  # Recognizable but undecidable language recognizable but undecidable language Answer (1 of 2): Recognizable Decidable Formidable To compare equally are not Compatable! Now,first thing being first let's eliminate the ones that need to go first like unrecognizable meaning nobody knows what the heck it is,to do anything with so it's gone, bye If it's recognizable then we There are undecidable (and unrecognizable) languages over {0,1} {Turing Machines} {0,1}* {Sets of strings of 0s and 1s} {Languages over {0,1}} Set L Set of all subsets of L: 2L {Recognizable languages over {0,1}} There are (many) unrecognizable languages In the early 1900’s, logicians were trying to define consistent foundations for mathematics. This is a subject of interest in mathematics and computer programming, where the undecidable problem has significant implications. True or False: The language A= n hMi: L(M) is An Undecidable but Recognizable Language Complementation Decidable and Recognizable Languages Recall: De nition A Turing machine M is said torecognizea language L if L = L (M ). • ATM is not Turing May 07, 2019 · What is an undecidable language? (definition) Definition: A language for which the membership cannot be decided by an algorithm — equivalently, cannot be recognized by a Turing machine that halts for all inputs. This contradiction shows that M is undecidable. A decision problem P is called “undecidable” if the language L of all yes instances to P is not decidable. Define: = decidable languages 𝑬= Turing-recognizable languages co 𝑬=L തis Turing recognizable} 11/3/2020 CS332 - Theory of Computation 36 \item [5. 1 A simple undecidable language We will start with the following language: Diag = {hMi : M is a TM and hMi 2/ L(M)}. (40 points) Let INF = fhMijM accepts an in nite number of stringsg. , which is not recognizable. } Theorem 1 L⊲⊳ is not Turing-recognizable. A language is called recognizable if it is the language of some TM. (i. Examples May 06, 2019 · Last post we introduced the concept of Turing Machine, recognizable and decidable languages. Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept Language and Problems: any problem can be restated are a language recognition problem Lecture 18: Important Undecidable Problems 4 Program Halting Problem •Input: a program P in some programming language •Output: true if P terminates; false if P runs forever. We proved in an earlier lecture A TM is Turing recognized. • We say that a language is co-Turing-recognizable if it is the complement of a Turing-recognizable It was proved in  that it is undecidable whether a Büchirecognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). Nov 09, 2020 · Show that if a language is both Turing-recognizable and mapping reducible to its complement, then that language is decidable. Much of this material is in Chapter 6 of Sipser. See also undecidable language, decidable problem, recursively enumerable x, 1x. 11) Give an example of an undecidable language B where B ≤m B. Consider a fully-connected, weighted, undirected graph G (V, E) where the edge weights may not be unique. Proof: 3/10/2021 CS332 - Theory of Computation 24. . Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept • We have seen a language that is undecidable. If a language is decidable, then its complement is decidable (by closure under complementation). Thus, $$\atm$$ is recognizable. = ) could use decider for P to decide A TM. TM. Now we use the undecidability of A TM to prove that other languages are undecidable; Key idea - Prove by contradiction that language L is undecidable: Assume TM B decides L ; Then use B to create a decider for A TM (or another undecidable language) This is a contradiction, and so B must not exist and L must be undecidable 1. For Example: Use turing machine to decide language {a^nb^nc^n | n >= 0} Some classical Turing undecidable languages: RL subset of CFL subset of Decidable Languages subset of Turing-recognizable languages Undecidable Languages: No deciders exist (p. All questions are undecidable for languages recognized by general Turing machines (Rice’s theorem). Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept In this lecture, we will prove that certain languages are undecidable. Active 8 years, 5 months ago. 5 Rice’s Theorem Recall that we proved A_{TM} undecidable, which means that there’s no decider that can determine whether a Turing machine accepts a given input. Answer (1 of 2): No. We proved that ATM is undecidable using a diagonalization argu-ment. A language L is decidable if and only if L is decidable. Prove that the language it recognizes is equal to the given language and that the algorithm halts on all inputs. Let MP a Turing machine with that property. 3. For any w ∉ (ℒ M), M does not accept w. Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept C2. Hence A TM is often called the halting problem. This post continues the topic of decidability, and introduces several important undecidable languages by reduction . Figure 1 shows this visually, and lists sev-eral more examples of undecidable and/or unrecognizable languages. Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept Reducibility. Proof: ( <== ) If both L and complement of L are Turing recognizable, we let Machine M1 be the Turing recognizer for L and M2 be the TM recognizer for complement of Engineering; Computer Science; Computer Science questions and answers; Which one of the following languages is recognizable but undecidable? None of the languages are recognizable but undecidable LNOT-EVEN = {((M),x): M does not accept any input w where |w| is even} LSLOW-HALT {((M),x): M halts on x after more than 376 steps } LFAST-HALT = {((M),x): M halts on x in fewer than 203 steps LEVEN Aug 09, 2004 · undecidable language (definition) Definition: A language for which the membership cannot be decided by an algorithm --- equivalently, cannot be recognized by a Turing machine that halts for all inputs. 2MM 2 ‰. Theorem The following language is undecidable. Rice's theorem: Any nontrivial property about the language recognized by a Turing machine is undecidable. If there were D accepting L⊲⊳, then what would be the class of D? BU CS 332 – Theory of Computation Lecture 15: • Undecidable and Unrecognizable Languages • Reductions Reading: Sipser Ch 4. Only 1 b. The only exceptions are the trivial questions that have only one pos-sible answer for all inputs. Define: = decidable languages 𝑬= Turing-recognizable languages co 𝑬=L തis Turing recognizable} 11/3/2020 CS332 - Theory of Computation 36 It was proved in  that it is undecidable whether a Büchirecognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). All semi-decidable+ languages are undecidable, but we’ll see there are undecidable languages that aren’t semi-decidable+! Decidable and Undecidable Languages 30-3 Dec vs. 4. Is this language: L = { M : M is a Turing machine and L ( M) is decidable } which I know that is undecidable, turing-recognizable? Is its complement recognizable? Thanks is advance. We show here that these two decision problems are actually Π1 2-complete, hence located at the second level of the analytical hierarchy, and “highly undecidable”. (Please briefly describe the reduction. True or False: If 1 is a Turing recognizable language and . We show here that Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i. Oct 10, 2021 · How do you tell if a language is decidable or undecidable? A language is decidable if and only if it and its complement are recognizable. Support of series contains all finite words which are not mapped to zero. Use them to build a TM M that runs M1 and M2 in parallel until one of them accepts (which must happen). Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept Answer (1 of 2): No. I've seen in some cases where they ask: Prove that the language $A_{TM} = \{ \ <M,w> | \ M \mbox{ is a Turing Machine accepting } w\}$ is undecidable. if w ∉ L, then M does not accept w. set” (What does this mean?) • Q: Are these sets decidable? • Q: Are there undecidable languages? 2 Rice's theorem: Any nontrivial property about the language recognized by a Turing machine is undecidable. (Undecidable languages) For each of the parts, formulate the given problem as a language and prove it is undecidable. is T-recognizable but also undecidable . The Halting Problem H is Undecidable A Language that is not Turing-recognizable We have the following results: IA language L is decidable i both L and L are Turing-recognizable. ⇒There must be an undecidable language. Show that S is undecidable. These are functions from the set of finite words over an alphabet to the integers. Prove that L 1 ∪ L 2 is also Turing Recognizable. Let C be a language. On words not belonging to the language, the computation of the TM either rejects or goes on forever. Corollary The complement H of the halting problem H isnot Turing-recognizable. Two languages L1 and L2 such that L1 mapping reducest to L2, and L1 is decidable, and L2 is undecidable. , such undecidable languages may be recursively enumerable. •As it is impossible to determine if a TM will always halt on every possible input –Note that this is Turing recognizable! We can simulate M on input w If ≤𝑚 and is not T-recognizable, then is not Turing-recognizable (by mapping-reducibility to unrecognizable language). Rice’s theorem [Problem 5. A TM is Undecidable A TM = fhB;w i j B is a TM that accepts string w g Undecidable Languages. Turing recognizable languages are closed under union and intersection. Problem whether support of a recognizable series is recognizable is undecidable. For a given M, submit its encoding hMi to M as an input. 30/33 class of languages recognized by two-dimensional Turing machines is exactly the Turing-recognizable languages. Mark Bun. 9 3. TM, the ability to solve 𝐿𝐿 enables checking the existence of an accepting computation history for 𝑀𝑀on 𝑤𝑤. Lecture 18: Important Undecidable Problems 5 Examples halts(“2+2 ”) halts x, 1x. For example: Consider the language INFINITE TM = fhMi j M is a TM and L(M) is an in nite languageg. To prove that a given language is Turing-recognizable: Construct an algorithm that accepts exactly those strings that are in the language. RE accept pipe For every language L in Dec, there is a deciding machine M that for an input string w is guaranteed to deliver a ball to either the accept pipe or reject pipe. From what we’ve learned, which closure properties can we prove for the class of T-recognizable languages? Choose all that apply. A property about Turing machines can be represented as the language of all Turing machines, encoded as strings, that satisfy that property. 3. This will either have the same language as MP , or the empty language. because by Theorem: if L is Turing Decidable if and only if both L and complement of L are Turing recognizer. guages: that is, P is a subclass of the class of Turing-recognizable languages that is neither empty nor equal to the class of all Turing-recognizable languages. For every non-deterministic Turing machine, there exists an equivalent deterministic Turing machine. ) A language may be recognizable but undecidable. is undecidable –It can only be undecidable due to a loop of M on w. (A ew" language is one neither presented nor discussed in lecture, your recitation section, or the text. Then, this follows from the closure of Turing-recognizable languages under concatenation (this was again a homework problem !!) 7. Brie y explain your answer. (Assume language nonempty, else pick complement). If is not decidable, then or ̅ is not Turing-recognizable. Problems 1. BU CS 332 – Theory of Computation Lecture 15: • Undecidable and Unrecognizable Languages • Reductions Reading: Sipser Ch 4. (Hint: use mapping reducibility. Turing decidable problems are recursive but Turing recognizable (Turing acceptable) problems are only recursively enumerable. First, it is nontrivial because some TMs have in nite languages and Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i. Proof is based on a reduction of Postʼs correspondence problem. (a) Closed under union. Part 1 (For some undecidable languages) Any non-trivial property of the LANGUAGE recognizable by a Turing machine (recursively enumerable language) is undecidable Theory of Computation | Decidability and Undecidability. present B, prove B is undecidable, prove B ≤m B. It was proved in  that it is undecidable whether a Büchirecognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). Simply, Decidable ---- always halt Recognizable ---- halt or loop Let INFINITEDFA={<A> A is a DFA and L(A) is an infinite language} . Viewed 2k times. Justify your answer. If a language is a decidable there is a TM that accepts and halts strings that belong to the language and rejects and halts strings that do not belong to the language. Hence its complement $\bar{K} = \{\langle M \rangle : M(\langle M \rangle) \text{ does not halt}\}$ is not recognizable. Only 2 c. Altenbernd, Thomas and Wöhrle have considered acceptance of languages of infinite two-dimensional words (infinite pictures) by finite tiling systems, with usual acceptance conditions, such as the Büchi and Muller ones . Complement of T-recognizable =co-T-recognizable. Recognizable Language II Corollary. In other words, if A TM was decidable, then every Turing-recognizable language would also be decidable. For example, the problem of finding the area of a rectangle reduces to the problem of multiplying the length of the rectangle by the width of the rectangle. 25]Give an example of an undecidable language $B$, where $B \le _{m} \overline {B}$. computability turing-machines. March 23, 2020 Reduction from the undecidable language 𝐴𝐴. Lemma. Then there would exist a machine M B that recognized (but did not decide) B. All semi-decidable+ languages are undecidable, but we’ll see there are undecidable languages that aren’t semi-decidable+! Decidable and Undecidable Languages 32-3 Dec vs. How do you know if a Turing machine is decidable? If a language is a decidable there is a TM that accepts and halts strings that belong to the language and rejects and halts strings that do not belong to the language. –If we could determine if it will loop forever, then could reject. Theorem: A language is decidable if and only if and ത are both Turing-recognizable. In order to earn full credit, your proof must be detail 1. Now we demonstrate a language which is not even Turing-recognizable. Decidable • A language L is Turing recognizable if some Recognizable Language II Corollary. Construct characteristic sequence of language A 2 : bit bi = 1 ifsi 2 A , and bi = 0 ifsi 2= A . Class 1 Class 2 M accepts hMi M doesn’t accept hMi Deﬁnition 1 L⊲⊳ = {hMi : M does not accept its encoding. March 23, 2020 language and show that it is undecidable. • Recall that the complement of a language L is the language L consisting of all strings that are not in the language L. Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept Aug 09, 2004 · Definition: A language for which membership can be decided by an algorithm that halts on all inputs in a finite number of steps --- equivalently, can be recognized by a Turing machine that halts for all inputs. We can define a Language over {1}* in terms of a TM:L = { < M > | M is a TM and L(M) = empty } So we can show that L is not decidable, because a TM U that receive L as a input need to test all elements over {1}* and then decide to accept in case of M rejected all of them, so it will never halt and it means that L is not decidable, implies that the empty Language is not decidable. 1. Definition of hard and of complete for a class of languages, under reductions--we saw the case where the class of languages is recognizable by a Tuting machine, and the reductions are mapping reductions or Turing reductions. Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept Turing-recognizable. 2. There are undecidable (and unrecognizable) languages over {0,1} {Turing Machines} {0,1}* {Sets of strings of 0s and 1s} {Languages over {0,1}} Set L Set of all subsets of L: 2L {Recognizable languages over {0,1}} There are (many) unrecognizable languages In the early 1900’s, logicians were trying to define consistent foundations for mathematics. Give a new undecidable language, and a short proof that it is, in fact, undecidable. True. Does every Turing-recognizable undecidable language have a NP-complete subset? The question could be seen as a stronger version of the fact that every infinite Turing-recognizable language has an infinite decidable subset. A language L is in co-RE iff there is a TM M such that if w ∈ L, then M does not reject w. . Can be used to prove undecidability of 𝐸𝐸. Now we use the undecidability of A TM to prove that other languages are undecidable; Key idea - Prove by contradiction that language L is undecidable: Assume TM B decides L ; Then use B to create a decider for A TM (or another undecidable language) This is a contradiction, and so B must not exist and L must be undecidable Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i. Are there problems that cannot be solved by any algorithm? Consider the language: ATM = {<M,w> | M is a TM and M accepts w} NOTE: <A,B,… > is just a string encoding the objects A, B, … Feb 19, 2018 · I was wondering what is the difference difference undecidable language and Turing recognizable language. 12) Let S = {hMi | M is a TM that accepts wR whenever it accepts w}. (a)(Sort-checker TM) A TM is a sort-checker if it accepts a string if and only if this string Proof: 𝐴TM is T-recognizable but also undecidable decidable. Undecidable Problems 1 Lecture 15 Andrew Black Andrew Tolmach Monday, 24 May 2010. Or: Construct a (mapping) reduction from another language already known to be non-Turing-recognizable to the given language. \\ \textbf {Solution:} Any Turing-recognizable but not co-Turing-recognizable language works (or vice versa), such as $A_{TM}$. A reduction involves two problems, A A and B B . 9) Show that all Turing-recognizable problems mapping reduce to ATM. if the language L of all yes instances to P is decidable. It was proved in  that it is undecidable whether a Büchi-recognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). 2, 5. There are languages which are recognizable, but not decidable Recognizing A tm Decidable and Recognizable Languages IButnot all languages are decidable! We will show: I A tm = fhM;wijM is a TM and M accepts wgis undecidable IHowever A tm isTuring-recognizable! Proposition There are languages which are recognizable, but not decidable If ≤𝑚 and is not T-recognizable, then is not Turing-recognizable (by mapping-reducibility to unrecognizable language). Construct M 1 as follows: On input x: run A on w , (run forever or reject like A does) run MP on x, accept ifMP does. 1. Classes of Languages. All of 1, 2, and 3 11. Also known as recursive language, totally decidable language. , INF is not co-Turing-recognizable. 𝐴TM 𝐴TM Check-in 8. A language is Turing-recognizable (or recursively enumerable) if it is recognized by a TM. A TM recognizing C operates on input x by going through each possible string y and testing whether hx;yi2D. Prove that its complement is undecidable. First, it is nontrivial because some TMs have in nite languages and Note that, L might be recognized by other TM M’ that does not always halt. ". In other words, language B is decidable iff B is both Turing-recognizable and co-Turing-recognizable. There are more languages than Turing machines) some languages are not recognizable (thus also undecidable). A TM is not Turing recognized. True or False: Every subset of every regular language is recognizable. LBA Now we use the undecidability of A TM to prove that other languages are undecidable; Key idea - Prove by contradiction that language L is undecidable: Assume TM B decides L ; Then use B to create a decider for A TM (or another undecidable language) This is a contradiction, and so B must not exist and L must be undecidable Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i. 28] says that it is undecidable, given a Turing machine M, whether the language M recognizes has property P. Corollary: If is Turing-recognizable and undecidable then തis not Turing-recognizable. Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept language and show that it is undecidable. For Example: Use turing machine to decide language {a^nb^nc^n | n >= 0} Some classical Turing undecidable languages: \All" questions are decidable for regular languages { the known counter-examples are somewhat \arti cial" problems. The basic idea is: The set $K = \{\langle M \rangle : M(\langle M \rangle) \text{ halt }\}$ is recognizable but not decidable. For ⇐, we have TM’s M1 and M2 that recognize L, L respectively. Turing decidable languages are closed under intersection and complementation. 26 % 1 Undecidable Languages. 11 shows that recognizers are more powerful than deciders Requiring a TM to halt on all inputs restricts the kind of languages that it can recognize Decidable Problems Concerning Context-Free Languages – p. that undecidable language to the given language). (e) Closed under star. May 03, 2020 · What is an undecidable language? (definition) Definition: A language for which the membership cannot be decided by an algorithm — equivalently, cannot be recognized by a Turing machine that halts for all inputs. to a language 𝐿𝐿using the following idea: Given an input 〈𝑀𝑀,𝑤𝑤〉to 𝐴𝐴. It must either reject or loop on any string not in the language. In order to earn full credit, your proof must be detail oriented. If M1 accepts M accepts too; if M2 accepts, M rejects. Undecidable Languages. IThe halting problem H is Turing-recognizable but not decidable. 1 An Undecidable but Recognizable Language Decidable and Recognizable Languages But not all languages are decidable! In the next class we will see an example: { A tm = fhM;wijMis a TM and w2L(M)gis undecidable However A tm is Turing-recognizable! Proposition 2. 0 Undecidable Problems (unsolvable problems) Slide 2 Slide 3 Slide 4 Slide 5 Slide 6 Slide 7 Slide 8 Slide 9 Slide 10 Slide 11 Slide 12 Slide 13 Slide 14 Undecidable Languages Slide 16 Slide 17 Slide 18 Slide 19 Slide 20 Slide 21 Slide 22 Slide 23 Slide 24 Slide 25 Slide 26 Slide 27 Slide Sep 26, 2015 · All undecidable problems are NP-Hard, but all NP-Hard problems are not undecidable. Check-in 8. A language that is enumerable, but not enumerable in lexicographic order. is a decidable language, then 5 < must be Turing-recognizable. Only 3 d. PowerPoint Lecture 5; 4/9 Q/A A language that is co-Turing-recognizable but not Turing-recognizable. are both Turing-recognizable. The set of all FSAs that recognize undecidable languages. L is said to beTuring-recognizable(or simply recognizable) if there It was proved in  that it is undecidable whether a Buchirecognizable language of infinite pictures is E-recognizable (respectively, A-recognizable). 9 Prove that the language it recognizes is equal to the given language and that the algorithm halts on all inputs. For an undecidable language, there is no Turing Machine which accepts the language and makes a decision for every input string w (TM can make decision for some input string though). 16. A decider that recognizes language L is said to decide language L Recall: Language of a TM The language of a Turing machine M, denoted ℒ(M), is the set of all strings that M accepts: ℒ(M) = { wM accepts w} For any w ∈ (ℒ M), M accepts w. 3 . ) Apr 30, 2011 · Research highlights We consider recognizable series over the semiring of the integers. Either w ∈ L, or w ∈ L . COT 4210 Final Exam Part C: TMs, Decidable/Undecidable Languages 5/4/2021 Regular Start Time: 2:20 pm (EDT) Regular End Time: 3:00 pm (EDT) Regular Late Time: 3:10 pm (EDT) 1) (8 pts) Let L 1 and L 2 be Turing Recognizable languages. RE Turing Machine M for a language L in Dec accept pipe reject pipe input string w Turing Machine M for a language L in RE Engineering; Computer Science; Computer Science questions and answers; Which one of the following languages is recognizable but undecidable? None of the languages are recognizable but undecidable LNOT-EVEN = {((M),x): M does not accept any input w where |w| is even} LSLOW-HALT {((M),x): M halts on x after more than 376 steps } LFAST-HALT = {((M),x): M halts on x in fewer than 203 steps LEVEN 2. Clearly this is a bijection. • Corollary: “The Turing Machines that halt on some input are an r. A Turing machine M is said todecidea language L if L = L (M ) and M halts on every input. • Corollary: HALT TM = {hM,wi : M eventually halts on w} (“The Halting Problem”) is Turing-recognizable. Prove the following results about INF. Recall: Recognizable vs. Aug 09, 2004 · undecidable language (definition) Definition: A language for which the membership cannot be decided by an algorithm --- equivalently, cannot be recognized by a Turing machine that halts for all inputs. ) 2. Proof: in class Theorem. It satis es the two conditions of Rice’s theorem. An Undecidable but Recognizable Language Complementation Decidable and Recognizable Languages Recall: De nition A Turing machine M is said torecognizea language L if L = L (M ). That is, all words in the language are accepted by the TM. Proof: ( <== ) If both L and complement of L are Turing recognizable, we let Machine M1 be the Turing recognizer for L and M2 be the TM recognizer for complement of COT 4210 Final Exam Part C: TMs, Decidable/Undecidable Languages 5/4/2021 Solution 1) (8 pts) Let L 1 and L 2 be Turing Recognizable languages. In fact, we saw that the language of a universal Turing machine (an interpreter) $$U$$ is $$L(U) = \atm$$. ) (a) (20 points) INF is not Turing-recognizable (i. Times New Roman Comic Sans MS class Microsoft Equation 3. Proof: 𝐴TM is T-recognizable but also undecidable decidable. If L is recognizable, then there might be such TM M that recognizes L but run forever, rather than rejecting, some inputs not in L. Recognizable language A language L for which there exists a Turing Machine that will accept when given input w[in]L. 25) Give an example of an undecidable language B, where B m B . There are two main techniques for doing so: the ﬁrst is a technique called diagonalization, and the second is called reductions . T-recognizable. Observe that the requirement for a recognizable language is less stringent than the decidability requirements. Only 1 and 3 e. Flipping the accept and reject states generates a TM to decide the complement of this language. (Sipser 5. 202) A TM = {<M,w>| M is a TM and M accepts w} Notional Proof: Assume H a decider for A TM Accept if M accepts w Reject if M does not accept w Define D as machine with inputs <M> Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i. (30 points) We can use Rice’s theorem to show that a language is undecidable. I The E TM problem is a special case of the EQ TM problem wherein one of Any language outside Dec is undecidable. • ATM is not Turing But this intersection is exactly L, the language shown to be unrecognizable, and thus surely undecidable, in the previous proof. So L is uncountable. L is said to beTuring-recognizable(or simply recognizable) if there Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i. is undecidable is however Turing-recognizable Hence, Theorem 4. TM was decidable, but that some other undecidable language Bwas Turing-recognizable. Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept We give also the answer to two other questions which were raised by Altenbernd, Thomas and Wohrle, showing that it is undecidable whether a Buchi recognizable language of infinite pictures is E \All" questions are decidable for regular languages { the known counter-examples are somewhat \arti cial" problems. Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept Undecidable Languages. It might loop forever, or it might explicitly reject. Identifying languages (or problems*) as decidable, undecidable or partially decidable is a very common question in GATE. ‣Proof: ⇒is obvious. Undecidable Languages The Question: Are there languages that are not decidable by any Turing machine (TM)? i. Proof. Hierarchy of undecidable languages. if w ∉ L, then M rejects w. We show here that these two decision problems are actually P$^{1}_{2}$-complete, hence located at the second level of the analytical hierarchy, and "highly undecidable". If 𝐽 is undecidable and 𝐽≤𝑚𝐽, then both ̅ 𝐽 and 𝐽 ̅are not Turing-recognizable. % 5. (b) Closed under intersection. • A language L is decidable ⇔both L and L are Turing-recognizable. Are all undecidable languages recognizable? In other words, if we give up the requirement of always halting, is there a machine associated with every possible Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i. Reducibility refers to the act of using the solution to one problem as a means to solve another. (20 points) (Sipser 5. So if A TM were Turing recognized, then A TM would be decidable giving a contradiction with the halting problem being undecidable. a. The set of languages over the alphabet {0, 1}. Prove that C is Turing-recognizable i a decidable language D exists such that C = fxj9y(hx;yi2D)g Solution: We need to prove both directions. Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept Q1: Which of the following claims are true?* 1 point The recognizable languages are closed under union and intersection The decidable languages are closed under union and intersection The class of undecidable languages contains the class of recognizable languages For every language A, at least one of A or A*c is recognizable Other: This is a Co-Turing Recognizable Language is co-Turing recognizable if it is the complement of a Turing-recognizable language Theorem: Language is decidable if it is Turing-recognizable and co-Turing recognizable Thus, for any undecidable language L, either L or L’ is not Turing-recognizable •Is A TM’ Turing-recognizable? 30 Reducibility Theorem: A language is decidable if and only if and ത are both Turing-recognizable. Solution Since Let L 1 and L 2 be Turing Recognizable languages, there must exist Language B is decidable iff both B and B are T-recognizable. In this lecture, we will prove that certain languages are undecidable. With correct knowledge and ample experience, this question becomes very easy to solve. Not all Recognizable languages are closed under complement. Turing recognizable languages are closed under union and complementation. (c) Closed under complement. (d) Closed under concatenation. Language B is decidable iff both B and B are T-recognizable. e. Let $M_x$ denote the Turing machine with code $x$. Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i. “Decidable” L(M) –“language recognized by M” is set of strings M accepts Language is Turing recognizable if some Turing machine recognizes it •Also called “recursively enumerable” Machine that halts on all inputs is a decider. Researchers with an interest in Turing machines, for example, have tackled the issue of the halting problem, looking at . ) 3. is, in particular, recognizable. A TM M of this sort is called a recognizer, and L is called recognizable. The only subset that is not decidable in {1}* is the empty set. 6. To handle the easier one rst, assume that the decidable language D exists. We could clearly construct a decider for Bby running M A TM on hM B;wi. Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept Then P is undecidable. Closed under intersection. An undecidable problem is a question that cannot be resolved with the use of one algorithm. A language L is in RE iff there is a TM M such that if w ∈ L, then M accepts w. To prove that a given language is non-Turing-recognizable: Either do both of these: Prove that its complement is Turing-recognizable. Undecidable Problems from Language Theory: EQ TM The idea is simple: if EQ TM were decidable, E TM also would be decidable, by giving a reduction from E TM to EQ TM. “Turing recognizable” vs. EQ TM = fhM 1;M 2ijM 1;M 2 are TMs and L(M 1) = L(M 2)g. An existential proof. Corollary (the acceptance problem ): The language AP of all strings (v,w) where v is an algorithm accepting the word w is recognizable but undecidable. Closed under union. Nov 08, 2016 · Turing Recognizable Language (recursive enumerable): for all words in this language L, we can use a turing machine to accept all the words in L; and for all the words not in L, the turing machine might reject it or not halt. recognizable but undecidable language