key: cord-0060370-as6tmymh authors: Fukihara, Yōji; Katsumata, Shin-ya title: Generalized Bounded Linear Logic and its Categorical Semantics date: 2021-03-23 journal: Foundations of Software Science and Computation Structures DOI: 10.1007/978-3-030-71995-1_12 sha: 1a5ce5c0d8729bf97457145927113b5c792c8785 doc_id: 60370 cord_uid: as6tmymh We introduce a generalization of Girard et al.’s BLL called GBLL (and its affine variant GBAL). It is designed to capture the core mechanism of dependency in BLL, while it is also able to separate complexity aspects of BLL. The main feature of GBLL is to adopt a multi-object pseudo-semiring as a grading system of the !-modality. We analyze the complexity of cut-elimination in GBLL, and give a translation from BLL with constraints to GBAL with positivity axiom. We then introduce indexed linear exponential comonads (ILEC for short) as a categorical structure for interpreting the [Formula: see text]-modality of GBLL. We give an elementary example of ILEC using folding product, and a technique to modify ILECs with symmetric monoidal comonads. We then consider a semantics of BLL using the folding product on the category of assemblies of a BCI-algebra, and relate the semantics with the realizability category studied by Hofmann, Scott and Dal Lago. Girard's linear logic is a refinement of propositional logic by restricting weakening and contraction in proofs [15] . Linear logic also has an of-course modality !, which restores these structural rules to formulas of the form !A. Later, Girard et al. extended the !-modality with quantitative information so that usage of !-modal formulas in proofs can be quantitatively controlled [16] . This extension, called bounded linear logic (BLL for short), is successfully applied to a logical characterization of P-time computations. Their extension takes two steps. First, the !-modality is extended to the form ! r A, where the index r is an element of a semiring [16, Section 2.4] . The index r is called grade in modern terminology [11, 13] . This extension and its variants have been employed in various logics and programming languages [7, 30, 14, 26, 28] . The categorical structure corresponding to ! r A is identified as graded linear exponential comonad [7, 13, 22] . Second, the ! r -modality is further extended to the form ! x
|π |(δ)), or 2) for all index δ ∈ ∆, the weight keeps (that is, |π|(δ) = |π |(δ)). The reduction of the former type is called symmetric or axiom reduction [ In the case where the weight keeps, we introduce another measure called the cut size π : ∆ → N of a proof π. Its definition is the same as the definition of weight except for Cut rule. For a proof π obtained by Cut rule from π 1 and π 2 , the cut size π (δ) is defined to be π 1 (δ) + π 2 (δ) + |π 1 |(δ) + |π 2 |(δ). In each commutative reduction from π to π the cut size decrease at all index (that is, for all δ ∈ ∆, ||π||(δ) > ||π ||(δ)), and the cut size is at most the square of the weight (that is, for all δ ∈ ∆, ||π||(δ) ≤ (|π|(δ)) 2 ). Therefore, the total number of steps is at most the cube of the weight. The number of reduction steps of a proof π and its weight depend on the length of lists computed by the Idx-morphisms occurring in π. However, to discuss the actual time complexity of cut-elimination, we further need to take into account the time complexity of the computation of Idx-morphisms. This would be achieved by looking at a subcategory of Idx computable within a certain time complexity. We leave this argument of analyzing the actual time complexity of cut-elimination as a future work. We show that GBLL can express BLL via a translation. This translation is actually given to variants of these calculi, namely from BLL with constraints (called CBLL) to GBAL with positivity axioms (called GBAL + ). CBLL is an extension of BLL with constraints, which are one of the features of Dal Lago and Hofmann's QBAL [10] . Constraints explicitly specify conditions imposed on resource variables, and it is natural to explicitly maintain these conditions throughout proofs. We also remark that in CBLL, weakening of !formulas ! x