Enumerable Research Articles

Classification of divisible design graphs with at most 39 vertices

AbstractA ‐regular graph is called a divisible design graph (DDG) if its vertex set can be partitioned into classes of size , such that two distinct vertices from the same class have exactly common neighbours, and two vertices from different classes have exactly common neighbours. A DDG with , , or is called improper, otherwise it is called proper. We present new constructions of DDGs and, using a computer enumeration algorithm, we find all proper connected DDGs with at most 39 vertices, except for three tuples of parameters: , , and .

Insertion-deletion systems with substitutions I

In this paper, we discuss the addition of substitutions as a further type of operations to (in particular, context-free) insertion-deletion systems, i.e., in addition to insertions and deletions we allow single letter replacements to occur. We investigate the effect of the addition of substitution rules on the context dependency of such systems, thereby also obtaining new characterizations of and even normal forms for context-sensitive (CS) and recursively enumerable (RE) languages and their phrase-structure grammars. More specifically, we prove that for each RE language, there is a system generating this language that only inserts and deletes strings of length two without considering the context of the insertion or deletion site, but which may change symbols (by a substitution operation) by checking a single symbol to the left of the substitution site. When we allow checking left and right single-letter context in substitutions, even context-free insertions and deletions of single letters suffice to reach computational completeness. When allowing context-free insertions only, checking left and right single-letter context in substitutions gives a new characterization of CS. This clearly shows the power of this new type of rules.

A MINIMAL SET LOW FOR SPEED

AbstractAn oracle A is low-for-speed if it is unable to speed up the computation of a set which is already computable: if a decidable language can be decided in time $t(n)$ using A as an oracle, then it can be decided without an oracle in time $p(t(n))$ for some polynomial p. The existence of a set which is low-for-speed was first shown by Bayer and Slaman who constructed a non-computable computably enumerable set which is low-for-speed. In this paper we answer a question previously raised by Bienvenu and Downey, who asked whether there is a minimal degree which is low-for-speed. The standard method of constructing a set of minimal degree via forcing is incompatible with making the set low-for-speed; but we are able to use an interesting new combination of forcing and full approximation to construct a set which is both of minimal degree and low-for-speed.

On Elliptic Modular Foliations, II

We give an example of a one dimensional foliation $\cal F$ of degree two in a Zariski open set of a four dimensional weighted projective space which has only an enumerable set of algebraic leaves. These are defined over rational numbers and are isomorphic to modular curves $X_0(d), d\in\Bbb B$ minus cusp points. As a by-product we get new models for modular curves for which we slightly modify an argument due to J. V. Pereira and give closed formulas for elements in their defining ideals. The general belief has been that such formulas do not exist and the emphasis in the literature has been on introducing faster algorithms to compute equations for small values of $d$.

Not all Kripke models of [formula omitted] are locally [formula omitted

Let K be an arbitrary Kripke model of Heyting Arithmetic, HA. For every node k in K, we can view the classical structure of k, Mk as a model of some classical theory of arithmetic. Let T be a classical theory in the language of arithmetic. We say K is locally T, iff for every k in K, Mk⊨T. One of the most important problems in the model theory of HA is the following question: Is every Kripke model ofHAlocallyPA? We answer this question negatively. We introduce two new Kripke model constructions to this end. The first construction actually characterizes the arithmetical structures that can be the root of a Kripke model K⊩HA+ECT0 (ECT0 stands for Extended Church Thesis). The characterization says that for every arithmetical structure M, there exists a rooted Kripke model K⊩HA+ECT0 with the root r such that Mr=M iff M⊨ThΠ2(PA). One of the consequences of this characterization is that there is a rooted Kripke model K⊩HA+ECT0 with the root r such that Mr⊭IΔ1 and hence K is not even locally IΔ1. The second Kripke model construction is an implicit way of doing the first construction which works for any reasonable consistent intuitionistic arithmetical theory T with a recursively enumerable set of axioms that has the existence property. We get a sufficient condition from this construction that describes when for an arithmetical structure M, there exists a rooted Kripke model K⊩T with the root r such that Mr=M. As applications of this sufficient condition, we construct two new Kripke models. The first one is a Kripke model K⊩HA+¬θ+MP (θ is an instance of ECT0 and MP is Markov's principle) which is not locally IΔ1. The second one is a Kripke model K⊩HA such that K forces exactly the sentences that are provable from HA, but it is not locally IΔ1. Also, we will prove that every countable Kripke model of intuitionistic first-order logic can be transformed into another Kripke model with the full infinite binary tree as the Kripke frame such that both Kripke models force the same sentences. So with the previous result, there is a binary Kripke model K of HA such that K is not locally IΔ1.

Towards Finding a Lattice that Characterizes the -Fickle Recursively Enumerable Turing Degrees

AbstractGiven a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees $\langle \mathcal {R}_{\mathrm {T}},\leq _{\mathrm {T}}\rangle $ , we do not in general know how to characterize the degrees $\mathbf {d}\in \mathcal {R}_{\mathrm {T}}$ below which L can be bounded. The important characterizations known are of the $L_7$ and $M_3$ lattices, where the lattices are bounded below $\mathbf {d}$ if and only if $\mathbf {d}$ contains sets of “fickleness” $>\omega $ and $\geq \omega ^\omega $ respectively. We work towards finding a lattice that characterizes the levels above $\omega ^2$ , the first non-trivial level after $\omega $ . We introduced a lattice-theoretic property called “ $3$ -directness” to describe lattices that are no “wider” or “taller” than $L_7$ and $M_3$ . We exhaust the 3-direct lattices L, but they turn out to also characterize the $>\omega $ or $\geq \omega ^\omega $ levels, if L is not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some 3-direct lattices, but the removals did not change the levels characterized. This leads us to conjecture that a USL characterizes the same r.e. degrees as the lattice on which the USL is based. We discovered three 3-direct lattices besides $M_3$ that also characterize the $\geq \omega ^\omega $ -levels. Our search for a $>\omega ^2$ -candidate therefore involves the lattice-theoretic problem of finding lattices that do not contain any of the four $\geq \omega ^\omega $ -lattices as sublattices.Abstract prepared by Liling Ko.E-mail: ko.390@osu.eduURL: http://sites.nd.edu/liling-ko/

Variants of derivation modes for which catalytic P systems with one catalyst are computationally complete

Catalytic P systems are among the first variants of membrane systems ever considered in this area. This variant of systems also features some prominent computational complexity questions, and in particular the problem of using only one catalyst: is one catalyst enough to allow for generating all recursively enumerable sets of multisets? Several additional ingredients have been shown to be sufficient for obtaining even computational completeness with only one catalyst. Last year we could show that the derivation mode max_{objects}, 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 this paper we follow this way of research and show that one catalyst is also sufficient for obtaining computational completeness when using specific variants of derivation modes based on non-extendable multisets of rules: we only take those non-extendable multisets whose application yields the maximal number of generated objects or else those non-extendable multisets whose application yields the maximal difference in the number of objects between the newly generated configuration and the current configuration. A similar computational completeness result can even be obtained when omitting the condition of non-extendability of the applied multisets when taking the maximal difference of objects or the maximal number of generated objects. Moreover, we reconsider simple P system with energy control—both symbol and rule energy-controlled P systems equipped with these new variants of derivation modes yield computational completeness.

Mixed computation

AbstractProof-theoretic models of grammar are based on the view that an explicit characterization of a language comes in the form of the recursive enumeration of strings in that language. That recursive enumeration is carried out by a procedure which strongly generates a set of structural descriptions Σ and weakly generates a set of strings S; a grammar is thus a function that pairs an element of Σ with elements of S. Structural descriptions are obtained by means of Context-Free phrase structure rules or via recursive combinatorics and structure is assumed to beuniform: binary branching trees all the way down. In this work we will analyse natural language constructions for which such a rigid conception of phrase structure is descriptively inadequate and propose a solution for the problem of phrase structure grammars assigning too much or too little structure to natural language strings: we propose that the grammar can oscillate between levels of computational complexity in local domains, which correspond to elementary trees in a lexicalised Tree Adjoining Grammar.

Nonlowness is independent from fickleness

It was recently shown that the computably enumerable (c.e.) degrees that embed the critical triple and the M 5 lattice structure are exactly those that are sufficiently fickle. Therefore the embeddability strength of a c.e. degree has much to do with the degree’s fickleness. Nonlowness is another common measure of degree strength, with nonlow degrees expected to compute more degrees than low ones. We ask if nonlowness and fickleness are independent measures of strength. Downey and Greenberg (A Hierarchy of Turing Degrees: A Transfinite Hierarchy of Lowness Notions in the Computably Enumerable Degrees, Unifying Classes, and Natural Definability (AMS-206) (2020) Princeton University Press) claimed this to be true without proof, so we present one here. We prove the claim by building low and nonlow c.e. sets with arbitrary fickle degrees. Our construction is uniform so the degrees built turn out to be uniformly fickle. We base our proof on our direct construction of a nonlow array computable set. Such sets were always known to exist, but also never constructed directly in any publication we know.

Hierarchy of Computably Enumerable Degrees II

A transfinite hierarchy of Turing degrees of c.e.\ sets has been used to calibrate the dynamics of families of constructions in computability theory, and yields natural definability results. We review the main results of the area, and discuss splittings of c.e.\ degrees, and finding maximal degrees in upper cones.

Quasi-universal k-regular sequences

We study the k-regular sequences introduced by Allouche and Shallit. We call a k-regular integer sequence s quasi-universal, if for every recursively enumerable set A of positive integers, the k-kernel of s contains a sequence t such that A equals the set of positive terms of t. We show that there are quasi-universal k-regular sequences for every integer k≥2. As a consequence, no nontrivial property is decidable for the set of positive terms of a k-regular sequence.

Some properties of the upper semilattice of computable families of computably enumerable sets

Some properties of the upper semilattice of computable families of computably enumerable sets

Iterated Virtual Screening-Assisted Antiviral and Enzyme Inhibition Assays Reveal the Discovery of Novel Promising Anti-SARS-CoV-2 with Dual Activity

Unfortunately, COVID-19 is still a threat to humankind and has a dramatic impact on human health, social life, the world economy, and food security. With the limited number of suggested therapies under clinical trials, the discovery of novel therapeutic agents is essential. Here, a previously identified anti-SARS-CoV-2 compound named Compound 13 (1,2,5-Oxadiazole-3-carboximidic acid, 4,4′-(methylenediimino) bis,bis[[(2-hydroxyphenyl)methylene]hydrazide) was subjected to an iterated virtual screening against SARS-CoV-2 Mpro using a combination of Ligand Designer and PathFinder. PathFinder, a computational reaction enumeration tool, was used for the rapid generation of enumerated structures via default reaction library. Ligand designer was employed for the computerized lead optimization and selection of the best structural modification that resulted in a favorable ligand–protein complex. The obtained compounds that showed the best binding to Mpro were re-screened against TMPRSS2, leading to the identification of 20 shared compounds. The compounds were further visually inspected, which resulted in the identification of five shared compounds M1–5 with dual binding affinity. In vitro evaluation and enzyme inhibition assay indicated that M3, an analogue of Compound 13 afforded by replacing the phenolic moiety with pyridinyl, possesses an improved antiviral activity and safety. M3 displayed in vitro antiviral activity with IC50 0.016 µM and Mpro inhibition activity with IC50 0.013 µM, 7-fold more potent than the parent Compound 13 and potent than the antivirals drugs that are currently under clinical trials. Moreover, M3 showed potent activity against human TMPRSS2 and furin enzymes with IC50 0.05, and 0.08 µM, respectively. Molecular docking, WaterMap analysis, molecular dynamics simulation, and R-group analysis confirmed the superiority of the binding fit to M3 with the target enzymes. WaterMap analysis calculated the thermodynamic properties of the hydration site in the binding pocket that significantly affects the biological activity. Loading M3 on zinc oxide nanoparticles (ZnO NPs) increased the antiviral activity of the compound 1.5-fold, while maintaining a higher safety profile. In conclusion, lead optimized discovery following an iterated virtual screening in association with molecular docking and biological evaluation revealed a novel compound named M3 with promising dual activity against SARS-CoV-2. The compound deserves further investigation for potential clinical-based studies.

When catalytic P systems with one catalyst can be computationally complete

Catalytic P systems are among the first variants of membrane systems ever considered in this area. This variant of systems also features some prominent computational complexity questions, and in particular the problem of using only one catalyst in the whole system: is one catalyst enough to allow for generating all recursively enumerable sets of multisets? Several additional ingredients have been shown to be sufficient for obtaining computational completeness even with only one catalyst. In this paper, we show that one catalyst is sufficient for obtaining computational completeness if either catalytic rules have weak priority over non-catalytic rules or else instead of the standard maximally parallel derivation mode, we use the derivation mode maxobjects, i.e., we only take those multisets of rules which affect the maximal number of objects in the underlying configuration.

Maximality and collapse in the hierarchy of α-c.a. degrees

In (A Hierarchy of Turing Degrees: A Transfinite Hierarchy of Lowness Notions in the Computably Enumerable Degrees, Unifying Classes, and Natural Definability (2020), Annals of Mathematics Studies, Princeton University Press), Downey and Greenberg define a transfinite hierarchy of low 2 c.e. degrees – the totally α-c.a. degrees, for appropriately small ordinals α. This new hierarchy is of particular interest because it has already given rise to several natural definability results, and provides a new definable antichain in the c.e. degrees. Several levels of this hierarchy contain maximal degrees. We discuss how maximality interacts with upper cones, and the related notion of hierarchy collapse in upper cones. For example, we show that there is a totally ω-c.a. degree above which there is no maximal totally ω-c.a. degree.

A MATHEMATICAL COMMITMENT WITHOUT COMPUTATIONAL STRENGTH

AbstractWe present a new manifestation of Gödel’s second incompleteness theorem and discuss its foundational significance, in particular with respect to Hilbert’s program. Specifically, we consider a proper extension of Peano arithmetic ( $\mathbf {PA}$ ) by a mathematically meaningful axiom scheme that consists of $\Sigma ^0_2$ -sentences. These sentences assert that each computably enumerable ( $\Sigma ^0_1$ -definable without parameters) property of finite binary trees has a finite basis. Since this fact entails the existence of polynomial time algorithms, it is relevant for computer science. On a technical level, our axiom scheme is a variant of an independence result due to Harvey Friedman. At the same time, the meta-mathematical properties of our axiom scheme distinguish it from most known independence results: Due to its logical complexity, our axiom scheme does not add computational strength. The only known method to establish its independence relies on Gödel’s second incompleteness theorem. In contrast, Gödel’s theorem is not needed for typical examples of $\Pi ^0_2$ -independence (such as the Paris–Harrington principle), since computational strength provides an extensional invariant on the level of $\Pi ^0_2$ -sentences.

Graph transformation and shortest paths algorithms for finite Markov chains.

The graph transformation (GT) algorithm robustly computes the mean first-passage time to an absorbing state in a finite Markov chain. Here we present a concise overview of the iterative and block formulations of the GT procedure and generalize the GT formalism to the case of any path property that is a sum of contributions from individual transitions. In particular, we examine the path action, which directly relates to the path probability, and analyze the first-passage path ensemble for a model Markov chain that is metastable and therefore numerically challenging. We compare the mean first-passage path action, obtained using GT, with the full path action probability distribution simulated efficiently using kinetic path sampling, and with values for the highest-probability paths determined by the recursive enumeration algorithm (REA). In Markov chains representing realistic dynamical processes, the probability distributions of first-passage path properties are typically fat-tailed and therefore difficult to converge by sampling, which motivates the use of exact and numerically stable approaches to compute the expectation. We find that the kinetic relevance of the set of highest-probability paths depends strongly on the metastability of the Markov chain, and so the properties of the dominant first-passage paths may be unrepresentative of the global dynamics. Use of a global measure for edge costs in the REA, based on net productive fluxes, allows the total reactive flux to be decomposed into a finite set of contributions from simple flux paths. By considering transition flux paths, a detailed quantitative analysis of the relative importance of competing dynamical processes is possible even in the metastable regime.

Some Properties of the Upper Semilattice of Computable Families of Computably Enumerable Sets

Some Properties of the Upper Semilattice of Computable Families of Computably Enumerable Sets

Time Varying Splicing Recognizers

In this work, we construct splicing recognizers of radius 2 capable of accepting recursively enumerable languages. Cavaliere, Jonoska and Leupold introduced the concept of splicing recognizers in 2006 and constructed splicing recognizers of radius 4 which can accept recursively enumerable languages. It was conjectured that the radius of splicing recognizers, accepting recursively enumerable languages can be decreased. We prove in this paper that the conjecture is true and in fact, radius 2 is sufficient. We also introduce the concept of time varying splicing recognizers and show that these recognizers of radius 1 are capable of accepting recursively enumerable languages.

On null 3-hypergraphs

Given a 3-uniform hypergraph H consisting of a set V of vertices, and T⊆V3 triples, a null labelling is an assignment of ±1 to the triples such that each vertex is contained in an equal number of triples labelled +1 and −1. Thus, the signed degree of each vertex is zero. A necessary condition for a null labelling is that the degree of every vertex of H is even. The Null Labelling Problem is to determine whether H has a null labelling. It is proved that this problem is NP-complete. Computer enumerations suggest that most hypergraphs which satisfy the necessary condition do have a null labelling. Some constructions are given which produce hypergraphs satisfying the necessary condition, but which do not have a null labelling. A self complementary 3-hypergraph with this property is also constructed.