A Focused Sequent Calculus for Semi-Substructural Logics with Additive Connectives

Core Concepts

This work introduces a focused sequent calculus for semi-substructural logics with additive conjunction and disjunction, providing a sound and complete normalization procedure for these logics.

Abstract

The content discusses the proof theory of semi-substructural logics, which are logical systems that relax some of the structural rules of intuitionistic linear logic. The authors focus on extending the sequent calculi for skew monoidal categories, previously studied by the authors and collaborators, with additive conjunction and disjunction.
The key highlights are:

The authors describe a cut-free sequent calculus for a fragment of non-commutative linear logic consisting of the skew multiplicative unit I, conjunction ⊗, and the additive connectives ∧and ∨. They introduce a congruence relation ⊜ that identifies derivations up-to η-equivalence and permutative conversions.

The authors provide a categorical semantics for this logic in terms of distributive skew monoidal categories, which are skew monoidal categories with binary products and coproducts satisfying a left-distributivity condition.

The authors introduce a focused sequent calculus with tag annotations that provides a sound and complete normalization procedure for the original sequent calculus. The focused calculus employs a root-first proof search strategy that uses tags to control the non-deterministic choices in the application of non-invertible rules.

The authors discuss extensions of the logic with additive units, a form of skew exchange, and linear implication. They argue that the normalization technique is modular and potentially scalable to other richer substructural logics.

The sequent calculi and the normalization procedure have been fully formalized in the Agda proof assistant and are available online.

Stats

None.

Quotes

None.

Key Insights Distilled From

Semi-Substructural Logics with Additives

by Nicc... at arxiv.org 04-24-2024

https://arxiv.org/pdf/2404.14922.pdf

Deeper Inquiries

How can the normalization technique presented in this work be extended to handle classical linear logic or other richer substructural logics beyond the fragment considered here

The normalization technique presented in the work can be extended to handle classical linear logic or other richer substructural logics by adapting the focused sequent calculus with tag annotations. In classical linear logic, additional rules and connectives such as the multiplicative additive conjunction and disjunction can be incorporated into the sequent calculus. By introducing new tags and rules for these additional connectives, the normalization procedure can be extended to handle the complexities of classical linear logic.
For other richer substructural logics beyond the fragment considered in the work, similar extensions can be made by introducing new rules and tags specific to the logic in question. By carefully defining the rules for the new connectives and ensuring the validity of tag annotations, the normalization technique can be scaled to accommodate the intricacies of various substructural logics.

What are the potential applications of semi-substructural logics with additive connectives, and how do they differ from the applications of traditional substructural logics

Semi-substructural logics with additive connectives have potential applications in various fields such as computational linguistics, resource-sensitive programming languages, and semantic analysis of programming languages. These logics offer a middle ground between traditional substructural logics and classical logics, allowing for more nuanced reasoning about resources and dependencies in computational systems.
The applications of semi-substructural logics with additive connectives differ from traditional substructural logics in that they provide a more flexible framework for capturing the interactions between resources and the logical structure of computations. By incorporating additive connectives like conjunction and disjunction, these logics can model more complex relationships and dependencies in computational processes. This makes them suitable for analyzing systems where both structural rules and additive operations play a crucial role.

Can the tag annotation mechanism used in the focused sequent calculus be generalized to provide a unifying framework for normalization procedures across different substructural logics

The tag annotation mechanism used in the focused sequent calculus can be generalized to provide a unifying framework for normalization procedures across different substructural logics. By defining a set of tags that correspond to different types of rules and operations in various logics, the tag annotation system can be extended to capture the specific features of each logic.
This generalized tag annotation framework can help in developing a systematic approach to normalization techniques for a wide range of substructural logics. By adapting the tags and rules to the specific characteristics of each logic, the framework can provide a unified method for achieving normalization and cut elimination in diverse substructural logics. This approach can streamline the development and analysis of proof systems for different logics, making it easier to compare and contrast their properties and behaviors.