Reflection in Logic

We model self-modification in AI by introducing “tiling” agents whose decision systems will approve the construction of highly similar agents, creating a repeating pattern (including similarity of the offspring’s goals). Constructing a formalism in the most straightforward way produces a Gödelian difficulty, the “Löbian obstacle.” By technical methods we demonstrates the possibility of avoiding this obstacle, but the underlying puzzles of rational coherence are thus only partially addressed. We extend the formalism to partially unknown deterministic environments, and show a very crude extension to probabilistic environments and expected utility; but the problem of finding a fundamental decision criterion for self-modifying probabilistic agents remains open.