Subexponentials in non-commutative linear logic

Abstract

Linear logical frameworks with subexponentials have been used for the specification of among other systems, proof systems, concurrent programming languages and linear authorization 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 organized as sets and multisets of formulae not being possible to organize 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 local and non-local contraction rules. We also obtain some new undecidability and decidability results on non-commutative linear logic with subexponentials.

Reconciling Lambek’s restriction, cut-elimination and substitution in the presence of exponential modalities

“…These modalities are called subexponentials. A systematic study of subexponentials in the non-commutative case, under the umbrella of the Lambek calculus and cyclic linear logic, is performed in [26]. We also plan to study systems with (sub)exponentials, built on top of light [27] and soft [28] linear logic.…”

Undecidability of the Lambek Calculus with Subexponential and Bracket Modalities

“…Namely, if we contract the formula A being cut, then after propagation we get two cut applications, one under another. For the lower cut, we fail to maintain the decrease of induction parameters, see Kanovich et al (2019a). The standard strategy, going back to Gentzen (1935) and applied to linear logic with exponentials by Girard (1987) and Lincoln et al (1992), replaces the cut rule with a more general rule called mix.…”

“…Recall the well-known proof of undecidability for !L * , the Lambek calculus (without brackets) enriched with a full-power exponential modality (Lincoln et al, 1992;Kanazawa, 1999;Kanovich et al, 2019a), via encoding of semi-Thue systems.…”

“…In comparison with a series of our papers on the Lambek calculus and non-commutative linear logic with subexponential modalities (Kanovich et al, 2016a(Kanovich et al, , 2020(Kanovich et al, , 2016b(Kanovich et al, , 2019a(Kanovich et al, , 2018, the principal difference of this paper is the presence of brackets and bracket modalities. Contraction rules used by Morrill in the bracketed calculi essentially interact with brackets and become disfunctional in the bracket-free fragment.…”

“…Medial extraction can be handled by adding a subexponential modality (cf. Kanovich et al (2019a)), denoted by !, which allows permutation. In general, the Lambek calculus is non-commutative, thus, the order of the words in a sentence matters.…”