# (1,0)-Super Solutions of (k,s)-CNF Formula

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Notations

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Lemma**

**1**

**.**An assignment is a (1,0)-super solution of F if and only if it satisfies $\pi \left(F\right)$.

**Lemma**

**2**

**.**If $k\ge 3$ and s are such that an unsatisfiable $(k,s)$-CNF formula exists, then $(k,s)$-SAT is NP-complete.

**Lemma**

**4**

**Lemma**

**5.**

## 4. The Existence of (1,0)-Super Solution of $(\mathit{k},\mathit{s})$-CNF Formula

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Corollary**

**3.**

**Theorem**

**3.**

**Proof.**

## 5. The NP-Completeness of (1,0)-$(\mathit{k},\mathit{s})$-SAT

**Theorem**

**4.**

**Proof.**

**Step 1**Let $f={C}_{1}\wedge {C}_{2}$. Here

**Step 2**Let ${f}_{i},1\le i\le m$ be disjoint copies of the formula f with the variables ${x}_{j},1\le j\le k+1$ of f being renamed as ${x}_{i,j}$ in ${f}_{i}$. Let $X=\{{x}_{i,j},1\le i\le m,1\le j\le k+1\}$ and ${\Psi}_{1}={\wedge}_{{C}_{i}\in \Psi}({C}_{i}\vee {x}_{i,1})$.

**Step 3**We construct the formula $\Phi ={\Psi}_{1}\wedge {f}_{1}\wedge {f}_{2}\wedge \dots \wedge {f}_{m}$.

**Corollary**

**4.**

**Proof.**

**Corollary**

**5.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Case 1:**the formula F is unsatisfiable. This illustrates that $f(k+1)<s$. By Lemma 3, $f\left(k\right)<s$. That is to say, there exists an unsatisfiable instance of ($k,s$)-SAT. Corollary 5 entails that (1,0)-($k+1,s$)-SAT is NP-complete.

**Case 2:**the formula F is satisfiable but does not have a (1,0)-super solution. Lemma 6 entails that we can construct a forced-$(k+1,s)$-CNF formula $\Phi $. Next we present a polynomial time reduction method from an instance of k-SAT to an instance of (1,0)-($k+1,s$)-SAT.

**Step 1**Let ${\Phi}_{ij},1\le i\le mk,1\le j\le k-2$ be disjoint copies of the formula $\Phi $ with the variables $x,y$ of $\Phi $ being renamed as ${x}_{i,j},{y}_{i,j}$ in ${\Phi}_{ij}$. These formulas and the formula $\Psi $ have pairwise disjoint sets of variables. Let ${\Psi}_{1}={\wedge}_{1\le i\le mk}{\wedge}_{1\le j\le k-2}{\Phi}_{ij}$ and $X=\left\{{x}_{i,j}\right\}(1\le i\le mk,1\le j\le k-2)$.

**Step 2**We introduce a new variable set $Z=\left\{{z}_{i,j}\right\}(1\le i\le m,1\le j\le k)$ to replace $mk$ literals in $\Psi $ in order to construct a new formula ${\Psi}_{2}$.

**Step 3**Let ${\Psi}_{3}={\bigwedge}_{1\le i\le mk}{d}_{i}$, and ${d}_{i}={z}_{i}\vee \neg {z}_{j}\vee \neg {x}_{i,1}\vee \neg {x}_{i,2}\vee \dots \vee \neg {x}_{i,k-2}\vee {x}_{j,1}$. Here ${z}_{i},{z}_{j}\in Z$ and the variables of Z are sorted by their subscripts. In addition, ${z}_{i}$ replaces a variable v in $\Psi $ and ${z}_{j}$ be the next variable of Z which replaces v (if ${z}_{i}$ is the last variable in the variable set Z which replaces the variable v, ${z}_{j}$ is set to be the first).

**Step 4**We construct the formula ${\Psi}^{\prime}=\{{\Psi}_{1},{\Psi}_{2},{\Psi}_{3}\}$.

## 6. The Transition Phenomenon of (1,0)-$(\mathit{k},\mathit{s})$-SAT

**Theorem**

**6.**

**Proof.**

**Corollary**

**6.**

**Corollary**

**7.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Fu, Z.; Xu, D.; Wang, Y.
(1,0)-Super Solutions of (*k*,*s*)-CNF Formula. *Entropy* **2020**, *22*, 253.
https://doi.org/10.3390/e22020253

**AMA Style**

Fu Z, Xu D, Wang Y.
(1,0)-Super Solutions of (*k*,*s*)-CNF Formula. *Entropy*. 2020; 22(2):253.
https://doi.org/10.3390/e22020253

**Chicago/Turabian Style**

Fu, Zufeng, Daoyun Xu, and Yongping Wang.
2020. "(1,0)-Super Solutions of (*k*,*s*)-CNF Formula" *Entropy* 22, no. 2: 253.
https://doi.org/10.3390/e22020253