*3.5. Interpolation in Stratified Institutions*

Interpolation is a notoriously important logical property which is easy to understand but difficult to establish. It also has a number of important applications in computing science, especially in formal specification theory [65,70–74] but also in databases (ontologies) [75], automated reasoning [76,77], type checking [78], model checking [79], structured theorem proving [80,81], etc. Computing science and model theoretic motivations have led to a very general approach to interpolation [30] within the theory of institutions that is completely independent of any concrete logical system. This direction of study and research has produced a substantial body of results reported in works such as [30,42,43,45,65,74,82–86]. In this context, the institution theoretic concept to interpolation had suffered a gradual evolution. At the level of ordinary institution theory, one way to express the end result of this evolution is that of 'interpolation square'. In its Craig interpolation version, this is as follows. In any given institution I, a commutative square of signature morphisms as below

$$\begin{array}{c} \Sigma \xrightarrow{\begin{subarray}{c} \varphi\_{1} \\ \varphi\_{2} \\ \varphi\_{3} \end{subarray}} \xrightarrow{\begin{subarray}{c} \varphi\_{1} \\ \varphi\_{2} \\ \varphi\_{3} \end{subarray}} \begin{array}{c} \Sigma\_{1} \\ \vdots \\ \Sigma\_{2} \xrightarrow{\begin{subarray}{c} \\ \varphi\_{2} \end{subarray}} \Sigma' \end{array} \end{array} \tag{7}$$

is a *Craig interpolation square* when for each finite set *Ek* of Σ*k*-sentences, *k* = 1, 2, such that when *θ*1*E*<sup>1</sup> |= *θ*2*E*2, there exists a finite set *E* of Σ-sentences such that

$$E\_1 \mid = \varrho\_1 E \text{ and } \varrho\_2 E \mid = E\_2.$$

How can we lift this concept of interpolation square to stratified institutions? The obvious answer is to maintain the concept by apply it to a flattening of the respective stratified institution <sup>S</sup>. However, here, we run into a problem: which of <sup>S</sup><sup>∗</sup> and <sup>S</sup> is the most appropriate for this? The answer is that this may be actually a wrong question, as both the local (|<sup>=</sup> ) and the global (|=∗) semantic consequences can be used legitimately to define interpolation concepts in stratified institutions. So, we naturally end up with two concepts of interpolation in stratified institutions.

Then, a natural question arises: what is the causal relationship between local and global interpolation? In [87], we have provided an answer to this question. Without some additional infrastructure, none of the two interpolation concepts causes the other one. However, the main result of [87] shows that local causes global interpolation when the respective stratified institution has some nominals infrastructure including universal quantification over the nominals. In [87], these properties are given precise mathematical sense through some rather intricate technicalities which we do not present here. This is only the first step toward a proper theory of interpolation specific to stratified institutions. More steps are needed in order to mature it at a level comparable to that of interpolation in ordinary institution model theory.

#### *3.6. Diagrams in Stratified Institutions*

In conventional model theory, the method of diagrams is one of the most important methods. The institution-independent method of diagrams plays a significant role in the development of a lot of model theoretic results at the level of abstract institutions, many of its applications being presented in [44]. These include the existence of co-limits of models, free models along theory morphisms, axiomatisability results, elementary homomorphisms results, filtered power embeddings results, saturated models results (including an abstract version of Keisler–Shelah isomorphism theorem), the equivalence between initial semantics and quasi-varieties, Robinson consistency results, interpolation theory, definability theory, proof systems, predefined types, etc.

In institution theory, diagrams had been introduced for the first time by Tarlecki in [31,32] in a form different from ours. In the form presented here, it has been introduced at the level of institution-independent model theory in [33] as a categorical property which formalises the idea that

the class of model homomorphisms from a model *M* can be represented (by a natural isomorphism) as a class of models of a theory in a signature extending the original signature with syntactic entities determined by *M*.

Let us recall from [33,44] the main concept of the institution theoretic method of diagrams. An institution I has *diagrams* when for each signature Σ and each Σ model *M*, there exists a signature Σ*<sup>M</sup>* and a signature morphism *ι*Σ(*M*) : Σ → Σ*M*, functorial in Σ and *M*, and a set *EM* of Σ*<sup>M</sup>* sentences such that *Mod*(Σ*M*, *EM*) and the comma category *M*/*Mod*(Σ) are naturally isomorphic, i.e., the following diagram commutes by the isomorphism *i*Σ,*<sup>M</sup>* that is natural in Σ and *M*

$$\operatorname{Mod}(\Sigma\_M, E\_M) \xrightarrow{i\_{\Sigma, M}} M/\operatorname{Mod}(\Sigma) \tag{8}$$

$$\operatorname{Mod}(\iota\_{\Sigma}(M)) \xrightarrow[M \operatorname{Mod}(\Sigma)]{} \operatorname{forgetful}$$

The signature morphism *ι*Σ(*M*) : Σ → Σ*<sup>M</sup>* is called the *elementary extension of* Σ *via M*, and the set *EM* of Σ*<sup>M</sup>* sentences is called the *diagram* of the model *M*.

This can be seen as a coherence property between the semantic and the syntactic structures of the institution. By following the basic principle that a structure is rather defined by its homomorphisms (arrows) than by its objects, the semantic structure of an institution is given by its model homomorphisms. On the other hand, the syntactic structure of an(y concrete) institution is based upon its corresponding concept of atomic sentence.

In [57], it has been proposed that the concept of a diagram in stratified institutions should be transferred to the flattenings:

*the diagrams in a stratified institution* <sup>S</sup> *are the diagrams in* <sup>S</sup> *(or in* <sup>S</sup>∗*).*

Based on this principle, in [57], we have developed a general result on the existence of diagrams at the level of abstract stratified institutions that is applicable to a wide class of concrete situations. Its underlying idea is to combine the diagrams in the two components of a decomposition. However, again, this requires some nominal infrastructure. Let us present briefly how we can obtain diagrams in S when this comes with a decomposition as in Section 3.2.


$$\operatorname{Sign}^0 \xleftarrow{\Phi^{\phi}} \operatorname{Sign}^S \xleftarrow{\Phi} \operatorname{Sign}^{\mathcal{S}}$$

is a product in **CAT**. This is a rather easy condition in concrete applications, typical examples being given by HPL and HFOL.


$$\iota\_{\Sigma}M = (\iota\_{\Sigma\_0}M\_{0\prime}\iota\_{\Sigma\_1}M\_1).$$

• Furthermore, we let

$$E\_M = \alpha^0\_{\Sigma\_M} E\_{M\mathbb{0}} \downarrow \bigcup\_{i \in \{M\mathbb{I}\}} \otimes\_i (\alpha\_{\Sigma\_M} E\_{M^i\_1})$$

where @*i*(*α*Σ*<sup>M</sup> EMi* 1 ) abbreviates {@*n*Σ,*M*(*i*)*α*Σ*<sup>M</sup> ρ* | *ρ* ∈ *EMi* 1 }. This gives the diagram *M* in S∗.

• In order to obtain the diagram of a model (*M*, *<sup>w</sup>*) in <sup>S</sup> , it is enough to add the syntactic designation of *w* as a sentence to *EM*.

Particular typical consequences of this general result are the existence of diagrams in hybrid logic institutions such as HPL, HFOL. The limitation of this result is represented by the general assumption on the availability of a nominals infrastructure. However, this seems to be an inherent limitation that has to do with the existence of diagrams; in other words, it is not a limitation of the way we have constructed the diagrams. This conclusion is supported toward the end of [57] by a proof showing that MPL and MFOL do not admit institution theoretic diagrams.
