scholarly journals A MALL geometry of interaction based on indexed linear logic

2021 ◽  
pp. 1-29
Author(s):  
Masahiro Hamano
Keyword(s):  
Monoidal Category ◽  
Linear Logic ◽  
Dynamic Modelling ◽  
Cut Elimination ◽  
Elimination Procedure ◽  
Dynamic Level

Abstract 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.

scholarly journals On geometry of interaction for polarized linear logic

2017 ◽  
Vol 28 (10) ◽  
pp. 1639-1694
Author(s):  
MASAHIRO HAMANO ◽  
PHILIP SCOTT
Keyword(s):  
Proof Theory ◽  
Monoidal Category ◽  
Linear Logic ◽  
Cut Elimination ◽  
List Type ◽  
Monoidal Categories ◽  
Categorical Structure ◽  
Categorical Models ◽  
Focusing Property ◽  
Special Case

We present Geometry of Interaction (GoI) models for Multiplicative Polarized Linear Logic, MLLP, which is the multiplicative fragment of Olivier Laurent's Polarized Linear Logic. This is done by uniformly adding multi-points to various categorical models of GoI. Multi-points are shown to play an essential role in semantically characterizing the dynamics of proof networks in polarized proof theory. For example, they permit us to characterize the key feature of polarization, focusing, as well as being fundamental to our construction of concrete polarized GoI models.Our approach to polarized GoI involves following two independent studies, based on different categorical perspectives of GoI: (i)Inspired by the work of Abramsky, Haghverdi and Scott, a polarized GoI situation is defined in which multi-points are added to a traced monoidal category equipped with a reflexive object U. Using this framework, categorical versions of Girard's execution formula are defined, as well as the GoI interpretation of MLLP proofs. Running the execution formula is shown to characterize the focusing property (and thus polarities) as well as the dynamics of cut elimination.(ii)The Int construction of Joyal–Street–Verity is another fundamental categorical structure for modelling GoI. Here, we investigate it in a multi-pointed setting. Our presentation yields a compact version of Hamano–Scott's polarized categories, and thus denotational models of MLLP. These arise from a contravariant duality between monoidal categories of positive and negative objects, along with an appropriate bimodule structure (representing ‘non-focused proofs’) between them.Finally, as a special case of (ii) above, a compact model of MLLP is also presented based on Rel (the category of sets and relations) equipped with multi-points.

Multi-focused cut elimination

2017 ◽  
Vol 28 (5) ◽  
pp. 614-650
Author(s):  
TAUS BROCK-NANNESTAD ◽  
NICOLAS GUENOT
Keyword(s):  
Simple Proof ◽  
Linear Logic ◽  
Cut Elimination ◽  
Sequent Calculi ◽  
Elimination Procedure ◽  
The Impact

We investigate cut elimination in multi-focused sequent calculi and the impact on the cut elimination proof of design choices in such calculi. The particular design we advocate is illustrated by a multi-focused calculus for full linear logic using an explicitly polarised syntax and incremental focus handling, for which we provide a syntactic cut elimination procedure. We discuss the effect of cut elimination on the structure of proofs, leading to a conceptually simple proof exploiting the strong structure of multi-focused proofs.

Double categories: a modular model of multiplicative linear logic

2002 ◽  
Vol 12 (4) ◽  
pp. 449-479 ◽  
Author(s):  
PAUL-ANDRÉ MELLIÈS
Keyword(s):  
Side Effects ◽  
Linear Logic ◽  
Cut Elimination ◽  
Elimination Procedure ◽  
Modular Model ◽  
Categorical Model ◽  
Double Categories ◽  
Proof Nets ◽  
Composite Module

We construct a double category [Dscr ] of proof-nets in multiplicative linear logic (MLL). Its horizontal arrows are MLL modules (subnets of well-formed nets), its vertical arrows model side-effects, and its double cells interpret the cut-elimination procedure. The categorical model is modular in the sense that every computation of a composite module (π1; π2) factors out as the separate and interacting computations of the two subcomponents π1 and π2. This enables us to trace MLL modules in the course of cut-elimination, and analyze their behaviour in time.

scholarly journals Generalized Bounded Linear Logic and its Categorical Semantics

2021 ◽  
pp. 226-246
Author(s):  
Yōji Fukihara ◽  
Shin-ya Katsumata
Keyword(s):  
Linear Logic ◽  
Cut Elimination ◽  
Grading System ◽  
Bounded Linear ◽  
The Core ◽  
Categorical Structure ◽  
Categorical Semantics

AbstractWe introduce a generalization of Girard et al.’s called (and its affine variant ). It is designed to capture the core mechanism of dependency in , while it is also able to separate complexity aspects of . The main feature of is to adopt a multi-object pseudo-semiring as a grading system of the !-modality. We analyze the complexity of cut-elimination in , and give a translation from with constraints to with positivity axiom. We then introduce indexed linear exponential comonads (ILEC for short) as a categorical structure for interpreting the $${!}$$ ! -modality of . 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 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.

scholarly journals Possession as Linear Knowledge

10.29007/ntkm ◽  
2018 ◽  
Author(s):  
Frank Pfenning
Keyword(s):  
Distributed System ◽  
Epistemic Logic ◽  
Linear Logic ◽  
Expressive Power ◽  
Cut Elimination ◽  
Multi Agent Systems ◽  
Agent Systems ◽  
Ongoing Effort ◽  
Multi Agent ◽  
Localized Knowledge

Epistemic logic analyzes reasoning governing localized knowledge, and is thus fundamental to multi- agent systems. Linear logic treats hypotheses as consumable resources, allowing us to model evolution of state. Combining principles from these two separate traditions into a single coherent logic allows us to represent localized consumable resources and their flow in a distributed system. The slogan “possession is linear knowledge” summarizes the underlying idea. We walk through the design of a linear epistemic logic and discuss its basic metatheoretic properties such as cut elimination. We illustrate its expressive power with several examples drawn from an ongoing effort to design and implement a linear epistemic logic programming language for multi-agent distributed systems.

scholarly journals Reconciling Lambek’s restriction, cut-elimination and substitution in the presence of exponential modalities

2020 ◽  
Vol 30 (1) ◽  
pp. 239-256 ◽  
Author(s):  
Max Kanovich ◽  
Stepan Kuznetsov ◽  
Andre Scedrov
Keyword(s):  
Linear Logic ◽  
Cut Elimination ◽  
Lambek Calculus ◽  
Full Extent ◽  
Intuitionistic Linear Logic

Abstract The Lambek calculus can be considered as a version of non-commutative intuitionistic linear logic. One of the interesting features of the Lambek calculus is the so-called ‘Lambek’s restriction’, i.e. the antecedent of any provable sequent should be non-empty. In this paper, we discuss ways of extending the Lambek calculus with the linear logic exponential modality while keeping Lambek’s restriction. Interestingly enough, we show that for any system equipped with a reasonable exponential modality the following holds: if the system enjoys cut elimination and substitution to the full extent, then the system necessarily violates Lambek’s restriction. Nevertheless, we show that two of the three conditions can be implemented. Namely, we design a system with Lambek’s restriction and cut elimination and another system with Lambek’s restriction and substitution. For both calculi, we prove that they are undecidable, even if we take only one of the two divisions provided by the Lambek calculus. The system with cut elimination and substitution and without Lambek’s restriction is folklore and known to be undecidable.

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