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Effective Algorithms for Solving Group Equations with Constraints in Virtually Abelian Groups


Core Concepts
In virtually abelian groups, it is effectively decidable whether a finite system of group equations has solutions satisfying various constraints, including linear length, abelianization, context-free, and lexicographic order constraints. The key is that these seemingly non-rational constraints can be effectively translated into rational sets in virtually abelian groups.
Abstract
The paper studies the satisfiability and solutions of group equations when combinatorial, algebraic, and language-theoretic constraints are imposed on the solutions in virtually abelian groups. The main results are: Theorem 1.1: In any finitely generated virtually abelian group, it is effectively decidable whether a finite system of equations with the following kinds of constraints has solutions: Linear length constraints (with respect to any weighted word metric) Abelianization constraints Context-free constraints Lexicographic order constraints Theorem 1.2: In any finitely generated virtually abelian group, there is an effective way to construct a rational set from the above constraints. Corollary 1.3: The set of solutions to a finite system of equations with the above constraints can be expressed as an EDT0L language in at least two different ways. Proposition 1.4: The weighted growth series (with respect to any finite generating set) of a finitely generated virtually abelian group can be computed explicitly. This was previously known to be rational, but the proof was non-constructive. The paper combines group theory, theoretical computer science, and combinatorics to study these equation solving problems in virtually abelian groups. The key insight is that seemingly non-rational constraints like length, abelianization, and context-free constraints turn out to be rational in this setting, allowing for decidability results.
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Deeper Inquiries

How do the decidability results for equation solving with constraints in virtually abelian groups compare to the undecidability results for similar problems in other classes of groups, such as non-abelian free groups or hyperbolic groups

The decidability results for equation solving with constraints in virtually abelian groups contrast with the undecidability results for similar problems in non-abelian free groups or hyperbolic groups. In non-abelian free groups and free monoids, constraints like length, abelianization, and context-free constraints are typically non-rational. The undecidability of the Diophantine Problem with abelianization constraints in non-abelian right-angled Artin groups or hyperbolic groups with abelianization rank ≥2 illustrates the complexity of solving equations with constraints in these groups. On the other hand, in virtually abelian groups, these constraints are shown to be rational, leading to decidability in solving equations with constraints. This difference highlights the unique properties of virtually abelian groups that make them more amenable to effective equation solving with constraints compared to other classes of groups.

What are some potential applications of the ability to effectively solve group equations with constraints in virtually abelian groups, beyond the string solving context mentioned in the introduction

The ability to effectively solve group equations with constraints in virtually abelian groups has various potential applications beyond the context of string solvers for software engineering and security analysis. One significant application is in cryptography, where group equations with constraints play a crucial role in designing secure cryptographic protocols. By being able to solve these equations effectively, researchers and practitioners can develop more robust and efficient cryptographic systems. Additionally, in computational biology, the ability to solve group equations with constraints can aid in analyzing biological sequences and structures, leading to advancements in genomics and bioinformatics. Furthermore, in optimization problems and artificial intelligence, the techniques developed for solving equations in virtually abelian groups can be utilized to enhance algorithmic efficiency and decision-making processes.

Can the techniques developed in this paper be extended to other classes of groups beyond virtually abelian groups, where similar constraints may also turn out to be rational

The techniques developed in this paper for solving group equations with constraints in virtually abelian groups may have potential extensions to other classes of groups. While the constraints considered in this study are shown to be rational in virtually abelian groups, similar constraints in other classes of groups may also exhibit rationality under certain conditions. By adapting the methods and algorithms used for virtually abelian groups to other group structures, it is possible to explore the decidability of equation solving with constraints in a broader range of group settings. This extension could lead to new insights into the computational complexity of group equations with constraints and their applications across diverse mathematical and computational domains.
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