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Subexponentials in non-commutative linear logic

Published online by Cambridge University Press:  02 May 2018

MAX KANOVICH ,
STEPAN KUZNETSOV ,
VIVEK NIGAM  and
ANDRE SCEDROV
MAX KANOVICH
Affiliation:
National Research University Higher School of Economics, Moscow, Russia Email: m.kanovich@qmul.ac.uk
STEPAN KUZNETSOV
Affiliation:
Steklov Mathematical Institute of RAS, Moscow, Russia Email: skuzn@inbox.ru
VIVEK NIGAM
Affiliation:
Federal University of Paraíba, João Pessoa, Brazil Email: vivek.nigam@gmail.com Fortiss GmbH, Munich, Germany
ANDRE SCEDROV
Affiliation:
National Research University Higher School of Economics, Moscow, Russia Email: m.kanovich@qmul.ac.uk University of Pennsylvania, Philadelphia, U.S.A. E-mail: scedrov@math.upenn.edu
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Abstract

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.

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