# Uniquely Satisfiable d-Regular (k,s)-SAT Instances

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- (i)
- all random k-CNF instances with $\alpha <\alpha \left(k\right)$ are satisfiable with high probability;
- (ii)
- all random k-CNF instances with $\alpha >\alpha \left(k\right)$ are unsatisfiable with high probability.

## 2. Related Works

## 3. Notations

**Definition**

**1.**

**Definition**

**2.**

- (i)
- there exist k variables ${x}_{1},{x}_{2},\dots ,{x}_{k}$ that only occur once;
- (ii)
- except for the k variables, every variable occurs in exactly s clauses, and the absolute value of the difference between positive and negative occurrences of every variable is no more than the nonnegative integer d.
- (iii)
- F is satisfiable and for any truth assignment τ satisfying F, it holds that$$\tau \left({x}_{1}\right)=\tau \left({x}_{2}\right)=\dots =\tau \left({x}_{k}\right)=true.$$

**Definition**

**3.**

**Lemma**

**1**

**.**Let ($k,s$)-CNF be a class of satisfiable formulas, then all ($k+r,s+r[s/k]$)-CNF formulas are satisfiable for any nonnegative integer r ($\left[x\right]$ denotes the integral part of x).

**Lemma**

**2**

**.**If the representation matrix of a formula F is

## 4. Uniquely Satisfiable d-Regular ($k,s$)-CNF Formula

**Theorem**

**1.**

**Proof.**

**Lemma**

**3.**

**Proof.**

- (i)
- Every variable of Z occurs positively in $\lceil s/2\rceil $ clauses and negatively in $\lfloor s/2\rfloor $ clauses;
- (ii)
- All literals of ${\mathsf{\Phi}}_{2i}$ and $\neg {y}_{i}$ occur exactly once in ${\mathsf{\Phi}}_{3}$, $1\le i\le k$;
- (iii)
- Every clause of ${\mathsf{\Phi}}_{3}$ must have at least one positive occurrence of any one of Z.

**Lemma**

**4.**

**Proof.**

- (i)
- every variable of X and Y occurs in exactly $s-k-1$ clauses of $\mathsf{\Psi}$,$$\mathrm{if}s\le 2k,pos(\mathsf{\Psi},{x}_{i})=s-k-1,pos(\mathsf{\Psi},{y}_{j,i})=s-k-1;$$$$\mathrm{if}s>2k,pos(\mathsf{\Psi},{x}_{i})=\lceil s/2\rceil -1,pos(\mathsf{\Psi},{y}_{j,i})=\lceil s/2\rceil -1.$$
- (ii)
- Every clause of $\mathsf{\Psi}$ must have at least one positive occurrence of any one of these variables.

**Theorem**

**2.**

**Proof.**

**Step 1**Divide the variables ${y}_{1},{y}_{2},\dots ,{y}_{n}$ arbitrarily into t variable sets ${Y}_{1},{Y}_{2},\dots ,{Y}_{t}$ of size $k-1$. Some variables of $\mathsf{\Psi}$ forced to be $true$ are added, so that every variable set contains exactly $k-1$ variables (a variable forced to be $false$ can be transformed to a variable forced to be $true$ by flipping all occurrences of the variable). The variables ${x}_{1},{x}_{2},\dots ,{x}_{(m+1)k}$ are arbitrarily divided into $4t+1$ variable sets ${X}_{1},{X}_{2},\dots ,{X}_{4t+1}$. Moreover, it should be guaranteed that any one of ${X}_{1},{X}_{2},\dots ,{X}_{3t}$ has $k-2$ variables, any one of ${X}_{3t+1},\dots ,{X}_{4t}$ has $k-1$ variables and ${X}_{4t+1}$ includes the rest. When m is appropriately chosen, the partition is feasible. Now assume ${X}_{4t+1}$ contains r variables.

**Step 2**For each $1\le i\le t$, we will construct a formula ${H}_{i}$ using the variable sets ${Y}_{i},{X}_{i},{X}_{t+i},{X}_{2t+i}$ and ${X}_{3t+i}$.

- (i)
- Let ${z}_{j,0}$ replace any one of positive occurrences of ${y}_{j}$, and $\neg {z}_{j,1}$ replace any one of negative occurrences of ${y}_{j}$ in $\mathsf{\Psi}$, for $j=1,2,\dots ,k-1$. If ${y}_{j}$ does not occur as a positive literal, then we let ${z}_{j,0}$ replace one of other negative occurrences of ${y}_{j}$ in $\mathsf{\Psi}$ and flip all occurrences of ${z}_{j,0}$ in the following formulas ${H}_{i1},{H}_{i2},{H}_{i3},{H}_{i4}$. If ${y}_{j}$ does not occur as a negative literal, then we perform similar operations.
- (ii)
- Let$$\begin{array}{cc}\hfill {H}_{i1}& ={\wedge}_{j=1}^{k-1}({y}_{j}\vee \neg {z}_{j,0}\vee \neg {x}_{1}\vee \dots \vee \neg {x}_{k-2}),\hfill \\ \hfill {H}_{i2}& ={\wedge}_{j=1}^{k-1}({z}_{j,0}\vee \neg {z}_{j,1}\vee \neg {x}_{t+1}\vee \dots \vee \neg {x}_{t+k-2}),\hfill \\ \hfill {H}_{i3}& ={\wedge}_{j=1}^{k-1}({z}_{j,1}\vee \neg {y}_{j}\vee \neg {x}_{2t+1}\vee \dots \vee \neg {x}_{2t+k-2}),\hfill \\ \hfill {H}_{i4}& ={\wedge}_{j=1}^{k-1}({z}_{j,1}\vee \neg {x}_{3t+1}\vee \dots \vee \neg {x}_{3t+k-1}).\hfill \end{array}$$

**Step 3**We will make up the gap of the number of occurrences of every variable. Using the variables in sets X and $Z=\{{Z}_{i},i=1,2,\dots ,t\}$, we construct a formula ${\mathsf{\Psi}}_{2}$ that satisfies the following conditions.

- (i)
- For $i=1,\dots ,t$, each ${z}_{j,0},j=1,2,\dots ,k-1$ in the variable set ${Z}_{i}$ occurs in exactly $s-3$ clause of ${\mathsf{\Psi}}_{2}$ and $pos({\mathsf{\Psi}}_{2},{z}_{j,0})+1-neg({\mathsf{\Psi}}_{2},{z}_{j,0})=min(d,1)$.
- (ii)
- For $i=1,\dots ,t$, each ${z}_{j,1},j=1,2,\dots ,k-1$ in the variable set ${Z}_{i}$ occurs in exactly $s-4$ clauses of ${\mathsf{\Psi}}_{2}$ and $pos({\mathsf{\Psi}}_{2},{z}_{j,1})-neg({\mathsf{\Psi}}_{2},{z}_{j,1})=min(d,1)$.
- (iii)
- Each variable x in ${X}_{1},{X}_{2},\dots ,{X}_{4t}$ occurs in exactly $s-k$ clauses of ${\mathsf{\Psi}}_{2}$,$$pos({\mathsf{\Psi}}_{2},x)=s-k\text{}\mathrm{for}\text{}s2k\text{}\mathrm{or}\text{}pos({\mathsf{\Psi}}_{2},x)=\u2308s/2\u2309-1\text{}\mathrm{for}s\ge 2k.$$
- (iv)
- Each variable x in ${X}_{4t+1}$ occurs in exactly $s-1$ clauses of ${\mathsf{\Psi}}_{2}$ and$$pos({\mathsf{\Psi}}_{2},x)+1-neg({\mathsf{\Psi}}_{2},x)=min(d,1).$$
- (v)
- Every clause of ${\mathsf{\Psi}}_{2}$ must have at least one positive occurrence of any one of the variables.

**Step 4**Let $\mathsf{\Phi}={\mathsf{\Psi}}_{1}\wedge \left({\wedge}_{i=1}^{t}{H}_{i}\right)\wedge {\mathsf{\Psi}}_{2}$.

**Lemma**

**5.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**1.**

**Proof.**

## 5. A Parsimonious Polynomial Time Reduction

**Theorem**

**5.**

**Proof.**

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

**Step 2**Let ${\mathsf{\Phi}}_{i},1\le i\le m(k-1)$ be disjoint copies of the formula $\mathsf{\Phi}$ with the variables ${x}_{j},1\le j\le k$ of $\mathsf{\Phi}$ being renamed as ${x}_{i,j}$ in ${\mathsf{\Phi}}_{i}$. All of ${x}_{i,j}$ are renumbered and formed a variable set $X=\{{x}_{i},1\le i\le mk(k-1)\}$. Let ${\mathsf{\Psi}}_{2}={\wedge}_{1\le i\le m(k-1)}{\mathsf{\Phi}}_{i}$.

**Step 3**Let ${\mathsf{\Psi}}_{3}={\wedge}_{1\le i\le m,1\le j\le k}{d}_{i,j}$, and ${d}_{i,j}={z}_{i,j}\vee \neg {{z}^{\prime}}_{i,j}{\vee}_{l=1}^{k-2}\neg {x}_{\left(\right(i-1)m-j-1)(k-2)+l}$. Here ${z}_{i,j},{z}_{i,j}^{\prime}\in Z$ and if ${z}_{i,j}$ replaces a variable v in $\mathsf{\Psi}$, then ${z}_{i,j}^{\prime}$ will point to the next variable in Z that replaces v (if ${z}_{i,j}$ is the last variable in Z that replaces v, then ${z}_{i,j}^{\prime}$ will point to the first variable in Z that replaces v). The variables in Z are sorted by their subscripts.

**Step 4**We construct a k-CNF formula ${\mathsf{\Psi}}_{4}$ with two variable sets X and Z, satisfying the following conditions.

- (i)
- Every variable ${z}_{i,j}$ of the variable set Z occurs in exactly $s-3$ clauses of ${\mathsf{\Psi}}_{4}$, and if ${z}_{i,j}$ occurs negatively in ${\mathsf{\Psi}}_{1}$,$$pos({\mathsf{\Psi}}_{4},{z}_{i,j})-neg({\mathsf{\Psi}}_{4},{z}_{i,j})=min(d,1).$$Otherwise$$neg({\mathsf{\Psi}}_{4},{z}_{i,j})-pos({\mathsf{\Psi}}_{4},{z}_{i,j})=min(d,1).$$
- (ii)
- For $1\le i\le mk(k-2)$, every variable ${x}_{i}$ of X occurs in exactly $s-2$ clauses of the formula ${\mathsf{\Psi}}_{4}$, and$$pos({\mathsf{\Psi}}_{4},{x}_{i})-neg({\mathsf{\Psi}}_{4},{x}_{i})=min(d,1).$$
- (iii)
- For $mk(k-2)+1\le i\le mk(k-1)$, every variable ${x}_{i}$ of the variable set X occurs in exactly $s-1$ clauses of the formula ${\mathsf{\Psi}}_{4}$, and$$pos({\mathsf{\Psi}}_{4},{x}_{i})+1-neg({\mathsf{\Psi}}_{4},{x}_{i})=min(d,1).$$
- (iv)
- Every clause of ${\mathsf{\Psi}}_{4}$ must have at least one positive occurrence of any one of the variable set X.

**Step 5**We construct the formula ${\mathsf{\Psi}}^{\prime}=\{{\mathsf{\Psi}}_{1},{\mathsf{\Psi}}_{2},{\mathsf{\Psi}}_{3},{\mathsf{\Psi}}_{4}\}$.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Cook, S.A. The complexity of theorem-proving procedures. In Proceedings of the Third Annual ACM Symposium on Theory of Computing, Shaker Heights, OH, USA, 3–5 May 1971; pp. 151–158. [Google Scholar] [CrossRef]
- Eén, N.; Sorenssön, N. An Extensible SAT-solver. In Theory and Applications of Satisfiability Testing; Springer: Berlin/Heidelberg, Germany, 2003. [Google Scholar] [CrossRef]
- Audemard, G.; Simon, L. GLUCOSE2.1: Aggressive-but Reactive-Clause Database Management, Dynamic Restarts. In Proceedings of the International Workshop of Pragmatics of SAT, Trento, Italy, 16 June 2012. [Google Scholar]
- Luo, M.; Minli, M.; Xiao, F.; Manyá, F.; Zhipeng, L. An Effective Learnt Clause Minimization Approach for CDCL SAT Solvers. In Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, Melbourne, Australia, 19–25 August 2017. [Google Scholar] [CrossRef] [Green Version]
- Calabro, C.; Paturi, R. k-SAT Is No Harder Than Decision-Unique-k-SAT. In Computer Science Symposium in Russia; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar] [CrossRef]
- Tovey, C.A. A simplified NP-complete satisfiability problem. Discret. Appl. Math.
**1984**, 8, 85–89. [Google Scholar] [CrossRef] - Daoyun, X.; Xiaofeng, W. A Regular NP-Complete Problem and Its Inapproximability. J. Front. Comput. Sci. Technol.
**2013**, 7, 691–697. [Google Scholar] [CrossRef] - Crawford, J.M.; Auton, L.D. Experimental Results on the Crossover Point in Satisfiability Problems. Artif. Intell.
**1996**, 81, 31–57. [Google Scholar] [CrossRef] [Green Version] - Kirkpatrick, S.; Selman, B. Critical behavior in the satisfiability of random boolean expressions. Science
**1994**, 264, 1297–1301. [Google Scholar] [CrossRef] - Jincheng, Z.; Daoyun, X.; Youjun, L. Satisfiability Threshold of the Regular Random (k,r)-SAT Problem. J. Softw.
**2016**, 27, 2985–2993. [Google Scholar] [CrossRef] - Jincheng, Z.; Daoyun, X.; Youjun, L. Satisfiability threshold of regular (k,r)-SAT problem via 1RSB theory. J. Huazhong Univ. Sci. Technol.
**2017**, 45, 7–13. [Google Scholar] [CrossRef] - Mézard, M.; Parisi, G.; Zecchina, R. Analytic and algorithmic solution of random satisfiability problems. Science
**2002**, 297, 812–815. [Google Scholar] [CrossRef] - Wahlström, M. Faster exact solving of SAT formulae with a low number of occurrences per variable. In Theory and Applications of Satisfiability Testing (SAT-2005); Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar] [CrossRef]
- Wahlström, M. An algorithm for the SAT problem for formulae of linear length. In European Conference on Algorithms; Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar] [CrossRef]
- Johannsen, D.; Razgon, I.; Wahlström, M. Solving SAT for CNF Formulas with a One-Sided Restriction on Variable Occurrences. In Theory and Applications of Satisfiability Testing; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar] [CrossRef] [Green Version]
- Fu, Z.; Xu, D. The NP-completeness of d-regular (k,s)-SAT problem. J. Softw.
**2020**, 31, 1113–1123. [Google Scholar] [CrossRef] - Fu, Z.; Xu, D. (1,0)-Super Solutions of (k,s)-CNF Formula. Entropy
**2020**, 22, 253. [Google Scholar] [CrossRef] [Green Version] - Valiant, L.; Vazirani, V. NP is as easy as detecting unique solutions. Theor. Comput. Sci.
**1986**, 47, 85–93. [Google Scholar] [CrossRef] [Green Version] - Calabro, C.; Impagliazzo, R.; Kabanets, V.; Paturi, R. The complexity of unique k-SAT: An isolation lemma for k-CNFs. Comput. Syst. Sci.
**2008**, 74, 386–393. [Google Scholar] [CrossRef] [Green Version] - Matthews, W.; Paturi, R. Uniquely Satisfiable k-SAT Instances with Almost Minimal Occurrences of Each Variable. In Theory and Applications of Satisfiability Testing; Springer: Berlin/Heidelberg, Germany, 2010. [Google Scholar] [CrossRef]
- Kratochvíl, J.; Savický, P.; Tuza, Z. One more occurrence of variables makes satisfiability jump from trivial to NP-complete. Acta Inform.
**1993**, 22, 203–210. [Google Scholar] [CrossRef] - Hoory, S.; Szeider, S. Computing unsatisfiable k-SAT instances with few occurrences per variable. Theor. Comput. Sci.
**2004**, 337, 347–359. [Google Scholar] [CrossRef] [Green Version] - Hoory, S.; Szeider, S. Families of unsatisfiable k-CNF formulas with few occurrences per variable. SIAM J. Discret. Math.
**2006**, 20, 523–528. [Google Scholar] [CrossRef] [Green Version] - Savický, P.; Sgall, J. DNF tautologies with a limited number of occurrences of every variable. Theor. Comput. Sci.
**2007**, 238, 495–498. [Google Scholar] [CrossRef] [Green Version] - Gebauer, H.; Szabo, T.; Tardos, G. The Local Lemma is asymptotically tight for SAT. ACM
**2016**, 63, 664–674. [Google Scholar] [CrossRef] [Green Version] - Markström, K. Locality and Hard SAT-Instances. J. Satisf. Boolean Modeling Comput.
**2006**, 2, 221–227. [Google Scholar] [CrossRef] [Green Version] - Giráldez-cru, J.; Levy, J. Generating SAT instances with community structure. Artif. Intell.
**2016**, 238, 119–134. [Google Scholar] [CrossRef] - Giráldez-cru, J.; Levy, J. Locality in Random SAT Instances. In Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, Melbourne, Australia, 19–25 August 2017. [Google Scholar] [CrossRef] [Green Version]
- Clark, D.; Frank, J.; Gent, I.; MacIntyre, E.; Tomov, N.; Walsh, T. Local search and the number of solutions. In The Principles and Practices of Contraint Programming (CP96); Springer: Berlin/Heidelberg, Germany, 1996. [Google Scholar] [CrossRef]
- Singer, J.; Gent, I.P.; Smaill, A. Backbone fragility and the local search cost peak. J. Artif. Intell. Res.
**2000**, 12, 235–270. [Google Scholar] [CrossRef] - Znidaric, M. Single-solution Random 3-SAT Instances. arXiv
**2005**, arXiv:cs/0504101. [Google Scholar] - Dubois, O. On the r,s-SAT satisfiability problem and a conjecture of Tovey. Discret. Appl. Math.
**1990**, 26, 51–60. [Google Scholar] [CrossRef] [Green Version]

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Fu, Z.; Xu, D.
Uniquely Satisfiable *d*-Regular (*k*,*s*)-SAT Instances. *Entropy* **2020**, *22*, 569.
https://doi.org/10.3390/e22050569

**AMA Style**

Fu Z, Xu D.
Uniquely Satisfiable *d*-Regular (*k*,*s*)-SAT Instances. *Entropy*. 2020; 22(5):569.
https://doi.org/10.3390/e22050569

**Chicago/Turabian Style**

Fu, Zufeng, and Daoyun Xu.
2020. "Uniquely Satisfiable *d*-Regular (*k*,*s*)-SAT Instances" *Entropy* 22, no. 5: 569.
https://doi.org/10.3390/e22050569