Questions tagged [undecidability]

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

88 questions with no upvoted or accepted answers
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Modern presentation of Ackermann's “Solvable Cases?”

Ackermann's book "Solvable Cases of the Decision Problem" discusses decidable instances of first order logic, particularly monadic logic, and so called "equality formulas". However, the book is from ...
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1answer
142 views

Proving a certain superset the halting language is not recursive

Let $\Sigma =\{ 0, 1\}$. Let $val:\Sigma^* \rightarrow \mathbb{N}$ be a function that given a string returns its decimal value, and $L_{halt} = \{\langle M\rangle \langle w\rangle \mid M $ halts on $w ...
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92 views

is it decidable whether a grammar in Chomsky normal form has useless rules?

Given a context-free grammar in Chomsky normal form, is it decidable whether that grammar has any useless rules? By "useless", I mean a rule that can be omitted from the grammar without ...
3
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1answer
300 views

A variation of the halting problem

Given an infinite set $S \subseteq \mathbb{N}$, define the language: $L_S = \{ \langle M \rangle : M $ is a deterministic TM that does not halt on $\epsilon$, or, $T_M \in S\}$ where $T_M$ is the ...
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106 views

Undecidability of language involving two TMs

I am currently browsing the lecture notes on computability/decidability and I have encountered the following exercise I am unable to solve. Given $M_1$, $M_2$ Turing machines, is it true that for ...
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57 views

Hamming connectivity of regular languages

Call a language $L$ Hamming connected iff, for every pair of strings $x, y \in L^2$, where $|x|=|y|$, $x$ may be transformed into $y$ by a sequence of single symbol in-place replacements, so that ...
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35 views

What is example of Kahr formula $[\forall\exists\forall, (\omega, 1), (0)]$ and what to do if such undecidable formula is encountered in practice?

There are mentioned many classes of undecidable formulas in the book "The Classical decision problem" http://www.springer.com/la/book/9783540423249. Kahr formulae is one class of undecidable formuls ...
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43 views

Does Rice's theorem apply to sequential logic circuits?

I am wondering if Rice's theorem (or something similar to that) applies also to sequential circuits. I.e. given any finite sequential circuit, can there be an algorithm that can formally verify any ...
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90 views

Decide whether a polynomial has a root

Let $A$ be a ring such that all elements of $A$ are complex computable numbers. I'm interested in knowing whether the decision problem that asks, given $P\in A[X]$, if $P$ has a root in $A$ is ...
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28 views

Do undecidable problems have no HO query? If so, could I have an example?

In descriptive complexity, HO corresponds to ELEMENTARY. ELEMENTARY is a subset of R, so therefore all HO queries are decidable. Then undecidable problems have no corresponding HO query. Is my ...
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48 views

The decidability of a problem involving univariate integer polynomials

Suppose that we are given $f_1(x),...,f_n(x) \in \mathbb{Z}[x]$. Decide whether there exist $a_1,...,a_n \in \mathbb{Z}$ such that $\sum_{i=1}^{n} a_if_i(x) = p(x)^2 $ $p(x) \in \mathbb{Z}[x]$. ...
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31 views

Nearest codeword in a translation-invariant code over $\mathbb{Z}^d$

Let $c_1,...,c_n,c':\mathbb{Z^d}\rightarrow \{0,1\}$ all have finite support. Let $C$ be the linear, shift-invariant code generated by $c_1,..,c_n$. It is possible to calculate the nearest codeword $...
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45 views

Decidability/recognizability of languages with strings lacking alphabet characters

Suppose that $\Sigma = \{c_1, \dots, c_m\}$ is some finite alphabet and supposing $s \in \Sigma^*$, let $\mathcal{I}_j(s)$ denote the number of instances of character $c_j$ in $s$. Call a string $s$ ...
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166 views

How to prove a Language is neither a Computably enumerable nor Co-Computably enumerable?

What would be the general approach for that? And what are the things that generally overlooked while proving such things? For example, I have a Language, L ={e:$L(M_e)$ such that it accepts only 'a ...
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17 views

Is undecidability contained in $PSPACE / o(exp(n))$?

It is not hard to show that $DSPACE(n+1)/2^n$ contains undecidability. But is it possible to make the advice string subexponentially long (while the machine is allowed to have any $poly(n)$ space) ...
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40 views

Decidability for $ \exists w´, w´´\in L:$ so that |w´´| - |w´| is prime

I tried to decide wheter the given Language $ L = \{ \langle M \rangle | M \space is \space TM \space and \space \exists \space w´,w´´\in L(M):|w´´|-|w´| \space is \space prime \} $ is recursive or ...
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41 views

How to show that these two disjoint sets $A$ and $B$ exist

I came across this problem which asks to show the existence of two disjoint Turing-recognizable sets $A$ and $B$ such that no decidable set $C$ can separate them... In this case, a set $C$ is said to ...
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41 views

difference between two uncomputable functions

for $L1$ regular language, $L2$ some language, $L1$ \ $L2$ is regular, decidable. yet, the next transition function might not even be computable: $F′=${$\,q:δ(q,w)∈F \, for\, some\, w∈ L2\,$}$.$ $F$ ...
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1answer
24 views

Many-one reductions between the set of true sentences and a particular arithmetical set

Never used this site before so not sure the best way to get help. However, I've been looking at many-one reductions in relations to sentences in logic. Let TH(N) = {ϕ : ϕ is a first order sentence ...
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24 views

Is it semidecidable to test whether a Turing decidable language is empty?

I'm not sure how to go about solving this. I tried this: Suppose L is a Turing decidable language. Turing Machine M1 is a decider of L and M2 is a decider of the complement L We construct a TM U ...
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75 views

How does a PDA compare two configurations of accepting histories?

In Michael Sipser's book, they prove that ALL_CFG = { G | G is a CFG and L(G) = Σ∗ } is undecidable using accepting computation histories and PDAs. My question is how exactly (with details of ...
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2answers
385 views

Can I apply Rice's theorem to decide decidability status of these languages?

I came across these languages: A Turing machine prints a specific letter. A Turing machine computes the products of two numbers I was guessing whether I can apply Rice's theorem to decide upon above ...
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1answer
531 views

Undecidable: $w$ on which a TM M $M$ halts after $\leq w$ steps

The detailed question is: Is there a word $w$ on which a TM M $M$ halts after a maximum of $|w|$ (word length) steps? I highly assume, that this problem is not decidable because in the worst case ...
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104 views

Why full Chomsky hierarchy is so detailed, if there are decidable recursive languages?

One can have a look on the Chomsky hierarchy https://en.wikipedia.org/wiki/Chomsky_hierarchy , especially the inset named "Automata theory: formal languages and formal grammars" at the bottom of the ...
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77 views

Can a CFG generate an accepting configuration? - or is there a turing-recognizable CFG language that is not decidable

I could not think of a way to concisely write down my question clearly, but I'd like to ask, from Sipser's book, $ALLCFG$ is an undecidable language (where $ALLCFG$ means that $G$ is a $CFG$ that ...
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634 views

decidability about intersection of regular language and context free language

Is "Given a CFL L and a regular language R, is intersection of L and R an empty set?" decidable? What if we replace L with the complement of L? One of them is decidable and the other is not. For the ...
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168 views

Rice theorem to prove Emptiness problem

Is it possible to use the theorem of Rice to prove that the emptiness problem is undecidable? With the emptiness problem I mean the question if a certain machine doens't accept any input ? If you ...
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58 views

A Turing Machine with undecideable Halting Status not relying on Open Problems

Is there any turing machine that it is proovably known to be undecideable, it has to fullfit the following characteristics: Not rely on Open Problems/ Conjectures Should not use the same machine used ...
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23 views

Computing shifted fix point in the BSS model

Let $p \colon \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ be a one-dimensional function that fulfills $p(0)=0$. Moreover, we are given some value $u \in \mathbb{R}_{> 0}$ such that $p$ is ...
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75 views

Undecidability of an existential theory

$F[u, u^{-1}]$ is a ring that contains the polynomials in $u$ and $u^{-1}$ with coefficients in the field $F$. Some theorems (from https://math.stackexchange.com/questions/1382120/ft-has-undecidable-...
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19 views

Is deciding whether there is a non-constant solution to a functional inequality with polynomial arguments decidable, with 2 variables?

So suppose we have a functional inequality with polynomial arguments in $2$ variables, $\sum_i c_i f(p_i(x,y)) \geq 0$, where $c_i$ are say given integer constants and $p_i$ are given polynomials, say ...
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1answer
125 views

Check if language is decidable

I would like to determine if the following language is decidable or not. L = { w $\in$ $\Sigma^*$ | $T(M_w)$ is recognized by a Turing machine with at most 42 states}. I know that every finite ...
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36 views

halting problem vs watchdog

I have a theory that all finite state machines can be monitored by a second turing machine with infinite tape to determine if the state of the first machine was repeated thus reaching the conclusion ...
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36 views

Let $f$ be a computable and injective function. Is $f^{-1}$ computable and injective?

So I just started learning about computability, undecidability and Turing machines. And I wonder if: Given a computable and injective function $f$, is $f^{-1}$ also computable and injective? I don't ...
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2answers
40 views

Is it decidable when a TM M gets another as inputs and checks if it fullfiills certain property?

I was asking myself if it is not possible to decide the language where a TM M gets the Godel number of a TM M' as input and the checks if, let us say, the TM M' has a certain amount of transitions. My ...
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1answer
175 views

Determine if a language is Decidable or semi decidable

Consider the language $L = \{\langle M \rangle: \text{ $M$ accepts at most two single-letter words}\}$, where $\langle M\rangle$ is the encoding of Turing machine $M$. We need to determine, without ...
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38 views

Show that this language is undecidable

Given the language $K$ $=\{<M> $ where $M$ is a turing machine ( that is on the alphabet {0,1}) and $L(M)$ contains at least one word of form $0^k1^l$ with $k,l\geq 0\}$ I would like to know if ...
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134 views

reduction from ALLTM to ETM

I am trying to understand why this "proof" of ETM undecidability is wrong. ALLTM={ < M >|M is a TM, L(M)=∑*} ETM={< M >|M is a TM, L(M)=∅} We know that ALLTM is undecidable, lets ...
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1answer
94 views

Undecidability of closure under reverse of language accepted by TM

Prove that the following problem is undecidable using a reduction: Given a Turing machine $S$, does $S$ accept a word $w$ iff it accepts its reverse $w^R$? There is a solution here, which I don't ...
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64 views

Halting problem. Decider “recognising itself” in the input?

This is about the halting problem. My questions are: where do you think are logical flaws in what I am going to write? How do you think this does not invalidate the proof for the undecidability of the ...
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63 views

How would I prove that nondeterministic Turing machines are undecidable?

How would I go about proving that the language: $$A_{NTM }= \{\langle N, w\rangle | N \text{ is a nondeterministic TM and } N \text{ accepts }w\}$$ is undecidable? I looked at the proof for $A_{TM}$ ...
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1answer
122 views

Undecidability of the language of PDAs that accept some ww

I'm trying to solve problem 5.33 from Sipser's Introduction to the Theory of Computation, "Consider the problem of determining whether a PDA accepts some string of the form $\{ww|w\in \{0,1\}^∗\}$...
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1answer
58 views

Are all Recursively Enumerable languages which are not Recursive also Undecidable?

Knowing that all Recursive languanges are Decidable and All Not R.E. Languages are Undecidable (correct me if I am wrong), Are all languages which are R.E. but not Recursive also Undecidable? R.E. ==&...
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30 views

TM for Language With a Specific Cardinality

I'm curious about how to build TM that decide and recognize languages defined by cardinality. For example, with the language $L_1$ = $\{w \in \{0,1\}^* | |w| = 1\}$ this is the language with a single ...
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47 views

Is the undecidability of a given problem undecidable?

Given an input problem P, can you construct an algorithm A to compute whether or not P is decidable or undecidable? In other words, is the undecidabiliy of a problem undecidable? My initial guess is ...
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1answer
84 views

Mapping reduction from $A_{TM}$ to $INFINITE_{TM}$ same as to $ALL_{TM}$?

I was trying to solve a problem with a mapping reduction from $A_{TM}$ to $INFINITE_{TM}$, and came across a solution that was 100% identical to another solution I saw for $A_{TM} \leq_M ALL_{TM}$. ...
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33 views

Is it decidable to know the number of positions used by a Turing machine for a fixed input?

I'm having trouble proving if the following language is recursive, recursively enumerable, or not r.e. at all: the set of all encodings of Turing machines $M$ such that the number of positions in the ...
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42 views

A Turing machine for which it is impossible to predict whether it halts or not on a fixed input

The halting problem is undecidable, i.e. $\not \exists$ $M$ Turing machine s.t. for every $(M_0,w_0)$ input where $M$ is the description of a Turing machine and $w_0$ is an input word, the output of $...
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27 views

Find decidable sets such that $A$ reduces to $B$ but not vice versa

I am stuck in this problem, so any help is appreciated. The problem asks to show that there exists decidable sets $A$ and $B$ such that $A \leq_{m}^{p} B$ but $B \not \leq_{m}^{p} A$, and that $A$, $B$...
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36 views

A TM that doesn't decide Σ*, and a TM that doesn't decide the empty set?

I was wondering if it was possible to create a TM that semi-decides (but doesn't decide) either of the following two languages: $\emptyset$ $\Sigma^{*}$ I assume for 2, a one-state TM that just ...