# Decidability Preservation and Complexity Bounds for Combined Logics

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Logics and Their Combination

#### 2.1. Syntax

#### 2.2. Propositional Logics and Theories

**Definition**

**1.**

- (R)
- $\Gamma \u22a2A$ whenever $A\in \Gamma $ (reflexivity);
- (M)
- $\Gamma \u22a2A$ whenever ${\Gamma}^{\prime}\u22a2A$ for ${\Gamma}^{\prime}\subseteq \Gamma $ (monoticity);
- (T)
- $\Gamma \u22a2A$ whenever $\Theta \u22a2A$ and $\Gamma \u22a2B$ for every $B\in \Theta $ (transitivity);
- (S)
- $\Gamma \u22a2A$ implies ${\Gamma}^{\sigma}\u22a2{A}^{\sigma}$ for any substitution σ (subst. invariance).

- (F)
- $\Gamma \u22a2A$ implies there is a finite ${\Gamma}_{0}\subseteq \Gamma $ such that ${\Gamma}_{0}\u22a2A$.

**Example**

**1.**

**Proposition**

**1.**

**Proof.**

#### 2.3. Combining Logics

**Definition**

**2.**

**Example**

**2.**

**Theorem**

**1.**

**Proof.**

- (T)
- Assume that $\Theta \u22a2A$ and $\Gamma \u22a2B$ for every $B\in \Theta $. This means that $A\in {\Delta}_{1}$ for every $\Theta \subseteq {\Delta}_{1}\in \mathsf{Th}(\langle {\Sigma}_{12},{\u22a2}_{1}^{{\Sigma}_{12}}\rangle )\cap \mathsf{Th}(\langle {\Sigma}_{12},{\u22a2}_{2}^{{\Sigma}_{12}}\rangle )$, and $\Theta \subseteq {\Delta}_{2}$ for every $\Gamma \subseteq {\Delta}_{2}\in \mathsf{Th}(\langle {\Sigma}_{12},{\u22a2}_{1}^{{\Sigma}_{12}}\rangle )\cap \mathsf{Th}(\langle {\Sigma}_{12},{\u22a2}_{2}^{{\Sigma}_{12}}\rangle )$. By (M) for each ${\u22a2}_{i}^{{\Sigma}_{12}}$, we conclude that $A\in {\Theta}_{3}$ for for every $\Gamma \subseteq {\Theta}_{3}\in \mathsf{Th}(\langle {\Sigma}_{12},{\u22a2}_{1}^{{\Sigma}_{12}}\rangle )\cap \mathsf{Th}(\langle {\Sigma}_{12},{\u22a2}_{2}^{{\Sigma}_{12}}\rangle )$. Therefore, $\Gamma \u22a2A$.
- (S)
- Assume that $\Gamma \u22a2A$, and thus $A\in {\Delta}_{1}$ for every $\Gamma \subseteq {\Delta}_{1}\in \mathsf{Th}(\langle {\Sigma}_{12},{\u22a2}_{1}^{{\Sigma}_{12}}\rangle )\cap \mathsf{Th}(\langle {\Sigma}_{12},{\u22a2}_{2}^{{\Sigma}_{12}}\rangle )$. By (S), for each ${\u22a2}_{i}^{{\Sigma}_{12}}$, we conclude that ${A}^{\sigma}\in {\Delta}_{2}$ for every ${\Gamma}^{\sigma}\subseteq {\Delta}_{2}\in \mathsf{Th}(\langle {\Sigma}_{12},{\u22a2}_{1}^{{\Sigma}_{12}}\rangle )\cap \mathsf{Th}(\langle {\Sigma}_{12},{\u22a2}_{2}^{{\Sigma}_{12}}\rangle )$. Therefore, ${\Gamma}^{\sigma}\u22a2{A}^{\sigma}$.

#### 2.4. Contextual Extensibility and Decidability Preservation

**Definition**

**3.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

`no`output happens when a fixed point is reached, meaning that $A\notin \Omega $, and thus $A\notin {\Gamma}^{{\u22a2}_{12}}$. When the

`yes`output is reached, we are sure that $A\in {\Gamma}^{{\u22a2}_{12}}$, as A was reached by departing from $\Omega =\Gamma $ and iteratively adding formulas in $\mathsf{ctx}(\Gamma \cup \{A\})$ to $\Omega $ if they follow either

`yes`precisely when $\Gamma {\u22ac}_{12}A$ by guessing correctly a set $\Omega $ for which $\Omega {\u22ac}_{1}B$ and $\Omega {\u22ac}_{2}B$, and, hence, with ${\mathtt{N}}_{\mathtt{1}}(\Omega ,B)=\mathtt{yes}$ and ${\mathtt{N}}_{\mathtt{2}}(\Omega ,B)=\mathtt{yes}$ for every $B\in \Theta \backslash \Omega $.

#### 2.5. Applications

#### 2.5.1. Combining Logics with Disjoint Signatures

**Proposition**

**2.**

**Proof.**

#### 2.5.2. Fusion of Modal Logics

**Proposition**

**3.**

**Proof.**

- If there is some $C\in {\Omega}^{\u266f}$ with ${v}_{i}(C)={\top}_{i}$, then $\mathrm{{\rm Y}}=\varnothing $, as actually we must have ${v}_{i}({\mathsf{skel}}_{i}(A))\in \{{\top}_{i},{\perp}_{i}\}$ for all $A\in {\Omega}^{\u266f}$. It is known that any two ciaBs are isomorphic. Since the top and bottom elements must be identified, we have ${v}_{1}({\mathsf{skel}}_{1}(A))$ and ${v}_{2}({\mathsf{skel}}_{2}(A))$ for every $A\in {\Omega}^{\u266f}$.
- Using the boolean-valid equation $x=(x\wedge y)\vee (x\wedge \neg y)$, when $\mathrm{{\rm Y}}\ne \varnothing $, it is clear that ${v}_{i}({\mathsf{skel}}_{i}(\bigvee \mathrm{{\rm Y}}))={v}_{i}({\mathsf{skel}}_{i}(\bigvee {\Omega}^{\u266f}))={\top}_{i}$. Furthermore, if ${C}_{1},{C}_{2}\in \mathrm{{\rm Y}}$ and ${C}_{1}\ne {C}_{2}$, then the boolean-valid equation $x\wedge \neg x=\perp $ implies that ${v}_{i}({\mathsf{skel}}_{i}({C}_{1}\wedge {C}_{2}))={\perp}_{i}$. This means that the set of values ${v}_{i}({\mathsf{skel}}_{i}(\mathrm{{\rm Y}}))$ is a partition of ${\mathbb{A}}_{i}$. Again, it is known that there is an isomorphism of the two ciaBs that identifies ${v}_{1}({\mathsf{skel}}_{1}(A))$ and ${v}_{2}({\mathsf{skel}}_{2}(A))$ for every $A\in \mathrm{{\rm Y}}$ (and, by the same argument as in the previous case, for every $A\in {\Omega}^{\u266f}\backslash \mathrm{{\rm Y}}$.

## 3. Beyond Propositional Logics

#### 3.1. Syntax

#### 3.2. 2-Logics, Equational Logics, and Their Theories

**Definition**

**4.**

- (R${}_{\approx}$)
- $\Gamma \u22a2A\approx B$ whenever $A\approx B\in \Gamma $;
- (M${}_{\approx}$)
- $\Gamma \u22a2A\approx B$ whenever ${\Gamma}^{\prime}\u22a2A\approx B$ for ${\Gamma}^{\prime}\subseteq \Gamma $;
- (T${}_{\approx}$)
- $\Gamma \u22a2A\approx B$ whenever $\Delta \u22a2A\approx B$, and $\Gamma \u22a2C\approx D$ for every $C\approx D\in \Delta $;
- (S${}_{\approx}$)
- $\Gamma \u22a2A\approx B$ implies ${\Gamma}^{\sigma}\u22a2{A}^{\sigma}\approx {B}^{\sigma}$ for any substitution $\sigma :P\to {L}_{\Sigma}(P)$.

- (F${}_{\approx}$)
- $\Gamma \u22a2A\approx B$ implies there is finite ${\Gamma}_{0}\subseteq \Gamma $ such that ${\Gamma}_{0}\u22a2A\approx B$.

**Example**

**3.**

**Proposition**

**4.**

**Proof.**

#### 3.3. Combining 2-Logics

**Definition**

**5.**

**Example**

**4.**

**Theorem**

**3.**

**Proof.**

#### 3.4. Contextual Extensibility and Decidability Preservation Revisited

**Definition**

**6.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**4.**

**Proof.**

#### 3.5. Applications

#### 3.5.1. Splitting the Smallest Equational Logic

**Proposition**

**5.**

**Proof.**

**Corollary**

**1.**

#### 3.5.2. Combining Equational Logics with Disjoint Signatures

**Proposition**

**6.**

**Proof.**

- -
- ${\Xi}_{k+1}={\Xi}_{k}\cup \Sigma \left[{\Xi}_{k}\right],\mathrm{and}$
- -
- ${\Omega}_{k+1}=\mathsf{Eqs}({\Xi}_{k+1}).$

- -
- ${\Gamma}_{i}^{k}={\Delta}_{i}^{k}\cap {\Omega}_{k}$,;
- -
- ${\Theta}_{i}^{k}={({\Gamma}_{i}^{k})}^{{\u22a2}_{i}^{\Sigma}}\cap {\Omega}_{k+1}$;
- -
- ${\Delta}_{i}^{k+1}={({\Theta}_{1}^{k}\cup {\Theta}_{2}^{k})}^{{\u22a2}_{i}^{\Sigma}}$.

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

## 4. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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C | ${\mathbf{C}}^{\prime}$ |
---|---|

$\mathbf{TIME}$$(t(n))$ | $\mathbf{TIME}$$(c(n)+d(n)\times t(d(n)))$ |

$\mathbf{SPACE}$$(s(n))$ | $\mathbf{SPACE}$$(d(n)+s(d(n)))$ |

$\mathbf{coNTIME}$$({t}^{\prime}(n))$ | $\mathbf{coNTIME}$$(c(n)+d(n)\times {t}^{\prime}(d(n)))$ |

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**MDPI and ACS Style**

Caleiro, C.; Marcelino, S.
Decidability Preservation and Complexity Bounds for Combined Logics. *Mathematics* **2022**, *10*, 3481.
https://doi.org/10.3390/math10193481

**AMA Style**

Caleiro C, Marcelino S.
Decidability Preservation and Complexity Bounds for Combined Logics. *Mathematics*. 2022; 10(19):3481.
https://doi.org/10.3390/math10193481

**Chicago/Turabian Style**

Caleiro, Carlos, and Sérgio Marcelino.
2022. "Decidability Preservation and Complexity Bounds for Combined Logics" *Mathematics* 10, no. 19: 3481.
https://doi.org/10.3390/math10193481