Calibrating word problems of groups via the complexity of equivalence relations

Word problems and ceers

This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called "computable reducibility"), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a computable permutation of the natural numbers). We observe for instance that every ce…

通过可计算归约性对有限生成代数的字问题进行分类

Classifying word problems of finitely generated algebras via computable reducibility

本文通过可计算归约性研究有限生成代数字问题的复杂性，发现部分ceers的c-degrees无法被有限型代数实现，同时证明有限生成半群的字问题能实现嵌入丰富结构且规模庞大的c-degrees集合。

Hyperbolicity and Cubulability Are Preserved Under Elementary Equivalence

The following properties are preserved under elementary equivalence, among finitely generated groups: being hyperbolic (possibly with torsion), being hyperbolic and cubulable, and being a subgroup of a hyperbolic group. In other words, if a finitely generated group G has the same first-order theory as a group possessing one of the previous property, then G enjoys this property as well.

On the word problem for SP-categories, and the properties of two-way communication

The word problem for categories with free products and coproducts (sums), SP-categories, is directly related to the problem of determining the equivalence of certain processes. Indeed, the maps in these categories may be directly interpreted as processes which communicate by two-way channels. The maps of an SP-category may also be viewed as a proof theory for a simple logic with a game theoretic intepretation. The cut-elimination procedure for this logic determines equality only up to certain permuting conversions. As the equality classes under these permuting conversions are finite, it is eas…

The combinatorial equivalence of a computability theoretic question

We show that a question of Miller and Solomon -- that whether there exists a coloring $c:d^{<ω}\rightarrow k$ that does not admit a $c$-computable variable word infinite solution, is equivalent to a natural, nontrivial combinatorial question. The combinatorial question asked whether there is an infinite sequence of integers such that each of its initial segment satisfies a Ramsian type property. This is the first computability theoretic question known to be equivalent to a natural, nontrivial question that does not concern complexity notions. It turns out that the negation of the combinatorial…

转换器层级中的钻石结构

A Diamond Structure in the Transducer Hierarchy

本文研究了转换器层级中钻石结构的存在性，通过构造证明其存在并引入新函数以进一步研究转换器层级。