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arXiv 2024年

作者： Atkey, Robert Kokke, Wen Mathematically Structured Programming Group Computer and Information Sciences University of Strathclyde Glasgow United Kingdom

Multiplicative-Additive System Virtual (MAV) is a logic that extends Multiplicative-Additive Linear Logic with a self-dual non-commutative operator expressing the concept of "before" or "sequencing". MAV is also an extenson of the the logic Basic System Virtual (BV) with additives. Formulas in BV have an appealing reading as processes with parallel and sequential composition. MAV adds internal and external choice operators. BV and MAV are also closely related to Concurrent Kleene Algebras. Proof systems for MAV and BV are Deep Inference systems, which allow inference rules to be applied anywhere inside a structure. As with any proof system, a key question is whether proofs in MAV can be reduced to a normal form, removing detours and the introduction of structures not present in the original goal. In Sequent Calcluli systems, this property is referred to as Cut Elimination. Deep Inference systems have an analogous Cut rule and other rules that are not present in normalised proofs. Cut Elimination for Deep Inference systems has the same metatheoretic benefits as for Sequent Calculi systems, including consistency and decidability. Proofs of Cut Elimination for BV, MAV, and other Deep Inference systems present in the literature have relied on intrincate syntactic reasoning and complex termination measures. We present a concise semantic proof that all MAV proofs can be reduced to a normal form avoiding the Cut rule and other "non analytic" rules. We also develop soundness and completeness proofs of MAV (and BV) with respect to a class of models. We have mechanised all our proofs in the Agda proof assistant, which provides both assurance of their correctness as well as yielding an executable normalisation procedure. Our technique extends to include exponentials and the additive units. © 2024, CC BY.

关键词： Algebra

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Actegories for the Working Amthematician

arXiv

引用

arXiv 2022年

作者： Capucci, Matteo Gavranović, Bruno Mathematically Structured Programming Group Department of Computer and Information Sciences University of Strathclyde Glasgow United Kingdom

Actions of monoidal categories on categories, also known as actegories, have been familiar to category theorists for a long time, and yet a comprehensive overview of this topic seems to be missing from the literature. Recently, actegories have been increasingly employed in applied category theory, thereby encouraging an effort to fill this gap according to the new needs of these applications. This work started as an investigation of the notion of monoidal actegory, a compatible pair of monoidal and actegorical structures, and ended up including a sizable reference on the elementary theory of actegories. We cover basic definitions and results on actegories and biactegories, spelling out explicitly many folkloric definitions, including their tensor product and their hom-tensor adjunction. We give new definitions of actegories with monoidal, braided monoidal and symmetric monoidal structure. In the last section, we provide three Cayley-like classification results for these *** Codes 18-02 © 2022, CC BY-SA.

关键词： Tensors

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