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

Kanovich, Max I.; Kuznetsov, Stepan L.; Scedrov, Andre

doi:10.1093/logcom/exaa010

Cited by 7 publications

(1 citation statement)

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“…In the view of the above, Lambek's non-emptiness restriction is a desirable feature of a logical system underlying categorial grammars. Unfortunately, as shown in [Kanovich et al 2020], this restriction conflicts with one of the extensions of the Lambek calculus we study in this paper, namely, (sub)exponential modalities (see below)-so we opt for the system without Lambek's restriction. This system is called the Lambek calculus allowing empty antecedents [Lambek 1961] and is a fragment of the broader system !ACT ω , which is defined in the next section.…”

Section: Linguistic Introductionmentioning

confidence: 99%

Infinitary Action Logic with Exponentiation

Kuznetsov¹,

Speranski²

2020

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We introduce infinitary action logic with exponentiation-that is, the multiplicativeadditive Lambek calculus extended with Kleene star and with a family of subexponential modalities, which allows some of the structural rules (contraction, weakening, permutation). The logic is presented in the form of an infinitary sequent calculus. We prove cut elimination and, in the case where at least one subexponential allows non-local contraction, establish exact complexity boundaries in two senses. First, we show that the derivability problem for this logic is Π 1 1 -complete. Second, we show that the closure ordinal of its derivability operator is ω CK 1 .(N * \ N ) / N . The Kleene star, as shown in the next session, is axiomatized by means of an ωrule, which raises algorithmic complexity of the system to very high levels. Morrill and Valentín, however, in order to avoid undecidability, formulate an incomplete set of rules for Kleene star. Historically, Kleene star first appeared in the study of events, or actions within a transition system: in the original paper [Kleene 1956] it was used when describing events in neural networks. If A denotes a class of actions, then A * means actions of class A repeated several (possibly zero) times; [Pratt 1991] proposes action algebras, an extension of Kleene algebras with residuals. Though Pratt's work was independent from Lambek, these residuals actually coincide with Lambek divisions. In the presence of residuals, usage of infinitary systems for axiomatizing Kleene star becomes inevitable, due to complexity reasons (see below). 1 The second family of connectives we use to extend the Lambek calculus is the family of subexponential modalities, or subexponentials for short. Their linguistic motivation, going back to Morrill, is as follows. The Lambek calculus itself has a limited capability of treating relativization, or dependent clauses. For example, "that" in "book that John read" gets syntactic type (CN \ CN ) /(S / N ) (CN stands for "common noun," a noun without article), because the dependent clause "John read" lacks a noun phrase ("John read the book") to become a complete sentence (S). The place where the lacking noun phrase should be placed is called gap: "John read []." In more complicated situations, however, this does not work: in the phrase "book that John read yesterday" the dependent clause "John read yesterday" has a gap in the middle: "John read [] yesterday," and is neither of type S / N , nor N \ S. This syntactic phenomenon is called medial extraction and can be handled by adding a special modality, denoted by !, which allows permutation rules. Now "that" receives syntactic type (CN \ CN ) /(S / !N ), and "John read [] yesterday" is indeed of type S / !N , since by permutation !N reaches its place to fill the gap.There is a more sophisticated phenomenon called parasitic extraction: in the example "paper that John signed without reading" the dependent clause includes two gaps: "John signed [] without reading []" which should both be filled with the same "paper." ...

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Section: Linguistic Introductionmentioning

confidence: 99%