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# Questions tagged [halting-problem]

Questions concerning the Halting problem which is to decide whether a given a program halts on a given input.

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### Why, really, is the Halting Problem so important?

I don't understand why the Halting Problem is so often used to dismiss the possibility of determining whether a program halts. The Wikipedia article correctly explains that a deterministic machine ...
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### What are the simplest examples of programs that we do not know whether they terminate?

The halting problem states there is no algorithm that will determine if a given program halts. As a consequence, there should be programs about which we can not tell whether they terminate or not. ...
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### Human computing power: Can humans decide the halting problem on Turing Machines?

We know the halting problem (on Turing Machines) is undecidable for Turing Machines. Is there some research into how well the human mind can deal with this problem, possibly aided by Turing Machines ...
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### Algorithm to solve Turing's "Halting problem‍​"

"Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist" Can I find a general algorithm to solve the halting problem for ...
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### Is there a more intuitive proof of the halting problem's undecidability than diagonalization?

I understand the proof of the undecidability of the halting problem (given for example in Papadimitriou's textbook), based on diagonalization. While the proof is convincing (I understand each step of ...
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### Are there any existing problems that wouldn't be solvable with a halting oracle?

I understand that most problems are trivial if a halting oracle is available (or, I think equivalently, hyper-computation). However, applying the argument that shows the Halting Problem is impossible ...
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### Halting problem reducing to the blank tape halting problem

I was going through my book of proof and I find very confusing its definition, so I would like someone to help me in understanding this. The blank tape problem takes a machine and an empty tape and ...
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### Why can't we solve the Halting Problem by using Artificial Intelligence? [duplicate]

Yesterday I was reading about Computability and they mention the Halting Problem. It got stuck in mind all day until I remember that some weeks ago, when learning Java, the IDE (Netbeans) show me a ...
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### Are all undecidable/uncomputable problems reducible to the Halting problem? [duplicate]

Theory of computation tells us that there are some languages that cannot be recognized by a Turing machine. That is, there are well-defined problems for which no Turing machines can provide an ...
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### Is the language of Turing Machines that halt on every input recognizable?

I am trying to reduce the complement of the HALTING problem (WLOG, the complement of the HALTING problem is the language of TMs that loop on some string w)to this language in order to show that it is ...
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### Is there an always-halting, limited model of computation accepting $R$ but not $RE$?

So, I know that the halting problem is undecidable for Turing machines. The trick is that TMs can decide recursive languages, and can accept Recursively Enumerable (RE) languages. I'm wondering, is ...
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### Is there any concrete relation between Gödel's incompleteness theorem, the halting problem and universal Turing machines?

I've always thought vaguely that the answer to the above question was affirmative along the following lines. Gödel's incompleteness theorem and the undecidability of the halting problem both being ...
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### Is there a connection between the halting problem and thermodynamic entropy?

Alan Turing proposed a model for a machine (the Turing Machine, TM) which computes (numbers, functions, etc.) and proved the Halting Theorem. A TM is an abstract concept of a machine (or engine if ...
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### Are there programs that never halt and have no non-termination proof?

Like black holes in computer science. We can only know they exist but when we have one of them we will never know it's one of them.
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### Does a never-halting machine always loop?

A Turing machine that returns to a previously encountered state with its read/write head on the same cell of the exact same tape will be caught in a loop. Such a machine doesn't halt. Can someone ...
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### Why is the halting problem decidable for LBA?

I have read in Wikipedia and some other texts that The halting problem is [...] decidable for linear bounded automata (LBAs) [and] deterministic machines with finite memory. But earlier it is ...
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### Program synthesis, decidability and the halting problem

I was reading an answer to a recent question, and sort of a strange, ephemeral thought came to mind. My asking this might betray either that my theory chops are seriously lacking (mostly true) or that ...
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### How to prove the existence of a number which cannot be written by any algorithm?

I have the problem: Show that there exists a real number for which no program exists that runs infinitely long and writes that number's decimal digits. I suppose it can be solved by reducing ...
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### Does the proof of undecidability of the Halting Problem cheat by reversing results?

I have trouble understanding Turing's halting problem. His proof assumes that there exists a magical machine $H$ which could determine whether a computer would halt or loop forever for a given input. ...
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### Why is the halting problem semi-decidable?

This is what is known about the halting problem and semi-decidability :- The halting problem says that for a given input x and a machine H, we can't say whether the machine H halts or not on input x. ...
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### Halting problem - one issue that's bothering me

To my knowledge, halting problem asks if there exists a program that decides whether a program being tested, given some input data (no matter what program it is, or what input data we give) will ...
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### If the Halting Problem was solvable, and we solved it, what would be its implications?

Perhaps a way to better understand the Halting Problem's importance is to know what would happen or what could be possible if this was solved. What would be the Halting Problem's implications in today'...
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### Is the halting problem specific to Turing machines?

The proofs that the halting problem is undecidable seem to make very few assumptions about the kind of program/machine under consideration: just that the programs take one input and either loop or ...
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### Possible to construct a probabilistic halting problem solver?

I'm a CS undergrad so my math/CS knowledge is not that deep so please correct me if my premise is flawed or I have made some incorrect assumptions. So I was thinking, much in the way that some ...
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### Does Church-Turing thesis also apply to artificial intelligence?

By Church-Turing's thesis, it is impossible to design an algorithm to decide the halting problem. Does the word algorithm in this context include artificial intelligence or not, that is, does ...
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### Why is the halting problem unsolvable by a turing machine? [duplicate]

So my knowledge of CS is amateurish at best but to me, logically, it seems like the halting problem is solvable. So any human can determine if a problem halts with rigorous inspection, so why can't a ...
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### Does the Halting Problem prove that true Artificial Intelligence is impossible?

The Halting Problem demonstrates that there are things that a machine can never be programmed to do. Is this proof that true Artificial Intelligence - that is, the ability for a machine to think and ...
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### Is the below language Non R.E?

$L_0=\{\langle M,w,0\rangle\mid M \text{ halts on } w\}$ $L_1=\{⟨M,w,1⟩\mid M \text{ does not halts on } w\}$ Here $\langle M,w,i \rangle$ is a triplet, whose first component $M$ is an encoding of a ...
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1 vote
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### Prove that $H$ reduces to $H\varepsilon$

I have to prove that $H_\varepsilon = \{<M> \mid M\ \text{halts on input }\varepsilon\}$ reduces to $H$ (the halting problem). I am very confused how to PROVE it, I mean it is clear that we can ...
1 vote
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### Reducing the infinite language problem to halting problem

Let: $INF = \{ w \in \Sigma^* | \quad |L(M_w)| = \infty \}$. It is easy to show with Rices theorem that $INF$ is not decidable. ($INF$ is non-trivial because of $\emptyset$ and $\Sigma^*$). How ...
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### Complement of halting set is not r.e

suppose we don't know that Halting problem is not recursive. I want to prove that complement of halting set is not r.e. then we can find halting problem is not recursive. Can you direct prove that ...
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### Is the halting problem decidable for pure programs on an ideal computer?

It's fairly simple to understand why the halting problem is undecidable for impure programs (i.e., ones that have I/O and/or states dependent on the machine-global state); but intuitively, it seems ...
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### Turing machine + time dilation = solve the halting problem?

There are relativistic spacetimes (e.g. M-H spacetimes; see Hogarth 1994) where a worldline of infinite duration can be contained in the past of a finite observer. This means that a normal observer ...
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### Is it provably true/false that for a program, there exists a proof whether it halts or not?

A standalone statement of my question Given a program that takes no argument, we are interested in whether the program will eventually terminate. My question is this: Theoretically speaking, can we ...
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### Halting problem without self-reference

In the halting problem, we are interested if there is a Turing machine $T$ that can tell whether a given Turing machine $M$ halts or not on a given input $i$. Usually, the proof starts assuming such a ...
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### Can a Turing Machine (TM) decide whether the halting problem applies to all TMs?

On this site there are many variants on the question whether TMs can decide the halting problem, whether for all other TMs or certain subsets. This question is somewhat different. It asks whether ...
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### Halting Problem without self-reference: why does this argument not suffice (or does it)?

I'm trying to find a way to explain the idea of the Halting Problem proof in as accessible a manner as possible (to undergrad CS students). The simplest argument I have found is this one; this is ...
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### Gödels (first) incompleteness Theorem and the Halting Problem - How limiting is it?

When I first heard of these things I was very fascinated as I thought it sets really a limit to mathematics and science in general. But how practically relevant are these things? For the Halting ...
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### How is Turing's Solution to the Halting Problem Not Simply "Failure By Design"?

I'm having a hard time viewing Turing's solution to the Halting Problem as a logician, rather than as an engineer. Here is my understanding of the Halting Problem: Let $M$ be the set of all ...
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### Detecting loops in NPDAs

I'm aware that the halting problem is solvable for PDAs, but I have recently discovered that I am wrong about how to actually do it. I used to think that you could detect an infinite loop by meeting ...
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### Understanding proof for Busy Beaver being uncomputable

I found this proof on http://jeremykun.com/2012/02/08/busy-beavers-and-the-quest-for-big-numbers/ and have highlighted the part I don't understand in bold. (BB(n) is defined as the number of steps ...
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### If modern computers aren't actually Turing-complete, does that mean that it is possible to determine if a program run on such a computer halts?

The halting problem says that it is impossible to create a general algorithm which can for all inputs and programs determine whether they halt. However, this assumes that the programs and/or the ...
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