Weighted Turing machines over strong bimonoids

Given a strong bimonoid P, we introduce P-valued non-deterministic Turing machines (P-NTM), P-valued non-deterministic Turing machines with classical transition (P-NTMc), P-valued deterministic Turing machines (P-DTM) and P-valued non-deterministic Turing machines with the tape-head no remove (P-NTM^S). We adapt depth-first and width-first methods for defining the weight of acceptance of languages recognized by Turning machines. And we study some basic properties of P-NTM, P-NTMc, P-NTM^S. Moreover, We prove that variants of P-NTM with changing its P-valued initial function have the same power as P-NTM. These conclusions still hold for P-NTMc and P-NTM^S.