A focused linear logical framework and its application to metatheory of object logics

Abstract

Linear logic (LL) has been used as a foundation (and inspiration) for the development of programming languages, logical frameworks, and models for concurrency. LL’s cut-elimination and the completeness of focusing are two of its fundamental properties that have been exploited in such applications. This paper formalizes the proof of cut-elimination for focused LL. For that, we propose a set of five cut-rules that allows us to prove cut-elimination directly on the focused system. We also encode the inference rules of other logics as LL theories and formalize the necessary conditions for those logics to have cut-elimination. We then obtain, for free, cut-elimination for first-order classical, intuitionistic, and variants of LL. We also use the LL metatheory to formalize the relative completeness of natural deduction and sequent calculus in first-order minimal logic. Hence, we propose a framework that can be used to formalize fundamental properties of logical systems specified as LL theories.

A linear logic framework for multimodal logics

“…There are several alternatives for representing binders in proof assistants (see e.g., Chlipala 2008; Heberlin et al 2017; Laurent 2021; Pitts 2003). The current formalization builds on our previous work (Felty et al 2021), where the Hybrid system (Felty and Momigliano 2009) is used to represent the syntax. From the positive side, this choice has the immediate effect of allowing us to define binders at two levels: the meta-logic ( specification logic in the terminology of the two-level approach in Hybrid) and the object logic , the logical system to be encoded as a theory in the specification logic.…”

“…As far as we know, it is not possible to represent in Coq quantifiers at the specification level and also at the object level and keep the definition of sequents in . The reader can find in Felty et al (2021) details about the representation of syntax in Hybrid. The repository (Xavier and Olarte 2021) also includes further documentation on how to define the subexponential signature in as well as some tactics to improve automatization when using the library in Coq.…”

“…That work also formalized the adequacy of the encoding of different logical systems as theories in . This formalization effort was extended in Felty et al (2021), by introducing first-order object logics and using the cut coherence criteria in Miller and Pimentel (2013) to show cut admissibility for some OLs. Defining quantifiers at the -level and also at the OL level was only possible by replacing the PHOAS representation of syntax with a definitional representation using a de Bruijn style supported by the Hybrid library (Felty and Momigliano 2012).…”

“…While such a more expressive formalism enables calculi for a broader class of logics, the greater bureaucracy makes it harder to prove logical properties, such as analyticity itself (Bruscoli and Guglielmi 2006). We hence lift to the method developed by Miller and Pimentel (2013) and revisited by Felty et al (2021). The idea is to use a meta-level framework in order to specify and reason about, in a uniform way, different logical (object-level) proof systems.…”

Facilitating Meta-Theory Reasoning (Invited Paper)

“…To have a more smooth treatment for binders, the work in [33] was adapted to use the Hybrid framework [10,9]. This has allowed the formalization of a completely internal proof of cut elimination for focused linear logic.…”