We investigate strong versions of enumeration reducibility, the most important one being s-reducibility. We prove that every countable distributive lattice is embeddable into the local structure $L(\mathfrak D_s)$ of the s-degrees. However, $L(\mathfrak D_s)$ is not distributive. We show that on $\Delta^{0}_{2}$ sets s-reducibility coincides with its finite branch version; the same holds of e-reducibility. We prove some density results for $L(\mathfrak D_s)$ . In particular $L(\mathfrak D_s)$ is upwards dense. Among the results about reducibilities (...) that are stronger than s-reducibility, we show that the structure of the $\Delta^{0}_{2}$ bs-degrees is dense. Many of these results on s-reducibility yield interesting corollaries for Q-reducibility as well. (shrink)

We study a strong enumeration reducibility, called bounded enumeration reducibility and denoted by ≤be, which is a natural extension of s-reducibility ≤s. We show that ≤s, ≤be, and enumeration reducibility do not coincide on the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^0_1}$$\end{document} –sets, and the structure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\mathcal{D}_{\rm be}}}$$\end{document} of the be-degrees is not elementarily equivalent to the structure of the s-degrees. We (...) show also that the first order theory of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\boldsymbol{\mathcal{D}_{\rm be}}}$$\end{document} is computably isomorphic to true second order arithmetic: this answers an open question raised by Cooper (Z Math Logik Grundlag Math 33:537–560, 1987). (shrink)

Enumeration reducibility is a notion of relative computability between sets of natural numbers where only positive information about the sets is used or produced. Extending e‐reducibility to partial functions characterises relative computability between partial functions. We define a polynomial time enumeration reducibility that retains the character of enumeration reducibility and show that it is equivalent to conjunctive non‐deterministic polynomial time reducibility. We define the polynomial time e‐degrees as the equivalence classes under this (...) reducibility and investigate their structure on the recursive sets, showing in particular that the pe‐degrees of the computable sets are dense and do not form a lattice, but that minimal pairs exist. We define a jump operator and use it to produce a characterisation of the polynomial hierarchy. (shrink)

Ash, C.J., Generalizations of enumeration reducibility using recursive infinitary propositional sentences, Annals of Pure and Applied Logic 58 173–184. We consider the relation between sets A and B that for every set S if A is Σ0α in S then B is Σ0β in S. We show that this is equivalent to the condition that B is definable from A in a particular way involving recursive infinitary propositional sentences. When α = β = 1, this condition is that (...) B is enumeration reducible to A. We establish further generalizations involving infinitely many sets and ordinals. (shrink)

A closed λ-term E is called an enumerator if M ε /gL/dg /gTn ε N E/drn/dl = β M. Here Λ° is the set of closed λ-terms, N is the set of natural numbers and the /drn/dl are the Church numerals λfx./tfnx. Such an E is called reducing if moreover M ε /gL/dg /gTn ε N E/drn/dl /a/gb M. In 1983 I conjectured that every enumerator is reducing. An ingenious recursion theoretic proof of this conjecture by Statman is presented in (...) Barendregt . The proof is not intuitionistically valid, however. Dirk van Dalen has encouraged me to find intuitionistic proofs whenever possible. In the lambda calculus this is usually not difficult. In this paper an intuitionistic version of Statmans proof will be given. It took me somewhat longer to find it than in other cases. (shrink)

We consider the computably enumerable sets under the relation of Q-reducibility. We first give several results comparing the upper semilattice of c.e. Q-degrees, RQ, Q, under this reducibility with the more familiar structure of the c.e. Turing degrees. In our final section, we use coding methods to show that the elementary theory of RQ, Q is undecidable.

A set A is symmetric enumeration (se-) reducible to a set B (A ≤\sb se B) if A is enumeration reducible to B and \barA is enumeration reducible to \barB. This reducibility gives rise to a degree structure (D\sb se) whose least element is the class of computable sets. We give a classification of ≤\sb se in terms of other standard reducibilities and we show that the natural embedding of the Turing degrees (D\sb T) into the (...) enumeration degrees (D\sb e) translates to an embedding (ι\sb se) into D\sb se that preserves least element, suprema, and infima. We define a weak and a strong jump and we observe that ι\sb se preserves the jump operator relative to the latter definition. We prove various (global) results concerning branching, exact pairs, minimal covers, and diamond embeddings in D\sb se. We show that certain classes of se-degrees are first-order definable, in particular, the classes of semirecursive, Σ\sb n ⋃ Π\sb n, Δ\sb n (for any n \in ω), and embedded Turing degrees. This last result allows us to conclude that the theory of D\sb se has the same 1-degree as the theory of Second-Order Arithmetic. (shrink)

We investigate and extend the notion of a good approximation with respect to the enumeration ${({\mathcal D}_{\rm e})}$ and singleton ${({\mathcal D}_{\rm s})}$ degrees. We refine two results by Griffith, on the inversion of the jump of sets with a good approximation, and we consider the relation between the double jump and index sets, in the context of enumeration reducibility. We study partial order embeddings ${\iota_s}$ and ${\hat{\iota}_s}$ of, respectively, ${{\mathcal D}_{\rm e}}$ and ${{\mathcal D}_{\rm T}}$ (the (...) Turing degrees) into ${{\mathcal D}_{\rm s}}$ , and we show that the image of ${{\mathcal D}_{\rm T}}$ under ${\hat{\iota}_s}$ is precisely the class of retraceable singleton degrees. We define the notion of a good enumeration, or singleton, degree to be the property of containing the set of good stages of some good approximation, and we show that ${\iota_s}$ preserves the latter, as also other naturally arising properties such as that of totality or of being ${\Gamma^0_n}$ , for ${\Gamma \in \{\Sigma,\Pi,\Delta\}}$ and n > 0. We prove that the good enumeration and singleton degrees are immune and that the good ${\Sigma^0_2}$ singleton degrees are hyperimmune. Finally we show that, for singleton degrees a s < b s such that b s is good, any countable partial order can be embedded in the interval (a s, b s). (shrink)

This paper continues the investigation into the relationship between good approximations and jump inversion initiated by Griffith. Firstly it is shown that there is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi^{0}_{2}}$$\end{document} set A whose enumeration degree a is bad—i.e. such that no set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${X \in a}$$\end{document} is good approximable—and whose complement \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\overline{A}}$$\end{document} has lowest possible jump, in other (...) words is low2. This also ensures that the degrees y ≤ a only contain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^{0}_{3}}$$\end{document} sets and thus yields a tight lower bound for the complexity of both a set of bad enumeration degree, and of its complement, in terms of the high/low jump hierarchy. Extending the author’s previous characterisation of the double jump of good approximable sets, the triple jump of a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma^{0}_{2}}$$\end{document} set A is characterised in terms of the index set of coinfinite sets enumeration reducible to A. The paper concludes by using Griffith’s jump interpolation technique to show that there exists a high quasiminimal \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^{0}_{2}}$$\end{document} enumeration degree. (shrink)

. We prove a jump inversion theorem for the enumeration jump and a minimal pair type theorem for the enumeration reducibilty. As an application some results of Selman, Case and Ash are obtained.

We study computably enumerable equivalence relations (ceers) on N and unravel a rich structural theory for a strong notion of reducibility among ceers.

A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitely many elements including h(x), f is a k(n)-enumerator if for every input x of length n, h(x) is among the first k(n) elements enumerated by f. If there is a k(n)-enumerator for h then h is called k(n)-enumerable. We also consider enumerators which are only A-recursive for some oracle A. We determine exactly how hard it is to enumerate the Kolmogorov function, which (...) assigns to each string x its Kolmogorov complexity: • For every underlying universal machine U, there is a constant a such that C is k(n)-enumerable only if k(n) ≥ n/a for almost all n. • For any given constant k, the Kolmogorov function is k-enumerable relative to an oracle A if and only if A is at least as hard as the halting problem. • There exists an r.e., Turing-incomplete set A such for every non-decreasing and unbounded recursive function k, the Kolmogorov function is k(n)-enumerable relative to A. The last result is obtained by using a relativizable construction for a nonrecursive set A relative to which the prefix-free Kolmogorov complexity differs only by a constant from the unrelativized prefix-free Kolmogorov complexity. Although every 2-enumerator for C is Turing hard for K, we show that reductions must depend on the specific choice of the 2-enumerator and there is no bound on the quantity of their queries. We show our negative results even for strong 2-enumerators as an oracle where the querying machine for any x gets directly an explicit list of all hypotheses of the enumerator for this input. The limitations are very general and we show them for any recursively bounded function g: • For every Turing reduction M and every non-recursive set B, there is a strong 2-enumerator f for g such that M does not Turing reduce B to f. • For every non-recursive set B, there is a strong 2-enumerator f for g such that B is not wtt-reducible to f. Furthermore, we deal with the resource-bounded case and give characterizations for the class ${\rm S}_{2}^{{\rm P}}$ introduced by Canetti and independently Russell and Sundaram and the classes PSPACE, EXP. • ${\rm S}_{2}^{{\rm P}}$ is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time tt-reduction which reduces A to every strong 2-enumerator for g. • PSPACE is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time Turing reduction which reduces A to every strong 2-enumerator for g. Interestingly, g can be taken to be the Kolmogorov function for the conditional space bounded Kolmogorov complexity. • EXP is the class of all sets A for which there is a polynomially bounded function g and a machine M which witnesses A ∈ PSPACEf for all strong 2-enumerators f for g. Finally, we show that any strong O(log n)-enumerator for the conditional space bounded Kolmogorov function must be PSPACE-hard if P = NP. (shrink)

We investigate the upper semilattice Eq* of recursively enumerable equivalence relations modulo finite differences. Several natural subclasses are shown to be first-order definable in Eq*. Building on this we define a copy of the structure of recursively enumerable many-one degrees in Eq*, thereby showing that Th has the same computational complexity as the true first-order arithmetic.

We study the relationship between a computably enumerable real and its presentations. A set A presents a computably enumerable real α if A is a computably enumerable prefix-free set of strings such that equation image. Note that equation image is precisely the measure of the set of reals that have a string in A as an initial segment. So we will simply abbreviate equation image by μ. It is known that whenever A so presents α then A ≤wttα, where ≤wtt (...) denotes weak truth table reducibility, and that the wtt-degrees of presentations form an ideal ℐ in the computably enumerable wtt-degrees. We prove that any such ideal is equation image, and conversely that if ℐ is any nonempty equation image ideal in the computably enumerable wtt-degrees then there is a computable enumerable real α such that ℐ = ℐ. We also prove a kind of Rice Theorem for these ideals, namely that if the index set of such a equation image ideal is not empty or equal to ω then it is equation image-complete. (shrink)

We give an alternative and more informative proof that every incomplete ${\Sigma^{0}_{2}}$ -enumeration degree is the meet of two incomparable ${\Sigma^{0}_{2}}$ -degrees, which allows us to show the stronger result that for every incomplete ${\Sigma^{0}_{2}}$ -enumeration degree a, there exist enumeration degrees x 1 and x 2 such that a, x 1, x 2 are incomparable, and for all b ≤ a, b = (b ∨ x 1 ) ∧ (b ∨ x 2 ).

We show that there is a limit lemma for enumeration reducibility to 0 e ', analogous to the Shoenfield Limit Lemma in the Turing degrees, which relativises for total enumeration degrees. Using this and `good approximations' we prove a jump inversion result: for any set W with a good approximation and any set X< e W such that W≤ e X' there is a set A such that X≤ e A< e W and A'=W'. (All jumps are (...) enumeration degree jumps.) The degrees of sets with good approximations include the Σ0 2 degrees and the n-CEA degrees. (shrink)

If r is a reducibility between sets of numbers, a natural question to ask about the structure ? r of the r-degrees containing computably enumerable sets is whether every element not equal to the greatest one is branching (i.e., the meet of two elements strictly above it). For the commonly studied reducibilities, the answer to this question is known except for the case of truth-table (tt) reducibility. In this paper, we answer the question in the tt case by (...) showing that every tt-incomplete computably enumerable truth-table degree a is branching in ? tt . The fact that every Turing-incomplete computably enumerable truth-table degree is branching has been known for some time. This fact can be shown using a technique of Ambos-Spies and, as noticed by Nies, also follows from a relativization of a result of Degtev. We give a proof here using the Ambos-Spies technique because it has not yet appeared in the literature. The proof uses an infinite injury argument. Our main result is the proof when a is Turing-complete but tt-incomplete. Here we are able to exploit the Turing-completeness of a in a novel way to give a finite injury proof. (shrink)

Using computable reducibility ⩽ 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 ceers is Σ3‐complete, whereas the index set of the weakly precomplete ceers is Π3‐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 Σ3‐complete. (shrink)

We consider the strongest forms of enumeration reducibility, those that occur between 1- and npm-reducibility inclusive. By defining two new reducibilities which are counterparts to 1- and i-reducibility, respectively, in the same way that nm- and npm-reducibility are counterparts to m- and pm-reducibility, respectively, we bring out the structure of the strong reducibilities. By further restricting n1- and nm-reducibility we are able to define infinite families of reducibilities which isomorphically embed the r. e. (...) Turing degrees. Thus the many well-known results in the theory of the r. e. Turing degrees have counterparts in the theory of strong reducibilities. We are also able to positively answer the question of whether there exist distinct reducibilities ≤y and ≤a between ≤e and ≤m such that there exists a non-trivial ≤y-contiguous ≤z degree. (shrink)

A strong reducibility relation between partial numberings is introduced which is such that the reduction function transfers exactly the numbers which are indices under the numbering to be reduced into corresponding indices of the other numbering. The degrees of partial numberings of a given set with respect to this relation form an upper semilattice.In addition, Ershov’s completion construction for total numberings is extended to the partial case: every partially numbered set can be embedded in a set which results from (...) the given set by adding one point and which is enumerated by a total and complete numbering. As is shown, the degrees of complete numberings of the extended set also form an upper semilattice. Moreover, both semilattices are isomorphic.This is not so in the case of the usual, weaker reducibility relation for partial numberings which allows the reduction function to transfer arbitrary numbers into indices. (shrink)

We discuss a notion of forcing that characterizes enumeration 1-genericity, and we investigate the immunity, lowness, and quasiminimality properties of enumeration 1-generic sets and their degrees. We construct an enumeration operator Δ such that, for any A, the set ΔA is enumeration 1-generic and has the same jump complexity as A. We deduce from this and other recent results from the literature that not only does every degree a bound an enumeration 1-generic degree b such (...) that a'=b', but also that, if a is nonzero, then we can find such b satisfying 0eenumeration 1-generic degree, hence proving that the class of 1-generic degrees is properly subsumed by the class of enumeration 1-generic degrees. (shrink)

An effectively closed set, or ${\Pi^{0}_{1}}$ class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of ${\Pi^{0}_{1}}$ classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of ${\Pi^{0}_{1}}$ classes (...) and for the subclasses of decidable and of homogeneous ${\Pi^{0}_{1}}$ classes. However no computable numberings exist for small or thin classes. No computable numbering of trees exists that includes all computable trees without dead ends. (shrink)

In this paper we consider the we known method by E. Post of solving the problem of construction of recursively enumerable sets that have a degree intermediate between the degrees of recursive and complete sets with respect to a given reducibility. Post considered reducibilities ≤m, ≤btt, ≤tt and ≤T and solved the problem for al of them except ≤T. Here we extend Post's original method of construction of incomplete sets onto two wide classes of sub-Turing reducibilities what were studying (...) in [1, 2]. (shrink)

We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).

Classical reducibilities have complete sets U that any recursively enumerable set can be reduced to U. This paper investigates existence of complete sets for reducibilities with limited oracle access. Three characteristics of classical complete sets are selected and a natural hierarchy of the bounds on oracle access is built. As the bounds become stricter, complete sets lose certain characteristics and eventually vanish.

We announce and explain recent results on the computably enumerable (c.e.) sets, especially their definability properties (as sets in the spirit of Cantor), their automorphisms (in the spirit of Felix Klein's Erlanger Programm), their dynamic properties, expressed in terms of how quickly elements enter them relative to elements entering other sets, and the Martin Invariance Conjecture on their Turing degrees, i.e., their information content with respect to relative computability (Turing reducibility).

We prove that a enumerable degree is contiguous iff it is locally distributive. This settles a twenty-year old question going back to Ladner and Sasso. We also prove that strong contiguity and contiguity coincide, settling a question of the first author, and prove that no $m$-topped degree is contiguous, settling a question of the first author and Carl Jockusch [11]. Finally, we prove some results concerning local distributivity and relativized weak truth table reducibility.

We study the degree structure of the ω-c.e., n-c.e., and Π10 equivalence relations under the computable many-one reducibility. In particular, we investigate for each of these classes of degrees the most basic questions about the structure of the partial order. We prove the existence of the greatest element for the ω-c.e. and n-computably enumerable equivalence relations. We provide computable enumerations of the degrees of ω-c.e., n-c.e., and Π10 equivalence relations. We prove that for all the degree classes considered, upward (...) density holds and downward density fails. (shrink)

The computable Lipschitz reducibility was introduced by Downey, Hirschfeldt and LaForte under the name of strong weak truth-table reducibility [6]). This reducibility measures both the relative randomness and the relative computational power of real numbers. This paper proves that the computable Lipschitz degrees of computably enumerable sets are not dense. An immediate corollary is that the Solovay degrees of strongly c.e. reals are not dense. There are similarities to Barmpalias and Lewis’ proof that the identity bounded Turing (...) degrees of c.e. sets are not dense [2]), however the problem for the computable Lipschitz degrees is more complex. (shrink)

Let$ \le _c $be computable the reducibility on computably enumerable equivalence relations. We show that for every ceerRwith infinitely many equivalence classes, the index sets$\left\{ {i:R_i \le _c R} \right\}$,$\left\{ {i:R_i \ge _c R} \right\}$, and$\left\{ {i:R_i \equiv _c R} \right\}$are${\rm{\Sigma }}_3^0$complete, whereas in caseRhas only finitely many equivalence classes, we have that$\left\{ {i:R_i \le _c R} \right\}$is${\rm{\Pi }}_2^0$complete, and$\left\{ {i:R \ge _c R} \right\}$ is${\rm{\Sigma }}_2^0$complete. Next, solving an open problem from [1], we prove that the index set (...) of the effectively inseparable ceers is${\rm{\Pi }}_4^0$complete. Finally, we prove that the 1-reducibility preordering on c.e. sets is a${\rm{\Sigma }}_3^0$complete preordering relation, a fact that is used to show that the preordering relation$ \le _c $on ceers is a${\rm{\Sigma }}_3^0$complete preordering relation. (shrink)

We examine how degrees of computably enumerable equivalence relations under computable reduction break down into isomorphism classes. Two ceers are isomorphic if there is a computable permutation of ω which reduces one to the other. As a method of focusing on nontrivial differences in isomorphism classes, we give special attention to weakly precomplete ceers. For any degree, we consider the number of isomorphism types contained in the degree and the number of isomorphism types of weakly precomplete ceers contained in the (...) degree. We show that the number of isomorphism types must be 1 or ω, and it is 1 if and only if the ceer is self-full and has no computable classes. On the other hand, we show that the number of isomorphism types of weakly precomplete ceers contained in the degree can be any member of $[0,\omega ]$. In fact, for any $n \in [0,\omega ]$, there is a degree d and weakly precomplete ceers ${E_1}, \ldots,{E_n}$ in d so that any ceer R in d is isomorphic to ${E_i} \oplus D$ for some $i \le n$ and D a ceer with domain either finite or ω comprised of finitely many computable classes. Thus, up to a trivial equivalence, the degree d splits into exactly n classes.We conclude by answering some lingering open questions from the literature: Gao and Gerdes [11] define the collection of essentially FC ceers to be those which are reducible to a ceer all of whose classes are finite. They show that the index set of essentially FC ceers is ${\rm{\Pi }}_3^0$-hard, though the definition is ${\rm{\Sigma }}_4^0$. We close the gap by showing that the index set is ${\rm{\Sigma }}_4^0$-complete. They also use index sets to show that there is a ceer all of whose classes are computable, but which is not essentially FC, and they ask for an explicit construction, which we provide.Andrews and Sorbi [4] examined strong minimal covers of downwards-closed sets of degrees of ceers. We show that if $\left$ is a uniform c.e. sequence of non universal ceers, then $\left\{ {{ \oplus _{i \le j}}{E_i}|j \in \omega } \right\}$ has infinitely many incomparable strong minimal covers, which we use to answer some open questions from [4].Lastly, we show that there exists an infinite antichain of weakly precomplete ceers. (shrink)

Informal construction -- Formal construction -- Limiting results.

The project was partially supported by a NSF grant of China. The author was grateful to Professor S. Lempp for his encouragement and suggestion while the author was visiting the Department of Mathematics at the University of Wisconsin.

The structure of the l-degrees included in an m-degree with a maximal set together with the l-reducibility relation is characterized. For this a special sublattice of the lattice of recursively enumerable sets under the set-inclusion is used.

Let ≤r and ≤sbe two binary relations on 2ℕ which are meant as reducibilities. Let both relations be closed under finite variation and consider the uniform distribution on 2ℕ, which is obtained by choosing elements of 2ℕ by independent tosses of a fair coin.Then we might ask for the probability that the lower ≤r-cone of a randomly chosen set X, that is, the class of all sets A with A ≤rX, differs from the lower ≤s-cone of X. By c osure (...) under finite variation, the Kolmogorov 0-1 aw yields immediately that this probability is either 0 or 1; in case it is 1, the relations are said to be separable by random oracles.Again by closure under finite variation, for every given set A, the probability that a randomly chosen set X is in the upper ≤r-cone of A is either 0 or 1; let Almostr be the class of sets for which the upper ≤r-cone has measure 1. In the following, results about separations by random oracles and about Almost classes are obtained in the context of generalized reducibilities, that is, for binary relations on 2ℕ which can be defined by a countable set of total continuous functionals on 2ℕ in the same way as the usual resource-bounded reducibilities are defined by an enumeration of appropriate oracle Turing machines. The concept of generalized reducibility comprises a natura resource-bounded reducibilities, but is more general; in particular, it does not involve any kind of specific machine model or even effectivity. The results on generalized reducibilities yield corollaries about specific resource-bounded reducibilities, including several results which have been shown previously in the setting of time or space bounded Turing machine computations. (shrink)

Degtev, A.N., On p-reducibility of numerations, Annals of Pure and Applied Logic 63 57–60. If α and β are two numerations of a set S, then αpβ if there exists a total recursive function f such that [s ε S][α-1=[x:[y ε Df][Dyβ-1]]], where Dn is a finite set with canonical number n. It is proved that if α and β are two computable numerations of some family of recursively enumerable sets A and αpβ, then there is a computable numeration, (...) γ, of S such that αpγ and βshrink)

To each computable enumerable (c.e.) set A with a particular enumeration {As}s∈ω, there is associated a settling function mA(x), where mA(x) is the last stage when a number less than or equal to x was enumerated into A. One c.e. set A is settling time dominated by another set B (B >st A) if for every computable function f, for all but finitely many x, mB(x) > f(m₄(x)). This settling-time ordering, which is a natural extension to an ordering of (...) the idea of domination, was first introduced by Nabutovsky and Weinberger in [3] and Soare [6]. They desired a sequence of sets descending in this relationship to give results in differential geometry. In this paper we examine properties of the <st ordering. We show that it is not invariant under computable isomorphism, that any countable partial ordering embeds into it, that there are maximal and minimal sets, and that two c.e. sets need not have an inf or sup in the ordering. We also examine a related ordering, the strong settling-time ordering where we require for all computable f and g, for almost all x, mB(x) > f(mA(g(x))). (shrink)

We examine the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ of equivalence relations on $\omega $ under computable reducibility. We examine when pairs of degrees have a least upper bound. In particular, we show that sufficiently incomparable pairs of degrees do not have a least upper bound but that some incomparable degrees do, and we characterize the degrees which have a least upper bound with every finite equivalence relation. We show that the natural classes of finite, light, and dark degrees (...) are definable in $\operatorname {\mathrm {\mathbf {ER}}}$. We show that every equivalence relation has continuum many self-full strong minimal covers, and that $\mathbf {d}\oplus \mathbf {\operatorname {\mathrm {\mathbf {Id}}}_1}$ needn’t be a strong minimal cover of a self-full degree $\mathbf {d}$. Finally, we show that the theory of the degree structure $\operatorname {\mathrm {\mathbf {ER}}}$ as well as the theories of the substructures of light degrees and of dark degrees are each computably isomorphic with second-order arithmetic. (shrink)

Various restrictions on transformational grammars have been investigated in order to reduce their generative power from recursively enumerable languages to recursive languages.It will be shown that any restriction on transformational grammars defining a recursively enumerable subset of the set of all transformational grammars, is either too weak (in the sense that there does not exist a general decision procedure for all languages generated under such a restriction) or too strong (in the sense that there exists a recursive language that cannot (...) be generated by any transformational grammar thus restricted). In addition, some related problems will be discussed. (shrink)

Perhaps nowhere better than, "On the Names of God," can readers discern Laclau's appreciation of theology, specifically, negative theology, and the radical potencies of political theology. // It is Laclau's close attention to Eckhart and Dionysius in this essay that reveals a core theological strategy to be learned by populist reasons or social logics and applied in politics or democracies to come. // This mode of algorithmically informed negative political theology is not mathematically inert. It aspires to relate a fraction (...) or ratio to a series ... It strains to reduce the decided determinateness of such seriality ever condemned to the naive metaphysics of bad infinity. // It is worth considering that it is the specific 'number' of Dionysius in differential identification with an ineffable god (and, as such, a singular becoming between theology and numbers) that is floating in at least two dimensions [of signification] (be it political Demand on the horizontal dimension or theological Desire on [a] floating dimension) that cannot but *perform the link that relinks* names of god with any political life, populist reason, social justice, or radical democracy straining toward peace. (shrink)

We show that the _sQ_-degree of a hypersimple set includes an infinite collection of \(sQ_1\) -degrees linearly ordered under \(\le _{sQ_1}\) with order type of the integers and each c.e. set in these _sQ_-degrees is a hypersimple set. Also, we prove that there exist two c.e. sets having no least upper bound on the \(sQ_1\) -reducibility ordering. We show that the c.e. \(sQ_1\) -degrees are not dense and if _a_ is a c.e. \(sQ_1\) -degree such that \(o_{sQ_1}, then there (...) exist infinitely many pairwise _sQ_-incomputable c.e. _sQ_-degrees \(\{c_i\}_{i\in \omega }\) such that \((\forall \,i)\;(a. (shrink)