# The undecidability of second order linear affine logic

## Abstract

Quantifier-free prepositional linear affine logic (i.e. linear logic with weakening) is decidable [Kop, Laf2]. Lafont and Scedrov proved that the multiplicative fragment of *second*-order linear logic is undecidable [LS]. In this paper we show that second order linear affine logic is undecidable as well. At the same time it turns out that even its multiplicative fragment is undecidable. Moreover, we obtain a whide class of undecidabile second order logics which lie between the Lambek calculus (LC) and linear affine logic. The proof is based on an encoding of two-counter Minsky machines in second order linear affine logic. The faithfulness of the encoding is proved by means of the phase semantics.

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