Questions tagged [decision-problem]

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

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30
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2answers
11k views

Optimization version of decision problems

It is known that each optimization/search problem has an equivalent decision problem. For example the shortest path problem optimization/search version: Given an undirected unweighted graph $G ...
13
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2answers
622 views

Is there an efficient algorithm for expression equivalence?

e.g. $xy+x+y=x+y(x+1)$ ? The expressions are from ordinary high-school algebra, but restricted to arithmetic addition and multiplication (e.g. $2+2=4; 2.3=6$), with no inverses, subtraction or ...
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4answers
9k views

Language of Turing machines that loop on all inputs, recognizable?

Prove that the language Loop Turning Machine = { < M > | M is a TM that loops on all inputs} is recognizable. I feel like $M$ would never halt. To make $M$ recognizable it needs to accept or ...
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1answer
1k views

Algorithm to test whether a language is regular

Is there an algorithm/systematic procedure to test whether a language is regular? In other words, given a language specified in algebraic form (think of something like $L=\{a^n b^n : n \in \mathbb{N}\...
5
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2answers
245 views

How does one find out whether $N = a^b$ for some $b$?

I was trying to find out how to find whether $N$ is a perfect power or not for some $a$ and $b$ (so the algorithm should discover that its not a perfect power if its not expressable in the form $a^b$)...
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3answers
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.
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2answers
6k views

Do any decision problems exist outside NP and NP-Hard?

This question asks about NP-hard problems that are not NP-complete. I'm wondering if there exist any decision problems that are neither NP nor NP-hard. In order to be in NP, problems have to have a ...
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2answers
3k 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 $\...
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3answers
4k views

Proof that whether a regular language is finite is decidable [duplicate]

I have this question for a homework. The question stems from the fact that you can determine whether a regular language is empty by using a Turing machine to count the states n in the given FSM. When ...
7
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3answers
7k views

Why are Chess, Mario, and Go not NP-complete?

I have a hole in my understanding of what makes a problem NP. I understand that Mario, for example, is NP-hard - it can be reduced to the NP-complete problem of 3SAT (see https://www.youtube.com/...
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3answers
2k views

Is there an efficient test for if an NFA accepts a subset of another NFA?

So, I know that testing if a regular language $R$ is a subset of regular language $S$ is decidable, since we can convert them both to DFAs, compute $R \cap \bar{S}$, and then test if this language is ...
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3answers
2k views

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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3answers
10k views

All NP problems reduce to NP-complete problems: so how can NP problems not be NP-complete?

My book states this If a decision problem B is in P and A reduces to B, then decision problem A is in P. A decision problem B is NP-complete if B is in NP and for every problem in A in ...
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1answer
4k views

Proving that a problem is in NP

I have an assignment in which the problem, $D$, is simple but, once found, easy to check. Is it enough to prove that a solution $x$ can be checked in polynomial time to prove that $D \in NP$? (The ...
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2answers
1k views

3-SAT where variables occur equally many times as a positive literal and as a negative literal

Let $\phi$ be a 3-CNF formula over variables $x_1,x_2,\ldots,x_n$. Every variable $x_i$, $i \in [n]$, occurs equally many times as a positive literal and as a negative literal in $\phi$. Is it NP-...
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3answers
1k views

Does every computational problem have a decision version?

Is the following claim correct?: Every computational problem has a decision version of roughly equal computational difficulty. If the above claim is correct, please give a reference for it. (I ...
7
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2answers
422 views

primitive recursive functional equivalence

Given two primitive recursive functions is it decidable whether or not they are the same function? For example lets take sorting algorithms A, and B which are primitive recursive. While there are many ...
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2answers
3k views

Optimization problem vs decision problem - reduction

Assume we have an optimization problem with function $f$ to maximize. Then, the corresponding decision problem 'Does there exist a solution with $f\ge k$ for a given $k$?' can easily be reduced to ...
3
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1answer
125 views

Questions on Graph and Hamiltonian [closed]

From this book and other study in complexity theory, I have seen the following statement: The definition of NP is not symmetric with respect to yes-instances and no-instances. For example, it is ...
3
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1answer
624 views

When is splitting a collection coins two ways NP-complete?

Suppose we have a set $D$ of denominations of coins and a our input is a "tip jar" containing some finite number of these coins (e.g., five nickels, a dime and three quarters). In the first two ...
3
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3answers
516 views

NP-hardness of an optimization problem with real value

I have an optimization problem, whose answer is a real value, not an integer such as vertex cover and set cover. Therefore, the decision version of my problem is given an input and a real value $r$. ...
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1answer
1k views

Can the decision version of an optimization problem in NP, be in P?

It is well known that a optimization problem can be turned into a decision problem with an extra parameter: e.g. in TSP we are looking for the lowest cost for a tour, the decision version therefore ...
3
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1answer
614 views

How exactly does a Max 2 Sat reduce to a 3 Sat?

I've been reading this article which tries and explains how the max 2 sat problem is essentially a 3-sat problem and is NP-hard. However, if you see the article, I'm not able to understand why, after <...
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2answers
2k views

Decision version of the traveling salesman problem and NP-hardness

Wikipedia says: The problem has been shown to be NP-hard and the decision problem version ("given the costs and a number x, decide whether there is a round-trip route cheaper than x") is NP-...
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1answer
731 views

Is equivalence of a CFG and an RG undecidable?

I know that the equivalence of two context-free grammars is undecidable, but what about the equivalence of a regular grammar and a context-free grammar?
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1answer
935 views

Equivalence of regular grammars

I know that proving context free grammars equivalent is undecidable. I also know that proving if a context free grammar recognizes a regular language is undecidable. Here is my question: is proving ...
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1answer
2k views

Decision Problem vs. Optimization Problem [duplicate]

Is the following statement correct? If a decision problem is NP-complete, the corresponding optimization problem can not be solved in polynomial time.
64
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7answers
15k views

Is legislation NP-complete?

I would like to know if there has been any work relating legal code to complexity. In particular, suppose we have the decision problem "Given this law book and this particular set of circumstances, is ...
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5answers
2k views

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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5answers
12k 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 ...
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2answers
5k views

Is the equality of two DFAs a decidable problem?

So given two DFAs, is the problem of finding if they generate the same language a Decidable problem? I already know that Equality of two CFL is not Decidable but what about Equality of two DFAs? ...
6
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1answer
279 views

Complexity of deciding the satisfiability of a quasi-monotone CNF formula

A quasi-monotone CNF formula is a formula where each variable appears at most once as a positive literal (and any number of times as a negative literal). What is the complexity of deciding its ...
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3answers
13k views

Prove that Hitting Set is NP-Complete

The Hitting Set Problem (HS) is defined as follow. Let $(C,k$) $C = \{ S_1, S_2, ..., S_m \}$ collection of subset of S i.e. $ S_i \subseteq S , \forall i$ $k \in \mathbb{N}$ We want to know if ...
4
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1answer
13k views

Is every regular language Turing-decidable, and how can we prove this?

I know every regular language is Turing-acceptable, but does that imply it is Turing-decidable?
4
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1answer
105 views

Maximal class for which function equivalence is decidable

I previously asked if it's decidable whether two primitive recursive functions are equivalent: "primitive recursive functional equivalence". The answer was no. Here is my followup. What is the most ...
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2answers
2k views

What is the difference between "Decision" and "Verification" in complexity theory?

In Michael Sipser's Theory of Computation on page 270 he writes: P = the class of languages for which membership can be decided quickly. NP = the class of languages for which membership can be ...
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3answers
619 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 ...
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2answers
1k views

Is there any strategy to brute force search?

I don't know how to state it elegantly, but basically, I want to implement a brute force search algorithm, but there are many different ways that I could enumerate through the search space. This ...
5
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2answers
734 views

Is the following Subset Sum variant NP-complete?

Is the following problem NP-hard: Input: $A\subset\mathbb Z, k\in\mathbb N$ Question: is there a multiset of indices $I$, such that $|I|=k$ and $\sum_{i\in I} a_i=0$? For example, on the input $A=\{-...
4
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2answers
404 views

Is deciding if there's a solution to a single multivariate quadratic equation NP-hard?

I know that given a system of multivariate quadratic equations (i.e, of the form $x^T Ax+b^T x=c$), deciding if there's a solution is NP-hard. Is deciding if there's a solution to a single ...
3
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1answer
2k views

Reduction: Vertex Cover to Binary Integer Program (Decision Problem)

I am stuck with the following task: Show that the Decision Problem "Vertex Cover" is polynomial-time reducible to the Decision Problem "Binary Integer programming". I have the feeling that there must ...
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0answers
102 views

Is there a generic procedure to produce (hard enough) decision problems?

This comment by @DerekElkins suggests a general method of constructing decision problems for problems with bit-strings as output, of which a slightly formalised version is the following: Given a ...
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2answers
640 views

Why are decision problems easier than the equivallent optimization problems?

Suppose that we have an optimization problem defined as follows: $OPT$ = Given an input string defining a set of feasible solutions $F$ and an objective function $f$, find $x\in F$ maximizing $f(x)$ ...
3
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2answers
717 views

Do all algorithms aim to solve decision problems?

An algorithm is a “unambiguous specification of how to solve a class of problems... calculation, data processing and automated reasoning tasks” Are the class of problems an algorithm can solve more ...
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2answers
872 views

Finding an exactly weighted st-path in a digraph

I have a weighted digraph graph $G = (V,E)$ where the weights are positive and negative integers. The graph $G$ is not necessarily acyclic. The question is: given 2 nodes $v_1$ and $v_2$, is there a ...
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1answer
39 views

Is there any interesting consequence of $\mathrm{DLogTime}$-uniform ${\mathrm{Mod}_6}^0=\mathrm{NP}$

$\mathrm{NP}$ has not been separated from constant-depth circuits consisting of solely $\mathrm{Mod}_6$ gates. So, the question is whether current techniques are enough to deduce interestingly ...
5
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1answer
271 views

How does the problem of having a coffee-machine close relate to vertex cover?

Meeting rooms on university campuses may or may not contain coffee machines. We would like to ensure that every meeting room either has a coffee machine or is close enough to a meeting room that ...
5
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1answer
4k views

Examples of undecidable problems whose intersection is decidable

I know that given two problems are undecidable it does not follow that their intersection must be undecidable. For example, take a property of languages $P$ such that it is undecidable whether the ...
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2answers
1k views

What is a Turing Machine in class coNP

On the wikipedia article about the polynomial hierarchy http://en.wikipedia.org/wiki/Polynomial_hierarchy it says "$A^B$ is the set of decision problems solvable by a Turing machine in class A ...
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0answers
39 views

Decision problems with complex input validation

In an answer to a question regarding input validation in decision problems, @Apass Jack wrote It is easy to check whether a problem instance is a valid instance or not for almost all decision ...