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Subexponentials in non-commutative linear logic [article]

Max Kanovich, Stepan Kuznetsov, Vivek Nigam, Andre Scedrov
2017 arXiv   pre-print
We also obtain some new undecidability and decidability results on non-commutative linear logic with subexponentials.  ...  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.  ...  This paper investigates the proof theory of subexponentials in non-commutative linear logic. Our contribution is as follows: 1.  ... 
arXiv:1709.03607v1 fatcat:ft5zuquzsza25a3csdz6hxxqhe

A Logical Framework with Commutative and Non-commutative Subexponentials [chapter]

Max Kanovich, Stepan Kuznetsov, Vivek Nigam, Andre Scedrov
2018 Lecture Notes in Computer Science  
Motivated by applications in distributed systems and in type-logical grammar, we propose a linear logical framework containing both commutative and non-commutative subexponentials.  ...  Commutative subexponentials enhance the expressiveness of linear logic frameworks by allowing the distinction of multiple contexts.  ...  In our previous work [10] , we proposed the proof system SNILL Σ (Subexponential Non-Commutative Intuitionistic Linear Logic) 7 which extends propositional Lambek Calculus with subexponentials.  ... 
doi:10.1007/978-3-319-94205-6_16 fatcat:lscl6zdtvfc33oxrsu43unsudm

Soft Subexponentials and Multiplexing [chapter]

Max Kanovich, Stepan Kuznetsov, Vivek Nigam, Andre Scedrov
2020 Lecture Notes in Computer Science  
We bring together these refinements of linear logic in a non-commutative setting.  ...  One of the subexponentials allows the multiplexing rule in the style of soft linear logic and light linear logic. The second subexponential provides the exchange rule.  ...  Drawing inspiration from commutative logics such as linear logic [6] , light linear logic [7] , soft linear logic [16] , and easy linear logic [10] , here we introduce our primitive non-commutative  ... 
doi:10.1007/978-3-030-51074-9_29 fatcat:rx3qbmznpzdh7fme4xiv27bqgu

A Proof Theoretic Study of Soft Concurrent Constraint Programming

ELAINE PIMENTEL, CARLOS OLARTE, VIVEK NIGAM
2014 Theory and Practice of Logic Programming  
In fact, the underlying constraint system can be built from a suitable fragment of intuitionistic (linear) logic -ILL- and processes can be interpreted as formulas in ILL.  ...  In this work, we give a proof theoretical meaning to soft constraints: they can be defined as formulas in a suitable fragment of ILL with subexponentials (SELL) where subexponentials, ordered in a c-semiring  ...  Soft-CCP as Theories in Linear Logic with Subexponential In this section we build soft constraints from formulas in a suitable fragment of intuitionistic linear logic (ILL) with subexponentials (Danos  ... 
doi:10.1017/s147106841400026x fatcat:z5cdsnb3hbd6jmmjqzrfhyneai

(Mathematical) Logic for Systems Biology (Invited Paper) [chapter]

Joëlle Despeyroux
2016 Lecture Notes in Computer Science  
We present two candidate logics (two modal extensions of linear logic, called HyLL and SELL), along with biological examples.  ...  Future works includes using automatic provers, which would extend existing automatic provers for linear logic.  ...  We note that ↓ and at commute freely with all non-hybrid connectives [8] .  ... 
doi:10.1007/978-3-319-45177-0_1 fatcat:imfgsz4r4jdfdi5et7jkrw7gwy

Subexponential concurrent constraint programming

Carlos Olarte, Elaine Pimentel, Vivek Nigam
2015 Theoretical Computer Science  
In previous works we have shown that linear logic with subexponentials (SELL), a refinement of linear logic, can be used to specify emergent features of concurrent constraint programming (CCP) languages  ...  In order to provide the proof theoretic foundations for such a language, we propose a unified logical framework (SELLS ) combining the extensions of linear logic with subexponentials mentioned above, and  ...  We believe that linear logic together with the strong levels of adequacy may help us develop more general techniques for verifying CCP programs.  ... 
doi:10.1016/j.tcs.2015.06.031 fatcat:otxe47mfjnf75i4nov2yn7ifya

An Adequate Compositional Encoding of Bigraph Structure in Linear Logic with Subexponentials [chapter]

Kaustuv Chaudhuri, Giselle Reis
2015 Lecture Notes in Computer Science  
In linear logic, formulas can be split into two sets: classical (those that can be used as many times as necessary) or linear (those that are consumed and no longer available after being used).  ...  This flexibility increases its expressiveness: we already have adequate encodings of a number of other proof systems, and for computational models such as concurrent constraint programming, in linear logic  ...  resource in linear logic.  ... 
doi:10.1007/978-3-662-48899-7_11 fatcat:g2twkmcfrjdmbkjdi7s3ri27by

Hybrid and Subexponential Linear Logics

Joëlle Despeyroux, Carlos Olarte, Elaine Pimentel
2017 Electronical Notes in Theoretical Computer Science  
HyLL (Hybrid Linear Logic) and SELL (Subexponential Linear Logic) are logical frameworks that have been extensively used for specifying systems that exhibit modalities such as temporal or spatial ones.  ...  In HyLL, truth judgments are labelled by worlds and hybrid connectives relate worlds with formulas. In SELL, the linear logic exponentials (!, ?)  ...  We note that ↓ and at commute freely with all non-hybrid connectives [11] .  ... 
doi:10.1016/j.entcs.2017.04.007 fatcat:ov52p5wcmjcurlpnlcuznebccm

Infinitary Action Logic with Exponentiation [article]

Stepan L. Kuznetsov, Stanislav O. Speranski
2021 arXiv   pre-print
We prove cut elimination and, in the case where at least one subexponential allows non-local contraction, establish exact complexity boundaries in two senses.  ...  In the case where no subexponential allows contraction, we show that complexity is the same as for infinitary action logic itself.  ...  The work was supported by the Russian Science Foundation, in cooperation with the Austrian Science Fund, under grant RSF-FWF 20-41-05002.  ... 
arXiv:2001.06863v3 fatcat:gu7sx7ot5rhvfb4z7ozhyj3bym

Quantale semantics of Lambek calculus with subexponential modalities [article]

Daniel Rogozin
2019 arXiv   pre-print
In our approach, subexponential modalities have an interpretation in terms of quantic conuclei. We show that this extension of Lambek calculus is complete w.r.t quantales with quantic conuclei.  ...  In this paper, we consider the polymodal version of Lambek calculus with subexponential modalities initially introduced by Kanovich, Kuznetsov, Nigam, and Scedrov and its quantale semantics.  ...  The initial idea to consider quantales within linear logic belongs to David Yetter [24] .  ... 
arXiv:1908.01055v2 fatcat:gpmrbxxedzhy3o5ih4oapfh5iu

Hybrid linear logic, revisited

KAUSTUV CHAUDHURI, JOËLLE DESPEYROUX, CARLOS OLARTE, ELAINE PIMENTEL
2019 Mathematical Structures in Computer Science  
Another extension of LL that has extensively been used for specifying systems with modalities is Subexponential Linear Logic (SELL). In SELL, the LL exponentials (!, ?)  ...  HyLL (Hybrid Linear Logic) is an extension of intuitionistic linear logic (ILL) that has been used as a framework for specifying systems that exhibit certain modalities.  ...  We note that ↓ and at commute freely with all non-hybrid connectives [DC14] .  ... 
doi:10.1017/s0960129518000439 fatcat:ovn75fwk4rfq3pmuybe4luwrtm

Hybrid and Subexponential Linear Logics Technical Report [article]

Joëlle Despeyroux, Elaine Pimentel
2016 arXiv   pre-print
HyLL (Hybrid Linear Logic) and SELL (Subexponential Linear Logic) are logical frameworks that have been extensively used for specifying systems that exhibit modalities such as temporal or spatial ones.  ...  In HyLL, truth judgments are labelled by worlds and hybrid connectives relate worlds with formulas. In SELL, the linear logic exponentials (!, ?)  ...  We note that ↓ and at commute freely with all non-hybrid connectives [DC14] .  ... 
arXiv:1608.08779v2 fatcat:pqnhffzanfamlgwix2ncdmdroa

Reconciling Lambek's restriction, cut-elimination, and substitution in the presence of exponential modalities [article]

Max Kanovich, Stepan Kuznetsov, Andre Scedrov
2019 arXiv   pre-print
The Lambek calculus can be considered as a version of non-commutative intuitionistic linear logic.  ...  In this paper we discuss ways of extending the Lambek calculus with the linear logic exponential modality while keeping Lambek's restriction.  ...  Acknowledgments This research was performed in part during visits of Stepan Kuznetsov and Max Kanovich to the University of Pennsylvania.  ... 
arXiv:1608.02254v2 fatcat:65wdvaugyzfo5fketvaifg5dc4

Undecidability of the Lambek calculus with subexponential and bracket modalities [article]

Max Kanovich, Stepan Kuznetsov, Andre Scedrov
2017 arXiv   pre-print
Their extension is based on a non-standard contraction rule for the exponential that interacts with the bracket structure in an intricate way.  ...  The Lambek calculus is a well-known logical formalism for modelling natural language syntax.  ...  Linguistic Introduction The Lambek calculus [23] is a substructural, non-commutative logical system (a variant of linear logic [15] in its intuitionistic non-commutative version [1] ) that serves  ... 
arXiv:1608.04020v2 fatcat:q2awwnol7jetzom5mzih23u2ou

Undecidability of the Lambek Calculus with Subexponential and Bracket Modalities [chapter]

Max Kanovich, Stepan Kuznetsov, Andre Scedrov
2017 Lecture Notes in Computer Science  
Their extension is based on a non-standard contraction rule for the exponential that interacts with the bracket structure in an intricate way.  ...  The Lambek calculus is a well-known logical formalism for modelling natural language syntax.  ...  Linguistic Introduction The Lambek calculus [23] is a substructural, non-commutative logical system (a variant of linear logic [15] in its intuitionistic non-commutative version [1] ) that serves  ... 
doi:10.1007/978-3-662-55751-8_26 fatcat:3gaf4jurf5bddf6kftqpqnkz2y
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