A regular function is given by a deterministic 2way transducer: a deterministic 2way automaton with outputs. Over the word \(w\in \Sigma^* \) such an automaton has an input tape with content \({\vdash} w {\dashv}\) and can move left and right in a read-only fashion (accepting runs have to end on \(\dashv\)). On every transition an output word in \(\Gamma^*\) is written on a right-only output tape. Such a machine realizes a partial function \(f:\Sigma^*\rightarrow\Gamma^*\), defined on words with an accepting run.
Such a function \(f\) is monotone if for any \(u,v\in\Sigma^*\), if \(u\leq v\) then \(f(u)\leq f(v)\), where \(\leq \) denotes the prefix order.
A sequential 2way transducer is a syntactic restriction of deterministic transducers where no transition is allowed on symbol \(\dashv\). Such a machine necessarily realizes a monotone function since the run over word \(u\) is a prefix of the run over word \(uv\). </p>
Question: Can every monotone regular function be realized by a sequential 2way transducer?