Universal recursion theoretic properties of r.e. preordered structures

Montagna, Franco; Sorbi, Andrea

doi:10.2307/2274228

Cited by 23 publications

(42 citation statements)

References 9 publications

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“…Then, equivalent to the definition above, a Σ 0 k Boolean algebra is given in the form F/I, where I is a Σ 0 k ideal of F. We rely on this view for coding a Σ 0 k preorder into a Σ 0 k Boolean algebra. We slightly extend concepts and results of [25] where k = 1.…”

Section: σmentioning

confidence: 62%

“…Boolean algebra was introduced by Pour-El and Kripke [29, Lemmas 1,2] when they studied the complexity of logical derivability for recursively axiomatizable theories. The method was developed in more generality by Montagna and Sorbi [25]. For instance, in their Theorem 3.1 they showed that the preorder associated with any effectively inseparable Boolean algebra is Σ 0 1 -complete.…”

Section: σmentioning

confidence: 99%

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Complexity of Equivalence Relations and Preorders From Computability Theory

Ianovski¹,

Miller

Ng³

et al. 2014

J. symb. log.

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We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations R, S, a componentwise reducibility is defined byHere, f is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and f must be computable. We show that there is a Π 0 1 -complete equivalence relation, but no Π 0 k -complete for k ≥ 2. We show that Σ 0 k preorders arising naturally in the above-mentioned areas are Σ 0 k -complete. This includes polynomial time m-reducibility on exponential time sets, which is Σ 0 2 , almost inclusion on r.e. sets, which is Σ 0 3 , and Turing reducibility on r.e. sets, which is Σ 0 4 . 859 860 EGOR IANOVSKI, RUSSELL MILLER, KENG MENG NG, AND ANDRÉ NIES so that we can actually learn something from the reduction. In our example, one should ask how hard it is to compute the dimension of a rational vector space. It is natural to restrict the question to computable vector spaces over Q (i.e., those where the vector addition is given as a Turing-computable function on the domain of the space). Yet even when its domain D E such vector spaces, computing the function which maps each one to its dimension requires a 0 -oracle, hence is not as simple as one might have hoped. (The reasons why 0 is required can be gleaned from [7] or [8].) 1.1. Effective reductions. Reductions are normally ranked by the ease of computing them. In the context of Borel theory, for instance, a large body of research is devoted to the study of Borel reductions (the standard book reference is [17]). Here, the domains D E and D F are the set 2 or some other standard Borel space, and a Borel reduction f is a reduction (from E to F , these being equivalence relations on 2 ) which, viewed as a function from 2 to 2 , is Borel. If such a reduction exists, one says that E is Borel reducible to F , and writes E ≤ B F . A stronger possible requirement is that f be continuous, in which case we have (of course) a continuous reduction. In case the reduction is given by a Turing functional from reals to reals, it is a (type-2) computable reduction.A further body of research is devoted to the study of the same question for equivalence relations E and F on , and reductions f : → between them which are computable. If such a reduction from E to F exists, we say that E is computably reducible to F , and write E ≤ c F , or often just E ≤ F . These reductions will be the focus of this paper. Computable reducibility on equivalence relations was perhaps first studied by Ershov [12] in a category theoretic setting.The main purpose of this paper is to investigate the complexity of equivalence relations under these reducibilities. In certain cases we will generalize from equivalence relations to preorders on . We restrict most of our discussion to relatively low levels of the hierarchy, usually to Π 0 n and Σ 0 n with n ≤ 4. One can focus more closely on very low levels: Such articles as [3,5,18], for instance, have dealt exclusively with Σ 0 1 equivalen...

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Section: σmentioning

confidence: 62%

Section: σmentioning

confidence: 99%

Complexity of Equivalence Relations and Preorders From Computability Theory

Ianovski¹,

Miller

Ng³

et al. 2014

J. symb. log.

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“…In particular, the BA's Lk embed all positive partial orderings. In [MS85] the following strengthening of this property was obtained. In this assertion I is a numeral denoting the number i. there is a computable function 9 such that Ai ~ 9-1 (Bi) for all i 5:.…”

Section: Effectively Dense Lattices In Logicmentioning

confidence: 79%

Positive Structures

Selivanov¹

2003

Computability and Models

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The paper reviews the current state of the theory of positive (or computably enumerable) structures. This theory is relevant to several branches of algebra, logic and computer science, e.g. to algorithmic problems in algebra, theory of quasi varieties, logic programming, algebraic specifications of data types. Along with the general theory of positive structures we consider also positive structures of special kind (the most interesting results are obtained for positive boolean algebras) and some appliations of positive structures to logic and computability theory.

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“…lattice L is precomplete if its associated (r.e.) equivalence relation is (see [11,Section 2] or Selivanov [17, 4.4]). By -we denote the corresponding (r.e.)…”

Section: Orderings Of † N -Sentences and Precomplete Latticesmentioning

confidence: 99%

Uniform Density in Lindenbaum Algebras

Shavrukov¹,

Visser²

2014

Notre Dame J. Formal Logic

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In this paper we prove that the preordering . of provable implication over any recursively enumerable theory T containing a modicum of arithmetic is uniformly dense. This means that we can find a recursive extensional density function F for .. A recursive function F is a density function if it computes, for A and B with A ' B, an element C such that A ' C ' B. The function is extensional if it preserves T -provable equivalence. Secondly, we prove a general result that implies that, for extensions of elementary arithmetic, the ordering . restricted to † n -sentences is uniformly dense. In the last section we provide historical notes and background material.

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Universal recursion theoretic properties of r.e. preordered structures

Cited by 23 publications

References 9 publications

Complexity of Equivalence Relations and Preorders From Computability Theory

Complexity of Equivalence Relations and Preorders From Computability Theory

Positive Structures

Uniform Density in Lindenbaum Algebras

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