sense of thenotation?
AmBmeans \Aproblems are no harder to solve thanBproblems." APBmeans \Aproblems are no harder to solve in polytime thanBproblems." AmBmeans \Being able to solve anyBproblem)being able to solve anyAproblem." APBmeans \Being able to solve anyBproblem in polytime)being able to solve anyA problem in polytime."Examples of decision problem instances:
AnATMproblem instance is (an encoding of) a given TM and a given string. The associated yes/no question is:Does the given TM accept the given string? AREGULARTMproblem instance is (an encoding of) a given TM. The associated yes/no question is:Is the language recognized by the given TM regular? ACLIQUEproblem instance is (an encoding of) a given undirected graph and a given number. The associated yes/no question is:Does the given graph have a clique of the given size? A3SATproblem instance is (an encoding of) a 3CNF Boolean formula. The associatedyes/no question is:Does the given Boolean formula have a satisfying assignment?1If the answer is just the opposite, for most purposes that's okay too. In this case the mapping reduction is
actually fromAtoB. 1What are mapping reductions used for?
To prove undecidability: IfAmBandAis undecidable, thenBis undecidable. To prove non-Turing-recognizability: IfAmBandAis non-Turing-recognizable, thenB is non-Turing-recognizable. To prove NP-completeness: IfAPBandAis NP-complete (andB2NP), thenBisNP-complete.
Other, less common, uses for mapping reductions:
To prove decidability: IfAmBandBis decidable, thenAis decidable. To prove Turing-recognizability: IfAmBandBis Turing-recognizable, thenAis Turing- recognizable. To prove membership in P: IfAPBandBis in P, thenAis in P. Important considerations when constructing mapping reductions1. Make sure your reduction goes in the correct direction. For example:
2 If you're trying to proveAis undecidable, will it help to construct a reductionAmB for some undecidable languageB? If you're trying to proveAis NP-complete, will it help to construct a reductionAPB for some NP-complete languageB?2.Type match:A reduction (transformation) is an algorithm whose input and output must be
of the right type. Examples:ATMmREGULARTM
{ATMproblem instances have the formhM;wi, whereMis a TM andwis a string. {REGULARTMproblem instances have the formhM0i, whereM0is a TM. {Therefore the input to the reducing function (transformation) must be (an encoding of) an arbitrary TM and an arbitrary string, and the output must be (an encoding of) some TM.3SATPCLIQUE
{3SATproblem instances have the formhi, whereis a 3CNF Boolean formula. {CLIQUEproblem instances have the formhG;ki, whereGis an undirected graph andkis a number. {Therefore the input to the reducing function (transformation) must be (an encoding of) an arbitrary 3CNF formulaand the output must be (an encoding of) someundirected graphGand some numberk.2Analogously, if you're trying to prove a number is very large, will it help to prove that it's less than or equal to
some very large number? 23.Answer correspondence:The answer to the transformed problem instance should be the same
as the answer to the original problem instance, foranyinstance of the original problem.3Examples:
ATMmREGULARTM
{LetM0denote the corresponding TM whose encoding is produced as output by the reducing function when given as input (the encoding of) a TMMand a stringw. {If the answer isyesto the questionDoes TMMaccept stringw?, then the answer must beyesto the questionDoes the TMM0recognize a regular language? {If the answer isnoto the questionDoes TMMaccept stringw?, then the answer must benoto the questionDoes the TMM0recognize a regular language?3SATPCLIQUE
{LetGandkdenote the corresponding graph and number, respectively, whose en- coding is produced as output by the reducing function when given as input (the encoding of) a 3CNF formula. {If the answer isyesto the questionDoeshave a satisfying assignment?, then the answer must beyesto the questionDoes the graphGhave ak-clique? {If the answer isnoto the questionDoeshave a satisfying assignment?, then the answer must benoto the questionDoes the graphGhave ak-clique?Proving answer correspondence always involves an \if and only if" proof.3As observed earlier, for most purposes it's okay if the answers are always opposite. This just means the mapping
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