Geometrical semantics for linear logic (multiplicative fragment)

Linear logic was described by Girard as a logic of dynamic interactions. On the other hand, Girard suggested an analogy between LL and quantum theory. Following these two intuitions we give an interpretation of linear logic in the language, which is common for both dynamical systems and quantization. Thus, we propose a denotational semantics for multiplicative linear logic using the language of symplectic geometry.We construct a category of coherent phase spaces and show that this category provides a model for MLL. A coherent phase space is a pair: a symplectic manifold and a distinguished field of contact cones on this manifold. The category of coherent phase spaces is a refinement of the symplectic “category” introduced by Weinstein. A morphism between two coherent phase spaces is a Lagrangian submanifold of their product, which is tangent to some distinguished field of contact cones. Thus, we interpret formulas of MLL as fields of contact cones on symplectic manifolds, and proofs as integral submanifolds of corresponding fields.In geometric and asymptotic quantization symplectic manifolds are phase spaces of classical systems, and Lagrangian submanifolds represent asymptotically states of quantized systems. Typically, a Lagrangian submanifold is the best possible localization of a quantum system in the classical phase space, as follows from the Heisenberg uncertainty principle. Lagrangian submanifolds are called sometimes “quantum points”.From this point of view we interpret linear logic proofs as (geometric approximations to) quantum states and formulas as specifications for these states. In particular, the interpretation of linear negation suggests that the dual formulas A and A⊥ stand in the same relationship as the position and momentum observables. These two observables cannot simultaneously have definite values, much like the case of two dual formulas, which cannot simultaneously have proofs.