On the average complexity of partial derivative transducers

2D regular expressions represent rational relations over two alphabets Σ and Δ. In standard 2D expressions (S2D-RE) the basic terms are generators of Σ⋆×Δ⋆, while in generalised 2D expressions (2D-RE) the basic terms are pairs of (ordinary) regular expressions over one alphabet (1D). In this paper we study the average state complexity of partial derivative standard transducers (TPD) for both S2D-RE and 2D-RE. For S2D-RE we obtain the same asymptotic bounds as for partial derivative automata. For 2D-RE, while in the worst case the number of states of TPD can be O(n2), where n is the size of the expression, asymptotically and on average that value is bounded from above by O(n32). We also show that asymptotically and on average the alphabetic size of a 2D-RE is half of its size. All results are obtained in the framework of analytic combinatorics considering generating functions of parametrised combinatorial classes defined implicitly by algebraic curves. In particular, we generalise the methods developed in previous work to a broad class of analytic functions.