Questions tagged [undecidability]

Questions about problems which cannot be solved by any Turing machine.

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146 votes
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How can it be decidable whether $\pi$ has some sequence of digits?

We were given the following exercise. Let $\qquad \displaystyle f(n) = \begin{cases} 1 & 0^n \text{ occurs in the decimal representation of } \pi \\ 0 & \text{else}\end{cases}$ ...
Raphael's user avatar
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50 votes
1 answer
8k views

What makes type inference for dependent types undecidable?

I have seen it mentioned that dependent type systems are not inferable, but are checkable. I was wondering if there is a simple explanation of why that is so, and whether or not there is there a limit ...
Victor's user avatar
  • 705
46 votes
2 answers
9k views

Perplexed by Rice's theorem

Summary: According to Rice's theorem, everything is impossible. And yet, I do this supposedly impossible stuff all the time! Of course, Rice's theorem doesn't simply say "everything is impossible". ...
MathematicalOrchid's user avatar
45 votes
2 answers
21k views

How to show that a function is not computable? How to show a language is not computably enumerable?

I know that there exists a Turing Machine, if a function is computable. Then how to show that the function is not computable or there aren't any Turing Machine for that. Is there anything like a ...
user5507's user avatar
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33 votes
7 answers
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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 ...
user118967's user avatar
33 votes
1 answer
2k views

Rice's theorem for non-semantic properties

Rice's theorem tell us that the only semantic properties of Turing Machines (i.e. the properties of the function computed by the machine) that we can decide are the two trivial properties (i.e. always ...
Kaveh's user avatar
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29 votes
3 answers
1k views

Are there any specific problems known to be undecidable for reasons other than diagonalization, self-reference, or reducibility?

Every undecidable problem that I know of falls into one of the following categories: Problems that are undecidable because of diagonalization (indirect self-reference). These problems, like the ...
templatetypedef's user avatar
27 votes
5 answers
14k views

Why isn't this undecidable problem in NP?

Clearly there aren't any undecidable problems in NP. However, according to Wikipedia: NP is the set of all decision problems for which the instances where the answer is "yes" have [.. proofs that ...
BlueRaja - Danny Pflughoeft's user avatar
27 votes
1 answer
3k views

What are the strongest known type systems for which inference is decidable?

It's well known that Hindley–Milner type inference (the simply-typed $\lambda$-calculus with polymorphism) has decidable type inference: you can reconstruct principle types for any programs without ...
Joey Eremondi's user avatar
23 votes
2 answers
1k views

Is there a "natural" undecidable language?

Is there any "natural" language which is undecidable? by "natural" I mean a language defined directly by properties of strings, and not via machines and their equivalent. In other words, if the ...
Ran G.'s user avatar
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23 votes
4 answers
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Do undecidable languages exist in constructivist logic?

Constructivist logic is a system which removes the Law of the Excluded Middle, as well as Double Negation, as axioms. It's described on Wikipedia here and here. In particular, the system doesn't ...
Joey Eremondi's user avatar
22 votes
6 answers
3k views

Halting problem theory vs. practice

It is often asserted that the halting problem is undecidable. And proving it is indeed trivial. But that only applies to an arbitrary program. Has there been any study regarding classes of programs ...
Jack Fleming's user avatar
21 votes
1 answer
562 views

Ratio of decidable problems

Consider decision problems stated in some “reasonable” formal language. Let's say formulae in higher-order Peano arithmetic with one free variable as a frame of reference, but I'm equally interested ...
Gilles 'SO- stop being evil''s user avatar
19 votes
3 answers
7k 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 ...
user5507's user avatar
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19 votes
2 answers
16k views

Is the set of Turing machines which stops in at most 50 steps on all inputs, decidable?

Let $F = \{⟨M⟩:\text{M is a TM which stops for every input in at most 50 steps}\}$. I need to decide whether F is decidable or recursively enumerable. I think it's decidable, but I don't know how to ...
Jozef's user avatar
  • 1,707
19 votes
1 answer
2k views

Is the unsolvability of the N-Body Problem equivalent to the Halting Problem

There is no general analytic solution to the n-body problem that can produce an analytic function which can be used to give an n-body system's state at arbitrary time t with exact precision. However, ...
Shufflepants's user avatar
19 votes
1 answer
387 views

Regular expressions with backreferences over unary alphabet

Setting: regular expressions with backreferences unary language (1-symbol alphabet) Is the following problem decidable in this setting: Given a regular expression with backreferences, does it ...
Jukka Suomela's user avatar
18 votes
4 answers
1k views

Is this finite graph problem decidable? What factors make a problem decidable?

I want to know if the following problem is decidable and how to find out. Every problem I see I can say "yes" or "no" to it, so are most problems and algorithms decidable except a few (which is ...
Gigili's user avatar
  • 2,193
18 votes
5 answers
1k views

Is it possible to solve the halting problem if you have a constrained or a predictable input?

The halting problem cannot be solved in the general case. It is possible to come up with defined rules that restrict allowed inputs and can the halting problem be solved for that special case? For ...
Ken Li's user avatar
  • 3,078
17 votes
5 answers
4k views

Proof of the undecidability of compiler code optimization

While reading Compilers by Alfred Aho, I came across this statement: The problem of generating the optimal target code from a source program is undecidable in general. The Wikipedia entry on ...
user avatar
17 votes
1 answer
6k views

Is it decidable whether a pushdown automaton recognizes a given regular language?

The problem whether two pushdown automaton recognize the same language is undecidable. The problem whether a pushdown automaton recognizes the empty language is decidable, hence it is also decidable ...
Thomas Klimpel's user avatar
15 votes
5 answers
2k views

Are there undecidable properties of non-turing-complete automata?

Are there undecidable properties of linear bounded automata (avoiding the empty set language trick)? What about for a deterministic finite automaton? (put aside intractability). I would like to get ...
Hernan_eche's user avatar
15 votes
3 answers
2k views

Is Deciding Decidability Decidable?

I am wondering if deciding the decidability of problem is a decidable problem. I am guessing not, but after initial searches I cannot find any literature on this problem.
sync's user avatar
  • 195
15 votes
2 answers
11k views

Is it decidable whether a TM reaches some position on the tape?

I have these questions from an old exam I'm trying to solve. For each problem, the input is an encoding of some Turing machine $M$. For an integer $c>1$, and the following three problems: ...
Jozef's user avatar
  • 1,707
15 votes
2 answers
460 views

Is it decidable if a language described by number of occurences is regular?

It is known that the language of words containing equal number of 0 and 1 is not regular, while the language of words containing equal number of 001 and 100 is regular (see here). Given two words $...
sdcvvc's user avatar
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14 votes
3 answers
3k views

undecidable problem and its negation is undecidable

A lot of "famous" undecidable problems are nonetheless at least semidecidable, with their complement being undecidable. One example above all can be the halting problem and its complement. However, ...
Giulia Frascaria's user avatar
14 votes
1 answer
453 views

For a Turing Machine $M_1$, how is the set of machines $M_2$ which are "shorter" than $M_1$ and which accept the same language decidable?

I wonder how come that the following language is in $\mathrm R$. $L_{M_1}=\Bigl\{\langle M_2\rangle \;\Big|\;\; M_2 \text{ is a TM, and } L(M_1)=L(M_2), \text{ and } |\langle M_1\rangle| > | \...
Jozef's user avatar
  • 1,707
14 votes
1 answer
22k views

What is the difference between halting, accepting, and deciding in the context of Turing machines?

Does accepting mean that the TM will read and recognize a char from the cell it's currently reading from? And is it the case that a TM halts iff the input is decidable?
sdfasdgasg's user avatar
14 votes
1 answer
930 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 ...
Patrick87's user avatar
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13 votes
2 answers
2k 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 ...
ike's user avatar
  • 235
13 votes
2 answers
2k 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 ...
zpavlinovic's user avatar
  • 1,644
13 votes
2 answers
3k views

Are all context-sensitive languages decidable?

I was going through the Wikipedia definition of context-sensitive language and I found this: Each category of languages is a proper subset of the category directly above it. Any automaton and any ...
bongubj's user avatar
  • 563
12 votes
4 answers
4k views

Operations under which the class of undecidable languages isn't closed

Do there exist undecidable languages such that their union/intersection/concatenated language is decidable? What is the physical interpretation of such example because in general, undecidable ...
user avatar
12 votes
6 answers
44k views

Can a Turing machine decide the language $L_\emptyset$ of machines with empty language?

Let $$L_\emptyset = \{\langle M\rangle \mid M \text{ is a Turing Machine and }L(M)=\emptyset\}.$$ Is there a Turing machine R that decides (I don't mean recognizes) the language $L_\emptyset$? It ...
Mahdi's user avatar
  • 579
12 votes
1 answer
1k views

Reductions among Undecidable Problems

Im sorry if this question has some trivial answer which I am missing. Whenever I study some problem which has been proven undecidable, I observe that the proof relies on a reduction to another problem ...
swarnim_narayan's user avatar
12 votes
5 answers
676 views

Undecidable problems limit physical theories

Does the existence of undecidable problems immediately imply the non-predictability of physical systems? Let us consider the halting problem, first we construct a physical UTM, say using the usual ...
user2663116's user avatar
11 votes
2 answers
426 views

Decidability of checking an antiderivative?

Let's suppose I have two functions $F$ and $G$ and I'm interested in determining whether $$F(x) = \int G(x)dx.$$ Let's suppose that my functions are composed of elementary functions (polynomials, ...
templatetypedef's user avatar
11 votes
2 answers
7k views

A Question relating to a Turing Machine with a useless state

OK, so here is a question from a past test in my Theory of Computation class: A useless state in a TM is one that is never entered on any input string. Let $$\mathrm{USELESS}_{\mathrm{TM}} = \{\...
BrotherJack's user avatar
  • 1,115
11 votes
3 answers
361 views

Decidability of a problem concerning polynomials

I have come across the following interesting problem: let $p,q$ be polynomials over the field of real numbers, and let us suppose that their coefficients are all integer (that is, there is a finite ...
042's user avatar
  • 696
11 votes
3 answers
714 views

Is it possible to decide if a given algorithm is asymptotically optimal?

Is there an algorithm for the following problem: Given a Turing machine $M_1$ that decides a language $L$, Is there a Turing machine $M_2$ deciding $L$ such that $t_2(n) = o(t_1(n))$? The ...
StaticBug's user avatar
  • 213
11 votes
2 answers
1k views

Can we show a language is not computably enumerable by showing there is no verifier for it?

One of the definitions of a computably enumerable (c.e., equivalent to recursively enumerable, equivalent to semidecidable) set is the following: $A \subseteq \Sigma^*$ is c.e. iff there is a ...
Anonymous's user avatar
  • 111
10 votes
4 answers
6k views

Is there an undecidable finite language of finite words?

Is there a need for $L\subseteq \Sigma^*$ to be infinite to be undecidable? I mean what if we choose a language $L'$ be a bounded finite version of $L\subseteq \Sigma^*$, that is $|L'|\leq N$, ($N \...
Hernan_eche's user avatar
10 votes
4 answers
4k views

The bounded halting problem is decidable. Why doesn't this conflict with Rice's theorem?

One statement of Rice's theorem is given on page 35 of "Computational Complexity: a Modern Approach" (Arora-Barak): A partial function from $\{0,1\}^*$ to $\{0,1\}^*$ is a function that is not ...
ttbo's user avatar
  • 113
10 votes
2 answers
11k views

Is it decidable whether a given context free grammar generates an infinite number of strings?

Is the decision problem "Does a given context free grammar generate an infinite number of strings" decidable? In order to test whether a context free grammar generates an infinite number of strings or ...
kauray's user avatar
  • 509
10 votes
4 answers
2k views

Probabilistic methods for undecidable problem

An undecidable problem is a decision problem proven to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. I wonder if there are examples of probabilistic ...
Student's user avatar
  • 219
10 votes
2 answers
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 ...
CS101's user avatar
  • 103
10 votes
1 answer
3k views

Why is deciding regularity of a context-free language undecidable?

As I have studied, deciding regularity of context-free languages is undecidable. However, we can test for regularity using the Myhill–Nerode theorem which provides a necessary and sufficient ...
Jiya's user avatar
  • 103
10 votes
3 answers
278 views

Constructive version of decidability?

Today at lunch, I brought up this issue with my colleagues, and to my surprise, Jeff E.'s argument that the problem is decidable did not convince them (here's a closely related post on mathoverflow). ...
G. Bach's user avatar
  • 2,019
9 votes
4 answers
4k 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 ...
yters's user avatar
  • 1,409
9 votes
3 answers
6k 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 ...
user13675's user avatar
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