IJCAR 2018: Automated Reasoning pp 228-245

# A Logical Framework with Commutative and Non-commutative Subexponentials

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10900)

## Abstract

Logical frameworks allow the specification of deductive systems using the same logical machinery. Linear logical frameworks have been successfully used for the specification of a number of computational, logics and proof systems. Its success relies on the fact that formulas can be distinguished as linear, which behave intuitively as resources, and unbounded, which behave intuitionistically. Commutative subexponentials enhance the expressiveness of linear logic frameworks by allowing the distinction of multiple contexts. These contexts may behave as multisets of formulas or sets of formulas. Motivated by applications in distributed systems and in type-logical grammar, we propose a linear logical framework containing both commutative and non-commutative subexponentials. Non-commutative subexponentials can be used to specify contexts which behave as lists, not multisets, of formulas. In addition, motivated by our applications in type-logical grammar, where the weakenening rule is disallowed, we investigate the proof theory of formulas that can only contract, but not weaken. In fact, our contraction is non-local. We demonstrate that under some conditions such formulas may be treated as unbounded formulas, which behave intuitionistically.

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© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• Max Kanovich
• 1
• Stepan Kuznetsov
• 1
• 2
• Vivek Nigam
• 3
• 4
Email author
• Andre Scedrov
• 1
• 5
1. 1.National Research University Higher School of EconomicsMoscowRussia
2. 2.Steklov Mathematical Institute of RASMoscowRussia
3. 3.Federal University of ParaíbaJoão PessoaBrazil
4. 4.fortissMunichGermany