*3.8. Abstract Connectives and Elementary Homomorphisms*

In the list of examples of stratified institutions, we have presented the example CON . We said that CON may provide foundations for an abstract theory of connectives. Let us see how this works by following some theory developed in [51]. The main idea is that we think of a stratified institution S as *having connectives* when we can 'extract' them from S. Technically, this means that there exists a functor *C* : *Sign*<sup>S</sup> → *Sign*CON and for each Σ ∈ |*Sign*<sup>S</sup> |, a function *β*<sup>Σ</sup> : |*Mod*<sup>S</sup> (Σ)|→|*Mod*CON (*C*Σ)| natural in Σ such that

$$\text{Sen} = T\_{-} \circ \mathbb{C}, \; \|M\|\_{\Sigma}^{\mathbb{S}} = \|\beta\_{\Sigma} M\|\_{C\Sigma}^{\text{CDN}}, \; M \coloneqq\_{\Sigma}^{\eta} \rho \text{ if and only if } \; \beta\_{\Sigma} M \doteq\_{C\Sigma}^{\eta} \rho.$$

This means that any sentence of S is formed from connectives, each S model has an underlying connective algebra, and the satisfaction in S is given by evaluating the connective terms. In a more sophisticated terminology, S having connectives provides an example of morphism of stratified institutions.

OFOL provides a good example of this situation by letting the null-ary connectives consist of the atoms, the unary connectives consist of negation and quantifiers, the binary connectives being ∧, ∨, ..., and that is all. Then, *β* maps to corresponding sets of valuations.

One of the consequences of these conceptual developments is the possibility of having a stratified institution theoretic alternative to the concepts of elementary homomorphism that is based on quasi-representability or on diagrams, such as in [35,44]. Thus, we say that a model homomorphism *h* : *M* → *N* in a stratified institution S is *elementary* when for each sentence *ρ* and each *η* ∈ [[*M*]], we have that

$$M \mathop{=}^{\eta}\_{\Sigma} \rho \text{ if and only if } N \mathop{=}^{\|h\|\_{\Sigma}\eta}\_{\Sigma} \rho.$$

The advantage of this concept of elementary homomorphism over the other ones from institution theory is that it does not depend on other properties that may be problematic in some cases. For instance, we have seen in Section 3.6 that diagrams are not always available especially in stratified contexts. So, in [51], there is a result that explains the common concept of elementary homomorphism in terms of stratified institution elementary homomorphism. Given a stratified institution with connectives, a Σ-homomorphism *h* : *M* → *N* is elementary if and only if [[*h*]] is a connective algebra homomorphism *β*Σ*M* → *β*Σ*N*.

In [51], this result had been used for providing a method for establishing Tarski's elementary chain/co-limit theorem for concrete model theories that can be captured as stratified institutions. This is one of the early model theoretic results in first-order logic [90] with manifold applications (these can be consulted in [2]), which has also received a proof in the abstract setting of arbitrary institutions in [35,44]. It says that the co-limit of a directed diagram of elementary homomorphisms consists of elementary homomorphisms, too. In the context of stratified institutions with connectives, this means that any co-limit of a directed diagram of elementary homomorphisms becomes mapped by the stratification to a co-limit in the category of connective algebra homomorphisms. Moreover, in [51], we can find examples on how this works in MPL and OFOL.

#### *3.9. Foundations for Formal Verification of Reconfigurable Systems*

In [56], the author employs stratified institutions with frame and nominals extraction (presented above in Section 3.4) (rebranded as 'hybrid institutions') as a general foundational framework for a formal verification methodology for reconfigurable systems. The envisaged methodology would thus constitute an alternative to the methodology implemented by the language *H* [68] based on the generic translation concept of [66]. While in the latter case, the verification process is exported to first-order logic, and the result of that is imported back to the source logic, in the former case, the verification process happens right in the respective stratified institution. However, both approaches share the same verification goal: that of reconfigurable systems.

The substance of [56] consists of the definition of a generic proof calculi applicable to a relevant class of stratified institutions with frame and nominals extraction, which is proved complete (apparently) with respect to the local satisfaction relation <sup>|</sup><sup>=</sup> . The method to prove completeness is Cohen's forcing [6,7] adapted to abstract institutions [38].
