A Logical Framework with Commutative and Non-commutative Subexponentials
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.
We are grateful to Glyn Morrill, Frank Pfenning, and the anonymous referees.
Financial Support: The work of Max Kanovich and Andre Scedrov was supported by the Russian Science Foundation under grant 17-11-01294 and performed at National Research University Higher School of Economics, Moscow, Russia. The work of Stepan Kuznetsov was supported by the Young Russian Mathematics award, by the Program of the Presidium of the Russian Academy of Sciences No. 01 ‘Fundamental Mathematics and Its Applications’ under grant PRAS-18-01, and by the Russian Foundation for Basic Research grant 18-01-00822. The work of Vivek Nigam was supported by CNPq grant number 304193/2015-1. Sections 1, 2, 3, 7 and 8 were contributed jointly and equally by all co-authors; Sect. 4 was contributed by Scedrov and Kanovich. Section 5 was contributed by Nigam. Section 6 was contributed by Kuznetsov.
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