Linear Logic and Sub-polynomial Classes of Complexity
Clément Aubert · 2013
This research in Theoretical Computer Science extends the gateways between Linear Logic and Complexity Theory by introducing two innovative models of computation. It focuses on sub-polynomial classes of complexity: AC and NC --the classes of efficiently parallelizable problems-- and L and NL --the deterministic and non-deterministic classes of problems efficiently solvable with low resources on space. Linear Logic is used through its Proof Net presentation to mimic with efficiency the parallel computation of Boolean Circuits, including but not restricted to their constant-depth variants. In a second moment, we investigate how operators on a von Neumann algebra can be used to model computation, thanks to the method provided by the Geometry of Interaction, a subtle reconstruction of Linear Logic. This characterization of computation in logarithmic space with matrices can naturally be understood as a wander on simple structures using pointers, parsing them without modifying themWe make this intuition formal by introducing Non Deterministic Pointer Machines and relating them to other well-known pointer-like-machines. We obtain by doing so new implicit characterizations of sub-polynomial classes of complexity.