Tag - Word problem

Laura Ciobanu: Group equations, constraints and decidability

8th March, 2024

In this talk I will discuss group equations with non-rational constraints, a topic inspired by the long line of work on word equations with length constraints. Deciding algorithmically whether a word equation has solutions satisfying linear length constraints is a major open question, with deep theoretical and practical implications. I will introduce equations in groups and several kinds of constraints, and show that equations with length, abelian or context-free constraints are decidable in virtually abelian groups (joint with Alex Evetts and Alex Levine). This contrasts the fact that solving equations with abelian constraints is undecidable for non-abelian right-angled Artin groups and hyperbolic groups with ‘large’ abelianisation (joint work with Albert Garreta).

Markus Lohrey: Streaming word problems

8th December, 2023

We are interested in highly efficient algorithms for word problems of groups: the algorithm should read the input word once from left to right symbol by symbol (such algorithms are known as streaming algorithms), spending ideally only constant time for each input letter. Moreover, the space used by the algorithm should be small, e.g. O(log n) if n is the length of the input word. To achieve these goals we need randomization: the algorithm is allowed to make random guesses and at the end it gives a correct answer (is the input word trivial in the underlying group?) with high probability. We show that for a large class of groups such algorithms exist, where in particular the space complexity is bounded by O(log n). These groups are obtained by starting with finitely generated linear groups and closing up under the following operations: finite extensions, graph products, and wreath products where the left factor is f.g. abelian. We also contrast this result with lower bounds. For instance, for Thompson’s group F every randomized streaming algorithm for the word problem for F has space complexity Ω(n) (n is again the length of the input word).