We construct a geometry of interaction (GoI: dynamic modelling of Gentzen-style cut elimination) for multiplicative-additive linear logic (MALL) by employing Bucciarelli–Ehrhard indexed linear logic MALL(I) to handle the additives. Our construction is an extension to the additives of the Haghverdi–Scott categorical formulation (a multiplicative GoI situation in a traced monoidal category) for Girard’s original GoI 1. The indices are shown to serve not only in their original denotational level, but also at a finer grained dynamic level so that the peculiarities of additive cut elimination such as superposition, erasure of subproofs, and additive (co-) contraction can be handled with the explicit use of indices. Proofs are interpreted as indexed subsets in the category Rel, but without the explicit relational composition; instead, execution formulas are run pointwise on the interpretation at each index, with respect to symmetries of cuts, in a traced monoidal category with a reflexive object and a zero morphism. The sets of indices diminish overall when an execution formula is run, corresponding to the additive cut-elimination procedure (erasure), and allowing recovery of the relational composition. The main theorem is the invariance of the execution formulas along cut elimination so that the formulas converge to the denotations of (cut-free) proofs.

中文翻译：

---

基于索引线性逻辑的 MALL 交互几何

我们为乘法-加法线性逻辑（购物中心) 通过采用 Bucciarelli-Ehrhard 索引线性逻辑购物中心(一世) 处理添加剂。我们的构造是对 Girard 的原始 GoI 1 的 Haghverdi-Scott 分类公式（追踪单曲面类别中的乘法 GoI 情况）的添加剂的扩展。这些指数不仅在其原始指称水平上服务，而且在更细粒度的动态级别，以便可以通过显式使用索引来处理加性切割消除的特性，例如叠加、子证明的擦除和加性（共）收缩。证明被解释为类别中的索引子集相对，但没有明确的关系组合；相反，执行公式在每个索引处的解释上逐点运行，关于切割的对称性，在具有自反对象和零态射的跟踪单曲面类别中。当执行公式运行时，索引集整体减少，对应于加法切割消除过程（擦除），并允许恢复关系组合。主要定理是执行公式沿切割消除的不变性，因此公式收敛到（无切割）证明的表示。