Nominal techniques have been praised for their ability to formalize grammars with binding structures closer to their informal developments. At its core, there lies the definition of nominal sets, which capture the notion of name (in)dependence through a simple, and uniform, metatheory based on name permutations. We present a formal constructive development of nominal sets in Rocq (formerly known as Coq), with its main design and project decisions. Furthermore, we formalize the concepts of freshness, nominal alpha-equivalence, name abstraction, and finitely supported functions. Our implementation relies on a type class hierarchy which, combined with Rocq generalized rewriting mechanism, achieves concise definitions and proofs, whilst easing the well-known "setoid hell" scenario. We conclude with a discussion on how to obtain the constructive alpha-structural recursion and induction combinators, towards a nominal framework.
Containers represent a wide class of type constructions that are relevant for functional programming and (co)inductive reasoning. Indexed containers generalize this notion to better fit the scope of dependently typed programming. When interpreting types to be sets, a container describes an endofunctor on the category of sets while an I-indexed container describes an endofunctor on the category Set(I) of I-indexed families of sets. We consider the monoidal structure on the category of I-indexed containers whose tensor product of containers describes the composition of the respective induced endofunctors. We then give a combinatorial characterization of monoids in this monoidal category, and we show how these monoids correspond precisely to monads on the induced endofunctors on Set(I). Lastly, we conclude by presenting some examples of monads on Set(I) that fall under our characterization, including the product of two monads, indexed variants of the state and the writer monads and an example of a free monad. The technical results of this work are accompanied by a formalization in the proof assistant Cubical Agda.
Representations are essential to mathematically model phenomena, but there are many options available. While each of those options provides useful properties with which to solve problems related to the phenomena in study, comparing results between these representations can be non-trivial, as different frameworks are used for different contexts. We present a general structure based on set-theoretic concepts that accommodates many situations related to logical and semantic frameworks. We show the versatility of this approach by presenting alternative constructions of modal logic; in particular, all modal logics can be represented within the framework.
This paper investigates the extension of lattice-based logics into modal languages. We observe that such extensions admit multiple approaches, as the interpretation of the necessity operator is not uniquely determined by the underlying lattice structure. The most natural interpretation defines necessity as the meet of the truth values of a formula across all accessible worlds-an approach we refer to as the normal interpretation. We examine the logical properties that emerge under this and other interpretations, including the conditions under which the resulting modal logic satisfies the axiom K and other common modal validities. Furthermore, we consider cases in which necessity is attributed exclusively to formulas that hold in all accessible worlds.
Data-aware modal logics offer a powerful formalism for reasoning about semi-structured queries in languages such as DataGL, XPath, and GQL. In brief, these logics can be viewed as modal systems capable of expressing both reachability statements and data-aware properties, such as value comparisons. One particularly expressive logic in this landscape is HXPath(D), a hybrid modal logic that captures not only the navigational core of XPath but also data comparisons, node labels (keys), and key-based navigation operators. While previous work on HXPath(D) has primarily focused on its model-theoretic properties, in this paper we approach HXPath(D) from a proof-theoretic perspective. Concretely, we present a sound and complete Gentzen-style sequent calculus for HXPathD. Moreover, we show all rules in this calculus are invertible, and that it enjoys cut elimination. Our work contributes to the proof-theoretic foundations of data-aware modal logics, and enables a deeper logical analysis of query languages over graph-structured data. Moreover, our results lay the groundwork for extending proof-theoretic techniques to a broader class of modal systems.
