  We study weighted basic parallel processes (WBPP), a nonlinear recursive
generalisation of weighted finite automata inspired from process algebra and
Petri net theory. Our main result is an algorithm of 2-EXPSPACE complexity for
the WBPP equivalence problem. While (unweighted) BPP language equivalence is
undecidable, we can use this algorithm to decide multiplicity equivalence of
BPP and language equivalence of unambiguous BPP, with the same complexity.
These are long-standing open problems for the related model of weighted
context-free grammars. Our second contribution is a connection between WBPP,
power series solutions of systems of polynomial differential equations, and
combinatorial enumeration. To this end we consider constructible differentially
finite power series (CDF), a class of multivariate differentially algebraic
series introduced by Bergeron and Reutenauer in order to provide a
combinatorial interpretation to differential equations. CDF series generalise
rational, algebraic, and a large class of D-finite (holonomic) series, for
which decidability of equivalence was an open problem. We show that CDF series
correspond to commutative WBPP series. As a consequence of our result on WBPP
and commutativity, we show that equivalence of CDF power series can be decided
with 2-EXPTIME complexity. The complexity analysis is based on effective bounds
from algebraic geometry, namely on the length of chains of polynomial ideals
constructed by repeated application of finitely many, not necessarily commuting
derivations of a multivariate polynomial ring. This is obtained by generalising
a result of Novikov and Yakovenko in the case of a single derivation, which is
noteworthy since generic bounds on ideal chains are non-primitive recursive in
general. On the way, we develop the theory of \WBPP~series and \CDF~power
series, exposing several of their appealing properties.
