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Specifying Proof Systems in Linear Logic with Subexponentials

Vivek Nigam, Elaine Pimentel, Giselle Reis
2011 Electronical Notes in Theoretical Computer Science  
This work shows that one can encode a wider range of proof systems by using focused linear logic with subexponentials.  ...  In the past years, linear logic has been successfully used as a general logical framework for encoding proof systems.  ...  This subexponential is used to store linear logic theories specifying proof systems.  ... 
doi:10.1016/j.entcs.2011.03.009 fatcat:wyzvjdecyvaebolnebwgliih6i

An extended framework for specifying and reasoning about proof systems

Vivek Nigam, Elaine Pimentel, Giselle Reis
2014 Journal of Logic and Computation  
First, in Section 4, we demonstrate how to declaratively specify proof systems with more involved structural and logical inferences rules using linear logic theories with subexponentials.  ...  Section 2 introduces the proof system for linear logic with subexponentials, called SELLF, which is the basis of the proposed logical framework.  ...  We are also grateful to Kaustuv Chaudhuri for pointing out an error in a previous version of this paper.  ... 
doi:10.1093/logcom/exu029 fatcat:uontgaw6gbfh3mcyxzpgdc7jqu

Quati: An Automated Tool for Proving Permutation Lemmas [chapter]

Vivek Nigam, Giselle Reis, Leonardo Lima
2014 Lecture Notes in Computer Science  
Given a proof system specification in the form of a theory in linear logic with subexponentials, Quati outputs in L A T E X the permutation transformations for which it was able to prove correctness and  ...  As illustrated in this paper, Quati's output is very similar to proof derivation figures one would normally find in a proof theory book.  ...  ) to derivations in linear logic with subexponentials.  ... 
doi:10.1007/978-3-319-08587-6_18 fatcat:ym2kxrefqrgylh7agegchqrgvy

(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.  ...  The examples we have considered so far are very simple ones-coming with completely formal (interactive) proofs in Coq.  ...  For instance, subexponentials can be used to represent contexts of proof systems [16] ; to specify systems with temporal, epistemic and spatial modalities [18] and to specify and verify biological systems  ... 
doi:10.1007/978-3-319-45177-0_1 fatcat:imfgsz4r4jdfdi5et7jkrw7gwy

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.  ...  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

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.  ...  Subexponentials in Linear Logic Linear logic with subexponentials 4 (SELL) shares with LL all its connectives except the exponentials: instead of having a single pair of exponentials ! and ?  ... 
doi:10.1016/j.entcs.2017.04.007 fatcat:ov52p5wcmjcurlpnlcuznebccm

Algorithmic specifications in linear logic with subexponentials

Vivek Nigam, Dale Miller
2009 Proceedings of the 11th ACM SIGPLAN conference on Principles and practice of declarative programming - PPDP '09  
In addition, we propose a new logical connective that allows for the creation of new subexponentials, thereby further augmenting the algorithmic expressiveness of logic.  ...  The linear logic exponentials !, ? are not canonical: one can add to linear logic other such operators, say ! l , ?  ...  Conclusions In this paper, we show that a wide range of algorithms can be specified in the linear logic system with subexponentials called SELL.  ... 
doi:10.1145/1599410.1599427 dblp:conf/ppdp/NigamM09 fatcat:2xzv7s5wlzh4polnubupd6divq

A framework for linear authorization logics

Vivek Nigam
2014 Theoretical Computer Science  
Linear authorization logics (LALs) are logics based on linear logic that can be used for modeling effect-based authentication policies.  ...  LALs have been used in the context of the Proof-Carrying Authorization framework, where formal proofs must be constructed in order for a principal to gain access to some resource elsewhere.  ...  We now review the proof system for intuitionistic linear logic with subexponentials in Section 2.1 and the focused proof system for this system, called SELLF, in Section 2.2.  ... 
doi:10.1016/j.tcs.2014.02.018 fatcat:22pxpnai5vflfhhq6sv6wmwvbi

A Proof Theoretic Study of Soft Concurrent Constraint Programming

ELAINE PIMENTEL, CARLOS OLARTE, VIVEK NIGAM
2014 Theory and Practice of Logic Programming  
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  ...  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 Section 3, we will specify such systems as formulas in intuitionistic linear logic (ILL (Girard 1987) ).  ... 
doi:10.1017/s147106841400026x fatcat:z5cdsnb3hbd6jmmjqzrfhyneai

Facilitating Meta-Theory Reasoning (Invited Paper)

Giselle Reis
2021 Electronic Proceedings in Theoretical Computer Science  
This is called the meta-theory of a proof system. Meta-theory proofs typically involve many cases on structures with lots of symbols.  ...  For this to be possible, proof systems must be designed as a set of rules such that proofs using those rules are correct by construction.  ...  It turns out that the linear logic framework cannot capture so easily sequent calculus systems with rules that have side conditions in the context, such as: Γ A Γ, Γ A, ∆ R To be able to specify rules  ... 
doi:10.4204/eptcs.337.1 fatcat:dobvkljpbraxfgyc5qtbdwoufq

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

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.  ...  L] Hybrid connectives Figure 5 : Focused linear logic system with (quantified) subexponentials.  ... 
arXiv:1608.08779v2 fatcat:pqnhffzanfamlgwix2ncdmdroa

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.  ...  logic system with (quantified) subexponentials.  ... 
doi:10.1017/s0960129518000439 fatcat:ovn75fwk4rfq3pmuybe4luwrtm

On the Complexity of Linear Authorization Logics

Vivek Nigam
2012 2012 27th Annual IEEE Symposium on Logic in Computer Science  
Linear authorization logics (LAL) are logics based on linear logic that can be used for modeling effect-based authentication policies.  ...  LAL has been used in the context of the Proof-Carrying Authorization framework, where formal proofs are constructed in order for a principal to gain access to some resource elsewhere.  ...  We now review the proof system for intuitionistic linear logic with subexponentials.  ... 
doi:10.1109/lics.2012.61 dblp:conf/lics/Nigam12 fatcat:mpujo73eovhhjaosobmvkr273a

A proof theoretic view of spatial and temporal dependencies in biochemical systems

C. Olarte, D. Chiarugi, M. Falaschi, D. Hermith
2016 Theoretical Computer Science  
To this aim, we consider a fragment of intuitionistic linear logic with subexponentials (SELL).  ...  In this paper we propose a formalism for specifying and verifying properties of biochemical systems that combines, coherently, temporal and spatial modalities.  ...  Linear logic with subexponentials In this section we recall the proof theory of intuitionistic linear logic (ILL) [6] with subexponentials [7, 8] .  ... 
doi:10.1016/j.tcs.2016.03.029 fatcat:t3t2p7qy5fgunda44sog6slufi
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