Related papers: Real Computational Universality: The Word Problem …
We introduce two new types of Dehn functions of group presentations which seem more suitable (than the standard Dehn function) for infinite group presentations and prove the fundamental equivalence between the solvability of the word…
We give lower bounds on the complexity of the word problem of certain non-solvable groups: for a large class of non-solvable infinite groups, including in particular free groups, Grigorchuk's group and Thompson's groups, we prove that their…
The worst-case complexity of group-theoretic algorithms has been studied for a long time. Generic-case complexity, or complexity on random inputs, was introduced and studied relatively recently. In this paper, we address the average-case…
In this work we introduce a new succinct variant of the word problem in a finitely generated group $G$, which we call the power word problem: the input word may contain powers $p^x$, where $p$ is a finite word over generators of $G$ and $x$…
Models of a generalized nondeterminism are defined by limitations on nonde- terministic behavior of a computing device. A regular realizability problem is a problem of verifying existence of a special sort word in a regular language. These…
In this paper, we introduce a family of residually finite groups that helps us to systematically study the residual finiteness growth function (RFG) from various perspectives. First, by strengthening results of Bou-Rabee and Seward and also…
In this paper we explore the connections between the class of Visibly Pushdown Languages ($\mathbf{VPL}$) and the natural sets of words one can associate to a finitely generated group. We show that the word problem of a finitely generated…
We show that every countable group H with solvable word problem (=computable group) can be subnormally embedded into a 2-generated group G which also has solvable word problem. Moreover, the membership problem for H < G is also solvable. We…
Modify the Blum-Shub-Smale model of computation replacing the permitted computational primitives (the real field operations) with any finite set $B$ of real functions semialgebraic over the rationals. Consider the class of boolean decision…
We investigate partial Equality and Word Problems for finitely generated groups. After introducing Upper Banach (UB) density on free groups, we prove that solvability of the Equality Problem on squares of UB-generic sets implies solvability…
We consider various decision problems for automatic semigroups, which involve the provision of an automatic structure as part of the problem instance. With mild restrictions on the automatic structure, which seem to be necessary to make the…
Infinite words, also known as streams, hold significant interest in computer science and mathematics, raising the natural question of how their complexity should be measured. We introduce cellular automaton reducibility as a measure of…
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…
Separability for groups refers to the question which subsets of a group can be detected in its finite quotients. Classically, separability is studied in terms of which classes have a certain separability property, and this question is…
Term algebras are important objects in computer science and are correspondingly well-studied. A natural generalization is to quotient these algebras by finitely many ground term equations, obtaining what we call almost free algebras. One of…
The absolute Galois group Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ of the field $\mathbb{Q}$ of rational numbers can be presented as a highly computable object, under the notion of type-2 Turing computation. We formalize such a presentation…
We study the complexity of computation in finitely generated free left, right and two-sided adequate semigroups and monoids. We present polynomial time (quadratic in the RAM model of computation) algorithms to solve the word problem and…
For the additive real BSS machines using only constants 0 and 1 and order tests we consider the corresponding Turing reducibility and characterize some semi-decidable decision problems over the reals. In order to refine, step-by-step, a…
Ramsey theory for words over a finite alphabet was unified in the work of Carlson and Furstenberg-Katznelson. Carlson, in the same work, outlined a method to extend the theory for words over an infinite alphabet, but subject to a fixed…
We prove that the word problem for the infinite cyclic group is not EDT0L, and obtain as a corollary that a finitely generated group with EDT0L word problem must be torsion. In addition, we show that the property of having an EDT0L word…