Computational irreducibility and compatibilism: towards a formalization.

If our actions are determined by the laws of nature, can we meaningfully claim to possess free will? Compatibilists argue that the answer is yes, and that free will is compatible with complete determinism. Previously, it has been suggested that the notion of computational irreducibility can shed light on this relation: it implies that there cannot in general be "shortcuts" to the decisions of agents, explaining why deterministic agents often appear to have free will. In this paper, we introduce a variant of computational irreducibility that intends to capture more accurately aspects of actual (as opposed to apparent) free will: computational sourcehood, i.e. the phenomenon that the successful prediction of a process' outputs must typically involve an almost-exact representation of the relevant features of that process, regardless of the time it takes to arrive at the prediction. We conjecture that many processes have this property, and we study different possibilities for how to formalize this conjecture in terms of universal Turing machines. While we are not able to settle the conjecture, we give several results and constructions that shed light on the quest for its correct formulation.