Equivalence relations induced by extensional formulae: classification by means of a new fixed point property

Bernardi, Claudio; Montagna, Franco

doi:10.4064/fm-124-3-221-233

“…Following a characterization given by Bernardi and Montagna [3], a ceer R is e-complete if R is u.f.p. and R has a total diagonal function, i.e., a computable function d such that d (x) R x, for all x.…”

Section: Introductionmentioning

confidence: 99%

Universal Computably Enumerable Equivalence Relations

Andrews

¹

,

Lempp

²

,

Miller

³

et al. 2014

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We study computably enumerable equivalence relations (ceers), under the reducibility $R \le S$ if there exists a computable function f such that $x\,R\,y$ if and only if $f\left( x \right)\,\,S\,f\left( y \right)$ , for every $x,y$ . We show that the degrees of ceers under the equivalence relation generated by $\le$ form a bounded poset that is neither a lower semilattice, nor an upper semilattice, and its first-order theory is undecidable. We then study the universal ceers. We show that 1) the uniformly effectively inseparable ceers are universal, but there are effectively inseparable ceers that are not universal; 2) a ceer R is universal if and only if $R\prime \le R$ , where $R\prime$ denotes the halting jump operator introduced by Gao and Gerdes (answering an open question of Gao and Gerdes); and 3) both the index set of the universal ceers and the index set of the uniformly effectively inseparable ceers are ${\rm{\Sigma }}_3^0$ -complete (the former answering an open question of Gao and Gerdes).

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“…Proof By a result by Bernardi and Montagna , the u.f.p. ceers coincide with the nontrivial quotients of any e‐complete ceer.…”

Section: Weakly Precomplete Ceers and Universalitymentioning

confidence: 90%

“…First, we recall, cf. : Definition A strong diagonal function d for an equivalence relation E , is a total computable function satisfying, for every finite set D ,

d (D) \notin {[D]}_{E} .

…”

Section: Weakly Precomplete Ceers and Universalitymentioning

confidence: 99%

“…Thus, if d is a strong diagonal function for E then there is an effective procedure for finding, given any finite set D , a number which is not E ‐equivalent to any number in D . It is known, , that every e‐complete ceer has a strong diagonal function. Theorem If E is a ceer, such that E has a strong diagonal function, then there exist infinitely many ceers

{E_{i} : i \in ω}

such that, for every

i, j

,

E \subseteq E_{i} & [i \neq j \Rightarrow E_{i} \neg ≃ E_{j}] .

…”

Section: Weakly Precomplete Ceers and Universalitymentioning

confidence: 99%

“…Precomplete ceers have been studied in several papers, of which, particularly relevant to our purposes are . For important examples of precomplete ceers that naturally arise in logic, cf., e.g., .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Weakly precomplete computably enumerable equivalence relations

Badaev

¹

,

Sorbi

²

2016

Mathematical Logic Qtrly

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Using computable reducibility $\le$ on equivalence relations, we investigate weakly precomplete ceers (a ``ceer'' is a computably enumerable equivalence relation on the natural numbers), and we compare their class with the more restricted class of precomplete ceers. We show that there are infinitely many isomorphism types of universal (in fact uniformly finitely precomplete) weakly precomplete ceers , that are not precomplete; and there are infinitely many isomorphism types of non-universal weakly precomplete ceers. Whereas the Visser space of a precomplete ceer always contains an infinite effectively discrete subset, there exist weakly precomplete ceers whose Visser spaces do not contain infinite effectively discrete subsets. As a consequence, contrary to precomplete ceers which always yield partitions into effectively inseparable sets, we show that although weakly precomplete ceers always yield partitions into computably inseparable sets, nevertheless there are weakly precomplete ceers for which no equivalence class is creative. Finally, we show that the index set of the precomplete\ud ceers is $\Sigma^0_3$-complete, whereas the index set of the weakly precomplete ceers is $\Pi^0_3$-complete. As a consequence of these results, we also show that the index sets of the uniformly precomplete ceers and of the $e$-complete ceers are $\Sigma_{3}$-complete

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Iterated Extensional Rosser's Fixed Points and Hyperhyperdiagonalizable Algebras

Montagna

¹

1987

Mathematical Logic Qtrly

Self Cite

2

0

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The study of self-reference by algebraic means was begun by R. MAGARI who introduced the notion of diagonalizable algebra (abbreviated as DA) (see [ 7 ] ) . A DA is a pair (8, t), where 8 is a Boolean algebra and t is a mapping from 8 into 8 satisfying:The main examples of DAs are the so called Lindenbaum DAs sT = (aT, tT), where T is an r.e. extension of Peano Arithmetic PA, 8, is the Lindenbaum sentence algebra of T, and tT is the mapping from gT into defined by t&p] = [PrT(p)] (where PrT is the standard provability predicate for T and [ p ] is the equivalence class of T modulo provable equivalence). Diagonalizable algebras provide a satisfactory algebraic translation of GODEL'S Diagonalization Lemma for formulas built up from PrT and Boolean connectives; indeed, D. H. J. DE JONUH and G. SAXVIBIN have independently found an explicit calculation of the (unique) fixed point in any DA of every term f (~, y )~) in which every occurrence of x is under the scope of t (see [ll]). However, DAs do not take any account of a very important kind of fixed points, namely the "Rosser's" fixed points; with this name are denoted the fixed points constructed by a ROSSER'S trick as, for instance, the fixed points of formula8 of the form PrT(A(x)) 5 Pr,(B(x)) or PrT(A(x)) < P T~( & x ) ) .~)The reason for the absence of an algebraic investigation of such fixed points is that formulas of the form introduced above are, in general, not extensional, even if A ( x ) , B(r) are.5) There are, however, two rather interesting con-1) I wish t o thank MASSIMO MIROLLI for his constribution to the second part of this paper; in particular, Claim c) of Theorem 2 is due t o him and is presented here with his permission.2, I n the following, + , 1, v denote the operations of join, meet and complementation respectively; moreover x 4 y is often used as an abbreviation for vx + y, and C a, and n a, denote the supremum and the infimum of the set (a,, . . . , a,), respectively. We recall that identities 1) and 2) follow

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Equivalence relations induced by extensional formulae: classification by means of a new fixed point property

Cited by 27 publications

References 0 publications

Universal Computably Enumerable Equivalence Relations

Universal Computably Enumerable Equivalence Relations

Weakly precomplete computably enumerable equivalence relations

Iterated Extensional Rosser's Fixed Points and Hyperhyperdiagonalizable Algebras

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