Turing completeness of unrestricted infinite-precision transformers

Determine whether a single infinite-precision transformer is Turing complete for inputs of arbitrary length without imposing assumptions such as positive output confidence, non-tameness restrictions, or conditions on the training distribution and chain-of-thought growth.

Background

The paper proves several negative results for single transformers on unbounded-length inputs, but each result relies on additional conditions. Finite-precision transformers cannot memorize non-converging Turing machines; infinite-precision transformers cannot memorize non-tame Turing machines when a fixed positive confidence margin is required; and, under absolute continuity and a growth restriction on the number of admissible chain-of-thoughts, random training produces a Turing-complete transformer with probability zero.

The authors do not resolve the general case in which an infinite-precision transformer is subject to none of these assumptions. Establishing whether such a transformer can memorize arbitrary Turing machines on inputs of unbounded length would clarify the ultimate computational power of the single-transformer architecture.