69 research outputs found
The Complexity of Orbits of Computably Enumerable Sets
The goal of this paper is to announce there is a single orbit of the c.e.
sets with inclusion, \E, such that the question of membership in this orbit
is -complete. This result and proof have a number of nice
corollaries: the Scott rank of \E is \wock +1; not all orbits are
elementarily definable; there is no arithmetic description of all orbits of
\E; for all finite , there is a properly
orbit (from the proof).
A few small corrections made in this versionComment: To appear in the Bulletion of Symbolic Logi
Relative order and spectrum in free and related groups
We consider a natural generalization of the concept of order of an element in a group:
an element g ¿ G is said to have order k in a subgroup H (resp., in a coset Hu) of a group G if k is the first strictly positive integer such that gk ¿ H (resp., gk ¿ Hu). We study this notion and its algorithmic properties in the realm of free groups and some related families.
Both positive and negative (algorithmic) results emerge in this setting. On the positive
side, among other results, we prove that the order of elements, the set of orders (called spectrum), and the set of preorders (i.e., the set of elements of a given order) w.r.t. finitely generated subgroups are always computable in free and free times free-abelian groups.
On the negative side, we provide examples of groups and subgroups having essentially any subset of natural numbers as relative spectrum; in particular, non-recursive and even non-recursively enumerable sets of natural numbers. Also, we take advantage of Mikhailova’s construction to see that the spectrum membership problem is unsolvable for direct products of nonabelian free groups.The first named author was partially supported by MINECO grant PID2019-107444GA-I00 and the Basque Government grant IT974-16. The second named author acknowledges partial support from the Spanish Agencia Estatal de Investigación, through grant MTM2017-82740-P (AEI/ FEDER, UE), and also from the Graduate School of Mathematics through the MarÃa de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445). The
third named author was partially supported by (Polish) Narodowe Centrum Nauki, grant
UMO-2018/31/G/ST1/02681.Peer ReviewedPostprint (author's final draft
The geometric subgroup membership problem
We show that every infinite graph which is locally finite and connected
admits a translation-like action by such that the distance between
a vertex and is uniformly bounded by 3. This action can be taken
to be transitive if and only if the graph has one or two ends. This strenghens
a theorem by Brandon Seward.
Our proof is constructive, and thus it can be made computable. More
precisely, we show that a finitely generated group with decidable word problem
admits a translation-like action by which is computable, and
satisfies an extra condition which we call decidable orbit membership problem.
As an application we show that on any finitely generated infinite group with
decidable word problem, effective subshifts attain all effectively closed
Medvedev degrees. This extends a classification proved by Joseph Miller for
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Computability Theory
Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science
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