Rules of the form q λ → w q () are -rules, or state-change rules, which substitute state q for state q without producing output, and stay at the current input subtree.
multiple initial states are not needed: we can use a single start state Q i , and instead of each initial state q with starting weight w add the rule Q i True → w q () (where We define the binary derivation relation for xT transducer m on partially transformed terms and derivation histories b is derived from a by application of a rule q λ → w rhs to an unprocessed input subtree a ↓ i which is in state q, replacing it by output given by rhs with variables (q , i ) replaced by the input subtree at relative path i in state q .