Unlocking the Connection Between Computational Complexity and Homotopy Theory 🤖

Examined herein is the possible correspondence between computational complexity classes in computational graphs and higher homotopy classes between computability paths via the application of two methods. The first method is the use of category theory for formalizing a model of (categorified) graphical Turing machines with a forgetful functor to a "computability \[Infinity]-group" (not a groupoid), which expresses computability in a manner that affords even greater abstraction than a Turing machine. The second method is the application of Homotopy Type Theory (HoTT) to the study of the properties of higher homotopy classes between machines (which are themselves simulations executed by higher machines), with the ultimate goal of calculating homotopy types that express invariances of computability (which is our abstract interpretation of the role of complexity classes). We conclude that such methods may make possible the formalization of 'higher complexity classes' and 'higher n-machines' currently unrepresented in theoretical computer science and computational complexity theory. The possibility of formalization as such can be investigated understood through the development of more rigorous proofs regarding homotopy types and homotopy groups, as well as computational experiments that demonstrate correspondences between well-known problems and our theoretical model.