On the Undecidability of Artificial Intelligence Alignment: Machines that Halt

The inner alignment problem, which asserts whether an arbitrary artificial intelligence (AI) model satisfices a non-trivial alignment function of its outputs given its inputs, is undecidable. This is rigorously proved by Rice's theorem, which is also equivalent to a reduction to Turing's Halting Problem, whose proof sketch is presented in this work. Nevertheless, there is an enumerable set of provenly aligned AIs that are constructed from a finite set of provenly aligned operations. Therefore, the alignment should be a guaranteed property from the AI architecture rather than a characteristic imposed post-hoc on an arbitrary AI model. Furthermore, while the outer alignment problem is the definition of a judge function that captures human values and preferences, such a function must also impose a halting constraint that guarantees that the AI model always reaches a terminal state in finite execution steps. This work presents examples and models that illustrate this constraint and argues for an intrinsically hard-aligned approach to AI systems architectures that ensures halting. Ensuring the decidability of the halting property of autonomous systems also ensures the decidability of other non-trivial properties, especially their alignment, so that verification becomes computationally possible, though not necessarily tractable. Therefore the halting constraint should be imposed over a tractable state space with few enough final states and loops that a group of human specialists can carefully analyse each of them. The requirement of always reaching a terminal state can be expressed as a utility function that penalizes the model for being in operation after the desired amount of time for halting, with the penalization higher than the highest reward the model could ever receive. Guaranteeing decidability of halting does not guarantee alignment; it only makes the alignment problem decidable, allowing further scrutiny of the model.