# Concepts of Interpolation in Stratified Institutions

## Abstract

**:**

## 1. Introduction

#### 1.1. Stratified Institutions

#### 1.2. Institution Theoretic Interpolation

#### 1.3. Our Contributions

## 2. Preliminaries

- category theory;
- institution theory; and
- stratified institution theory.

#### 2.1. Categories

#### 2.2. Institutions

- a category ${\mathrm{Sign}}^{\mathcal{I}}$ whose objects are called signatures,
- a sentence functor ${\mathrm{Sen}}^{\mathcal{I}}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{\mathrm{Sign}}^{\mathcal{I}}\to \mathbf{Set}$ defining for each signature a set whose elements are called sentences over that signature and defining for each signature morphism a sentence translation function,
- a model functor ${\mathrm{Mod}}^{\mathcal{I}}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{\left({\mathrm{Sign}}^{\mathcal{I}}\right)}^{\ominus}\to \mathbf{CAT}$ defining for each signature $\mathrm{\Sigma}$ the category ${\mathrm{Mod}}^{\mathcal{I}}\left(\mathrm{\Sigma}\right)$ of Σ-models and $\mathrm{\Sigma}$-model homomorphisms, and for each signature morphism $\phi $ the reduct= functor ${\mathrm{Mod}}^{\mathcal{I}}\left(\phi \right)$,
- for every signature $\mathrm{\Sigma}$, a binary Σ-satisfaction relation ${\vDash}_{\mathrm{\Sigma}}^{\mathcal{I}}\phantom{\rule{4pt}{0ex}}\subseteq \phantom{\rule{4pt}{0ex}}\left|{\mathrm{Mod}}^{\mathcal{I}}\left(\mathrm{\Sigma}\right)\right|\times {\mathrm{Sen}}^{\mathcal{I}}\left(\mathrm{\Sigma}\right)$,

if $\mathrm{\Gamma}{\vDash}_{\mathrm{\Sigma}}{\mathrm{\Gamma}}_{i}$ for each $i\in I$ then $\mathrm{\Gamma}{\vDash}_{\mathrm{\Sigma}}{\bigcup}_{i\in I}{\mathrm{\Gamma}}_{i}$ | union |

if ${\mathrm{\Gamma}}^{\prime}\supseteq \mathrm{\Gamma}$ then ${\mathrm{\Gamma}}^{\prime}{\vDash}_{\mathrm{\Sigma}}\mathrm{\Gamma}$ | monotonicity |

if $\mathrm{\Gamma}{\vDash}_{\mathrm{\Sigma}}{\mathrm{\Gamma}}^{\prime}$ and ${\mathrm{\Gamma}}^{\prime}{\vDash}_{\mathrm{\Sigma}}{\mathrm{\Gamma}}^{\u2033}$ then $\mathrm{\Gamma}{\vDash}_{\mathrm{\Sigma}}{\mathrm{\Gamma}}^{\u2033}$ | transitivity |

if $\mathrm{\Gamma}{\vDash}_{\mathrm{\Sigma}}{\mathrm{\Gamma}}^{\prime}$ then $\phi \mathrm{\Gamma}{\vDash}_{{\mathrm{\Sigma}}^{\prime}}\phi {\mathrm{\Gamma}}^{\prime}$ | translation. |

#### 2.3. Stratified Institutions

- category ${\mathrm{Sign}}^{\mathcal{S}}$ of signatures,
- a sentence functor ${\mathrm{Sen}}^{\mathcal{S}}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{\mathrm{Sign}}^{\mathcal{S}}\to \mathbf{Set}$;
- a model functor ${\mathrm{Mod}}^{\mathcal{S}}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{\left({\mathrm{Sign}}^{\mathcal{S}}\right)}^{\ominus}\to \mathbf{CAT}$;
- a “stratification” lax natural transformation ${\left[\phantom{\rule{-0.166667em}{0ex}}[\_]\phantom{\rule{-0.166667em}{0ex}}\right]}^{\mathcal{S}}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{\mathrm{Mod}}^{\mathcal{S}}\Rightarrow \mathit{SET}$, where $\mathit{SET}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{\mathrm{Sign}}^{\mathcal{S}}\to \mathbf{CAT}$ is a functor mapping each signature morphism to the identity functor on $\mathbf{Set}$; and
- a satisfaction relation between models and sentences which is parameterized by model states, $M\phantom{\rule{4pt}{0ex}}{\left({\vDash}^{\mathcal{S}}\right)}_{\mathrm{\Sigma}}^{w}\phantom{\rule{4pt}{0ex}}\rho $ where $w\in {\left[\phantom{\rule{-0.166667em}{0ex}}\left[M\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\mathrm{\Sigma}}^{\mathcal{S}}$ such that the following satisfaction condition$${\mathrm{Mod}}^{\mathcal{S}}\left(\phi \right){M}^{\prime}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\left({\vDash}^{\mathcal{S}}\right)}_{\mathrm{\Sigma}}^{{\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\phi}w}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\rho \phantom{\rule{4pt}{0ex}}\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}\phantom{\rule{4pt}{0ex}}{M}^{\prime}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\left({\vDash}^{\mathcal{S}}\right)}_{{\mathrm{\Sigma}}^{\prime}}^{w}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathrm{Sen}}^{\mathcal{S}}\left(\phi \right)\rho $$

- –
- for each signature $\mathrm{\Sigma}\in \left|\mathrm{Sign}\right|$, $\left[\phantom{\rule{-0.166667em}{0ex}}\right[\mathrm{Mod}\left(\mathrm{\Sigma}\right)\to \mathbf{Set}\left]\phantom{\rule{-0.166667em}{0ex}}\right]$ denotes the set of all the mappings $f\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}\left|\mathrm{Mod}\right(\mathrm{\Sigma}\left)\right|\to \mathbf{Set}$ such that $f\left(M\right)\subseteq {\left[\phantom{\rule{-0.166667em}{0ex}}\left[M\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\mathrm{\Sigma}}$; and
- –
- for each signature morphism $\phi \phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}\mathrm{\Sigma}\to {\mathrm{\Sigma}}^{\prime}$$$\left[\phantom{\rule{-0.166667em}{0ex}}[\mathrm{Mod}\left(\phi \right)\to \mathbf{Set}]\phantom{\rule{-0.166667em}{0ex}}\right]\left(f\right)\left({M}^{\prime}\right)={\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\phi}^{-1}\left(f\left(\mathrm{Mod}\left(\phi \right){M}^{\prime}\right)\right).$$

**Assumption**

**A1.**

- In modal propositional logic ($\mathcal{M}\phantom{\rule{-0.166667em}{0ex}}\mathcal{P}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$) the category of the signatures is $\mathbf{Set}$, $\mathrm{Sen}\left(P\right)$ is the set of the usual modal sentences formed with the atomic propositions from P, and the P-models are the Kripke structures $(W,M)$ where $W=\left(\right|W|,{W}_{\lambda})$ consists of a set of “possible worlds” $\left|W\right|$ and an accessibility relation ${W}_{\lambda}\subseteq \left|W\right|\times \left|W\right|$, and $M\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}\left|W\right|\to {2}^{P}$. The stratification is given by $\left[\phantom{\rule{-0.166667em}{0ex}}\right[(W,M)\left]\phantom{\rule{-0.166667em}{0ex}}\right]=\left|W\right|$.
- In first order modal logic ($\mathcal{M}\phantom{\rule{-0.166667em}{0ex}}\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$) the signatures are first-order logic ($\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$) signatures consisting of sets of operation and relation symbols structured by their arities. The sentences extend the usual construction of $\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$ sentences with the modal connectives □ and ⋄. The models for a signature $\mathrm{\Sigma}$ are Kripke structures $(W,M)$ where W is like in $\mathcal{M}\phantom{\rule{-0.166667em}{0ex}}\mathcal{P}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$ but $M\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}\left|W\right|\to |{\mathrm{Mod}}^{\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}}\left(\mathrm{\Sigma}\right)|$ subject to the constraint that the carrier sets and the interpretations of the constants are shared across the possible worlds. The stratification is like in $\mathcal{M}\phantom{\rule{-0.166667em}{0ex}}\mathcal{P}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$.
- Hybrid logics refine modal logics by adding explicit syntax for the possible worlds such as nominals and @. Stratified institutions of hybrid logics upgrade the syntactic and the semantic components of the stratified institutions of modal logics accordingly. For instance, in the stratified institution of hybrid propositional logic ($\mathcal{H}\phantom{\rule{-0.166667em}{0ex}}\mathcal{P}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$) the signatures are pairs of sets $(\mathrm{Nom},P)$, the $(\mathrm{Nom},P)$-models are Kripke structures $(W,M)$ like in $\mathcal{M}\phantom{\rule{-0.166667em}{0ex}}\mathcal{P}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$, but where W adds interpretations of the nominals, i.e., $W=\left(\right|W|,{\left({W}_{i}\right)}_{i\in \mathrm{Nom}},{W}_{\lambda})$, and at the level of the syntax, for each $i\in \mathrm{Nom}$ we have a new sentence $i-\mathrm{sen}$, a new unary connective ${@}_{i}$, and existential quantifications over nominals variables. Then $((W,M){\vDash}^{w}i-\mathrm{sen})=({W}_{i}=w),\left((W,M){\vDash}^{w}{@}_{i}\rho \right)=\left((W,M){\vDash}^{{W}_{i}}\rho \right)$, etc.
- Multi-modal logics exhibit several modalities instead of only the traditional ⋄ and □ and, moreover, these may have various arities. If one considers the sets of modalities to be variable then they have to be considered as part of the signatures. Each of the stratified institutions discussed in the previous examples admit an upgrade to the multi-modal case.
- In a series of works on modalization of institutions [38,39,40] modal logic and Kripke semantics are developed by abstracting away details that do not belong to modality, such as sorts, functions, predicates, etc. This is achieved by extensions of abstract institutions (in the standard situations meant in principle to encapsulate the atomic part of the logics) with the essential ingredients of modal logic and Kripke semantics. The result of this process, when instantiated to various concrete logics (or to their atomic parts only) generate a uniformly wide range of hierarchical combinations between various flavours of modal logic and various other logics. Concrete examples discussed in [38,39,40] include various modal logics over non-conventional structures of relevance in computing science, such as partial algebra, preordered algebra, etc. Various constraints on the respective Kripke models, many of them having to do with the underlying non-modal structures, have also been considered. All of these arise as examples of stratified institutions like the examples presented above in the paper. An interesting class of examples that has emerged quite smoothly out of the general works on hybridization (i.e., modalization including also hybrid logic features) of institutions is that of multi-layered hybrid logics that provide a logical base for specifying hierarchical transition systems (see [41]).
- Open first order logic ($\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$). This is a $\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$ instance of $St\left(\mathcal{I}\right)$, the “internal stratification” abstract example developed in [5]. An $\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$ signature is a pair $(\mathrm{\Sigma},X)$ consisting of $\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$ signature $\mathrm{\Sigma}$ and a finite block of variables. To any $\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$ signature $(\mathrm{\Sigma},X)$ it corresponds a $\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$ signature $\mathrm{\Sigma}+X$ that adjoins X to $\mathrm{\Sigma}$ as new constants. Then, ${\mathrm{Sen}}^{\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}}(\mathrm{\Sigma},X)={\mathrm{Sen}}^{\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}}(\mathrm{\Sigma}+X)$, ${\mathrm{Mod}}^{\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}}(\mathrm{\Sigma},X)={\mathrm{Mod}}^{\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}}\left(\mathrm{\Sigma}\right)$, ${\left[\phantom{\rule{-0.166667em}{0ex}}\left[M\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\mathrm{\Sigma},X}={M}^{X}$, i.e the set of the “valuations” of X to M and for each $(\mathrm{\Sigma},X)$-model M, each $w\in {M}^{X}$, and each $(\mathrm{\Sigma},X)$-sentence $\rho $ we define $\left(M{\left({\vDash}_{\mathrm{\Sigma},X}^{\mathcal{O}\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}}\right)}^{w}\rho \right)=\left({M}^{w}{\vDash}_{\mathrm{\Sigma}+X}^{\mathcal{F}\phantom{\rule{-0.166667em}{0ex}}\mathcal{O}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}}\rho \right)$ where ${M}^{w}$ is the expansion of M to $\mathrm{\Sigma}+X$ such that ${M}_{X}^{w}=w$ (i.e., the new constants of X are interpreted in ${M}^{w}$ according to the “valuation” w).
- Various kinds of automata theories can be presented as stratified institutions. For instance, the deterministic automata (for regular languages) have the set of the input symbols as signatures, the automata A are the models, and the words are the sentences. Then, $\left[\phantom{\rule{-0.166667em}{0ex}}\right[A\left]\phantom{\rule{-0.166667em}{0ex}}\right]$ is the set of the states of A and $A{\vDash}^{s}\alpha $ if and only if $\alpha $ is recognized by A from the state s.
- In [10], there is a development of a general representation theorem of $3/2$-institutions as stratified institutions. The theory of $3/2$-institutions [11] is an extension of ordinary institution theory that accommodates the partiality of the signature morphisms and its syntactic and semantic effects, motivated by applications to conceptual blending and software evolution. The representation theorem is based, for each $\phi \square $-model M, on setting $\left[\phantom{\rule{-0.166667em}{0ex}}\right[M\left]\phantom{\rule{-0.166667em}{0ex}}\right]$ to the set its $\phi $-reducts. This is possible because in $3/2$-institutions, unlike in ordinary institution theory, a model may have more than one reduct with respect to a fixed signature morphism, this being the semantic effect of the (implicit) partiality of the signature morphisms.

#### 2.4. Flattening Stratified Institutions to Ordinary Institutions

- –
- the objects of ${\mathrm{Mod}}^{\u266f}\left(\mathrm{\Sigma}\right)$ are the pairs $(M,w)$ such that $M\in \left|\mathrm{Mod}\right(\mathrm{\Sigma}\left)\right|$ and $w\in {\left[\phantom{\rule{-0.166667em}{0ex}}\left[M\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\mathrm{\Sigma}}$;
- –
- the $\mathrm{\Sigma}$-homomorphisms $(M,w)\to (N,v)$ are the pairs $(h,w)$ such that $h\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}M\to N$ and ${\left[\phantom{\rule{-0.166667em}{0ex}}\left[h\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\mathrm{\Sigma}}w=v$;
- –
- for any signature morphism $\phi \phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}\mathrm{\Sigma}\to {\mathrm{\Sigma}}^{\prime}$ and any ${\mathrm{\Sigma}}^{\prime}$-model $({M}^{\prime},{w}^{\prime})$$${\mathrm{Mod}}^{\u266f}\left(\phi \right)({M}^{\prime},{w}^{\prime})=(\mathrm{Mod}\left(\phi \right){M}^{\prime},{\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\phi}{w}^{\prime});$$
- –
- for each $\mathrm{\Sigma}$-model M, each $w\in {\left[\phantom{\rule{-0.166667em}{0ex}}\left[M\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\mathrm{\Sigma}}$, and each $\rho \in \mathrm{Sen}\left(\mathrm{\Sigma}\right)$$$\left((M,w){\vDash}_{\mathrm{\Sigma}}^{\u266f}\rho \right)=\left(M{\vDash}_{\mathrm{\Sigma}}^{w}\rho \right).$$

#### 2.5. Nominals in Stratified Institutions

- a $\mathrm{\Sigma}$-sentence $i-\mathrm{sen}$ is an i-sentence when$$(M{\vDash}^{w}i-\mathrm{sen})=({\left({\mathit{Nm}}_{\mathrm{\Sigma}}M\right)}_{i}=w);$$
- for any $\mathrm{\Sigma}$-sentence $\rho $, a $\mathrm{\Sigma}$-sentence ${@}_{i}\rho $ is the satisfaction of ρ at i when$$\left(M{\vDash}^{w}{@}_{i}\rho \right)=\left(M{\vDash}^{{\left({\mathit{Nm}}_{\mathrm{\Sigma}}M\right)}_{i}}\rho \right)$$

#### 2.6. Quantifications in Stratified Institutions

$M{\vDash}_{\mathrm{\Sigma}}\rho $ if and only if ${M}^{\prime}{\vDash}_{{\mathrm{\Sigma}}^{\prime}}{\rho}^{\prime}$ for each $\chi $-expansion ${M}^{\prime}$ of M.

#### 2.7. Concepts of Model Amalgamation in Stratified Institutions

- is a model amalgamation square when for each ${\mathrm{\Sigma}}_{k}$-model ${M}_{k}$, $k=1,2$ such that $\mathrm{Mod}\left({\phi}_{1}\right){M}_{1}=\mathrm{Mod}\left({\phi}_{2}\right){M}_{2}$ there exists an unique ${\mathrm{\Sigma}}^{\prime}$-model ${M}^{\prime}$ such that $\mathrm{Mod}\left({\theta}_{k}\right){M}^{\prime}={M}_{k}$, $k=1,2$, and
- is a stratified model amalgamation square when for each ${\mathrm{\Sigma}}_{k}$-model ${M}_{k}$ and each ${w}_{k}\in {\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}_{k}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\mathrm{\Sigma}}_{k}}$, $k=1,2$, such that $\mathrm{Mod}\left({\phi}_{1}\right){M}_{1}=\mathrm{Mod}\left({\phi}_{2}\right){M}_{2}$ and ${\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}_{1}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\phi}_{1}}{w}_{1}={\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}_{2}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\phi}_{2}}{w}_{2}$ there exists an unique ${\mathrm{\Sigma}}^{\prime}$-model ${M}^{\prime}$ and a unique ${w}^{\prime}\in {\left[\phantom{\rule{-0.166667em}{0ex}}\left[{N}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\mathrm{\Sigma}}^{\prime}}$ such that $\mathrm{Mod}\left({\theta}_{k}\right){M}^{\prime}={M}_{k}$ and ${\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\theta}_{k}}{w}^{\prime}={w}_{k}$, $k=1,2$.

**Fact**

**1**

## 3. Two Concepts of Interpolation

#### 3.1. Two Semantic Consequence Relations

**Definition**

**1.**

- the local semantic consequence relation ${\vDash}^{\u266f}$ is the semantic consequence relation of the institution ${\mathcal{S}}^{\u266f}$, and
- the global semantic consequence relation ${\vDash}^{*}$ is the semantic consequence relation of the institution ${\mathcal{S}}^{*}$.

**Fact**

**2.**

- $E{\vDash}^{\u266f}e$ if and only if for each Σ-model M and each $w\in {\left[\phantom{\rule{-0.166667em}{0ex}}\left[M\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\mathrm{\Sigma}}$,$$M{\vDash}^{w}E\phantom{\rule{4pt}{0ex}}implies\phantom{\rule{4pt}{0ex}}M{\vDash}^{w}e,$$
- $E{\vDash}^{*}e$ if and only if for each Σ-model M,$$foreach\phantom{\rule{4pt}{0ex}}w\in {\left[\phantom{\rule{-0.166667em}{0ex}}\left[M\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\mathrm{\Sigma}},\phantom{\rule{4pt}{0ex}}M{\vDash}^{w}E\phantom{\rule{4pt}{0ex}}impliesthatforeach\phantom{\rule{4pt}{0ex}}w\in {\left[\phantom{\rule{-0.166667em}{0ex}}\left[M\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\mathrm{\Sigma}},\phantom{\rule{4pt}{0ex}}M{\vDash}^{w}e.$$

**Proposition**

**1.**

**Proof.**

#### 3.2. Local Versus Global Interpolation in Stratified Institutions

**Proposition**

**2.**

**Proof.**

**Definition**

**2**

- With ordinary interpolation, the diagram (7) is restricted to an intersection-union square consisting of signature inclusions, i.e., $\mathrm{\Sigma}={\mathrm{\Sigma}}_{1}\cap {\mathrm{\Sigma}}_{2}$ and ${\mathrm{\Sigma}}^{\prime}={\mathrm{\Sigma}}_{1}\cup {\mathrm{\Sigma}}_{2}$.
- Concrete interpolation considers single sentences rather than sets of sentences. In the presence of logical conjunctions, there is no difference between these two approaches. However, there are important logics that lack conjunctions, such as equational and Horn logics, and many more. In these logics, interpolation is traditionally considered as failed. But works such as [29,48,49] show that this is a misunderstanding due to the rather faulty import of the single-sentence formulation of interpolation from some very prominent logics (such as classical propositional or first order logic) to other logical systems.

## 4. When Local Interpolation Causes Global Interpolation

#### 4.1. Signature Extensions with Nominals

**Definition**

**3**

- 1.
- $N\left(\iota \right)\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}N\left(\mathrm{\Sigma}\right)\to N\left({\mathrm{\Sigma}}^{\prime}\right)=N\left(\mathrm{\Sigma}\right)\cup \left\{i\right\}$ is the extension of $N\left(\mathrm{\Sigma}\right)$ with one new constant i, and,
- 2.
- for each Σ-model M and each $w\in {\left[\phantom{\rule{-0.166667em}{0ex}}\left[M\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\mathrm{\Sigma}}$ there exists a ι-expansion ${M}^{\prime}$ of M such that$${\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\iota}{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}^{\prime}}{M}^{\prime}\right)}_{i}=w.$$
- 3.
- For each signature morphism ${\theta}_{1}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{\mathrm{\Sigma}}_{1}\to {\mathrm{\Sigma}}^{\prime}$ and each signature extension ${\iota}^{\prime}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{\mathrm{\Sigma}}^{\prime}\to {\mathrm{\Sigma}}^{\u2033}$ with one nominal ${i}^{\prime}$ there exists a signature extension ${\iota}_{1}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{\mathrm{\Sigma}}_{1}\to {\mathrm{\Sigma}}_{1}^{\prime}$ with one nominal i and signature morphism ${\theta}_{1}^{\prime}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}{\mathrm{\Sigma}}_{1}^{\prime}\to {\mathrm{\Sigma}}^{\u2033}$ such that

- $N\left(\iota \right)$ is the set inclusion $\mathrm{Nom}\subseteq \mathrm{Nom}\cup \left\{i\right\}$.
- Let $(W,M)$ be any $(\mathrm{Nom},P)$-model. For any $w\in \left|W\right|$ we define the $\iota $-expansion $({W}^{\prime},M)$ of $(W,M)$ by ${W}_{i}^{\prime}=w$.
- Let ${\theta}_{1}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}({\mathrm{Nom}}_{1},{P}_{1})\to ({\mathrm{Nom}}^{\prime},{P}^{\prime})$, ${\iota}^{\prime}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}({\mathrm{Nom}}^{\prime},{P}^{\prime})\to ({\mathrm{Nom}}^{\prime}\cup \left\{{i}^{\prime}\right\},{P}^{\prime})$. Then we consider ${\iota}_{1}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}({\mathrm{Nom}}_{1},{P}_{1})\to ({\mathrm{Nom}}_{1}\cup \left\{i\right\},{P}_{1})$, $i\notin {\mathrm{Nom}}_{1}$, and ${\theta}_{1}^{\prime}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}({\mathrm{Nom}}_{1}\cup \left\{i\right\},{P}_{1})\to ({\mathrm{Nom}}^{\prime},\cup \left\{{i}^{\prime}\right\},{P}^{\prime})$ that just extends ${\theta}_{1}$ by letting ${\theta}_{1}^{\prime}i={i}^{\prime}$. Note that in this case the commutative square Equation (8) yields a pushout square in ${\mathrm{Sign}}^{\mathcal{H}\phantom{\rule{-0.166667em}{0ex}}\mathcal{P}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}}$. Since $\mathcal{H}\phantom{\rule{-0.166667em}{0ex}}\mathcal{P}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$ is semi-exact (cf. [9]) this is a model amalgamation square. Furthermore, since $\mathcal{H}\phantom{\rule{-0.166667em}{0ex}}\mathcal{P}\phantom{\rule{-0.166667em}{0ex}}\mathcal{L}$ is strictly stratified this is also a stratified model amalgamation square.

#### 4.2. Interpolation in Stratified Institutions with Universal Nominals

**Definition**

**4**

- it has signature extensions with nominals;
- it has universal quantifications corresponding to the signature extensions with nominals; and
- it has explicit local satisfaction.

**Notation**

**1.**

**Proposition**

**3.**

- $M{\vDash}^{*}e$ if and only if $(M,w){\vDash}^{\u266f}{e}^{\iota}$.
- $M{\vDash}^{*}e$ if and only if $M{\vDash}^{*}{e}^{\iota}$.
- $E{\vDash}^{*}e$ implies ${E}^{\iota}{\vDash}^{\u266f}{e}^{\iota}$.
- ${E}^{\iota}{\vDash}^{*}e$ implies $E{\vDash}^{*}e$.
- $E{\vDash}^{*}{e}^{\iota}$ implies $E{\vDash}^{*}e$.

**Proof.**

- For the implication from the left to the right we consider that $M{\vDash}^{*}e$. Let ${M}^{\prime}$ be any $\iota $-expansion of M and let ${w}^{\prime}\in {\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\mathrm{\Sigma}}^{\prime}}$ such that ${\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\iota}{w}^{\prime}=w$. We have to prove that ${M}^{\prime}{\vDash}^{{w}^{\prime}}{@}_{i}\left(\iota e\right)$. We have that

The latter satisfaction holds because $M{\vDash}^{*}e$.${M}^{\prime}{\vDash}^{{w}^{\prime}}{@}_{i}\left(\iota e\right)\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}{M}^{\prime}{\vDash}^{{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}^{\prime}}{M}^{\prime}\right)}_{i}}\iota e$ definition of local explicit satisfaction $\phantom{\rule{4pt}{0ex}}=\phantom{\rule{4pt}{0ex}}M{\vDash}^{{\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\iota}{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}^{\prime}}{M}^{\prime}\right)}_{i}}e$ satisfaction condition in $\mathcal{S}$. For the implication from the right to the left we assume $M{\vDash}^{w}(\forall \iota ){@}_{i}\left(\iota e\right)$. Let $v\in {\left[\phantom{\rule{-0.166667em}{0ex}}\left[M\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\mathrm{\Sigma}}$ be arbitrary. We have to prove that $M{\vDash}^{v}e$. Let us consider ${M}^{\prime}$ a $\iota $-expansion of M such that$${\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\iota}{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}^{\prime}}{M}^{\prime}\right)}_{i}=v.$$Then$M{\vDash}^{w}(\forall \iota ){@}_{i}\left(\iota e\right)$ $\Rightarrow {M}^{\prime}{\vDash}^{{w}^{\prime}}{@}_{i}\left(\iota e\right)$ for any ${w}^{\prime}\in {\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\iota}^{-1}$ definition of univ.

quantification$=\phantom{\rule{4pt}{0ex}}{M}^{\prime}{\vDash}^{{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}^{\prime}}{M}^{\prime}\right)}_{i}}\iota e$ definition of local explicit satisfaction $=\phantom{\rule{4pt}{0ex}}M{\vDash}^{v}e$ satisfaction condition. - Follows from 1. because $w\in {\left[\phantom{\rule{-0.166667em}{0ex}}\left[M\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{\mathrm{\Sigma}}$ is arbitrary.
- Let $M{\vDash}^{v}{E}^{\iota}$. By 1. it follows that $M{\vDash}^{*}E$ hence $M{\vDash}^{*}e$. By 1. (again) we obtain $M{\vDash}^{v}{e}^{\iota}$.
- Let $M{\vDash}^{*}E$. By 2. it follows that $M{\vDash}^{*}{E}^{\iota}$. Hence $M{\vDash}^{*}e$.
- Let $M{\vDash}^{*}E$. Then $M{\vDash}^{*}{e}^{\iota}$. By 2. it follows $M{\vDash}^{*}e$.

**Proposition**

**4.**

**Proof.**

- On the one hand, $({M}^{\prime},{w}^{\prime}){\vDash}^{\u266f}{\left({\theta}_{1}{E}_{1}\right)}^{{\iota}^{\prime}}$ means$$({M}^{\prime},{w}^{\prime}){\vDash}^{\u266f}(\forall {\iota}^{\prime}){@}_{{i}^{\prime}}\left({\iota}^{\prime}\left({\theta}_{1}{E}_{1}\right)\right)$$$$({M}^{\u2033},{w}^{\u2033}){\vDash}^{\u266f}{@}_{{i}^{\prime}}\left({\iota}^{\prime}\left({\theta}_{1}{E}_{1}\right)\right)$$

Because ${\mathit{Nm}}_{{\theta}_{1}^{\prime}}{M}^{\u2033}\phantom{\rule{0.166667em}{0ex}}:\phantom{\rule{0.277778em}{0ex}}N\left({\theta}_{1}^{\prime}\right)\left({\mathit{Nm}}_{{\mathrm{\Sigma}}^{\u2033}}{M}^{\u2033}\right)\to {\mathit{Nm}}_{{\mathrm{\Sigma}}_{1}^{\prime}}\left({\theta}_{1}^{\prime}{M}^{\u2033}\right)$ is a homomorphism of $S\phantom{\rule{-0.166667em}{0ex}}E\phantom{\rule{-0.166667em}{0ex}}T\phantom{\rule{-0.166667em}{0ex}}C$-models, because its underlying function is ${\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\u2033}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\theta}_{1}^{\prime}}$, and because $N\left({\theta}_{1}^{\prime}\right){i}_{1}={i}^{\prime}$, we have that1 $({M}^{\u2033},{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}^{\u2033}}{M}^{\u2033}\right)}_{{i}^{\prime}}){\vDash}^{\u266f}{\iota}^{\prime}\left({\theta}_{1}{E}_{1}\right)$ definition of explicit local satisfaction 2 $({\theta}_{1};{\iota}^{\prime})({M}^{\u2033},{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}^{\u2033}}{M}^{\u2033}\right)}_{{i}^{\prime}}){\vDash}^{\u266f}{E}_{1}$ satisfaction condition in $\mathcal{S}$ 3 $({\theta}_{1}{M}^{\prime},{\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\u2033}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\theta}_{1};{\iota}^{\prime}}{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}^{\u2033}}{M}^{\u2033}\right)}_{{i}^{\prime}}){\vDash}^{\u266f}{E}_{1}$ definition of reducts in ${\mathcal{S}}^{\u266f}$ 4 $({\theta}_{1}{M}^{\prime},{\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\u2033}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\iota}_{1};{\theta}_{1}^{\prime}}{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}^{\u2033}}{M}^{\u2033}\right)}_{{i}^{\prime}}){\vDash}^{\u266f}{E}_{1}$ ${\theta}_{1};{\iota}^{\prime}={\iota}_{1};{\theta}^{\prime}$ 5 $({\theta}_{1}{M}^{\prime},{\left[\phantom{\rule{-0.166667em}{0ex}}\left[{\theta}_{1}^{\prime}{M}^{\u2033}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\iota}_{1}}\left({\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\u2033}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\theta}_{1}^{\prime}}{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}^{\u2033}}{M}^{\u2033}\right)}_{{i}^{\prime}}\right)){\vDash}^{\u266f}{E}_{1}$ definition of reducts in ${\mathcal{S}}^{\u266f}$. $${\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\u2033}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\theta}_{1}^{\prime}}{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}^{\u2033}}{M}^{\u2033}\right)}_{{i}^{\prime}}={\left({\mathit{Nm}}_{{\mathrm{\Sigma}}_{1}^{\prime}}\left({\theta}_{1}^{\prime}{M}^{\u2033}\right)\right)}_{{i}_{1}}$$6 $({M}^{\u2033},{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}^{\u2033}}{M}^{\u2033}\right)}_{{i}^{\prime}}){\vDash}^{\u266f}{\iota}^{\prime}\left({\theta}_{1}{E}_{1}\right)$. - On the other hand, $({M}^{\prime},{w}^{\prime}){\vDash}^{\u266f}{\theta}_{1}\left({E}_{1}^{{\iota}_{1}}\right)$ is equivalent to the following satisfactions:

By the definition of universal ${\iota}_{1}$-quantifications the satisfaction 9 means that for all ${\iota}_{1}$-expansions $({M}_{1}^{\prime},{w}_{1}^{\prime})$ of $({\theta}_{1}{M}^{\prime},{\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\theta}_{1}}{w}^{\prime})$7 ${\theta}_{1}({M}^{\prime},{w}^{\prime}){\vDash}^{\u266f}{E}_{1}^{{\iota}_{1}}$ satisfaction condition in ${\mathcal{S}}^{\u266f}$ 8 $({\theta}_{1}{M}^{\prime},{\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\theta}_{1}}{w}^{\prime}){\vDash}^{\u266f}{{E}_{1}}^{{\iota}_{1}}$ definition of reducts in ${\mathcal{S}}^{\u266f}$ 9 $({\theta}_{1}{M}^{\prime},{\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\theta}_{1}}{w}^{\prime}){\vDash}^{\u266f}(\forall {\iota}_{1}){@}_{{i}_{1}}\left({\iota}_{1}{E}_{1}\right)$ definition of ${E}_{1}^{{\iota}_{1}}$.

which is successively equivalent to the following satisfactions:10 $({M}_{1}^{\prime},{w}_{1}^{\prime}){\vDash}^{\u266f}{@}_{{i}_{1}}\left({\iota}_{1}{E}_{1}\right)$ 11 $({M}_{1}^{\prime},{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}_{1}^{\prime}}{M}_{1}^{\prime}\right)}_{{i}_{1}}){\vDash}^{\u266f}{\iota}_{1}{E}_{1}$ definition of explicit local satisfaction 12 ${\iota}_{1}({M}_{1}^{\prime},{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}_{1}^{\prime}}{M}_{1}^{\prime}\right)}_{{i}_{1}}){\vDash}^{\u266f}{E}_{1}$ satisfaction condition in ${\mathcal{S}}^{\u266f}$ 13 $({\iota}_{1}{M}_{1}^{\prime},{\left[\phantom{\rule{-0.166667em}{0ex}}\left[{M}_{1}^{\prime}\right]\phantom{\rule{-0.166667em}{0ex}}\right]}_{{\iota}_{1}}{\left({\mathit{Nm}}_{{\mathrm{\Sigma}}_{1}^{\prime}}{M}_{1}^{\prime}\right)}_{{i}_{1}}){\vDash}^{\u266f}{E}_{1}$ definition of reducts in ${\mathcal{S}}^{\u266f}$.

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

## 5. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Concepts of Interpolation in Stratified Institutions. *Logics* **2023**, *1*, 80-96.
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