The Undecidability of Second Order Linear Logic Without Exponentials

Recently, Lincoln, Scedrov and Shankar showed that the multi-plicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicative-additive fragment, but it does not work in the classical case, because second order classical logic is decidable. Here we show that the multiplicative-additive fragment of second order classical linear logic is also undecidable, using an encoding of two-counter machines originally due to Kanovich. The faithfulness of this encoding is proved by means of the phase semantics. In this paper, we write LL for the full propositional fragment of linear logic, MLL for the multiplicative fragment, MALL for the multiplicative-additive fragment, and MELL for the multiplicative-exponential fragment. Similarly, we write ILL, IMLL, etc. for the fragments of intuitionistic linear logic, LL2, MLL2, etc. for the second order fragments of linear logic, and ILL2, IMLL2, etc. for the second order fragments of intuitionistic linear logic. If i j are integers, we write i; j] for the set fi; i + 1; : : : ; jg. If A is a formula and n 2 N, we write A n for the multiset consisting of n occurrences of A. If ? is a multiset of formulas A 1 ; : : : ; A n , we write !? for the multiset !A 1