Commutative algebras of series

2026-01-27

AI Summary

This study investigates product operations on formal power series that satisfy bilinearity, associativity, and commutativity, along with the decidability of equivalence for their corresponding automata. By introducing the notion of a $P$-product—recursively defined via a polynomial multiplication rule $P$—the work unifies and generalizes classical products such as Hadamard, shuffle, and infiltration. The paper provides the first complete characterization of all $P$-products fulfilling the aforementioned algebraic properties and establishes that equivalence of the associated (potentially infinite-state) $P$-automata is decidable. Leveraging coinductive definitions and formal language theory, the results are formally verified in Agda, thereby confirming both the existence of infinitely many such product structures and the decidability of their automaton equivalence.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Structured Learning

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• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• commutative algebras
• formal power series
• equivalence problem
• P-automata
• product rules

Innovation

• P-product
• commutative algebra
• formal power series
• P-automata
• equivalence problem