Let’s define a pushdown transducer as a 9-tuple $V = (A, B, S, Q_A, Q_S, \phi, \psi, \chi, q_0)$, where $A$ is the finite input alphabet, $B$ is the finite output alphabet, $S$ is the finite stack alphabet, $Q_A$ are the finite set of read-from-input states, $Q_S$ is the finite set of read-from-stack states, $\phi: (Q_A \times A) \cup (Q_S \times (S \cup \{ \epsilon \})) \to (Q_A \cup Q_S)$ (where $\epsilon \not\in S$) - is the state transition function, $\psi: (Q_A \times A) \cup (Q_S \times (S \cup \{ \epsilon \})) \to S^*$ (where $\epsilon \not\in S$) is stack transition function, $\chi: (Q_A \times A) \cup (Q_S \times (S \cup \{ \epsilon \})) \to B^*$ (where $\epsilon \not\in S$) is output function, $q_0 \in Q_A$ is the initial state. Now, let’s define the total transducer function of $V$ of $V$ as $f_V: A^* \to (Q_A \cup Q_S) \cup S^* \to B^*$ defined by recurrence relation
$$f_V(\Lambda, q, \sigma) = \Lambda$$
$$f_V(a\alpha, q, \Lambda) = \begin{cases} \chi(q, a) f_V(\alpha, \phi(q, a), \psi(q, a)) & \quad q \in Q_A \\ \chi(q, \epsilon) f_V(\alpha, \phi(q, \epsilon), \psi(q, \epsilon)) & \quad q \in Q_S \end{cases}$$
$$f_V(a\alpha, q, \sigma s) = \begin{cases} \chi(q, a) f_V(\alpha, \phi(q, a), \sigma s \psi(q, a)) & \quad q \in Q_A \\ \chi(q, s) f_V(\alpha, \phi(q, s), \sigma \psi(q, s)) & \quad q \in Q_S \end{cases}$$
and limited transduction function as $t_V(A^*) = f_V(A^*, q_0, \Lambda)$.
We call a deterministic function $A^* \to B^*$ a pushdown transduction iff it is a limited transduction function of some pushdown transducer.
Pushdown transducers are a more powerful computation model than finite state transducers, but less powerful than Turing machines.
Is the set of pushdown transductions closed under composition?
I know, how to prove that composition of a pushdown transduction and a regular transduction is a pushdown transduction, and that composition of a regular transduction and a regular transduction is a regular transduction, by explicitly constructing corresponding automata. But attempting to do this for two irregular pushdown transductions in the same straightforward way we get a transducer with two stacks (which is equivalent to Turing machine), which gives us nothing we have not known before…
