Examples of undecidable languages The language Let us dive into the Languages further. The language Language A DFA is Decidable Language. So I don't see how this question should be asked in this form. No algorithm exists to solve undecidable problems for all instances. Decidable and Undecidable Languages 37/38-4 Dec vs. Modified 4 years, 7 months ago. • If problem P reduces to problem Q, and P is undecidable, then Q is undecidable! ‣Otherwise, we could use Q to decide P. Recognition of context-free languages is provided as an example. 1,2,3,4 B. g. It is a well known and the most common example of an undecidable language. Dec 12, 2024 · Decidability in computer science refers to the ability of an algorithm to determine whether a given problem can be solved or not, while undecidability means that no such algorithm can decisively resolve the problem for all inputs. ” Formally, Definition 6. Now, let us define B as: Here, T is also a Turing Machine for some language. A non-Recursively Enumerable Language Diagonalization: Cantor Proposition 3. Thus the language is in R. Since SA TM is undecidable (Theorem(Turing We’ll soon see examples of languages th ar einREbu o D c. Let us call it $\mathcal{H}$. - Undecidable problems have no Turing machine that halts on every input. Finally, it Run!(<!>),thisresultsinacontradiction: ’ ’ !(<!>)’accepts’’⇒’’!(<!>)’rejects’ ’ !(<!>)’rejects’⇒’’!(<!>)’accepts’ Language Decidability: Determines whether a language can be decided by a Turing machine, involving concepts of recognition and decidability. Theorem 3. num=1; while(num=0) { num=num+1; } It counts up forever since it will never equal 0. Language Decidability: Determines whether a language can be decided by a Turing machine, involving concepts of recognition and decidability. - Semi-decidable problems are ones where a Turing machine halts on accepting inputs but may loop on rejecting inputs. Find whether the problem given below is Mar 17, 2025 · Otherwise, the class of problems is said to be unsolvable or undecidable. Problem. Decidable and Undecidable Languages: Decidable languages have a definitive decision process via Turing machines, whereas undecidable languages, like the Halting Problem, do not. RE languages or type-0 languages are generated by type-0 grammars. Koether (Hampden-Sydney College)The Acceptance Problem - Undecidable Languages Fri, Nov 7, 2014 22 / 25 In the previous lecture we saw a few examples of undecidable languages: DIAG is not semidecidable, and therefore not decidable, while A DTM and HALT are semide-cidable but not decidable. [2] Those inputs can be numbers (for example, the decision problem "is the input a prime number?") or values of some other kind, such as strings of a formal language. Is there a notable unrecognizable language, in the same sense that HALT is a notable undecidable language? Lets prove two theorems about closure to show the answer is yes. On the other hand, statically typed programming languages feature a type system which statically prevents type errors. Rice’s Theorem. e. LTMaccept is undecidable. A decision problem P is decidable if the language L of all yes instances to P is decidable. Apr 25, 2017 · Example of an undecidable language. By the Church-Turing thesis, any effective model of computation is equivalent in power to a Turing machine. We will also introduce the notion of a reduction from one language to another, which can Feb 24, 2021 · Examples of undecidable problems include the halting problem and determining if a Turing machine accepts a language that is not recursively enumerable. That isn't the case, though, and this question explores why. 1. In computability theory, an undecidable problem is a decision problem for which an effective method (algorithm) to derive the correct answer does not exist. many languages and only countably many recognizable languages, so most languages are unrecognizable. We will use this decider to come up with a decider for SA TM. Note: Every context-free language is decidable. Consider the following language \( L \) Dec 22, 2022 · A language is undecidable if it is not decidable. the set of strings (Turing Machine = M, string = s) such that M accepts s. Some of the other undecidable problems are: A decision problem is a question which, for every input in some infinite set of inputs, requires a "yes" or "no" answer. Example of an Undecidable Problem. Undecidable Undecidable Undecidable Undecidable Undecidable Lecture 17: Proving Undecidability 22 Next Class •Examples of some problems we actually care about that are undecidable •Are there any problems that we don’t know if they are decidable or undecidable? •PS5 Due next Tuesday (April 1) •Exam 2 in two weeks Feb 7, 2025 · Examples of undecidable languages include the Halting Problem and the Post Correspondence Problem. In summary, decidable problems in the theory of automata can be solved using an algorithm, while undecidable problems do not have an algorithmic solution. A TM is not only the rst language in this class that we’ve seen, but also the most important. Here are some examples of undecidable problems: Jun 28, 2021 · If L is recursive language then L’ is also recursive? A. Also M' with input T must run M on input (1,T) . Decidable languages, on the other hand, are languages for which there exists an algorithm that May 28, 2023 · Undecidable problems lack a definite answer for all instances. Jan 25, 2025 · Here is an example of using the many-one reducibility to show the undecidability of a language. Robb T. With correct knowledge and ample experience, this question becomes very easy to solve. languages and the set of all infinite binary strings. The question is, can we build a Turing Machine that, given the encoding of another Turing Many of the examples we've seen of undecidable languages involve checking for properties of Turing machines or computer programs, which might give you the sense that every question you might want to ask about TMs or programs is undecidable. Or: Construct a (mapping) reduction from another language already known to be non-Turing-recognizable to the given language. Here is a simple example of a proof by reduction. Feb 6, 2023 · Examples given include checking if a number is prime or if a string is in a regular language. Suppose, if a language is not even partially decidable, then there is no Turing machine that exists for the respective language. More This document discusses examples of undecidable languages in computability theory. A language is in R iff there is an algorithm for Aug 10, 2019 · Well, probably you have already heard about the halting problem. Algorithms: Decidable problems have algorithms that solve them efficiently. Lecture 12: Undecidable Languages example of this is the set of all infinite-length binary strings. This shows H is undecidable. We want to show that SA TM is undecidable. • So must show how a TM that decides HALTTM can be used to decide ATM. Universal Turing machines Please go through the below videos first, then the given examples and for more examples see the reference link. For example, consider the language $\{0, 1, 00\}$. By the Church-Turing Thesis, these results highlight the inherent limitations of computation. Theorem: L u is RE but not recursive. The crucial idea is diagonalization. Feb 14, 2025 · Rice’s Theorem: Every non-trivial property of the RE languages is undecidable. To prove that a given language is non-Turing-recognizable: Either do both of these: Prove that its complement is Turing-recognizable. . See full list on cs. Example 1. Prove that its complement is undecidable. com/@varunainashotsIn this video Decidability & Undecidability table in toc for all languages is explained The standard example of an undecidable language is: LTMaccept = f< M;w >j M is a TM and M accepts wg Theorem. Then there would exist a machine M that undecidable language to the given language). W l these languages semi-decidable+. Aug 3, 2023 · Identifying languages (or problems*) as decidable, undecidable or partially decidable is a very common question in GATE. The language SA TM is undecidable. The proof is by contradiction, so assume SA TM is decidable and let M SA be a decider for it. More formally, an undecidable problem is a problem whose language is not a recursive set; see the article Decidable language. 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 May 28, 2023 · Check out some more examples of Undecidable Problem. The most well-known undecidable problem is the Halting Problem, which asks whether a given program will stop running (halt) or continue running forever for a particular input. It proves that ADTM is Turing-recognizable but undecidable. Thus, the set of all languages is uncountable. Let’s be clear about what this does and does not say. Assume that ATM was decidable but that some other undecidable A “language” is a set of strings Any “problem” can be expressed as a set of all strings that are of the form: “<inputoutput><input, output>” e. We will study the Halting problem. Jul 17, 2020 · Thanks for contributing an answer to Computer Science Stack Exchange! Please be sure to answer the question. Undecidability of Universal Languages: The universal language L u is a recursively enumerable language and we have to prove that it is undecidable (non-recursive). Proof. Find out whether the following problem is decidable Feb 13, 2025 · Recursive Enumerable (RE) or Type -0 Language. If a language is not even partially decidable , then there exists no Turing machine for that language. edu Undecidable Languages in Automata Theory - Explore the concept of undecidable languages in automata theory, including key examples and their implications in computation. We call such languages ”properties of recognizable languages. In computer science, understanding decidable and undecidable languages helps identify what can be computed efficiently and what cannot. P is a property of recognizable languages if P ⊂{ M |M is a TM, and if L(M 1) = L(M Jan 31, 2016 · Wikipedia contains examples, and it's quite clear (assuming some mastery of the fundamentals) that the complents of all undecidable yet semi-decidable languages are examples as well. A less "constructive" proof that there exist unrecognizable languages is that there are uncountably many languages and countably many TMs. Compiler Design Playlist: https://www. Thus if there is any algorithm for deciding membership in the language, there must be a decider for it. Undecidable Languages The Question: Are there languages that are not decidable by any Turing machine (TM)? i. It Mar 10, 2025 · A recursively enumerable language is a formal language that can be generated by an algorithm or a Turing machine. Nov 22, 2019 · Example of two undecidable languages that cannot be mapping reduced to each other. If B is decidable, then a Turing Machine M for language B will show the existence of a TM M' which will decide H . com/playlist?list=PLXj4XH7LcRfC9pGMWuM6UWE3V4YZ9TZzM----- We can now conclude that the set of recognizable (but not decidable) languages contains at least one member. In this lecture we will see a few more examples. 3 More Undecidable Languages Theorem (SA TM is undecidable). #decidability_and_undecidability#abhilashav#automata Apr 17, 2020 · 👉Subscribe to our new channel:https://www. Post's correspondence problem and its modified version are presented with examples. De ne L d = fhMijhMi62L(M)g. This means that our Turing Machine is Recognizable, but it is not decidable. undecidable languages •We first introduce the diagonalization method, which is a powerful tool to show a language is undecidable •Afterwards, we give examples of undecidable languages that are –Turing recognizable but not decidable –Non-Turing recognizable Objectives undecidable languages •We first introduce the diagonalization method, which is a powerful tool to show a language is undecidable •Afterwards, we give examples of undecidable languages that are –Turing recognizable but not decidable –Non-Turing recognizable Objectives Apr 14, 2015 · But for strings not in the language (the first given machine cannot generate all the strings the second one can), our machine may halt and reject, or may never halt. Decidable; Undecidable; A problem is called decidable, when there is a solution to that problem and also can construct algorithms corresponding to that. Provide details and share your research! But avoid …. This is an example of the halting problem. 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, … In this chapter, we formally prove that almost all languages are undecidable using the countability and uncountability concepts from a previous chapter. 2,3,4 D. Ask Question Asked 5 years, 6 months ago. Language A DFA is defined as follows: { (M ,w) : M is a Deterministic Finite Automaton (DFA) that accepts the string w } Therefore, in this language, DFA is enough. Proof: Consider that language L u is recursively Apr 8, 2021 · Prerequisite - Undecidability, Decidable and undecidable problems Identifying languages (or problems*) as decidable, undecidable or partially decidable is a very common question in GATE. Undecidable problems often pose significant challenges in computer science, pushing the boundaries of what can be computed. Language A NFA. Common examples of undecidable problems include the Halting Problem, Post Correspondence Problem, and Word Problem for Groups. Asking for help, clarification, or responding to other answers. Part 1 (For some undecidable languages) Informally, if we show that being able to decide language \(L\) implies that we can decide another language that is already known to be undecidable, like \(A_{TM}\), then this proves that \(L\) must also be undecidable. Examples: Language membership, regular expression equivalence, sorting. HALTTM is undecidable • HALTTM = { M,w ⃒M is a TM and M halts on input w} • Proof is by reduction from ATM. 1 Diagonalization The Diagonal Language De nition 2. Oct 1, 2024 · Is a language Learning which is a CFL, regular: This is an Undecidable Problem as we can not find from the production rules of the CFL whether it is regular or not. Every TM for a semi-decidable+ language halts in the accept state for strings in the language but loops for some strings not in the language. The Halting problem is the most famous of the undecidable problems. That is, they are languages of the form { M |M is a TM and L(M) satisfies}. Example 8. An undecidable language maybe a partially decidable language or something else but not decidable. Thus, L d is the collection of Turing machines (programs) M such that M does not halt and accept when given itself as input. Halting problem, Post Correspondence Problem, Collatz Conjecture Any language outside Dec is undecidable. We will also introduce the notion of a reduction from one language to another, which can Nov 28, 2024 · 1. Assume that A TM was decidable, but that some other undecidable language Bwas Turing-recognizable. Option 1 is also undecidable as there is no TM to a language, that language is in R. For example, Rice's theorem implies that in dynamically typed programming languages which are Turing-complete, it is impossible to verify the absence of type errors. It begins by introducing the language ADTM, which is the set of pairs (M,w) where M is a Turing machine that accepts string w. We also present (with proofs) several explicit examples of undecidable languages. 1or when the context is clear just L(TM) 15: Undecidability by Reduction-1 There are two types of languages in the theory of computation (TOC), which are as follows −. An undecidable language may be partially decidable but not decidable. Language A DFA. Context free languages are not closed under complementation, option 2 is undecidable. EQ TM = { M1, M2 | where ℒ(M1) = ℒ(M1) } { M | M loops on at least five strings } – Note that we can write a verifier for the language { M | M accepts at least five strings }, using the certificate Oct 30, 2022 · undecidability || undecidability in theory of computation || undecidable problems || decidable and undecidable problems || decidable languages || decidable a In the previous lecture we saw a few examples of undecidable languages: DIAG is not semidecidable, and therefore not decidable, while A DTM and HALT are semide-cidable but not decidable. wellesley. Another example is the problem, "given two numbers x and y , does x evenly divide y ?" A decision procedure for a decision problem is an algorithmic method that answers the yes-no question on all inputs, and a decision problem is called decidable if there is a non-CE languages suppose the set of all CE languages is countable list characteristic vectors of all languages according to the bijection f: n f(n) 10101010::: 2 1010011::: 3 1110001::: 4 0100011::::: create language L with membership string x where i-th digit of x 6= i-th digit of f(i) x cannot be in the list! therefore, the language L is not Assume H : Language of all the Turing Machine which halts on empty input. It is recognizable and not co-recognizable. We do not need stronger models. youtube. L An example is the Halting Problem, which determines whether a program will halt. The proof (to be gone through in class) shows that, in fact, the more restricted language Lselfaccept = f< M;< M >>j M is a TM g is undecidable. , Problem (a+b) ≡ Language of strings of the form { “a#b, a+b” } ==> Every problem also corresponds to a language!! 11 Think of the language for a “problem” == a verifier for the (Lava Diagram Category “All Languages”) Example languages: The complements of undecidable RE languages. For a decidable language, for each input string, the TM halts either at the accept or the reject state as depicted in the following diagram −. RE Turing Machine M for a language L in Dec accept pipe Languages Figure 1: Relationship between classes of Languages 1. 3,4 Explanation: As regular and recursive languages are closed under complementation, option 3 and 4 are decidable problems. 1,2 C. Every decidable language is Turing-Acceptable. Jun 14, 2021 · A decision problem P is undecidable if the language L of all yes instances to P is not decidable. Sep 25, 2016 · An example of an undecidable language is the set "acceptance problem," i. See the Encyclopedia of Mathematics for more on recognizable and undecidable languages (specifically Note that many of the undecidable languages we have learned about fit a common pattern. An RE language can be accepted or recognized by Turing machine which means it will enter into final state for the strings of language and may or may not enter into rejecting state for the strings which are not part of the language. 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. enumerable languages is undecidable. Language A NFA is defined An example of a decision problem is deciding whether a given natural number is prime. In this topic you will see Decidability table and shortcut to learn them. It then introduces the halting problem language HALT and proves it is also undecidable but Turing-recognizable. The language may be infinite, and not all strings can be verified to belong in the language. Consider the code. This does not mean that all non-trivial questions about specific languages are undecidable. iubc haayaco anmtamn cykim ffjbfa beflih avkmnb cvp iujsts ddujr
© Copyright 2025 Williams Funeral Home Ltd.