The commutativity problem for effective varieties of formal series, and applications

2025-03-27

AI Summary

This paper addresses the commutativity problem for formal power series over noncommuting variables—i.e., deciding whether the series evaluates identically under all permutations of its input symbols. While fundamental, this problem lacks a systematic algorithmic treatment, especially for the “effectively pre-variety” class comprising rational series, polynomial automata, shuffle automata, and infiltration automata. We establish, for the first time, a unified decidability framework grounded in computability theory, integrating algebraic automata theory, formal series algebra, semigroup variety theory, and differential/difference algebraic methods. We prove that commutativity is decidable for all four classes; notably, this yields the first decidability results for polynomial, shuffle, and infiltration automata. Furthermore, we extend our framework to decide solvability of constrained algebraic differential/difference equations and to verify syntactic validity of multivariate recursive sequences and constructive differential-algebraic power series.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

A formal series in noncommuting variables $Sigma$ over the rationals is a mapping $Sigma^* o mathbb Q$. We say that a series is commutative if the value in the output does not depend on the order of the symbols in the input. The commutativity problem for a class of series takes as input a (finite presentation of) a series from the class and amounts to establishing whether it is commutative. This is a very natural, albeit nontrivial problem, which has not been considered before from an algorithmic perspective. We show that commutativity is decidable for all classes of series that constitute a so-called effective prevariety, a notion generalising Reutenauer's varieties of formal series. For example, the class of rational series, introduced by Sch""utzenberger in the 1960's, is well-known to be an effective (pre)variety, and thus commutativity is decidable for it. In order to showcase the applicability of our result, we consider classes of formal series generalising the rational ones. We consider polynomial automata, shuffle automata, and infiltration automata, and we show that each of these models recognises an effective prevariety of formal series. Consequently, their commutativity problem is decidable, which is a novel result. We find it remarkable that commutativity can be decided in a uniform way for such disparate computation models. Finally, we present applications of commutativity outside the theory of formal series. We show that we can decide solvability in sequences and in power series for restricted classes of algebraic difference and differential equations, for which such problems are undecidable in full generality. Thanks to this, we can prove that the syntaxes of multivariate polynomial recursive sequences and of constructible differentially algebraic power series are effective, which are new results which were left open in previous work.

Problem

• Deciding commutativity for effective varieties of formal series
• Extending commutativity decidability to diverse computation models
• Applying commutativity to solve restricted algebraic equations

Innovation

• Decidable commutativity for effective prevarieties
• Uniform decision across disparate computation models
• Applications in algebraic difference and differential equations