Related papers: A variety with solvable, but not uniformly solvabl…
For two given $\omega$-terms $\alpha$ and $\beta$, the word problem for $\omega$-terms over a variety $\boldsymbol{\mathrm{V}}$ asks whether $\alpha=\beta$ in all monoids in $\boldsymbol{\mathrm{V}}$. We show that the word problem for…
A special inverse monoid is one defined by a presentation where all the defining relations have the form $r = 1$. By a result of Ivanov Margolis and Meakin the word problem for such an inverse monoid can often be reduced to the word problem…
The square-free word problem relative to a system of two defining relations is decidable.
Mateescu, Salomaa, and Yu asked: is it decidable whether a given subword history assumes only non-negative values for all words over a given alphabet. In this paper, we solve this open problem by proving that this problem is undecidable…
For each Turing machine T, we construct an algebra A'(T) such that the variety generated by A'(T) has definable principal subcongruences if and only if T halts, thus proving that the property of having definable principal subcongruences is…
We explore a natural class of semigroups that have word problem decidable by finite state automata. Among the main results are invariance of this property under change of generators, invariance under basic algebraic constructions and…
We consider questions related to the structure of infinite words (over an integer alphabet) with bounded additive complexity, i.e., words with the property that the number of distinct sums exhibited by factors of the same length is bounded…
We construct a class of finitely presented groups where the isomorphism problem is solvable but the commensurability problem is unsolvable. Conversely, we construct a class of finitely presented groups within which the commensurability…
The additive square problem is a relatively famous open problem in the area of combinatorics on words: Does there exist an infinite word over a finite alphabet, such that no two consecutive blocks of the same length have the same sum? In…
We show that the problem `whether a finite set of regular-linear axioms defines a rigid theory' is undecidable.
A group-word $w$ is concise in a class of groups $\mathcal X$ if and only if the verbal subgroup $w(G)$ is finite whenever $w$ takes only finitely many values in a group $G\in \mathcal X$. It is a long-standing open problem whether every…
Let $\mathbf{A}$ be a finite algebra generating a finitely decidable variety and having nontrivial strongly solvable radical $\tau$. We provide an improved bound on the number of variables in which a term can be sensitive to changes within…
In this paper we address the decision problem for a fragment of set theory with restricted quantification which extends the language studied in [4] with pair related quantifiers and constructs, in view of possible applications in the field…
We show that for any $i > 0$, it is decidable, given a regular language, whether it is expressible in the $\Sigma_i[<]$ fragment of first-order logic FO[<]. This settles a question open since 1971. Our main technical result relies on the…
The classical decision problem, as it is understood today, is the quest for a delineation between the decidable and the undecidable parts of first-order logic based on elegant syntactic criteria. In this paper, we treat the concept of…
In this paper we explore fundamental concepts in computational complexity theory and the boundaries of algorithmic decidability. We examine the relationship between complexity classes \textbf{P} and \textbf{NP}, where $L \in \textbf{P}$…
When modeling an application of practical relevance as an instance of a combinatorial problem X, we are often interested not merely in finding one optimal solution for that instance, but in finding a sufficiently diverse collection of good…
We give an example of a finitely based locally finite variety which has uncountably many term clones. (Such varieties were known before.)
We solve the word problem for free double categories without equations between generators by translating it to the word problem for 2-categories. This yields a quadratic algorithm deciding the equality of diagrams in a free double category.…
An algorithm is presented that generates sets of size equal to the degree of a given variety defined by a homogeneous ideal. This algorithm suggests a versatile framework to study various problems in combinatorial algebraic geometry and…