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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.  ...  Linear logical frameworks have been successfully used for the specification of a number of computational, logics and proof systems.  ...  Conclusions This paper introduced the logical framework SNILLF which allows for both commutative and non-commutative subexponentials.  ... 
doi:10.1007/978-3-319-94205-6_16 fatcat:lscl6zdtvfc33oxrsu43unsudm

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.  ...  Linear logical frameworks with subexponentials have been used for the specification of among other systems, proof systems, concurrent programming languages and linear authorization logics.  ...  While their focused proof system contained a single unbounded, a single bounded and a single non-commutative modalities, focused proof systems with commutative and non-commutative subexponentials would  ... 
arXiv:1709.03607v1 fatcat:ft5zuquzsza25a3csdz6hxxqhe

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.  ...  We introduce a noncommutative substructural system with subexponential modalities controlled by a minimalistic set of rules.  ...  For example [12] , we proposed a logical framework with commutative and noncommutative subexponentials, applying it for applications in type-logical grammar.  ... 
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  
We hence achieve two goals: (1) obtain a CCP language where agents can tell and ask soft constraints and (2) prove that the language in (1) has a strong connection with logic.  ...  Hence we keep a declarative reading of processes as formulas while providing a logical framework for soft-CCP based systems.  ...  Nigam was supported by CNPq and Pimentel was supported by CNPq and CAPES.  ... 
doi:10.1017/s147106841400026x fatcat:z5cdsnb3hbd6jmmjqzrfhyneai

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

Joëlle Despeyroux
2016 Lecture Notes in Computer Science  
The potential candidate logics should have a traditional proof theoretic pedigree (including a sequent calculus presentation enjoying cut-elimination and focusing), and should come with (certified) proof  ...  We advocates here the use of (mathematical) logic for systems biology, as a unified framework well suited for both modeling the dynamic behaviour of biological systems, expressing properties of them, and  ...  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 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  ...  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  ...  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

Infinitary Action Logic with Exponentiation [article]

Stepan L. Kuznetsov, Stanislav O. Speranski
2021 arXiv   pre-print
We introduce infinitary action logic with exponentiation – that is, the multiplicative-additive Lambek calculus extended with Kleene star and with a family of subexponential modalities, which allows some  ...  We prove cut elimination and, in the case where at least one subexponential allows non-local contraction, establish exact complexity boundaries in two senses.  ...  The authors are grateful to the referees for valuable comments and suggestions.  ... 
arXiv:2001.06863v3 fatcat:gu7sx7ot5rhvfb4z7ozhyj3bym

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.  ...  Both frameworks have linear logic (LL) as a common ground and they admit (cut-free) complete focused proof systems. The difference relies on the way modalities are handled.  ...  We note that ↓ and at commute freely with all non-hybrid connectives [11] .  ... 
doi:10.1016/j.entcs.2017.04.007 fatcat:ov52p5wcmjcurlpnlcuznebccm

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

Kaustuv Chaudhuri, Giselle Reis
2015 Lecture Notes in Computer Science  
with subexponentials (SEL).  ...  In this work we present an encoding of the bigraph structure in SEL, thus giving an indication of the expressive power of this logic, and at the same time providing a framework for reasoning and operating  ...  Background Subexponential Logic (SEL) Subexponential logic (SEL) is a variant of linear logic with a family of exponential connectives-called subexponentials (a term coined in [12] )-that are indexed  ... 
doi:10.1007/978-3-662-48899-7_11 fatcat:g2twkmcfrjdmbkjdi7s3ri27by

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.  ...  Namely, we design a system with Lambek's restriction and cut elimination and another system with Lambek's restriction and substitution.  ...  A part of the work was also done during the stay of Andre Scedrov at the National Research University Higher School of Economics. We would like to thank S. O. Kuznetsov and I. A.  ... 
arXiv:1608.02254v2 fatcat:65wdvaugyzfo5fketvaifg5dc4

Hybrid linear logic, revisited

KAUSTUV CHAUDHURI, JOËLLE DESPEYROUX, CARLOS OLARTE, ELAINE PIMENTEL
2019 Mathematical Structures in Computer Science  
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.  ...  In HyLL, truth judgements are labelled by worlds (having a monoidal structure) and hybrid connectives (at and ↓) relate worlds with formulas.  ...  We would like to thank Dale Miller for being such a good mentor and colleague. We also thank the anonymous reviewers for their valuable comments on an earlier draft of this paper.  ... 
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.  ...  Both frameworks have linear logic (LL) as a common ground and they admit (cut-free) complete focused proof systems. The difference between the two logics relies on the way modalities are handled.  ...  We note that ↓ and at commute freely with all non-hybrid connectives [DC14] .  ... 
arXiv:1608.08779v2 fatcat:pqnhffzanfamlgwix2ncdmdroa

Page 9525 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
the identity element), addressing the framework of constraint logic programming (CLP) languages.  ...  In proposition 3.2 the author proves when a Y*4 sentence is satisfiable with respect to NW + AFA, and in proposition 4.1 the author also establishes when a V/A V* sentence is satisfiable with respect to  ... 

Witnessing matrix identities and proof complexity

Fu Li, Iddo Tzameret
2018 International journal of algebra and computation  
A matrix identity of d × d matrices over a field F is a non-commutative polynomial f (x 1 , . . . , x n ) over F, such that f vanishes on every d × d matrix assignment to its variables.  ...  Finally, we present several concrete open problems about non-commutative algebraic circuits and speed-ups in proof complexity, whose solution would establish stronger size lower bounds on PI proofs of  ...  We are also greatly indebted to Vesselin Drensky for his help with the bibliography and providing us with his monograph [10] .  ... 
doi:10.1142/s021819671850011x fatcat:aevrlmha6bd5djqugsialn55fu

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
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