Calibrating word problems of groups via the complexity of equivalence relations

Nies, André; Sorbi, Andrea

doi:10.1017/s0960129516000335

Cited by 15 publications

(9 citation statements)

References 31 publications

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“…Using this we show that they can be realized by computable permutations. This connects our approach with previous work in computability theory concerning algebraic and computabily structure of the group of computable permutations, see [11] and [12].…”

Section: Introductionsupporting

confidence: 62%

Sofic profiles of $S(ω)$ and computability

Ivanov

2020

Preprint

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Section: Introductionsupporting

confidence: 62%

Sofic profiles of $S(ω)$ and computability

Ivanov

2020

Preprint

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“…This was first proved by Miller III [26]. Another example, due to [29] refers to the computability theoretic notion of effective inseparability. We recall that a disjoint pair

(U, V)

of sets of numbers is effectively inseparable ( e.i .)…”

Section: Classes Of Algebras ≃ S‐realizing Provable Equivalence Of Pementioning

confidence: 99%

“…A f.p. group G is built in [29] such that

=_{G}

is uniformly effectively inseparable , i.e., uniformly in

x, y

one can find an index of a partial recursive function ψ witnessing that the pair of sets

({[x]}_{=_{G}}, {[y]}_{=_{G}})

is e.i., if

{[x]}_{=_{G}} \cap {[y]}_{=_{G}} = ⌀

. Such a f.p.…”

Section: Classes Of Algebras ≃ S‐realizing Provable Equivalence Of Pementioning

confidence: 99%

Word problems and ceers

Rose

Mauro

Sorbi

2020

Mathematical Logic Qtrly

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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, e.g., that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is it in the same reducibility degree of some non‐periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphism type as the word problem of some non‐commutative and non‐Boolean c.e. ring.

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“…Indeed, ceers have played a leading role in the tale of computable reducibility: they appeared as the main characters of what are perhaps the first results about ď (although before the notion appeared in the literature), namely Miller III's construction ( [21]) of a finitely presented group whose word problem is Σ 0 1 -complete with respect to ď; and Miller III's proof ( [21]) that the isomorphism problem for finitely presented groups is Σ 0 1 -complete with respect to ď. (For other applications of ď to word problems of finitely presented groups see [28].) The first work explicitly tackling ď on ceers was done by Ershov [8], who pointed out important examples of Σ 0 1 -complete ceers, showing also that their degree is join-irreducible.…”

Section: Introductionmentioning

confidence: 99%

The theory of ceers computes true arithmetic

Andrews

Schweber

Sorbi

2020

Annals of Pure and Applied Logic

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We show that the theory of the partial order of computably enumerable equivalence relations (ceers) under computable reduction is 1-equivalent to true arithmetic. We show the same result for the structure comprised of the dark ceers and the structure comprised of the light ceers. We also show the same for the structure of I-degrees in the dark, light, or complete structure. In each case, we show that there is an interpretable copy of pN,`,¨q.2010 Mathematics Subject Classification. 03D25.

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Calibrating word problems of groups via the complexity of equivalence relations

Cited by 15 publications

References 31 publications

Sofic profiles of $S(ω)$ and computability

Sofic profiles of $S(ω)$ and computability

Word problems and ceers

The theory of ceers computes true arithmetic

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