# Serial Composition Of Gotoh Transducers

### Serial composition of two Gotoh transducers

The composite transducer, , where is the Gotoh Transducer. (See Transducer Legend for key.)

The diagrams on this page are admittedly pretty horrendous. My motivation for looking at these is to see if there is some way to approximate the following transducer using the simpler Gotoh Transducer. This would essentially correspond to a "renormalization" of the transducer's parameter space, which in turn might illuminate a classic problem in statistical alignment: the derivation of convex gap penalties from an evolutionary indel model with affine gaps. See Renormalized Transducer Composition for more details.

We can also consider the jointly normalized HMM that is the composition of the Singlet Transducer with the above composed transducer (c.f. Gotoh Pair HMM). Notationally this is , where is the Singlet Transducer.

This diagram assumes S/M symmetry in the Gotoh Transducer and S/I symmetry in the Singlet Transducer (so e.g. "SSI" and "IMI" are the same state).

-- Ian Holmes - 28 Feb 2007