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.

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.

On categorical models of classical logic and the Geometry of Interaction

2007 ◽  
Vol 17 (5) ◽  
pp. 957-1027 ◽  
Author(s):  
CARSTEN FÜHRMANN ◽  
DAVID PYM
Keyword(s):  
Classical Logic ◽  
Disjoint Union ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Structural Theory ◽  
Cut Elimination ◽  
Categorical Models ◽  
Boolean Lattices ◽  
Deep Exploration ◽  
Straightforward Proof

It is well known that weakening and contraction cause naive categorical models of the classical sequent calculus to collapse to Boolean lattices. In previous work, summarised briefly herein, we have provided a class of models calledclassical categoriesthat is sound and complete and avoids this collapse by interpreting cut reduction by a poset enrichment. Examples of classical categories include boolean lattices and the category of sets and relations, where both conjunction and disjunction are modelled by the set-theoretic product. In this article, which is self-contained, we present an improved axiomatisation of classical categories, together with a deep exploration of their structural theory. Observing that the collapse already happens in the absence of negation, we start with negation-free models calledDummett categories. Examples of these include, besides the classical categories mentioned above, the category of sets and relations, where both conjunction and disjunction are modelled by the disjoint union. We prove that Dummett categories are MIX, and that the partial order can be derived from hom-semilattices, which have a straightforward proof-theoretic definition. Moreover, we show that the Geometry-of-Interaction construction can be extended from multiplicative linear logic to classical logic by applying it to obtain a classical category from a Dummett category.Along the way, we gain detailed insights into the changes that proofs undergo during cut elimination in the presence of weakening and contraction.

scholarly journals Towards a typed Geometry of Interaction

2010 ◽  
Vol 20 (3) ◽  
pp. 473-521 ◽  
Author(s):  
ESFANDIAR HAGHVERDI ◽  
PHILIP SCOTT
Keyword(s):  
Linear Logic ◽  
Independent Interest ◽  
Cut Elimination ◽  
Mathematical Framework ◽  
Monoidal Categories ◽  
Original Theory ◽  
Categorical Setting ◽  
Abstract Notion ◽  
The Relationship

Girard's Geometry of Interaction (GoI) develops a mathematical framework for modelling the dynamics of cut elimination. We introduce a typed version of GoI, called Multiobject GoI for both multiplicative linear logic (MLL) and multiplicative exponential linear logic (MELL) with units. We present a categorical setting that includes our previous (untyped) GoI models, as well as more general models based on monoidal *-categories. Our development of multiobject GoI depends on a new theory of partial traces and trace classes, which we believe is of independent interest, as well as an abstract notion of orthogonality (which is related to work of Hyland and Schalk). We develop Girard's original theory of types, data and algorithms in our setting, and show his execution formula to be an invariant of cut elimination (under some restrictions). We prove soundness theorems for the MGoI interpretation (for Multiplicative and Multiplicative Exponential Linear Logic) in partially traced *-categories with an orthogonality. Finally, we briefly discuss the relationship between our GoI interpretation and other categorical interpretations of GoI.

scholarly journals Subexponentials in non-commutative linear logic

2018 ◽  
Vol 29 (8) ◽  
pp. 1217-1249 ◽  
Author(s):  
MAX KANOVICH ◽  
STEPAN KUZNETSOV ◽  
VIVEK NIGAM ◽  
ANDRE SCEDROV
Keyword(s):  
Programming Languages ◽  
Proof Theory ◽  
Linear Logic ◽  
Cut Elimination ◽  
Proof Systems ◽  
Logical Frameworks ◽  
Local Contraction ◽  
Concurrent Programming Languages ◽  
Exchange Rule ◽  
Non Local

Linear logical frameworks with subexponentials have been used for the specification of, among other systems, proof systems, concurrent programming languages and linear authorisation logics. In these frameworks, subexponentials can be configured to allow or not for the application of the contraction and weakening rules while the exchange rule can always be applied. This means that formulae in such frameworks can only be organised as sets and multisets of formulae not being possible to organise formulae as lists of formulae. This paper investigates the proof theory of linear logic proof systems in the non-commutative variant. These systems can disallow the application of exchange rule on some subexponentials. We investigate conditions for when cut elimination is admissible in the presence of non-commutative subexponentials, investigating the interaction of the exchange rule with the local and non-local contraction rules. We also obtain some new undecidability and decidability results on non-commutative linear logic with subexponentials.

scholarly journals On traced monoidal closed categories

2009 ◽  
Vol 19 (2) ◽  
pp. 217-244 ◽  
Author(s):  
MASAHITO HASEGAWA
Keyword(s):  
Fixed Point ◽  
Full Subcategory ◽  
Monoidal Category ◽  
Structure Theorem ◽  
Linear Logic ◽  
Monoidal Categories ◽  
Simple Observation ◽  
Fixed Point Computation ◽  
Canonical Inclusion ◽  
Monoidal Closed Category

The structure theorem of Joyal, Street and Verity says that every traced monoidal category arises as a monoidal full subcategory of the tortile monoidal category Int. In this paper we focus on a simple observation that a traced monoidal category is closed if and only if the canonical inclusion from into Int has a right adjoint. Thus, every traced monoidal closed category arises as a monoidal co-reflexive full subcategory of a tortile monoidal category. From this, we derive a series of facts for traced models of linear logic, and some for models of fixed-point computation. To make the paper more self-contained, we also include various background results for traced monoidal categories.

Transcendental syntax I: deterministic case

2015 ◽  
Vol 27 (5) ◽  
pp. 827-849 ◽  
Author(s):  
JEAN-YVES GIRARD
Keyword(s):  
Proof Theory ◽  
A Priori ◽  
Cut Elimination ◽  
List Type ◽  
A Posteriori ◽  
Deterministic Case ◽  
Four Blocks ◽  
Image Position

We study logic in the light of the Kantian distinction between analytic (untyped, meaningless, locative) answers and synthetic (typed, meaningful, spiritual) questions. Which is specially relevant to proof-theory: in a proof-net, the upper part is locative, whereas the lower part is spiritual: a posteriori (explicit) as far as correctness is concerned, a priori (implicit) for questions dealing with consequence, typically cut-elimination. The divides locative/spiritual and explicit/implicit give rise to four blocks which are enough to explain the whole logical activity.

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 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.

Uniqueness, definability and interpolation

1988 ◽  
Vol 53 (2) ◽  
pp. 554-570 ◽  
Author(s):  
Kosta Došen ◽  
Peter Schroeder-Heister
Keyword(s):  
Proof Theory ◽  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
General Notion ◽  
Structural Part ◽  
First Order ◽  
First Case ◽  
Additional Constant ◽  
Special Case

This paper is meant to be a comment on Beth's definability theorem. In it we shall make the following points.Implicit definability as mentioned in Beth's theorem for first-order logic is a special case of a more general notion of uniqueness. If α is a nonlogical constant, Tα a set of sentences, α* an additional constant of the same syntactical category as α and Tα, a copy of Tα with α* instead of α, then for implicit definability of α in Tα one has, in the case of predicate constants, to derive α(x1,…,xn) ↔ α*(x1,…,xn) from Tα ∪ Tα*, and similarly for constants of other syntactical categories. For uniqueness one considers sets of schemata Sα and derivability from instances of Sα ∪ Sα* in the language with both α and α*, thus allowing mixing of α and α* not only in logical axioms and rules, but also in nonlogical assumptions. In the first case, but not necessarily in the second one, explicit definability follows. It is crucial for Beth's theorem that mixing of α and α* is allowed only inside logic, not outside. This topic will be treated in §1.Let the structural part of logic be understood roughly in the sense of Gentzen-style proof theory, i.e. as comprising only those rules which do not specifically involve logical constants. If we restrict mixing of α and α* to the structural part of logic which we shall specify precisely, we obtain a different notion of implicit definability for which we can demonstrate a general definability theorem, where a is not confined to the syntactical categories of nonlogical expressions of first-order logic. This definability theorem is a consequence of an equally general interpolation theorem. This topic will be treated in §§2, 3, and 4.

scholarly journals Maker–Breaker percolation games I: crossing grids

2020 ◽  
pp. 1-28
Author(s):  
A. Nicholas Day ◽  
Victor Falgas-Ravry
Keyword(s):  
Percolation Theory ◽  
Winning Strategy ◽  
List Type ◽  
Rectangular Grid ◽  
Grid Graph ◽  
Left Hand ◽  
Right Hand ◽  
Winning Strategies ◽  
Special Case ◽  
Switching Game

Abstract Motivated by problems in percolation theory, we study the following two-player positional game. Let Λm×n be a rectangular grid-graph with m vertices in each row and n vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims p (as yet unclaimed) edges of the board Λm×n, while on each of his turns Breaker claims q (as yet unclaimed) edges of the board and destroys them. Maker wins the game if she manages to claim all the edges of a crossing path joining the left-hand side of the board to its right-hand side, otherwise Breaker wins. We call this game the (p, q)-crossing game on Λm×n. Given m, n ∈ ℕ, for which pairs (p, q) does Maker have a winning strategy for the (p, q)-crossing game on Λm×n? The (1, 1)-case corresponds exactly to the popular game of Bridg-it, which is well understood due to it being a special case of the older Shannon switching game. In this paper we study the general (p, q)-case. Our main result is to establish the following transition. If p ≥ 2q, then Maker wins the game on arbitrarily long versions of the narrowest board possible, that is, Maker has a winning strategy for the (2q, q)-crossing game on Λm×(q+1) for any m ∈ ℕ. If p ≤ 2q − 1, then for every width n of the board, Breaker has a winning strategy for the (p, q)-crossing game on Λm×n for all sufficiently large board-lengths m. Our winning strategies in both cases adapt more generally to other grids and crossing games. In addition we pose many new questions and problems.

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