Extending a system in the calculus of structures with a self-dual quantifier

We recall that SBV, a proof system developed under the methodology of deep inference, extends multiplicative linear logic with the self-dual non-commutative logical operator Seq. We introduce SBVQ that extends SBV by adding the self-dual quantifier Sdq. The system SBVQ is consistent because we prove that (the analogous of) cut elimination holds for it. Its new logical operator Sdq operationally behaves as a binder, in a way that the interplay between Seq, and Sdq can model {\beta}-reduction of linear {\lambda}-calculus inside the cut-free subsystem BVQ of SBVQ. The long term aim is to keep developing a programme whose goal is to give pure logical accounts of computational primitives under the proof-search-as-computation analogy, by means of minimal, and incremental extensions of SBV.

💡 Research Summary

The paper introduces SBVQ, an extension of the deep‑inference proof system SBV, by adding a self‑dual quantifier called Sdq. SBV itself augments multiplicative linear logic (MLL) with a non‑commutative connective Seq, allowing inference rules to be applied at any depth within a formula. However, SBV lacks a mechanism for binding variables, which limits its ability to directly encode computational constructs such as those found in the λ‑calculus.

Sdq is designed to fill this gap. Unlike the traditional universal (∀) and existential (∃) quantifiers, Sdq is self‑dual: its dual is itself. This property enables a symmetric treatment of introduction and elimination rules and makes Sdq act as a genuine binder within the logical language. The authors define two principal rules for Sdq—an introduction rule that inserts a new Sdq binder into a formula, and an elimination rule that removes it. These rules coexist with the existing SBV rules for Seq, tensor (⊗), par (⅋), and other linear connectives, preserving the deep‑inference principle that any sub‑formula may be rewritten.

A central technical contribution is the proof of cut elimination for SBVQ. By adapting the classic cut‑reduction technique to the calculus of structures and exploiting the self‑duality of Sdq, the authors show that every cut can be systematically reduced and eventually eliminated without increasing the complexity of the proof. This result guarantees the consistency of SBVQ and establishes a solid foundation for using the system as a logical model of computation.

The paper then focuses on a cut‑free fragment called BVQ. Within BVQ, the interaction between Seq and Sdq can faithfully simulate β‑reduction of the linear λ‑calculus. Specifically, function application in the λ‑calculus corresponds to the Seq connective, while variable binding corresponds to the Sdq quantifier. The β‑reduction step—substituting an argument for a bound variable—is mirrored by a single logical transformation in BVQ. This correspondence exemplifies the “proof search as computation” paradigm: the process of finding a proof in BVQ directly implements the computational step of a λ‑term.

Finally, the authors position this work as the first step in a broader research agenda aimed at providing pure logical accounts of computational primitives through minimal, incremental extensions of SBV. Future directions include incorporating more sophisticated control structures, stateful operations, and non‑linear connectives, with the ultimate goal of capturing the full expressive power of modern programming languages within a clean, proof‑theoretic framework.