Non-associative, Non-commutative Multi-modal Linear Logic

Abstract

Adding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so to allow for commutativity as well. In this work, we close the cycle by considering associativity. We show that the resulting system ($$\mathsf {acLL}_\varSigma $$ acLL Σ ) admits the (multi)cut rule, and we prove two undecidability results for fragments/variations of $$\mathsf {acLL}_\varSigma $$ acLL Σ .

Explorations in Subexponential Non-associative Non-commutative Linear Logic

“…In [7], we went one step further and presented a non-commutative, non-associative linear logic based system with the possibility of assuming associativity…”

“…This is equivalent to flipping the structure around to put a formula into spot, in the same way we would select a key in a keychain. This works due to the following definitions and technical lemmas (the proofs are in the accompanying technical report [8]). Definition 3.…”

“…In [7] we extended the work in [19] by proposing acLL Σ , a non-associative analogue of the previous system. In the present work, we introduce the cut-admissible calculus CacLL Σ for classical nonassociative non-commutative multi-modal linear logic and show that this is conservative over acLL Σ .…”

“…Extensions of linear logic/Lambek calculi with subexponentials are considered in [19,20,7]. In [19] a cut-admissible framework for subexponentials in a non-commutative intuitionistic linear logic was introduced.…”

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In [7] we introduced a non-associative non-commutative linear logic extended by multimodalities, called subexponentials, licensing local application of structural rules. Here, we further explore this system, considering its classical one-sided multi-succedent classical version, following the exponentialfree calculi of [9] and [14], where the intuitionistic calculus is shown to embed faithfully into the classical fragment.

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