# Set-Blocked Clause and Extended Set-Blocked Clause in First-Order Logic

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Theorem**

**1.**

**Theorem**

**2.**

## 3. Set-Blocked Clause and Extended Set-Blocked Clause in First-Order Logic without Equality

**Definition**

**5.**

**Example**

**1.**

**Definition**

**6.**

**Example**

**2.**

**Lemma**

**1.**

**Proof.**

**Example**

**3.**

**Theorem**

**3.**

**Proof.**

**Definition**

**7.**

**Example**

**4.**

**Lemma**

**2.**

**Proof.**

**Case 1**: In the resolution environment $en{v}_{F}(C)$ of C, there are no external ground atoms. Since C is an E-SBC and there is no ground atoms in the resolution environment of C, it means that C is set-blocked in the formula F. Now that C is set-blocked in the formula F, there exists a subset $S\lambda =\{{L}_{1}\lambda ,{L}_{2}\lambda ,\dots ,{L}_{n}\lambda \}$ of $C\lambda $ such that the assignment ${\alpha}^{\prime}$, obtained from $\alpha $ by flipping all the truth values of ground literals ${L}_{1}\lambda ,{L}_{2}\lambda ,\dots ,{L}_{n}\lambda $ in $S\lambda $, still satisfies all the ground instances in $F\backslash \{C\}$ according to Lemma 1.

**Case 2**: In the resolution environment $en{v}_{F}(C)$ of C, there exist external ground atoms $\{{A}_{1},{A}_{2},\dots ,{A}_{m}\}$. Since $\alpha $ is an assignment which covers all the ground instances of clauses in $F\backslash \{C\}$, it also assigns the truth values of those external ground atoms $\{{A}_{1},{A}_{2},\dots ,{A}_{m}\}$ to be true or false. Assume that the assignment $\alpha $ to the external ground atoms $\{{A}_{1},{A}_{2},\dots ,{A}_{m}\}$ make the truth values of clauses $\{{C}_{1},{C}_{2},\dots ,{C}_{k}\}$ in the resolution environment of C are true. Since clause C is extended set-blocked in a formula F, there exist a subset $S=\{{L}_{1},{L}_{2},\dots ,{L}_{n}\}$ of C, C is set-blocked upon S in $F|\alpha $, which means C is set-blocked upon S in the formula $F\backslash \{{C}_{1},{C}_{2},\dots ,{C}_{k}\}$. Now that C is set-blocked upon S in the formula $F\backslash \{{C}_{1},{C}_{2},\dots ,{C}_{k}\}$, flipping all the truth values of ground literals ${L}_{1}\lambda ,{L}_{2}\lambda ,\dots ,{L}_{n}\lambda $ in $S\lambda $ will not falsify any ground instances of $F\backslash \{C,{C}_{1},{C}_{2},\dots ,{C}_{k}\}$. Furthermore, $\alpha $ already satisfies all the ground instances of $\{{C}_{1},{C}_{2},\dots ,{C}_{k}\}$ by its assignment to those external ground atoms $\{{A}_{1},{A}_{2},\dots ,{A}_{m}\}$ according to the assumption, flipping all the truth values of ground literals ${L}_{1}\lambda ,{L}_{2}\lambda ,\dots ,{L}_{n}\lambda $ in $S\lambda $ will not falsify any ground instances of $\{{C}_{1},{C}_{2},\dots ,{C}_{k}\}$. Hence, flipping all the truth values of ground literals ${L}_{1}\lambda ,{L}_{2}\lambda ,\dots ,{L}_{n}\lambda $ in $S\lambda $ will not falsify any ground instances of $F\backslash \{C\}$. Therefore, the assignment ${\alpha}^{\prime}$, obtained from $\alpha $ by flipping all the truth values of ground literals ${L}_{1}\lambda ,{L}_{2}\lambda ,\dots ,{L}_{n}\lambda $ in $S\lambda $, still satisfies all the ground instances in $F\backslash \{C\}$. □

**Theorem**

**4.**

**Proof.**

## 4. Equality-Set-Blocked Clause and Extended Equality-Blocked Clause in First-Order Logic Formulas with Equality

**Example**

**5.**

**Definition**

**8.**

**Example**

**6.**

**Definition**

**9.**

**Example**

**7.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Definition**

**10.**

**Example**

**8.**

**Lemma**

**4.**

**Proof.**

**Case 1**: In the resolution environment $en{v}_{{F}_{E}}(C)$ of C, there is no external ground atoms. Since there are no external ground atoms in the resolution environment of C and C is an extended equality-set-blocked clause in the formula F, it means that C is equality-set-blocked in the formula F. In addition, because C is equality-set-blocked in the formula F, there must exist a subset $S\lambda =\{{L}_{1}\lambda ,{L}_{2}\lambda ,\dots ,{L}_{n}\lambda \}$ of $C\lambda $ and the assignment ${\beta}^{\prime}$, obtained from $\beta $ by flipping all the truth values of ground literals ${L}_{1}\lambda ,{L}_{2}\lambda ,\dots ,{L}_{n}\lambda $ in $S\lambda $, still satisfies all the ground instances in $F\backslash \{C\}$ and all the ground instances of equality axioms according to Lemma 3.

**Case 2**: In the resolution environment $en{v}_{{F}_{E}}(C)$ of C, there exist external ground atoms $\{{A}_{1},{A}_{2},\dots ,{A}_{m}\}$. Since $\beta $ is an assignment which covers all the ground instances of clauses in $F\backslash \{C\}$, it also assigns the truth values of those external ground atoms $\{{A}_{1},{A}_{2},\dots ,{A}_{m}\}$. Assume that the assignment to the external ground atoms make the truth values of clauses $\{{C}_{1},{C}_{2},\dots ,{C}_{k}\}$ in the resolution environment of C are true. Therefore, there exist a subset $S=\{{L}_{1},{L}_{2},\dots ,{L}_{n}\}$ of C, C is equality-set-blocked upon S in the formula $F\backslash \{{C}_{1},{C}_{2},\dots ,{C}_{k}\}$. Now that C is equality-set-blocked upon S in the formula $F\backslash \{{C}_{1},{C}_{2},\dots ,{C}_{k}\}$, flipping all the truth values of ground literals ${L}_{1}\lambda ,{L}_{2}\lambda ,\dots ,{L}_{n}\lambda $ in $S\lambda $ will not falsify any ground instances of $F\backslash \{C,{C}_{1},{C}_{2},\dots ,{C}_{k}\}$. Furthermore, $\alpha $ satisfies all the ground instances of $\{{C}_{1},{C}_{2},\dots ,{C}_{k}\}$ by its assignment to those external ground atoms $\{{A}_{1},{A}_{2},\dots ,{A}_{m}\}$, as a result, equivalence flipping all the truth values of ground literals ${L}_{1}\lambda ,{L}_{2}\lambda ,\dots ,{L}_{n}\lambda $ in $S\lambda $ will not falsify any ground instances of $\{{C}_{1},{C}_{2},\dots ,{C}_{k}\}$. Hence, equivalence flipping all the truth values of ground literals ${L}_{1}\lambda ,{L}_{2}\lambda ,\dots ,{L}_{n}\lambda $ in $S\lambda $ will not falsify any ground instances of $F\backslash \{C\}$. Therefore, the assignment ${\beta}^{\prime}$, obtained from $\beta $ by flipping all the truth values of ground literals ${L}_{1}\lambda ,{L}_{2}\lambda ,\dots ,{L}_{n}\lambda $ in $S\lambda $, still satisfies all the ground instances in $F\backslash \{C\}$ and all the ground axioms of equality axioms. □

**Theorem**

**6.**

**Proof.**

## 5. Effectiveness and Confluence Property

#### 5.1. Comparison of Effectiveness

**Definition**

**11.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Case 1**: There is no external ground atoms of the clause C. When there are no external ground atoms, the clause C is trivially an E-SBC when it is an SBC.

**Case 2**: Some external ground atoms $\{{A}_{1},{A}_{2},\dots ,{A}_{m}\}$ exist in the resolution environment of C. For any assignment $\alpha $ over $\{{A}_{1},{A}_{2},\dots ,{A}_{m}\}$, it may assign some clauses $\{{C}_{1},{C}_{2},\dots ,{C}_{k}\}$ in $en{v}_{F}(C)$ as true and it only needs to consider $en{v}_{F}(C)\backslash \{{C}_{1},{C}_{2},\dots ,{C}_{k}\}$ as the resolution environment of C in $F|\alpha $. Since C is a SBC upon $S\subseteq C$, all the ${L}_{i}^{S}$-resovlents $(1\le i\le n)$ obtained by resolving C with clauses in $en{v}_{F}(C)$ are tautologies, then there is no doubt that all the ${L}_{i}^{S}$-resovlents $(1\le i\le n)$ obtained by resolving C with clauses in $en{v}_{F}(C)\backslash \{{C}_{1},{C}_{2},\dots ,{C}_{k}\}$ are tautologies. Therefore, C is an SBC upon S in $F|\alpha $. As a result, C is an E-SBC.

**Theorem**

**9.**

**Proof.**

**Theorem**

**10.**

**Proof.**

#### 5.2. Confluence Property

**Definition**

**12.**

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Proof.**

**Theorem**

**13.**

**Proof.**

**Theorem**

**14.**

**Proof.**

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Effectiveness among those clause elimination methods. A arrow from A to B means A is more effective than B.

Clause elimination method | Confluence |
---|---|

SBCE | Yes |

E-SBCE | Yes |

ESBCE | Yes |

E-ESBCE | Yes |

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**MDPI and ACS Style**

Ning, X.; Xu, Y.; Wu, G.; Fu, H.
Set-Blocked Clause and Extended Set-Blocked Clause in First-Order Logic. *Symmetry* **2018**, *10*, 553.
https://doi.org/10.3390/sym10110553

**AMA Style**

Ning X, Xu Y, Wu G, Fu H.
Set-Blocked Clause and Extended Set-Blocked Clause in First-Order Logic. *Symmetry*. 2018; 10(11):553.
https://doi.org/10.3390/sym10110553

**Chicago/Turabian Style**

Ning, Xinran, Yang Xu, Guanfeng Wu, and Huimin Fu.
2018. "Set-Blocked Clause and Extended Set-Blocked Clause in First-Order Logic" *Symmetry* 10, no. 11: 553.
https://doi.org/10.3390/sym10110553