eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-08-29
18:1
18:22
10.4230/LIPIcs.CONCUR.2024.18
article
Weighted Basic Parallel Processes and Combinatorial Enumeration
Clemente, Lorenzo
1
https://orcid.org/0000-0003-0578-9103
Department of Mathematics, Mechanics, and Computer Science, University of Warsaw, Poland
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 no complexity upper bound for equivalence was known. 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.
In order to showcase the CDF equivalence algorithm, we show that CDF power series naturally arise from combinatorial enumeration, namely as the exponential generating series of constructible species of structures. Examples of such species include sequences, binary trees, ordered trees, Cayley trees, set partitions, series-parallel graphs, and many others. As a consequence of this connection, we obtain an algorithm to decide multiplicity equivalence of constructible species, decidability of which was not known before.
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol311-concur2024/LIPIcs.CONCUR.2024.18/LIPIcs.CONCUR.2024.18.pdf
weighted automata
combinatorial enumeration
shuffle
algebraic differential equations
process algebra
basic parallel processes
species of structures