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

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1answer
32 views

Proof that L(M) = {accepts the string 1100 } is undecidable

Let $$L_\ = \{\langle M\rangle \mid M \text{ is a Turing Machine that accepts the string 1100}\}\, .$$ To proof that the language $L$ is undecidable I should reduce something to $L$, right? I tried ...
0
votes
0answers
20 views

How to show {n:U(n,x) is defined for all x} is not enumerable

U(n,x) is Gödel universal function, and we need to show {n:U(n,x) is defined for all x} is not enumerable. I do not have any clue right now. Anyone can give me some hint about this question.
1
vote
1answer
39 views

Satisfiability of first-order logic is undecidable?

I struggle with understanding why the satisfiability in the first-order logic is undecidable. Could you explain it with some examples? I've also seen that satisfiability in some first-order formulas ...
4
votes
1answer
71 views

Decide whether there exists a walk of weight exactly k

Consider the following problem: Input: a directed graph $G = (V,E,\omega)$ where $\omega : E \longrightarrow \mathbb{Z}$, two vertices $v_1, v_2 \in V$, and a weight $k \in \mathbb{Z}$ Question: ...
2
votes
2answers
251 views

Is this variant of ATM decidable?

Ok so I understand how $\mathrm{ATM} = \{\langle M,w \rangle \mid \text{$M$ is a TM and $M$ accepts $w$}\}$ is undecidable. Is this because $w$ is a variable? What if the parameter is fixed? ...
0
votes
1answer
21 views

Rice's Theorem: implication of having an undecidable property

I understand the assumptions that have to be true about a property or set of properties in a Turing machine description for Rice's Theorem to apply. But then what? If a set of Turing machines have ...
0
votes
2answers
55 views

Decidability of empty intersection of two languages accepted by Turing machines

I am really struggling with determining the decidability of languages and cant figure out whether this problem is decidable or not. I have a language $\qquad\displaystyle L = \{ (R(M_1), R(M_2)) ...
3
votes
2answers
89 views

Is it possible to ever define $L(M)$ of a given Turing Machine, $M$?

In class, we were discussing creating a Turing Machine $M$ based on the set of input strings it should accept, i.e. define a Turing Machine that accepts only the input $\{ w\ \#\ w\ |\ w \in ...
-1
votes
2answers
79 views

Extension of Rice's theorem

How can one prove that every nontrivial property of pairs of semi-decidable sets is undecidable? (This is an extension of Rice's theorem that "Every nontrivial property of the r.e. sets is ...
2
votes
1answer
51 views

Decide if a specific Turing machine halts on a specific string

Can you always decide if a specific Turing machine accepts a specific string? I started thinking about this after reading an answer to this question, Rice's theorem vs Turing completeness, which ...
0
votes
1answer
41 views

Clarification of Hopcroft's proof that “deciding whether a program halts on all inputs” is not R.E

$DoesNotHaltOn\_w=\{(M, w) : M$ does not halt on input w$\}$ $AlwaysHalt =\{ M : M$ halts on all inputs x $\}$ Hopcroft gives the following proof that $AlwaysHalt$ is not R.E. 1) Given an input ...
0
votes
2answers
100 views

The image of a recursive language under a computable function

Let $f:\Sigma^{*}\to\Sigma^{*}$ be a computable function and let $L$ be a recursive language. Is $f(L):=\left \{{f(w)|w\in L} \right\}$ recursive? Here, I see clearly, that $f^{-1}(L)$ is recursive ...
4
votes
2answers
184 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 ...
6
votes
1answer
70 views

Is Post's Correspondence Problem decidable with fixed word size?

So, it's known that PCP is undecidable even when we fix the number of tiles to $n \geq 7$. I'm wondering, can anything similar be said for when there is a fixed word length? To be precise, here's ...
3
votes
1answer
41 views

Is there an undecidable decision problem that computable algorithm for it leads to an algorithm for halting problem?

Suppose, to the contrary, that there exists a computable algorithm for some undecidable decision problem. Would this mean that halting problem would be solved by a computable algorithm? I know that ...
1
vote
2answers
74 views

Can an intersection of two context-free languages be an undecidable language?

I'm trying to prove that $\exists L_1, L_2 : L_1$ and $L_2$ are context-free languages $\land\;L_1 \cap L_2 = L_3$ is an undecidable language. I know that context-free languages are not closed ...
3
votes
1answer
54 views

The proportion of halting programs vs non-halting programs, of decidable programs vs undecidable languages

Can the following two statistics be bounded: the proportion of halting programs vs non-halting programs the proportion of decidable vs undecidable languages For example, can we say that one class ...
1
vote
1answer
34 views

Examples for incomparable semi-decidable but undecidable languages

In Schönig and Pruim's Gems of Theoretical Computer Science, the following statement is made: 'Post's Problem', as it has come to be known, is the question of whether there exist undecidable, ...
0
votes
1answer
57 views

Why is $A_{TM}$ reducible to $HALT_{TM}$?

In Sipser, there is a proof I don't understand. First he established the undecidability of $A_\mathrm{TM}$, the problem of determining whether a Turing machine accepts a given input. ...
0
votes
1answer
28 views

Intersection and partial quantity decidability [closed]

I'm still insecure in the section decidability (no proof needed, I want to divine it): X is decidable and Y is undecidable. Is the intersection of X and Y decidable or undecidable? X is decidable ...
3
votes
3answers
184 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 ...
2
votes
1answer
31 views

Why can't configuration counting decide undecideable problems?

I know it might sound like a silly question, I just can't get my head around it... I just read that $DSPACE(f(n))\subseteq DTIME(n\cdot 2^{O(f(n))})$. The proof for it relies heavily on the fact ...
1
vote
2answers
66 views

if (dis)proving a conjecture on graph theory can be done just by a counter example then can every (dis)proof be mapped actually to a counter-example?

Suppose we have a conjecture on graph theory that can be (dis)proved by means of a counter example, then, is it true that every alternative (dis)proof of the conjecture can be mapped to a counter ...
4
votes
1answer
109 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 ...
1
vote
2answers
66 views

Do all decidable problems lie in the class NP?

All decision problems (i.e.language membership problems), which are verifiable in polynomial time by a deterministic Turing machine are called NP problems. Further, these problems can be solved by a ...
1
vote
1answer
81 views

Turing Decidable [closed]

M = (Q, Σ, Γ, δ, q1, qaccept, qreject), where Q ={q1, q2, qaccept, qreject}, Σ = {0, 1}, Γ = {0, 1, U}, and transition function δ is as follows: ...
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vote
0answers
32 views

Undecidability instance on a “find a proof/disproof” machine

I'm following through the proof of the impossibility of the Halting problem for the umpteenth time. It all makes sense logically, but not intuitively. A question I got stuck on: Suppose we built the ...
0
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0answers
48 views

Can we recognize wheter a Turing machine is a decider?

Let $L=\{ \langle M \rangle \mid M \text{ is a Turing Machine which halts on all inputs}\}$. Is this a Turing-recognizable language? I guess that it is neither Turing-recognizable, nor ...
1
vote
2answers
64 views

Reducing A(TM) to some decidable problem

We know that A(TM) is undecidable, what if we reduce A(TM) to A(DFA) which is decidable? How will we prove that A(DFA) is decidable? I couldn't find an example or theory. Thanks
1
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0answers
30 views

Notable decidable operations on context-sensitive languages [closed]

It is not always so easy to determine which basic questions on languages are (un)decidable. Also due to Rice's theorem, many nontrivial questions on languages are undecidable. What are notable or ...
0
votes
1answer
51 views

How can a problem be undecidable yet enumerable? [duplicate]

How can something be enumerable but be un-decidable ie, this states the halting set is un-decidable and enumerable. Enumerable means it can be computed, ie has the same cardinality as natural numbers ...
0
votes
1answer
33 views

How can an LBA check legality of TM transitions without extra memory?

In Sipser's book there is a proof that an emptiness of LBA is undecidable, with the help of reduction to A_$_{\text{TM}}$. The reduction is proposed in the following manner: we receive a TM $M$ and a ...
2
votes
1answer
128 views

Can you recognize or decide if a Turing Machine has an infinite sized language?

That is, can you build a Turing Machine that, if given a Turing Machine as input, can decide (or at least recognize) if the inputted Turing Machine has an infinite number of strings in its language? ...
1
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0answers
17 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 ...
6
votes
2answers
89 views

Deciding the set of all Turing machines that halt in at most $k|x|$ steps $\forall x \in \Sigma^*$

Let $L = \{ <M> | M$ halts on every input $x$ in at most $200 * |x|$ steps $\}$. Is $L$ decidable? Recognizable? Given that membership in $L$ asserts something about $M$'s behavior on an ...
1
vote
1answer
36 views

How do you prove that this TM decides a language that is undecidable? [closed]

In Sipser's Introduction to the Theory of Computation, there is an exercise that asks to prove $T$ decides $A_{TM}$, which is the language $$A_{TM} = \{ \langle M,w \rangle | M \text{ is a TM and $w ...
1
vote
1answer
46 views

Proof that $A_{DFA}$ is decidable in Sipser

It seems like the proof that $A_{DFA}$ is decidable in Sipser (2nd ed.) assumes the computation will halt... and hence only really proves that $A_{DFA}$ is recognizable. The language $A_{DFA}$ is ...
5
votes
2answers
348 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 ...
0
votes
1answer
102 views

how do I find a undecidable subset of a set that's decidable? [closed]

Given that Let S = {a | |a| is odd}. I know that since S is decidable, but does there exist a subset within S that is undecidable?
0
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0answers
11 views

two undecidable languages with a decidable union/intersection? [duplicate]

does there exist two undecidable languages such that their union is decidable? what about a decidable intersection? One thing that I've been trying to figure out is if J and K are both undecidable ...
4
votes
2answers
472 views

Sandwiching Languages

I am studying for my algorithms final and came across the following problem: Find three languages $L_1 \subset L_2 \subset L_3$ over the same alphabet such that $L_2 \in P$ and $L_1,L_3$ are ...
4
votes
2answers
136 views

Is this language depending on P = NP recursive?

Nobody yet knows if ${\sf P}={\sf NP}$. Let us consider the following language $$L = \begin{cases} (0+1)^* & \text{ if ${\sf P}$ = ${\sf NP}$} \\ \emptyset &\text{ otherwise}. \end{cases}$$ ...
0
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1answer
55 views

Palindromes and linear grammars

Given a linear grammar G, is it possible to determine if L(G) contains a palindrome?
2
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1answer
63 views

Undecidability and Countability

This question is prompted by Undecidable unary languages (also known as Tally languages) How does the countability of a language imply (un)decidability?
2
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2answers
92 views

Is equivalence of CFGs decidable for finite sets of grammars?

Is there a way to show that for all finite sets $S$ of context free grammars, there exists a Turing Machine $M$ such that for all grammars $G_1, G_2 \in S$, we have that $M(G1,G2)$ terminates and ...
1
vote
1answer
156 views

Reduction to complement of Accept Problem

I am reducing a given Turing Machine to the complement of the known undecidable problem, $$ Complement(A_{TM}) = \{ \langle M,w \rangle \mid M \text{ is TM}, w \not\in L(M) \}$$ To this Turing ...
1
vote
3answers
244 views

Determining if a context-free grammar produces even-length strings [closed]

Given a context-free grammar, is there an algorithm to determine if the CFG will ever produce an even length string? Or is this undecidable?
1
vote
1answer
86 views

Decidability of an regular expression

I have this question about if the decidability of an regular expression and would appreciate if someone can check my answer and see if it makes sense, and if not, what is missing. Be A = {(R)|R it ...
4
votes
1answer
108 views

Is it possible to prove EQTM is undecidable by the Rice theorem?

Given the problem $EQ_{TM} = \{ \langle M_1, M_2\rangle \mid M_1 \text{ and } M_2 \text{ are } TM, L_{M_1} = L_{M_2}\}$, is it possible to prove that this is undecidable by using (a variant of) Rice ...
0
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1answer
20 views

Decidability of fullness of intersection of a CSL with a regular language

Let $L_r$ be a regular language with alphabet $\Sigma$ and $L_{\text{csl}}$ be a context sensitive language. Are any of the following questions decidable? $L_r \cap L_\text{csl} \stackrel{?}{=} L_r$ ...