Questions tagged [decision-problem]

A question in some formal system with a yes-or-no answer.

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18 views

1-Tape Non-deterministic Turing machine and non-palindromes

I have to design a Non-deterministic 1-Tape Turing machine that accepts only non-palindromes in O (n log n). But my best shot was only in O(n^2). How can I use the properties of NTM on a single tape ...
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145 views

Given the Turing machines M1 and M2, is L (M1) = L (M2)? is decidable?

I thought to reduce from the halting problem to conclude undecidability, yet I don't know how to do it. Perhaps the problem reduces to other decidable problem, and thus it is also decidable?
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63 views

show that NP is a subset of decidable languages

More specifically the problem says: "Let us call the set of decidable languages D. Show that NP ⊆ D" My problem is that I always assumed that NP is decidable, but to prove it, I never ...
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1answer
20 views

Is this “superset existence” problem NP-complete?

The "Superset Existence Problem": Let there be a set $S$, and $x$ subsets of $S$. Does there exist a set of size $y < |S|$, which is a superset of at least $z$ of those subsets? To me, ...
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1answer
52 views

Understanding P vs NP

I want to make sure my understanding on P vs NP is correct. I know that NP-complete problems cannot be solved in polynomial time, and if P != NP, then all problems in NP cannot be solved in polynomial ...
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1answer
32 views

Cyclic tour minimizing total weight

I asked the following question on math.se but it wasn't really answered so moved it over here as I feel it's more relevant. I saw the question below on an old stack exchange question when looking to ...
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29 views

Difference longest path problem and underlying decision problem [duplicate]

I am studying the longest path problem with the final objective to show that it is NP-complete. On wikipedia I read that the problem itself is NP-hard but the underlying decision problem is NP-...
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Algorithm to test whether a language is context-free

Is there an algorithm/systematic procedure to test whether a language is context-free? In other words, given a language specified in algebraic form (think of something like $L=\{a^n b^n a^n : n \in \...
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56 views

Another version of the 3-coloring decision problem?

Given a graph $G$, is there a 3-coloring with colors $c1$, $c2$ and $c3$ such that at most $k$ nodes are given the color $c1$ and that no two adjacent nodes are given the same color? Is there a ...
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1answer
33 views

PCP when upper and lower words have different length

The Post correspondence problem (PCP) asks, given two sets of words $a_1,\ldots,a_n$ and $b_1,\ldots,b_m$ over the same alphabet, whether there are indices $i_1,\ldots,i_s \in \{1,\ldots,n\}$ and $j_1,...
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1answer
248 views

Solve Max 3 color problem using 3 color decision problem

I've been stumped on this question for a while and can't find a solution. How can I find the max 3 colorability of a graph(optimization problem) with 3 colorability (decision problem) without brute ...
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58 views

Turing-completeness of Goto language with limited constants

This is taken from an old exam of my university that I am using to prepare myself for the coming exam: Given is a language $\text{Goto}_{17} \subseteq \text{Goto}$. This language includes exactly ...
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19 views

What's the union between a decision problem and its complement

I see the union of problems as something like this: $P=F\cup G$ $P(\omega):$ $\;\;\;\;if\; F(\omega)==True: return \;\; True$ $\;\;\;\;else\;if\; G(\omega)==True: return \;\; True$ $\;\;\;\;else: ...
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Are there any tetrational-time problems?

I know there exists problems decidable in polynomial-time, exponential-time, etc. I couldn't find any tetrational-time problems, however. Are there any and if not, why?
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Is there a #$P$-complete counting problem such that every (valid) instance of its decision version is a Yes-instance?

I want to know whether there is a decision problem, written EasyProblem, satisfying the follow property: For every valid instance $x$, $x$ is a Yes-instance for EasyProblem (if we construct ...
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294 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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1answer
23 views

Different Classes of NP [closed]

I was solving problems related to P and NP where I encountered the following problem: Given a standard definition of NP, if x belongs to L then there exists y such that |y| <= |x|^d and A(x, y) = ...
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65 views

3-Sat reduction to facility location problem

I'm learning about NP problems and I this problem which is a bit challenging for me. You are given an undirected, simple graph G = (V,E) and an integer k where nodes represent cities and edges ...
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Proving that DCONN is NL-Complete

I am having trouble with some homework regarding proving that DCONN is NL-Complete. As part of the exercise, the fact that RCH is NL-Complete can be assumed. Problem definitions: RCH: Given a ...
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2answers
204 views

NP-completeness of testing whether a SAT formula does not contain any redundant clause

This paper explains that testing the irredundancy of a SAT instance is NP-complete. But I don't understand the theorem/reduction. How would one reduce 3SAT or k-SAT to this problem, for example?
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153 views

Efficient algorithm to determine if a lambda calculus term is equivalent to one without a given free variable

Consider the following problem: given a lambda calculus term $t$ and free variable $v$ determine whether $\phi(t,v)$, where $\phi(t,v) := \exists t'. t' \equiv t \land v \notin FV(t')$. This problem ...
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1answer
85 views

How To Show That B is Semi-Decidable Given A

I am preparing for my Computational Theory final and ran into this exact problem : B={ x | there exists a prefix of x that is in A}. Show that B is semi-decidable. In other words, you need to ...
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43 views

If a poly-time solution exists for an NP-Complete (Decision) problem, then there exists a poly-time solution for the NP-hard (Optimization) flavor?

This question boils down to: If a polynomial time solution exists for a decision problem, is there also a polynomial time solution for the same problems optimization flavor? Let's take the Traveling ...
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3answers
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Show a TM-recognizable language of TMs can be expressed by TM-description language of equivalent TMs

I am studying "An Introduction to the Theory of Computation" by Sipser -- there is a problem *3.17 (p.161) which I can not solve. Any hints (not answers) from which side to attack it? Let $B=\{M_1,...
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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 $\...
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107 views

How to convert a decision tree to an automaton?

From what I know, a problem can be transformed to a yes/no answer, which can be described by a decision tree. Solution to a problem also can be represented by a set of strings (a language), which ...
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15 views

What does a kernel of size n,n^2 ,… mean?

So according to Wikipedia, In the Notation of [Flum and Grohe (2006)], a ''parameterized problem'' consists of a decision problem $L\subseteq\Sigma^*$ and a function $\kappa:\Sigma^*\to N$, the ...
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91 views

Enumerator for Word and Halting Problem

in theoretical computer science I learned for every recursive enumerable language there would be an enumerator and a grammar. So since word problem and halting problem are recursively enumerable, I ...
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188 views

Why don't passwords prove P != NP?

Pardon my ignorance on the matter but, Verifying passwords = Polynomial (linear) Guessing passwords = Exponential Since each guess has nothing to do with one another, exponential time is best possible ...
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1answer
53 views

Is $\{w~|~\forall x \in T(M_v):|w|>|x|~\}$ decidable?

I want to ask if $\{w|\forall x\in T(M_v):|w|>|x|\}$ is decidable if v is a Index of a random but fixed Turing Machine with $|T(M_v)|<\infty$. My idea: It is co-semi-decidable since as soon as i ...
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1answer
55 views

Is the problem that determines whenever the word member $\in$ L(M) decidable or not?

Given a Turing machine M on alphabet {m,e,b,r} we're asked to determine if member $\in$ L(M). You must realize that M is not one specific machine and can be any turing Machine with the same alphabet. ...
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1answer
51 views

Rice's Theorem for Turing machine with fixed output

So I was supposed to prove with the help of Rice's Theorem whether the language: $L_{5} = \{w \in \{0,1\}^{*}|\forall x \in \{0,1\}^{*}, M_{w}(w) =x\}$ is decidable. First of all: I don't understand, ...
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1answer
45 views

Why is the Halting problem decidable for Goto languages limited on the highest value of constants and variables?

This is taken from an old exam of my university that I am using to prepare myself for the coming exam: Given is a language $\text{Goto}_{17}^c \subseteq \text{Goto}$. This language contains exactly ...
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95 views

I have a decision problem with $2^n$ bit sized certificates, how would I verify my decision problem efficiently if it is in $NP$?

Decision Problem: Is $2^k$ + $M$ a prime? The inputs for both $K$ and $M$ are integers only. The solution is the sum of $2^k$+$M$. (Use AKS to decide prime) The powers of 2 have approximately $2^n$ ...
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60 views

Proof: is the language $L_1=\{\langle M\rangle\mid\emptyset \subseteq L(M)\}$ (un)-decidable?

I want to show that $L_1 = \{\langle M\rangle \mid \emptyset \subseteq L(M)\}$ is decidable/undecidable - without rice theorem (just for the case that I can apply it). Every language contain the $\...
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1answer
71 views

Nondeterministic polynomial time algorithm versus certificate/verifier for showing membership in NP

In this paper (https://arxiv.org/pdf/1706.06708.pdf) the authors prove that optimally solving the $n\times n\times n$ Rubik's Cube is an NP-complete problem. In the process, they must show that the ...
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64 views

Every decidable language $L$ has an infinite decidable subset $S \subset L$ such that $L \setminus S$ is infinite

Given an infinite decidable language $L$, then if $S \subset L$ such that $L \setminus S$ is finite, then $S$ must be decidable. This is true since given a decider of $L$ we contruct a decider for $S$:...
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Proof of Co-Problem being in NP if Problem is in NP using negated output

Given any problem $P$ that we know of being in $NP-\text{complete}$, where is the flaw in the following proof? Given a problem $Co-P$ which is the co-problem of $P \in NP-\text{complete}$, $Co-P$ is ...
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37 views

Computing automaton for $L(A) / L(B)$ gives ones for $A,B$

I'm trying to figure out whether infinite language change the answer. Show that the following language is decidable: $$L=\{\langle A,B \rangle : \text{$A,B$ are DFAs, $L(B)$ is finite, and $L(A)/ L(B)...
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37 views

Is $L_2:=${$<M>$|$L(M)=\overline{A_TM}$} (un-)decidable?

I have to prove that the language $L_2:=${$<M>$|$L(M)=\overline{A_TM}$} is (un-)decidable. In a previous assignment we proved that $L_1:=${$<M>$|$L(M)=A_TM$} is undecidable. I would say ...
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22 views

Pick elements that don't exhaust any set

The following is an NP-Complete problem: Suppose you have a collection $\mathcal{C}$ of sets, so that $A_i\in \mathcal{C}$ and $A_i$ is some set--we can suppose the elements of $A_i$ are integers. ...
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34 views

why is $\Pi_2$ smaller than $NP\cap coNP$

Consider the language $A=\{(\phi_1, \phi_2) | \phi_1 \in SAT, \phi_2\in \overline{SAT} \}$. What is the smallest class that $A$ is known to belong to? Apparently, the answer is $\Pi_2$, although I ...
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1answer
37 views

Why minimum vertex cover problem is in NP

I am referring to the definition of the minimum vertex cover problem from the book Approximation Algorithms by Vijay V. Vazirani (page 23): Is the size of the minimum vertex cover in $G$ at most $k$? ...
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1answer
39 views

checking whether turing machine passes at least k>2 states before accepting a word

$L=\{\langle M,k\rangle \mid\exists w\in L(M) \text{ such that $M$ passes at least $k>2$ distinct states before accepting $w$}\}$ I try to think of reduction to prove that this language is ...
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54 views

Variant of Subset-sum has an $O(1)$ algorithm if $Goldbach$ is true

Given $S$ of positive integers $>$ $1$ is there some combination with even $SUM$ > $2$ that is NOT the sum of two primes? $SUM$ = 10 $S$ = $[4,6]$ $No$, Sum of Two Primes $5 + 5 = 10$. ...
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80 views

Complexity of Integer Factorization

In Quantum Information and Quantum Computation by Nielsen and Chuang, they define the complexity class NP as follows (page 142): A language $L$ is in NP if there is a turing machine $M$ with the ...
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47 views

An optimization version of 2QBF: is it $\mathsf{NP}^{\mathsf{NP}}$-hard?

I am studying the computational complexity of the following decision problem related to 2QBF: Input: a 3-CNF formula $\varphi$ over $X \cup Y$, where $X$, $Y$ are disjoint sets of propositional ...
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1answer
38 views

Why Right-Division of regular language with RE\E language is regualr?

I think I can't understand the meaning of language being decidable. The next case makes no sense to me: Considering I have language L1 which is regular, and language L2 which is in RE\R (in ...
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34 views

Is necessarily the following language not decideable

For A,B that are not decidable, does AB U BA not necessarily decidable? I think that the answer is NO. Not necessarily. I thought about the following example, but it does not refute exactly: If we ...
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19 views

What decision problems are their that are outside of elementary but still decidable

What decision problems are their that are outside of ELEMENTARY but still decidable? I'm curious about problems that are still solveable, but take a very long time to do so.

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