# Questions tagged [reductions]

In computability and complexity, finding mappings between problems that allow solving one problem using a solution of another one. For reduction in programming language theory (e.g. beta-reduction), see [lambda-calculus] or [term-rewriting].

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### Show that a set is reducible to another set

Consider an operator $+$ defined on $P(\mathbb{N})$ as follows $$A + B = \{2x:\ x\in A\}\cup\{2x + 1:\ x\in B\}$$ Show that $A$ is $m$-reducible to $A+B$ and $B$ is $m$-reducible to $A+B$ As per the ...
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### Any problem in P can be reduced to the language of odd integers

Given $A=\left\{n\in \mathbb{N} \mid \text{$n$is odd}\right\}$, we want to prove that if $S \in P$ then there is a Karp reduction from $S$ to $A$. My attempt: If $S \in P$ we can solve $S$ with a ...
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### Find a Cook Reduction from $R_{Clique}$ to its determinist problem

The question is to find Find a Cook Reduction from $R_{k-Clique}$ to its determinist problem. Basically: k-Clique: a group of $k$ nodes in the graph such there is an edge between every two nodes. ...
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### If $f$ reduces $L_1$ to $L$ and also $L_2$ to $L$ is $L_1=L_2$

If the same $f$ reduces $L_1$ to $L$ and also $L_2$ to $L$ does it imply that $L_1=L_2$? My intuition says no, but I couldn't find a counterexample.
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### Reduction from $VC$ to $CD$

We define the vertex cover as the problem of finding for a graph $G$, a cover of size $k$. A cover is a set of vertices such that every vertex has an edge to this set. We define CD (cycles destructor),...
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### Reducing the Hamiltonian cycle to the travelling salesman problem and self loops

If this is my adjacency matrix for the hamiltonian cycle: $$\begin{pmatrix}0&1&0&1\\ 1&0&1&0\\ 0&1&0&1\\ 1&0&1&0\end{pmatrix}$$ Then a reduction ...
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### Cook–Levin theorem and reduction as injective function

I saw that the injectivity "derives directly from the theorem", but i can't see how it's happen, any explanation?
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### Sufficient condition for a complexity class's closure under NP-reductions?

Let us say that there exists a $\mathsf{NP}$-reduction from a problem $A$ to another problem $B$ when there exists a non-deterministic, polynomial-time Turing machine $T$ such that for each $a \in A$, ...
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### If$A \leq_T B$ is given, can you reduce $\overline{A}$ to $B$ and vice-versa

If you are given two languages $A$, $B$ and $$A \leq_T B.$$ Is it possible to $\overline{A} \leq_T B$ or $A \leq_T \overline{B}$? Here is my shot. Case 1: $\overline{A} \leq_T B$ This is only possible ...
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### Can you reduce every decidable language to a regular language?

One of my previous questions on an exam was the following Can you reduce a decidable language to a given regular language? (decidable language $\leq _m$ regular language). If so, does this mean that ...
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### For every non-trivial language $A$ and every finite strict subset $B \subsetneq A$, it's holds that $A \le_m A \setminus B$

It's claim 1 from Bader Abu Radi's solution to this question. My solution (have no idea how wrong it is): $B$ finite $\Rightarrow$ $B\in R \Rightarrow$ exists TM $\langle M_B\rangle$ that halts $B$. *...
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### Which of these properties hold for all FO theories? (but not regarding fragments thereof)

Which of these properties hold for all FO theories? (but not regarding fragments thereof) a. Decidable b. At least expressive as propositional logic c. NP-complete a) Decidable: no, some first order ...
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### 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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### 3Col reduction Variation, Special edges

I have a question concerning NP reduction. My question asks me to show that if I have a graph with Edges that connect 3 nodes together instead of 2, (Y style I assume). I need to prove that finding ...
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### Why do I only need to reduce from one problem in NP to prove NP-Hardness?

Suppose I wish to show that my decision problem $Q$ is NP-Hard. Why do I need to reduce from one problem $Q'$ of known hardness ? Consider for instance the following situation: Here, I have my set NP ...
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### If A is not in NP, and A reduces to B, does this mean B is not in NP?

I know it is true that if A is not in P, and A reduces B, then B is not in P. But is it true for NP as well? If A is not in NP, and A reduces to B, does this mean B is not in NP? Why or why not? ...
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### Check if a linear function or an affine function can be a pseudo random function

Let $G = \{0, \cdots , p-1 \}$ be a field. Let $K = G^{m \times n}$ and $F:K \times G^n \to G^m$ be a family of functions. For $A \in G^{m \times n}$ and $x \in G^n$ we have $F(A,x) = Ax$. I need to ...
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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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### 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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### Can an NP-Complete problem be reduced to an NP problem?

All NP problems can be reduced to NP-Complete problems, can an NP-Complete problem be reduced to a NP problem (non complete)?
### $A \leq_p {\overline{A}} \Leftrightarrow {\overline{A}} \leq_p A$
I want to prove that $$A \leq_p {\overline{A}} \Leftrightarrow {\overline{A}} \leq_p A$$. Does anyone have a Idea how to solve this ?