# Interpolation and Uniform Interpolation in Quantifier-Free Fragments of Combined First-Order Theories

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- $\varphi (\underline{x},\underline{y})\to \theta \left(\underline{x}\right)$ is T-valid;
- For any further quantifier-free formula $\psi (\underline{x},\underline{z})$ such that $\varphi (\underline{x},\underline{y})\to \psi (\underline{x},\underline{z})$ is T-valid, we have that the implication $\theta \left(\underline{x}\right)\to \psi (\underline{x},\underline{z})$ is T-valid, too.

#### Structure of the Paper

## 2. Preliminaries

#### 2.1. Combinations of Theories

#### 2.2. Interpolation Properties

**Definition**

**1.**

**Definition**

**2.**

#### 2.3. Amalgamation Properties

**Definition**

**3.**

## 3. Strong Amalgamation and Combined Interpolation

**Theorem**

**1**

**Theorem**

**2**

**.**Let ${T}_{1}$ and ${T}_{2}$ be two stably infinite theories over disjoint signatures ${\mathsf{\Sigma}}_{1}$ and ${\mathsf{\Sigma}}_{2}$. If both ${T}_{1}$ and ${T}_{2}$ have the strong sub-amalgamation property, then so does ${T}_{1}\cup {T}_{2}$. Thus, in view of Theorem 1, ${T}_{1}\cup {T}_{2}$ has quantifier-free interpolants.

**Theorem**

**3**

**.**Let T be a theory admitting quantifier-free interpolation and let Σ be a proper signature disjoint from the signature of T. Then, $T\cup \mathcal{EUF}\left(\mathsf{\Sigma}\right)$ has quantifier-free interpolation iff T has the strong sub-amalgamation property.

**Theorem**

**4**

**.**A theory T has the general quantifier-free interpolation property iff T has the strong sub-amalgamation property.

#### 3.1. Strong Amalgamation: A Syntactic Characterization

**Definition**

**4.**

- •
- for every triple $\underline{x},{\underline{y}}_{1},{\underline{y}}_{2}$ of tuples of variables and for every pair of quantifier-free formulae ${\delta}_{1}(\underline{x},{\underline{y}}_{1}),{\delta}_{2}(\underline{x},{\underline{y}}_{2})$ such that$${\delta}_{1}(\underline{x},{\underline{y}}_{1})\wedge {\delta}_{2}(\underline{x},{\underline{y}}_{2}){\u22a2}_{T}{\underline{y}}_{1}\cap {\underline{y}}_{2}\ne \varnothing $$there exists a tuple $\underline{v}\left(\underline{x}\right)$ of terms such that$${\delta}_{1}(\underline{x},{\underline{y}}_{1})\wedge {\delta}_{2}(\underline{x},{\underline{y}}_{2}){\u22a2}_{T}{\underline{y}}_{1}{\underline{y}}_{2}\cap \underline{v}\ne \varnothing \phantom{\rule{3.33333pt}{0ex}}.$$

**Theorem**

**5**

- (i)
- T is strongly sub-amalgamable;
- (ii)
- T is equality interpolating.

**Proof.**

**Example**

**1.**

**Example**

**2.**

- (a)
- $n\in \mathbb{Z}$ and $\bowtie \phantom{\rule{0.166667em}{0ex}}\in \{=,<\}$;
- (b)
- $i,j$ are variables or the constant 0;
- (c)
- ${f}^{0}\left(j\right)$ is j, ${f}^{k}\left(j\right)$ abbreviates $succ\left(suc{c}^{k-1}\left(j\right)\right)$ when $k>0$ or $pred\left(pre{d}^{k-1}\left(j\right)\right)$ when $k<0$.

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

#### 3.2. The Case of Convex Theories

**Proposition**

**1**

**.**The following conditions are equivalent for a convex theory T with quantifier-free interpolation:

- (i)
- T is equality interpolating;
- (ii)
- For every pair ${y}_{1},{y}_{2}$ of variables and for every pair of conjunctions of literals ${\delta}_{1}(\underline{x},{\underline{z}}_{1},{y}_{1})$, ${\delta}_{2}(\underline{x},{\underline{z}}_{2},{y}_{2})$ such that$${\delta}_{1}(\underline{x},{\underline{z}}_{1},{y}_{1})\wedge {\delta}_{2}(\underline{x},{\underline{z}}_{2},{y}_{2}){\u22a2}_{T}{y}_{1}={y}_{2}$$there exists a term $v\left(\underline{x}\right)$ such that$${\delta}_{1}(\underline{x},{\underline{z}}_{1},{y}_{1})\wedge {\delta}_{2}(\underline{x},{\underline{z}}_{2},{y}_{2}){\u22a2}_{T}{y}_{1}=v\left(\underline{x}\right)\wedge {y}_{2}=v\left(\underline{x}\right).$$
- (iii)
- For every tuple of variables $\underline{x}$, for every further variable y, and for every primitive formula $\delta (\underline{x},y)$ such that$$\delta (\underline{x},{y}^{\prime})\wedge \delta (\underline{x},{y}^{\u2033}){\u22a2}_{T}{y}^{\prime}={y}^{\u2033}$$there is a term $v\left(\underline{x}\right)$ such that$$\delta (\underline{x},y){\u22a2}_{T}y=v\left(\underline{x}\right)\phantom{\rule{3.33333pt}{0ex}}.$$

#### 3.3. Sketch of the Combined Interpolation Algorithm

## 4. Non-Disjoint Combinations

**Definition**

**5.**

- (i)
- ${T}_{0}\subseteq {T}_{0}^{\star}$;
- (ii)
- ${T}_{0}^{\star}$ is a model completion of ${T}_{0}$;
- (iii)
- Every model of T can be embedded, as a Σ-structure, into a model of $T\cup {T}_{0}^{\star}$.

#### Strong Amalgamation over a Horn Theory

**Definition**

**6.**

**Definition**

**7.**

**Theorem**

**7**

**.**If ${T}_{1},{T}_{2}$ are both ${T}_{0}$-compatible and ${T}_{0}$-strongly sub-amalgamable (for an amalgamable universal Horn theory ${T}_{0}$ in their common subsignature ${\mathsf{\Sigma}}_{0}$), then so is ${T}_{1}\cup {T}_{2}$.

**Definition**

**8.**

**Theorem**

**8**

**.**A BAO-equational theory T has the superamalgamation property iff it is $BA$-strongly amalgamable.

## 5. Uniform Interpolants

**Lemma**

**1**

**.**A formula $\psi \left(\underline{y}\right)$ is a T-cover of $\exists \underline{e}\phantom{\rule{0.166667em}{0ex}}\varphi (\underline{e},\underline{y})$ iff it satisfies the following two conditions:

- (i)
- $T\vDash \forall \underline{y}\phantom{\rule{0.166667em}{0ex}}(\exists \underline{e}\phantom{\rule{0.166667em}{0ex}}\varphi (\underline{e},\underline{y})\to \psi \left(\underline{y}\right))$;
- (ii)
- For every model $\mathcal{M}$ of T and for every tuple of elements $\underline{a}$ from the support of $\mathcal{M}$ such that $\mathcal{M}\vDash \psi \left(\underline{a}\right)$, it is possible to find another model $\mathcal{N}$ of T such that $\mathcal{M}$ embeds into $\mathcal{N}$ and $\mathcal{N}\vDash \exists \underline{e}\phantom{\rule{0.166667em}{0ex}}\varphi (\underline{e},\underline{a})$.

**Theorem**

**9**

**.**Suppose that T is a universal theory. Then, T has a model completion ${T}^{\star}$ iff T has uniform quantifier-free interpolation. If this happens, ${T}^{\star}$ is axiomatized by the infinitely many sentences

#### 5.1. Uniform Interpolants in $\mathcal{EUF}$

**Definition**

**9.**

- (1)
- Simplification Rules:
- (1.0)
- If an atom such as $t=t$ belongs to $\mathsf{\Psi}$, just remove it; if a literal such as $t\ne t$ occurs somewhere, delete $\mathsf{\Psi}$, replace $\Phi $ with ⊥, and stop;
- (1.i)
- If t is not a variable and $\mathsf{\Psi}$ contains both $t=a$ and $t=b$, remove the former and replace it with $a=b$.
- (1.ii)
- If $\mathsf{\Psi}$ contains ${e}_{i}={e}_{j}$ with $i>j$, remove it and replace ${e}_{i}$ with ${e}_{j}$ everywhere.

- (2)
- DAG Update Rule: If $\mathsf{\Psi}$ contains ${e}_{i}=t(\underline{y},\underline{z})$, remove it, rename ${e}_{i}$ as ${y}_{j}$ everywhere (for fresh ${y}_{j}$), and add ${y}_{j}=t(\underline{y},\underline{z})$ to $\mathtt{ExplDef}(\underline{y},\underline{z}).$ More formally:$$\mathtt{ExplDef}(\underline{y},\underline{z})\wedge \Phi (\underline{y},\underline{z})\wedge \left(\mathsf{\Psi}(\underline{e},{e}_{i},\underline{y},\underline{z})\wedge {e}_{i}=t(\underline{y},\underline{z})\right)$$$$\left(\mathtt{ExplDef}(\underline{y},\underline{z})\wedge {y}_{j}=t(\underline{y},\underline{z})\right)\wedge \Phi (\underline{y},\underline{z})\wedge \mathsf{\Psi}(\underline{e},{y}_{j},\underline{y},\underline{z})$$
- (3)
- $\underline{e}$-Free Literal Rule: If $\mathsf{\Psi}$ contains a literal $L(\underline{y},\underline{z})$, move it to $\Phi (\underline{y},\underline{z})$. More formally:$$\mathtt{ExplDef}(\underline{y},\underline{z})\wedge \Phi (\underline{y},\underline{z})\wedge \left(\mathsf{\Psi}(\underline{e},\underline{y},\underline{z})\wedge L(\underline{y},\underline{z})\right)$$$$\mathtt{ExplDef}(\underline{y},\underline{z})\wedge \left(\Phi (\underline{y},\underline{z})\wedge L(\underline{y},\underline{z})\right)\wedge \mathsf{\Psi}(\underline{e},\underline{y},\underline{z})$$
- (4)
- Splitting Rule: If $\mathsf{\Psi}$ contains a pair of atoms $t=a$ and $u=b$, where t and u are compatible flat terms as in (23) (thus, in particular, t and u are of the kinds $f({a}_{1},\cdots ,{a}_{n})$ and $f({b}_{1},\cdots ,{b}_{n})$, respectively), and no disequality from the difference set of $t,u$ belongs to $\Phi $, then apply one of the following alternatives:
- (4.0)
- Remove from $\mathsf{\Psi}$ the atom $f({b}_{1},\cdots ,{b}_{n})=b$, add to $\mathsf{\Psi}$ the atom $a=b$, and add to $\Phi $ all equalities ${a}_{i}={b}_{i}$ such that ${a}_{i}\ne {b}_{i}$ is in the difference set of $t,u$;
- (4.1)
- Add to $\Phi $ one of the disequalities from the difference set of $t,u$ (notice that the difference set cannot be empty; otherwise, Rule (1.i) applies).

**Theorem**

**10**

**.**Suppose that we apply the above algorithm to the primitive formula $\exists \underline{e}\left(\varphi (\underline{e},\underline{z})\right)$ and that the algorithm terminates with its branches in the states

**Example**

**6.**

#### 5.2. Combined Uniform Interpolants

**Lemma**

**2.**

**Theorem**

**11**

- (1)
- Update ${\psi}_{1}$ by adding to it a disjunct from the DNF of ${\bigwedge}_{{e}_{i}\in \underline{e}}\neg {\mathtt{ImplDef}}_{{\psi}_{1},{e}_{i}}^{{T}_{1}}(\underline{x},\underline{z})$ and ${\psi}_{2}$ by adding to it a disjunct from the DNF of ${\bigwedge}_{{e}_{i}\in \underline{e}}\neg {\mathtt{ImplDef}}_{{\psi}_{2},{e}_{i}}^{{T}_{2}}(\underline{x},\underline{z})$;
- (2.i)
- Select ${e}_{i}\in \underline{e}$ and $h\in \{1,2\}$; then, update ${\psi}_{h}$ by adding to it a disjunct ${L}_{ij}$ from the DNF of ${\mathtt{ImplDef}}_{{\psi}_{h},{e}_{i}}^{{T}_{h}}(\underline{x},\underline{z})$; the equality ${e}_{i}={t}_{ij}$ (where ${t}_{ij}$ is the term mentioned in Lemma 2) is added to $\mathtt{ExplDef}(\underline{z},\underline{x})$; the variable ${e}_{i}$ becomes, in this way, part of the defined variables.

**Proposition**

**2**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Ghilardi, S.; Gianola, A.
Interpolation and Uniform Interpolation in Quantifier-Free Fragments of Combined First-Order Theories. *Mathematics* **2022**, *10*, 461.
https://doi.org/10.3390/math10030461

**AMA Style**

Ghilardi S, Gianola A.
Interpolation and Uniform Interpolation in Quantifier-Free Fragments of Combined First-Order Theories. *Mathematics*. 2022; 10(3):461.
https://doi.org/10.3390/math10030461

**Chicago/Turabian Style**

Ghilardi, Silvio, and Alessandro Gianola.
2022. "Interpolation and Uniform Interpolation in Quantifier-Free Fragments of Combined First-Order Theories" *Mathematics* 10, no. 3: 461.
https://doi.org/10.3390/math10030461