Complexity of Equivalence Relations and Preorders From Computability Theory

Ianovski, Egor; Miller, Russell; Ng, Keng Meng; Nies, André

doi:10.1017/jsl.2013.33

Cited by 27 publications

(53 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…sets is Σ 0 3 -universal. Ianovski et al [20,Theorem 3.5] provide a natural example of a Π 0 1 -universal equivalence relation, namely equality of unary quadratic time computable functions. In contrast, they show [20,Corollary 3.8] that there is no Π 0 n -universal equivalence relation for n > 1.…”

Section: Introductionmentioning

confidence: 99%

Calibrating word problems of groups via the complexity of equivalence relations

Nies

Sorbi

2016

Math. Struct. Comp. Sci.

Self Cite

View full text Add to dashboard Cite

(1) There is a finitely presented group with a word problem which is a\ud uniformly effectively inseparable equivalence relation. (2) There is a\ud finitely generated group of computable permutations with a word problem\ud which is a universal co-computably enumerable equivalence relation. (3)\ud Each c.e.\ truth-table degree contains the word problem of a finitely\ud generated group of computable permutations

show abstract

Section: Introductionmentioning

confidence: 99%

Calibrating word problems of groups via the complexity of equivalence relations

Nies

Sorbi

2016

Math. Struct. Comp. Sci.

Self Cite

View full text Add to dashboard Cite

show abstract

“…pre-ordering relation relative to B, then ď B is universal. Ianovski et al [14] have extended this result throughout the arithmetical hierarchy by showing that for every n ě 1, the pre-ordering relation of a Σ 0 n -effectively inseparable Boolean pre-algebra (called Σ 0 n -effectively inseparable Boolean algebra) is a Σ 0 n -universal pre-ordering relation. Moving in this paper from e.i.…”

Section: Introductionmentioning

confidence: 95%

Effective Inseparability, Lattices, and Preordering Relations

Andrews¹,

Sorbi²

2019

The Review of Symbolic Logic

View full text Add to dashboard Cite

We study effectively inseparable (e.i.) pre-lattices (i.e. structures of the form L " xω,^, _, 0, 1, ďLy where ω denotes the set of natural numbers and the following hold:^, _ are binary computable operations; ďL is a c.e. pre-ordering relation, with 0 ďL x ďL 1 for every x; the equivalence relation "L originated by ďL is a congruence on L such that the corresponding quotient structure is a non-trivial bounded lattice; the "L-equivalence classes of 0 and 1 form an effectively inseparable pair), and show (Theorem 30, solving a problem in [17]), that if L is an e.i. pre-lattice then ďL is universal with respect to all c.e. pre-ordering relations, i.e. for every c.e. pre-ordering relation R there exists a computable function f such that, for all x, y, x R y if and only if f pxq ďL f pyq; in fact (Corollary 43) ďL is locally universal, i.e. for every pair a ăL b and every c.e. pre-ordering relation R one can find a reducing function f from R to ďL such that the range of f is contained in the interval tx : a ďL x ďL bu. Also (Theorem 47) ďL is uniformly dense, i.e. there exists a computable function f such that for every a, b if a ăL b then a ăL f pa, bq ăL b, and if a "L a 1 and b "L b 1 then f pa, bq "L f pa 1 , b 1 q. Some consequences and applications of these results are discussed: in particular (Corollary 55) for n ě 1 the c.e. pre-ordering relation on Σn sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson's Q or R is locally universal and uniformly dense; and (Corollary 56) the c.e. pre-ordering relation of provable implication of Heyting Arithmetic is locally universal and uniformly dense.2010 Mathematics Subject Classification. 03D25.

show abstract

“…They use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy. In doing so, we continue the program of work already set in motion in [6,2,11,5,1,13] and augment their results. However, the second and more important purpose of these results is to help explain how we came to develop the notion of finitary reducibility and why we find that notion to be both natural and useful.…”

Section: Background In Computable Reducibilitymentioning

confidence: 99%

“…For every Turing degree d, there exist equivalence relations E and F on ω such that E is finitarily reducible to F (via a computable function, of course), but there is no d-computable reduction from E to F .Proof. We again recall from[13] that there is no Π 0 2 -complete equivalence relation under ≤ c . The proof there relativizes to any degree d and any set D ∈ d, to show that no Π D 2 equivalence relation on ω can be complete among Π D 2 equivalence relations even under d-computable reducibility.…”

mentioning

confidence: 99%

“…The proof there relativizes to any degree d and any set D ∈ d, to show that no Π D 2 equivalence relation on ω can be complete among Π D 2 equivalence relations even under d-computable reducibility. (The authors of[13] use this relativization to show that there is no Π 0 3 -complete equivalence relation, for example, by taking D = ∅ ′ , but their proof really shows that for every Π 0 3 equivalence relation, there is another one which is not even 0 ′ -computably reducible to the first one. )Therefore, there exists some Π D 2 equivalence relation E such that E ≤ d E D = .…”

mentioning

confidence: 99%

See 1 more Smart Citation

Finitary Reducibility on Equivalence Relations

Miller

Ng²

2016

J. symb. log.

Self Cite

View full text Add to dashboard Cite

Abstract. We introduce the notion of finitary computable reducibility on equivalence relations on the domain . This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be Π 0 n+2 -complete under computable reducibility, we show that, for every n, there does exist a natural equivalence relation which is Π 0 n+2 -complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy. §1. Introduction. Computable reducibility provides a natural way of measuring and comparing the complexity of equivalence relations on the natural numbers. Like most notions of reducibility on sets of natural numbers, it relies on the concept of Turing computability to rank objects according to their complexity, even when those objects themselves may be far from computable. It has found particular usefulness in computable model theory, as a measurement of the classical property of being isomorphic: if one can computably reduce the isomorphism problem for computable models of a theory T 0 to the isomorphism problem for computable models of another theory T 1 , then it is reasonable to say that isomorphism on models of T 0 is no more difficult than on models of T 1 . The related notion of Borel reducibility was famously applied this way by Friedman and Stanley in [10], to study the isomorphism problem on all countable models of a theory. Yet computable reducibility has also become the subject of study in pure computability theory, as a way of ranking various well-known equivalence relations arising there.Recently, as part of our study of this topic, we came to consider certain reducibilities weaker than computable reducibility. This article introduces these new, finitary notions of reducibility on equivalence relations and explains some of their uses. We believe that researchers familiar with computable reducibility will find finitary reducibility to be a natural and appropriate measure of complexity, not to supplant computable reducibility but to enhance it and provide a finer analysis of situations in which computable reducibility fails to hold.Computable reducibility is readily defined. It has gone by many different names in the literature, having been called m-reducibility in [1, 2, 11] and FF-reducibility

show abstract

Complexity of Equivalence Relations and Preorders From Computability Theory

Cited by 27 publications

References 23 publications

Calibrating word problems of groups via the complexity of equivalence relations

Calibrating word problems of groups via the complexity of equivalence relations

Effective Inseparability, Lattices, and Preordering Relations

Finitary Reducibility on Equivalence Relations

Contact Info

Product

Resources

About