# Gotoh Transducer

### Gotoh transducer

A transducer, $a\stackrel{\Delta T}{\to}b$, capable of global alignment with affine gap penalty.

See Transducer Legend for explanations of symbols in this diagram.

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Probabilistic constraints:

• a + b + c = 1,
• d + e + f = 1,
• p + q + r = 1,
• v + w + x = 1.

Typically outgoing transitions from S are the same as those from M:

• a = d;
• b = e;
• c = f.

The D->I transition (weight p) is unnecessary for separable distributions over insertions & deletions, but may be desirable for several purposes:

1. it appears in TKF91;
2. without it, alignments like MDIM have zero likelihood, while those like MIDM do not;
• note that fp = dx required for strict symmetry, but this is NOT the case in TKF91;
1. (speculatively) it may be required to approximate the serial composition of gotoh transducers (see Renormalized Transducer Composition).
• apparent support for this is the observation that it's required for reversible affine gaps in the Gotoh Pair HMM.

The W and X states are equivalent if the condition b/c = e/f = q/r = w/x is satisfied (as is the case in TKF91).

A fuller discussions of the symmetry constraints imposed by reversibility may be found on the Gotoh Pair HMM page.