Commutative algebras of series
Lorenzo Clemente
TL;DR
This work studies a broad, coinductive family of binary products on formal power series in noncommuting indeterminates, defined by product rules $P$, and the resulting automata-theoretic classes. It provides a complete equational characterization: a BAC $P$-product arises exactly when $P$ is a special, simple rule with parameters satisfying $\alpha\gamma=\beta(\beta-1)$, yielding explicit instances such as Hadamard, shuffle, and infiltration. The authors prove that the corresponding $P$-automata form a robust, finitely-state-equivalent framework whose semantics is a homomorphism of commutative $\mathbb{Q}$-algebras, and they show decidability of the equivalence problem for finite-variable polynomial $P$-automata via Hilbert’s basis theorem, unifying and extending known results. The work also places the theory in a coalgebraic GSOS setting and formalizes the key results in Agda, highlighting both theoretical and practical implications for deciding equivalence in infinite-state weighted automata models.
Abstract
We consider a large family of product operations of formal power series in noncommuting indeterminates, the classes of automata they define, and the respective equivalence problems. A $P$-product of series is defined coinductively by a polynomial product rule $P$, which gives a recursive recipe to build the product of two series as a function of the series themselves and their derivatives. The first main result of the paper is a complete and decidable characterisation of all product rules $P$ giving rise to $P$-products which are bilinear, associative, and commutative. The characterisation shows that there are infinitely many such products, and in particular it applies to the notable Hadamard, shuffle, and infiltration products from the literature. Every $P$-product gives rise to the class of $P$-automata, an infinite-state model where states are terms. The second main result of the paper is that the equivalence problem for $P$-automata is decidable for $P$-products satisfying our characterisation. This explains, subsumes, and extends known results from the literature on the Hadamard, shuffle, and infiltration automata. We have formalised most results in the proof assistant Agda.