Towards a Complexity-through-Realizability Theory

We explain how recent developments in the fields of realizability models for linear logic [Sei14e] – or geometry of interaction – and implicit computational complexity [AS14, AS15] can lead to a new approach of implicit computational complexity. This semantic-based approach should apply uniformly to various computational paradigms, and enable the use of new mathematical methods and tools to attack problem in computational complexity. This paper provides the background, motivations and perspectives of this complexitythrough-realizability theory to be developed, and illustrates it with recent results [Sei15].