Correctness of Linear Logic Proof Structures is NL-Complete

23 pagesInternational audienceWe provide new correctness criteria for all fragments (multiplicative, exponential, additive) of linear logic. We use these criteria for proving that deciding the correctness of a linear logic proof structure is NL-complete

Correctness of Multiplicative (and Exponential) Proof Structures is NL-Complete

15 pagesInternational audienceWe provide a new correctness criterion for unit-free MLL proof structures and MELL proof structures with units. We prove that deciding the correctness of a MLL and of a MELL proof structure is NL-complete. We also prove that deciding the correctness of an intuitionistic multiplicative essential net is NL-complete

Constructing Fully Complete Models of Multiplicative Linear Logic

The multiplicative fragment of Linear Logic is the formal system in this family with the best understood proof theory, and the categorical models which best capture this theory are the fully complete ones. We demonstrate how the Hyland-Tan double glueing construction produces such categories, either with or without units, when applied to any of a large family of degenerate models. This process explains as special cases a number of such models from the literature. In order to achieve this result, we develop a tensor calculus for compact closed categories with finite biproducts. We show how the combinatorial properties required for a fully complete model are obtained by this glueing construction adding to the structure already available from the original category.Comment: 72 pages. An extended abstract of this work appeared in the proceedings of LICS 201

Quantum Turing automata

A denotational semantics of quantum Turing machines having a quantum control is defined in the dagger compact closed category of finite dimensional Hilbert spaces. Using the Moore-Penrose generalized inverse, a new additive trace is introduced on the restriction of this category to isometries, which trace is carried over to directed quantum Turing machines as monoidal automata. The Joyal-Street-Verity Int construction is then used to extend this structure to a reversible bidirectional one.Comment: In Proceedings DCM 2012, arXiv:1403.757