An Algebra of Transformers

On the theoretic analyses on transforms, they mainly focus on the computability analysis, such the Turing-completeness of unbounded precision transformers, the advantage of parallelizability of unbounded precision transformers and the limitations of bounded precision transformers. This paper give transformers an algebra for reasoning, which relies heavily on the functional decomposition results of transformers. This algebra uses functional decompositions as atomic actions, includes sequential composition and parallel composition, recursion for feedback and abstraction for hidden layers. As we know, it is the first attempt to model transformers logically and can be used to verify the correctness of transformer based applications.