**7. Conclusions**

The paper presents an abstract framework allowing us to construct, in a uniform and universal way, specification formalisms in arbitrary categories enabling us to specify semantic structures while employing the full expressive power of first-order logic.

The framework is based upon a formalization of "open formulas" as statements in contexts and offers a freshly new and abstract view of logics and specification formalisms.

Relying on the new framework, we present a general and universal account of "syntactic" encodings and representations of semantic structures generalizing the idea of elementary diagrams in traditional first-order logic.

Guided by the top-down principle, we consider at this first stage of extension of our framework just simple categories. To extend a specification formalism to a proper logic, we also have to develop, however, appropriate deduction calculi. To establish those deduction calculi, we should have features, like the translation of statements along variable substitutions, for example, at hand. As exemplified in the paper, we have to assume at least the existence of pushouts to support those features. We are not logicians, but the extension of our framework by general deduction calculi will be one of the main topics in our future work.

Another main topic will be operations. At the present stage, our abstract framework does not comprise operations since it is not clear for us how to generalize the concept of operation from set-based structures to semantic structures defined in an arbitrary category. Already, the step from operations on sets to operations on graphs is not that trivial, and even the concepts, constructions and results we developed for graph operations in [3] are not fully satisfactory yet.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Data Availability Statement:** Data sharing is not applicable.

**Acknowledgments:** I want to thank the guest editor of this special volume for encouraging me to write this paper.

**Conflicts of Interest:** The author declares no conflict of interest.

#### **Appendix A. Translation of Feature Expressions**

For a footprint Ξ and an object *X* in Var we denote by FE(Ξ, *X*) the set of all feature Ξexpressions on *X*. In Example 26, we discussed the replacement of auxiliary feature symbols by feature expressions. To formalize those replacements, we consider footprint morphisms. A **morphism** *η* : Ξ → Ξ between two footprints over the same category Var is given by a map *η* assigning to each feature symbol *F* ∈ Φ a feature Ξ -expression *η*(*F*) ∈ FE(Ξ , *α*(*F*)). *η* is called **simple** if *η*(*F*) = *F* (*idα*(*F*)), with *F* ∈ Φ and *α* (*F* ) = *α*(*F*), for all *F* ∈ Φ.

Any footprint morphism *η* : Ξ → Ξ induces an Var*Obj*-indexed family of maps *η<sup>X</sup>* : FE(Ξ, *X*) → FE(Ξ , *X*). To define these maps for non-simple footprint morphisms, we have to rely, however, on a mechanism translating feature expressions along variable

translations. Fortunately, we can establish such a mechanism, if Var has pushouts, and we fix a choice of pushouts in Var.

**Definition A1** ( Translation maps)**.** *We define inductively and in parallel a family of translation maps ψ*<sup>Ξ</sup> : FE(Ξ, *X*) → FE(Ξ, *Z*) *with ψ ranging over all variable translations ψ* : *X* → *Z:*


*for Q* ∈ {∃, ∀} *where Z <sup>ϕ</sup>*<sup>∗</sup> <sup>→</sup> *<sup>Y</sup><sup>ϕ</sup> ψ ψ*∗ <sup>←</sup> *Y is the chosen pushout of Z <sup>ψ</sup>* <sup>←</sup> *<sup>X</sup> <sup>ϕ</sup>* → *Y:*

Note that the pushout construction formalizes and generalizes the "introduction of

fresh variables" in traditional FOL! If we choose the cospan *<sup>Z</sup> <sup>ψ</sup>*−1;*<sup>ϕ</sup>* −→ *<sup>Y</sup>* ←− *idY <sup>Y</sup>*, whenever *<sup>ψ</sup>* is an isomorphism, we ensure, especially, that (*idX*)<sup>Ξ</sup> becomes the identity map on FE(Ξ, *X*). Since the composition of chosen pushouts does not result, in general, in a chosen pushout, the assignments *ψ* → *ψ*<sup>Ξ</sup> constitute only a pseudo functor from Var into Set. This may be a hint to develop future deduction calculi for Institutions of Statements rather in a fibred setting (compare [45])?

The translation *ψ*Ξ*CT* (*mon*) of the universal property *mon* of monomorphisms in Example 29 along the unique graph morphism *ψ* : (*xv*<sup>1</sup> *xe* −→ *xv*2) <sup>→</sup> *xv xe* . gives us, for example, a definition of monic loops at hand.

For any footprint morphism *η* : Ξ → Ξ , we can define inductively and in parallel for all variable declarations *X* a **substitution map** *η<sup>X</sup>* : FE(Ξ, *X*) → FE(Ξ , *X*) where the only non-trivial case is the base case :

*1*. *Atomic: ηX*(*F*(*β*)) := *β*<sup>Ξ</sup> (*η*(*F*)) for any *F* ∈ Φ and *β* : *αF* → *X* in Var.

If *η* is simple, this base case degenerates, according to Definition A1, to a simple replacement of feature symbols:

*1*'. *Atomic': ηX*(*F*(*β*)) := *β*<sup>Ξ</sup> (*F* (*idα*(*F*))) = *F* (*idα*(*F*); *β*) = *F* (*β*).

Thus, we do not need to employ translation maps to define substitution maps in case of simple footprint morphisms!
