Towards causality formalism of quantum mechanics

Transducers (in particular, Mealy automata) constitute a formal model for causality and discreteness. It will be shown that Hilbert space-based QM formalism can be (to some extend) deduced mathematically from that model. Interestingly, the model leads to finite-dimensional (rather to infinite-dimensional) Hilbert spaces, but of very high dimension about $10^{45}$ (actually of order $\ln 2/\tau$, where $\tau$ is Planck time). The model also gives some evidence that in $p$-adic QM it should be taken $p=2$.