Enumerable Research Articles

Logical specifications of effectively separable data models

It is established that any effectively separable many-sorted universal algebra has an enrichment that is the only (up to isomorphism) model constructed from constants for a suitable computably enumerable set of sentences

Enumerations of families closed under finite differences

Slaman and Wehner independently built a family of sets with the property that every non-computable degree can compute an enumeration of the family, but there is no computable enumeration of the family. We call such a family a Slaman–Wehner family. The original Slaman–Wehner argument relies on all sets in the family constructed being finite, and in particular, it diagonalizes against computably enumerated families using only finite differences. In this paper we ask whether this is a necessary feature, that is, whether there is a Slaman–Wehner family closed under finite differences. This question remains open but we obtain a number of interesting partial results which can be interpreted as saying that the question is quite hard. First of all, no Slaman–Wehner family closed under finite differences can contain a finite set, and the enumeration of the family from a non-computable degree cannot be uniform (whereas, in the Slaman–Wehner construction, it is uniform). On the other hand, we build the following examples of families closed under finite differences which show the impossibility of several natural attempts to show that no Slaman–Wehner family exists: (1) a family that can be enumerated by every non-low degree, but not by any low degree; (2) a family that can be enumerated by any set in a given uniform list of c.e. sets, but which cannot be enumerated computably; and (3) a family that can be enumerated by a given Δ 2 0 set, but which cannot be computably enumerated.

Lowness properties for strong reducibilities and the computational power of maximal sets

We introduce the notion of eventually uniformly weak truth table array computable (e.u.wtt-a.c.) sets. As our main result, we show that a computably enumerable (c.e.) set has this property iff it is weak truth table ( wtt-) reducible to a maximal set. Moreover, in this equivalence we may replace maximal sets by quasi-maximal sets, hyperhypersimple sets or dense simple sets and we may replace wtt-reducibility by identity-bounded Turing reducibility (or any intermediate reducibility). Here, a set A is e.u.wtt-a.c. if there is an effective procedure which, for any given partial wtt-functional Φ ˆ, yields a computable approximation g ( x , s ) of the domain of Φ ˆ A together with a computable indicator function k ( x , s ) and a computable order h ( x ) such that, once the indicator becomes positive, i.e., k ( x , s ) = 1, the number of the mind changes of the approximation g on x after stage s is bounded by h ( x ) where, for total Φ ˆ A , the indicator eventually becomes positive on almost all arguments x of Φ ˆ A . In addition to our main result, we show several properties of the computably enumerable e.u.wtt-a.c. sets. For instance, the class of these sets is closed downwards under wtt-reductions and closed under join. Moreover, we relate this class to – and separate it from – well known classes in the literature. On the one hand, the class of the wtt-degrees of the c.e. e.u.wtt-a.c. sets is strictly contained in the class of the array computable c.e. wtt-degrees. On the other hand, every bounded low set is e.u.wtt-a.c. but there are e.u.wtt-a.c. c.e. sets which are not bounded low. Here a set A is bounded low if A † ⩽ wtt ∅ † , i.e., if A † is ω-c.a., where A † is the wtt-jump of A (Anderson, Csima and Lange (Archive for Mathematical Logic 56(5–6) (2017) 507–521)). Finally, we prove that there is a strict hierarchy within the class of the bounded low c.e. sets A depending on the order h that bounds the number of mind changes of a computable approximation of A † , and we show that there exists a Turing complete set A such that A † is h-c.a. for any computable order h with h ( 0 ) > 0.

ТЕОРЕМА ПРО НЕПОВНОТУ ДЛЯ ОБЧИСЛЮВАНИХ ЗАДАЧ

In the theory of recursive functions, a recursively enumerable (but not recursive) set K = {x |x ∈ Wx } is obtained by Cantor’s diagonal method. Based on Diophantine sets, K is expressed by some polynomial that has positive roots. On the contrary, the set K = {x |x ∉ Wx } is not recursively enumerable. None of the computable functions can enumerate all the elements of the set K . As a result of the productivity of the set K , a parameter exists for which the polynomial has no positive roots. However, it is impossible to prove their absence since this parameter does not belong to any recursively enumerable set. Keywords: diagonal method, Diophantine set, recursively enumerable sets.

The Archimedean Origin of Modern Positional Number Systems

A symbolic analysis of Archimedes’s periodical number system is developed, from which a natural link emerges with the modern positional number systems with zero. After the publication of Fibonacci’s Liber Abaci, the decimal Indo-Arabic positional system was the basis of the algorithmic and algebraic trend of modern mathematics, but even if zero plays a crucial role in algebra and mathematical analysis, zeroless positional systems show the same capability of producing efficient arithmetical algorithms based on operation tables over digits. The crucial role of digits is assessed, by considering a representation of numbers based on strings in lexicographic order. A new algorithm for the determination of decimal periods is presented by remarking on the cruciality of this topic in number theory. Periods of ordinal numbers and enumerations of recursive enumerability are shortly recalled. Concluding remarks are formulated about the deep relationship between numbers and information, which shed new light on a red line passing through the whole history of mathematics.

Prediction and customized design of Curie temperature of Fe-based amorphous alloys based on interpretable machine learning

Predicting the Curie temperature (Tc) is a crucial problem in the field of amorphous alloys. In this study, Fe-based amorphous alloys are taken as an example, and the composition and corresponding Tc are collected through a literature review. Three feature construction strategies are employed to establish the relationship between the composition and Tc using machine learning. The research findings demonstrate that the combination of the Meredig rule and the GBT algorithm yields the highest prediction accuracy. The features are constructed using recursive elimination and enumeration methods, ultimately resulting in an optimal 8-dimensional feature subset. Furthermore, the Shapley Additive exPlanations (SHAP) values are introduced to analyze the interpretability of the prediction model, providing a feature importance ranking and their critical values. Finally, by replacing the Fe and P atoms with Mn and Si atoms, respectively, in the Fe80P13C7 alloy, Fe62Mn18P(13-x)C7Six (x = 4,5,6) alloys are successfully designed to exhibit a Tc close to room temperature (335 K), enabling customized Tc design for Fe-based amorphous alloys.

Conotoxin Prediction: New Features to Increase Prediction Accuracy.

Conotoxins are toxic, disulfide-bond-rich peptides from cone snail venom that target a wide range of receptors and ion channels with multiple pathophysiological effects. Conotoxins have extraordinary potential for medical therapeutics that include cancer, microbial infections, epilepsy, autoimmune diseases, neurological conditions, and cardiovascular disorders. Despite the potential for these compounds in novel therapeutic treatment development, the process of identifying and characterizing the toxicities of conotoxins is difficult, costly, and time-consuming. This challenge requires a series of diverse, complex, and labor-intensive biological, toxicological, and analytical techniques for effective characterization. While recent attempts, using machine learning based solely on primary amino acid sequences to predict biological toxins (e.g., conotoxins and animal venoms), have improved toxin identification, these methods are limited due to peptide conformational flexibility and the high frequency of cysteines present in toxin sequences. This results in an enumerable set of disulfide-bridged foldamers with different conformations of the same primary amino acid sequence that affect function and toxicity levels. Consequently, a given peptide may be toxic when its cysteine residues form a particular disulfide-bond pattern, while alternative bonding patterns (isoforms) or its reduced form (free cysteines with no disulfide bridges) may have little or no toxicological effects. Similarly, the same disulfide-bond pattern may be possible for other peptide sequences and result in different conformations that all exhibit varying toxicities to the same receptor or to different receptors. We present here new features, when combined with primary sequence features to train machine learning algorithms to predict conotoxins, that significantly increase prediction accuracy.

NOTES ON SACKS’ SPLITTING THEOREM

Abstract We explore the complexity of Sacks’ Splitting Theorem in terms of the mind change functions associated with the members of the splits. We prove that, for any c.e. set A, there are low computably enumerable sets $A_0\sqcup A_1=A$ splitting A with $A_0$ and $A_1$ both totally $\omega ^2$ -c.a. in terms of the Downey–Greenberg hierarchy, and this result cannot be improved to totally $\omega $ -c.a. as shown in [9]. We also show that if cone avoidance is added then there is no level below $\varepsilon _0$ which can be used to characterize the complexity of $A_1$ and $A_2$ .

The Higman operations and embeddings of recursive groups

Abstract In the context of Higman embeddings of recursive groups into finitely presented groups, we suggest an approach, termed the 𝐻-machine, which for certain wide classes of groups allows constructive Higman embeddings of recursive groups into finitely presented groups. The approach is based on Higman operations, and it explicitly constructs some specific recursively enumerable sets of integer sequences arising during the embeddings. Specific auxiliary operations are introduced to make the work with Higman operations a simpler and more intuitive procedure. Also, an automated mechanism of constructive embeddings of countable groups into 2-generator groups preserving certain “patterns” is mentioned.

Effective Concept Classes of PAC and PACi Incomparable Degrees, Joins and Embedding of Degrees

AbstractThe Probably Approximately Correct (PAC) learning is a machine learning model introduced by Leslie Valiant in 1984. The PACi reducibility refers to the PAC reducibility independent of size and computation time. This reducibility in PAC learning resembles the reducibility in Turing computability. The ordering of concept classes under PAC reducibility is nonlinear, even when restricted to particular concrete examples.Due to the resemblance to Turing Reducibility, we suspected that there could be incomparable PACi and PAC degrees for the PACi and PAC reducibilities as in Turing incomparable degrees. In 1957 Friedberg and in 1956 Muchnik independently solved the Post problem by constructing computably enumerable sets A and B of incomparable degrees using the priority construction method. We adapt this idea to PACi and PAC reducibilities and construct two effective concept classes C and D such that C is not reducible to D and vice versa. When considering PAC reducibility it was necessary to work on the size of an effective concept class, thus we use Kolmogorov complexity to obtain the size. The non-learnability of concept classes in the PAC learning model is explained by the existence of PAC incomparable degrees.Analogous to the Turing jump, we give a jump operation on effective concept classes for the zero jump. To define the zero jump operator for PACi degrees the join of all the effective concept classes is constructed and proved that it is a greatest element. There are many properties proven for existing degrees. Thus we can explore proving those properties to PACi and PAC degrees. But if we prove an embedding from those degrees to PACi and PAC degrees then those properties will be true for PACi and PAC degrees without explicitly proving them.Abstract prepared by Dodamgodage Gihnee M. Senadheera and taken directly from the thesisE-mail: senadheerad@winthrop.eduURL: https://www.proquest.com/docview/2717762461/abstract/ACD19F29A8774AF6PQ/1?accountid=13864

Abstract 1045: RePhyNER: Overcoming limitations of access to matched germline for serial liquid biopsy applications

Abstract Background: Comparison of a blood or tissue sample with a matched germline control is a key step for accurate detection of somatic mutations. This is particularly important for tracking minimal residual disease (MRD), since alterations erroneously considered to be tumor derived may lead to false detection. Use of blood leukocytes as a germline control is critical not only for appropriate censoring of constitutional allelic variants, but also for addressing those associated with clonal hematopoiesis (CH). However, such matched germline is not always available, or may be suboptimal when contaminated with tumor cells. We hypothesized that accurate tumor genotyping might be feasible without a matched germline, using a dedicated algorithmic framework relying on clonal relationships of alleles of interest (i.e., phylons). Methods: Here, we introduce RePhyNER (Recursive Phylon Nomination, Enumeration, and Recovery), an algorithm for addressing this challenge in serial liquid biopsies. RePhyNER relies on the assumption that when comparing two specimens from the same individual (e.g., before and after therapy), the genomic variants of interests (e.g., tumor somatic mutations) behave differently in their allelic levels when compared to variants needing to be censored (e.g., germline constitutional alleles or CH). Using the change in mean allelic level of all variants in ≥2 samples, RePhyNER relies on a recursive approach to test each candidate somatic variant’s corresponding change gradient against this distribution using count-based statistics. Results: We used simulations to assess the importance of multiple parameters including fraction of contaminating variants or alleles and mean allelic fold-change. These simulations showed substantially superior performance of a recursive approach and revealed that for ≥2-fold changes in the mean VAF in a pair of samples, RePhyNER accurately removed >99% of contaminating alleles while preserving ~100% of true variants. We then applied RePhyNER to 108 plasma cell-free DNA (cfDNA) samples from 47 patients with classic Hodgkin Lymphoma (cHL) profiled by PhasED-Seq and compared MRD-detection performance with or without matched germline. RePhyNER substantially improved specificity by ~30% (68% vs 99%), and Precision by ~50% (46% vs 96%), while only modestly reducing sensitivity (100% vs 91%). We next applied PhasED-Seq to a cohort of cHL patients (n=65) without matched germline information. MRD positivity within the first 2 cycles of therapy was associated with inferior outcome when using RePhyNER (P<0.05 vs ns), correctly reclassifying 26% of detected patients to undetected. Conclusions: RePhyNER enables germline-free and accurate genotyping for MRD detection. Notably, RePhyNER obviates the need for additional germline profiling overcoming key limitations described above and avoids the additional associated costs of sequencing. Citation Format: Mohammad Shahrokh Esfahani, Stefan Alig, Emily Hamilton, Joseph Schroers-Martin, Brian Sworder, Jan Boegeholz, Mari Olsen, Chih Long Liu, David Kurtz, Maximilian Diehn, Ash Alizadeh. RePhyNER: Overcoming limitations of access to matched germline for serial liquid biopsy applications [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2023; Part 1 (Regular and Invited Abstracts); 2023 Apr 14-19; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2023;83(7_Suppl):Abstract nr 1045.

ON THE EXISTENCE OF UNIVERSAL NUMBERINGS

The paper is devoted to research existence property of universal numberings for different computable families. A numbering α is reducible to a numbering β if there is computable function ƒ such that α = β ◦ ƒ. A computable numbering α for some family S is universal if any computable numbering β for the family S is reducible to α. It is well known that the family of all computably enumerable (c.e.) sets has a computable universal numbering. In this paper, we study families of almost all c.e. sets, recursive sets, and almost all differences of c.e. sets, namely questions about the existence of universal numberings for given families. We proved that there is no universal numbering for the family of all recursive sets. For families of c.e. sets without an empty set or a finite number of finite sets, there still exists a universal numbering. However, for families of all c.e. sets without an infinite set, there is no universal numbering. Also, we proved that family ∑2-1 \ Β and the family ∑1-1 has no universal ∑2-1-computable numbering for any Β ∈ ∑2-1.

All galls are divided into three or more parts: recursive enumeration of labeled histories for galled trees

ObjectiveIn mathematical phylogenetics, a labeled rooted binary tree topology can possess any of a number of labeled histories, each of which represents a possible temporal ordering of its coalescences. Labeled histories appear frequently in calculations that describe the combinatorics of phylogenetic trees. Here, we generalize the concept of labeled histories from rooted phylogenetic trees to rooted phylogenetic networks, specifically for the class of rooted phylogenetic networks known as rooted galled trees.ResultsExtending a recursive algorithm for enumerating the labeled histories of a labeled tree topology, we present a method to enumerate the labeled histories associated with a labeled rooted galled tree. The method relies on a recursive decomposition by which each gall in a galled tree possesses three or more descendant subtrees. We exhaustively provide the numbers of labeled histories for all small galled trees, finding that each gall reduces the number of labeled histories relative to a specified galled tree that does not contain it.ConclusionThe results expand the set of structures for which labeled histories can be enumerated, extending a well-known calculation for phylogenetic trees to a class of phylogenetic networks.

Uncovering the mechanism for aggregation in repeat expanded RNA reveals a reentrant transition

RNA molecules aggregate under certain conditions. The resulting condensates are implicated in human neurological disorders, and can potentially be designed towards specified bulk properties in vitro. However, the mechanism for aggregation—including how aggregation properties change with sequence and environmental conditions—remains poorly understood. To address this challenge, we introduce an analytical framework based on multimer enumeration. Our approach reveals the driving force for aggregation to be the increased configurational entropy associated with the multiplicity of ways to form bonds in the aggregate. Our model uncovers rich phase behavior, including a sequence-dependent reentrant phase transition, and repeat parity-dependent aggregation. We validate our results by comparison to a complete computational enumeration of the landscape, and to previously published molecular dynamics simulations. Our work unifies and extends published results, both explaining the behavior of CAG-repeat RNA aggregates implicated in Huntington’s disease, and enabling the rational design of programmable RNA condensates.

Relative order and spectrum in free and related groups

We consider a natural generalization of the concept of order of an element in a group [Formula: see text]: an element [Formula: see text] is said to have order [Formula: see text] in a subgroup [Formula: see text] (respectively, in a coset [Formula: see text]) of [Formula: see text] if [Formula: see text] is the first strictly positive integer such that [Formula: see text] (respectively, [Formula: see text]). We study this notion and its algorithmic properties in the realm of free groups and some related families. Both positive and negative (algorithmic) results emerge in this setting. On the positive side, among other results, we prove that the order of elements, the set of orders (called spectrum), and the set of preorders (i.e. the set of elements of a given order) with respect to finitely generated subgroups are always computable in free and free times free-abelian groups. On the negative side, we provide examples of groups and subgroups having essentially any subset of natural numbers as relative spectrum; in particular, non-recursive and even not computably enumerable sets of natural numbers. Also, we take advantage of Mikhailova’s construction to see that the spectrum membership problem is unsolvable for direct products of nonabelian free groups.

$$sQ_1$$-degrees of computably enumerable sets

$$sQ_1$$-degrees of computably enumerable sets

Demand-Aware Cooperative Content Caching in 5G/6G Networks With MEC-Enabled Edges

Today, billions of smart devices are interconnected via wireless networks, leading to large volumes of video contents circulating through the bandwidth-limited backhaul. This causes network performance to deteriorate. As a mitigation mechanism, caching of highly popular contents to network edges is deployed. We propose a cooperative and demand-aware caching strategy, which is modelled using the Separable Assignment Problem, to maximize the cache hit ratio. This problem is solved with a recursive enumeration method, where dynamic programming is used to fill each edge. The extensive application-level evaluations show that the proposed strategy outperforms existing caching policies.

Enumeration of binary trees compatible with a perfect phylogeny

Evolutionary models used for describing molecular sequence variation suppose that at a non-recombining genomic segment, sequences share ancestry that can be represented as a genealogy—a rooted, binary, timed tree, with tips corresponding to individual sequences. Under the infinitely-many-sites mutation model, mutations are randomly superimposed along the branches of the genealogy, so that every mutation occurs at a chromosomal site that has not previously mutated; if a mutation occurs at an interior branch, then all individuals descending from that branch carry the mutation. The implication is that observed patterns of molecular variation from this model impose combinatorial constraints on the hidden state space of genealogies. In particular, observed molecular variation can be represented in the form of a perfect phylogeny, a tree structure that fully encodes the mutational differences among sequences. For a sample of n sequences, a perfect phylogeny might not possess n distinct leaves, and hence might be compatible with many possible binary tree structures that could describe the evolutionary relationships among the n sequences. Here, we investigate enumerative properties of the set of binary ranked and unranked tree shapes that are compatible with a perfect phylogeny, and hence, the binary ranked and unranked tree shapes conditioned on an observed pattern of mutations under the infinitely-many-sites mutation model. We provide a recursive enumeration of these shapes. We consider both perfect phylogenies that can be represented as binary and those that are multifurcating. The results have implications for computational aspects of the statistical inference of evolutionary parameters that underlie sets of molecular sequences.

A Generalization of the Cylinder Conjecture for Divisible Codes

We extend the original cylinder conjecture on point sets in affine three-dimensional space to the more general framework of divisible linear codes over $\mathbb{F}_q$ and their classification. Through a mix of linear programming, combinatorial techniques and computer enumeration, we investigate the structural properties of these codes. In this way, we can prove a reduction theorem for a generalization of the cylinder conjecture, show some instances where it does not hold and prove its validity for small values of $q$. In particular, we correct a flawed proof for the original cylinder conjecture for $q = 5$ and present the first proof for $q = 7$.

Variants of derivation modes for which purely catalytic P systems are computationally complete

Catalytic P systems and purely catalytic P systems are among the first variants of membrane systems ever considered in this area. These variants of systems also feature some prominent computational complexity questions, and in particular the problem if only one catalyst in catalytic P systems and two catalysts in purely catalytic P systems are enough to allow for generating all recursively enumerable sets of multisets. Several additional ingredients have been shown to be sufficient for obtaining such results.Previously, we could show that using the derivation mode maxobjects, where we only take those multisets of rules which affect the maximal number of objects in the underlying configuration, one catalyst is sufficient for obtaining computational completeness without any other ingredients in catalytic P systems. In this paper we investigate the question whether we can obtain a similar result for purely catalytic P systems, i.e., we show that two catalysts in purely catalytic P systems are enough to allow for generating all recursively enumerable sets of multisets when using specific variants of the maximally parallel derivation mode: we take only those applicable multisets of rules which (i) generate the maximal number of objects, or (ii) yield the maximal difference in the number of objects between the newly generated configuration and the current configuration.In addition, we also consider non-extendable multisets of rules which (i) generate the minimal number of objects, or (ii) yield the minimal difference in the number of objects between the newly generated configuration and the current configuration.In all cases, we have also shown that register machines with n decrementable registers can be simulated by simple purely catalytic P systems working in any of these derivation modes using only n catalysts. Hence, simple purely catalytic P systems working in any of these derivation modes are computationally complete.