Hypercomputation and the Physical Church‐Turing Thesis

A version of the Church‐Turing Thesis states that every effectively realizable physical system can be defined by Turing Machines (‘Thesis P’); in this formulation the Thesis appears an empirical, more than a logico‐mathematical, proposition. We review the main approaches to computation beyond Turing definability (‘hypercomputation’): supertask, non‐well‐founded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be affected. Therefore, Thesis P is not essentially different from the standard Church‐Turing Thesis. 1Introduction 2Computability and incomputability 3The physical interpretation of the Church‐Turing Thesis 4Supertasks and infinite computation 5Computation on non‐well‐founded domains 6Analog computation 7Quantum computation 8Retrocausal computation 9Conclusions