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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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149
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12answers
49k views

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][1] correctly explains that a deterministic ...
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5answers
12k views

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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10answers
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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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7answers
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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 ...
29
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7answers
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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 ...
27
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7answers
6k views

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. ...
24
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4answers
4k views

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 ...
22
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6answers
3k views

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 ...
22
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2answers
2k views

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.
22
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1answer
2k views

Is possible to prove undecidability of the halting problem in Coq?

I was watching the "Five Stages of Accepting Constructive Mathematics" by Andrej Bauer and he says that there is two kinds of proof by contradiction (or two things that mathematicians call proof by ...
21
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4answers
3k views

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 ...
20
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5answers
4k views

Could the Halting Problem be “resolved” by escaping to a higher-level description of computation?

I've recently heard an interesting analogy which states that Turing's proof of the undecidability of the halting problem is very similar to Russell's barber paradox. So I got to wonder: ...
18
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6answers
3k views

Is halting problem computable for particular inputs/assumptions

From my understanding of the proof that halting problem is not computable, this problem is not computable because if we have a program P(x) which computes if the program x halts or not, we got a ...
18
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4answers
7k views

Can a runtime environment detect an infinite loop?

Would it be possible for a runtime environment to detect infinite loops and subsequently stop the associated process, or would implementing such logic be equivalent to solving the halting problem? ...
17
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4answers
2k views

Defining the halting problem for non-deterministic automata

The primary definition of Turing machine (TM), at least in my own reference textbook (Hopcroft+Ullman 1979) is deterministic. Hence my own understanding of the halting problem is primarily for ...
16
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5answers
4k views

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 ...
16
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3answers
699 views

Is there a TM that halts on all inputs but that property is not provable?

Does there exist a Turing machine that halts on all inputs but that property is not provable for some reason? I am wondering if this question has been studied. Note, "unprovable" could mean a "...
14
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5answers
4k views

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 ...
13
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3answers
5k views

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 ...
12
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3answers
2k views

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. ...
11
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1answer
627 views

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 ...
10
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4answers
3k views

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 ...
10
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1answer
940 views

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 ...
10
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2answers
929 views

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 ...
10
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1answer
443 views

Hilbert's 10th Problem and Chaitin's Diophantine Equation “Computer”?

In Chaitin's Meta Math! The Quest For Omega, he briefly talks about Hilbert's 10th Problem. He then says that any Diophantine Equation $p=0$ can be changed into two equal polynomials with positive ...
10
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1answer
407 views

Is the halting problem decidable for 3 symbol one dimensional cellular automata?

I've been trying to figure out if the halting problem is decidable for 3-symbol one-dimensional cellular automata. Definition Let $f(w,i)$ denote the configuration of the system at time step $i$. ...
9
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2answers
2k views

Is it possible that the halting problem is solvable for all input except the machine's code?

This question occurred to me about the halting problem and I couldn't find a good answer online, wondering if someone can help. Is it possible that the halting problem is decidable for any TM on any ...
8
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2answers
730 views

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 ...
8
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3answers
1k views

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 ...
8
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1answer
547 views

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 ...
8
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3answers
950 views

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 ...
8
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2answers
388 views

Chaitin's constant is normal?

According to this source, Chaitin's constant $\Omega$ is normal. Each halting probability is a normal and transcendental real number that is not computable, which means that there is no algorithm ...
7
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3answers
950 views

The 'directionality' of reductions?

I've been finding myself a bit confused with the direction of reductions used to show that certain languages are not recursive. For example, let us say we want to determine if the Halting Problem ($...
7
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6answers
1k views

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 ...
7
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1answer
4k views

Why is the halting problem semi-decidable?

This is what is know about halting problem and semi-decidability :- 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. A ...
7
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1answer
3k views

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 ...
7
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3answers
901 views

Undecidability of telling if a program returns true or false

Consider the problem of taking an input Turing machine and determining if the final cell is a $0$ or $1$ after computation halts. On cases where it writes something else or does not halt, you are ...
6
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2answers
2k views

Does it make sense to talk about the complexity of non-computable functions (such as the Halting problem)?

I have seen numerous proofs (such as this) that the Halting problem is in the class of NP. However, the Halting problem is non-computable. Does it make sense to discuss the complexity of computing a ...
6
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1answer
957 views

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 ...
6
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2answers
259 views

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 ...
6
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3answers
4k views

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 ...
6
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1answer
5k views

Relationship between Undecidable Problems and Recursively Enumerable languages

I have read the Wikipedia article on Recursively Enumerable languages. The article suggests that the halting problem is recursively enumerable but undecidable. My idea till today was that the halting ...
6
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1answer
2k views

Decidability of halting problem for DPDAs with $\epsilon$-transitions?

For LBAs it's rather easy to prove the decidability of the halting problem, as there can only be a finite number of different configurations when using limited space. But what about PDAs with $\...
6
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1answer
588 views

Machine Learning to predict the Halting Problem

I seem to recall an academic paper from some years ago which used machine learning (possibly genetic or evolutionary programming) to predict whether a Turing Machine would halt. By predict, I mean ...
6
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1answer
12k views

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 ...
6
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1answer
335 views

Halting problem

I have some concerns about the Halting problem. This is the proof I know: Let $h(M, i)$ be a function, $M$ being Turing machine and $i$ input for the Turing machine. Let $h(M, i)$ output true ...
6
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1answer
208 views

Is there any demonstrably uncomputable concrete problem which does not rely on diagonalization?

So diagonalization as we all know is an extremely productive way of showing uncomputability, the other main tool used by CS people for this task being reduction. But it has occurred to me that I do ...
6
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1answer
210 views

Showing the function=? is impossible

Here's a lab from a first-year computer science course, taught in Scheme: https://www.student.cs.uwaterloo.ca/~cs135/assns/a07/a07.pdf At the end of the lab, it basically presents the halting problem,...
5
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2answers
5k views

Why won't a Turing machine halt?

I am reading Sipsers. The book introduces halting problem and proves that is a turing recognisable language but not a turing decidable language. Thus giving a Turing machine which does not halt on ...
5
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2answers
710 views

What helpful solution does the Halting Problem give to computing?

What problem does the halting problem solve in computing, whether theoretical or practical? It is very easy to debug code which loops forever, just signal the debugger to break if the program is ...