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While context-free grammars are characterized by a simple proof-theoreticgrammatical formalism namely categorial grammar and its logic the Lambekcalculus, no such characterizations were known for tree-adjoining grammars, andeven for any mildly context-sensitive languages classes in the last forty yearsdespite some efforts. We settle this problem in this paper. On the basis of theexisting fragment of the Lambek-Grishin calculus which captures tree-adjoininglanguages, we present a logic called HLG: a proof-theoretic characterization oftree-adjoining languages based on the Lambek-Grishin calculus restricted toHyperedge-replacement grammar with rank two studied by Moot. HLG is defined indisplay calculus with cut-admissibility. Several new techniques are introducedfor the proofs, such as purely structural connectives, usefulness, and agraph-theoretic argument on proof nets for HLG.

The Logic for a Mildly Context-Sensitive Fragment of the Lambek-Grishin Calculus