Enumerable Research Articles

Quasoids in Knot Theory

This paper is devoted to the definition and construction of quasoids, which are algebraic objects generating invariants of oriented knots and links. Such an invariant can be expressed in terms of the number of proper colorings of the regions into which a knot diagram divides the 2-sphere. A coloring with elements of a set X is proper if the colors of all four regions in a neighborhood of each crossing point of the diagram are matched by means of a function Q: X×X×X → X. This function is called a quasoid over the set X. In this paper,we construct two infinite series of quasoids. The first series is formed by linear quasoids over finite rings. The second series consists of quasoids generated by finite biquasiles. The invariants of knots and links generated by quasoids are nontrivial and can be used to distinguish knots. We show that all knots and links admitting diagrams with at most six crossings are distinguished by linear quasoids over ℤn, where n ≤ 11. We give results of the computer enumeration of all different quasoids over sets whose cardinality does not exceed 4.

Immunogenic Heterogeneity of Renal Cell Carcinoma With Venous Tumor Thrombus

ObjectiveTo perform immune-cell enumeration and programmed death-ligand 1 (PD-L1) expression in clear cell renal cell carcinoma (cc-RCC) with tumor thrombus (TT) to guide therapeutic decisions. MethodsAfter obtaining IRB approval and surgical consent, 6 patients underwent radical nephrectomy with venous tumor thrombectomy. We utilized RNA Sequencing to obtain RNAseq expression profiles. Computational calculation and enumeration of immune cells were performed using CIBERSORT, xCell, and ingenuity pathway analysis software. Statistical assessment was conducted using a t test, chi-square, ANOVA and Spearman rank correlations using SPSS v21. ResultsWe observed a higher proportion of M1 macrophages in the primary tumor and tumor thrombus, while we noted no difference in M2 macrophages despite M2 representing a high number in thrombus samples. (ANOVA, P = .032, and P = .89, respectively). Validation with xCell and ingenuity pathway analysis analysis showed a high involvement of macrophages. We observed a higher number of M1 macrophages (CIBERSORT mean 0.11 vs 0.03, P < 0.01) and (nonactivated) resting Natural Killer (NK) cells (0.077 vs 0.017, P = .02) associated PD-L1 high expression of the primary tumor. PDL1 expression was variable without differences in tumor stage, level, or immune cell detection. We observed an inverse correlation of mean platelet volume with PD-L1 expression within the primary tumor (Spearman, −0.89, P = 02) and the TT (Spearman, −0.77, P = 0.07). ConclusionRenal tumor thrombus has higher levels of M1 macrophages that could be utilized as additional targets for future drug development. The PD-L1 expression on clear cell RCC biopsy may not represent its corresponding TT. Future studies are needed to confirm mean platelet volume as a potential blood-based biomarker for PD-L1 expression in RCC.

ENUMERATION OF DIMER CONFIGURATIONS ON A FRACTAL LATTICE

In this paper, we present a solution to the close-packed dimer problem on a fractal lattice. The dimer model is canonical model in statistical physics related with many physical phenomena. Originally, it was introduced as a model for adsorption of diatomic molecules on surfaces. Here we assume that the two dimensional substrate on which the adsorption occurs is nonhomogeneous and we represent it by the modified rectangular (MR) fractal lattice. Self-similarity of the fractal lattice enables exact recursive enumeration of all close-packed dimer configurations at every stage of fractal construction. Asymptotic form for the overall number of dimer coverings is determined and entropy per dimer in the thermodynamic limit is obtained.

Hyperresolution for Gödel logic with truth constants

In the paper, we generalise the well-known hyperresolution principle to the general first-order Gödel logic with explicit partial truth. We propose a hyperresolution calculus suitable for automated deduction in a useful expansion of Gödel logic by intermediate truth constants and the equality, ≖, strict order, ≺, projection, Δ, operators. We solve the deduction problem of a formula from a countable theory in this expansion. We expand Gödel logic by a countable set of intermediate truth constants of the form c¯, c∈(0,1). Our approach is based on translation of a formula to an equivalent satisfiable finite order clausal theory, consisting of order clauses. An order clause is a finite set of order literals of the form ε1⋄ε2 where εi is an atom or a truth constant or a quantified atom, and ⋄ is a connective ≖, equality, or ≺, strict order. ≖ and ≺ are interpreted by the equality and standard strict linear order on [0,1], respectively. We shall investigate the canonical standard completeness, where the semantics of Gödel logic is given by the standard G-algebra, and truth constants are interpreted by ‘themselves’. The hyperresolution calculus is refutation sound and complete for a countable order clausal theory under a certain condition for the set of truth constants occurring in the theory. As an interesting consequence, we get an affirmative solution to the open problem of recursive enumerability of unsatisfiable formulae in Gödel logic with truth constants and the equality, ≖, strict order, ≺, projection, Δ, operators.

Strong jump inversion

We say that a structure $\mathcal{A}$ admits \emph{strong jump inversion} provided that for every oracle $X$, if $X'$ computes $D(\mathcal{C})'$ for some $\mathcal{C}\cong\mathcal{A}$, then $X$ computes $D(\mathcal{B})$ for some $\mathcal{B}\cong\mathcal{A}$. Jockusch and Soare \cite{JS} showed that there are low linear orderings without computable copies, but Downey and Jockusch \cite{DJ} showed that every Boolean algebra admits strong jump inversion. More recently, D.\ Marker and R.\ Miller \cite{MM} have shown that all countable models of $DCF_0$ (the theory of differentially closed fields of characteristic $0$) admit strong jump inversion. We establish a general result with sufficient conditions for a structure $\mathcal{A}$ to admit strong jump inversion. Our conditions involve an enumeration of $B_1$-types, where these are made up of formulas that are Boolean combinations of existential formulas. Our general result applies to some familiar kinds of structures, including some classes of linear orderings and trees. We do not get the result of Downey and Jockusch for arbitrary Boolean algebras, but we do get a result for Boolean algebras with no $1$-atom, with some extra information on the complexity of the isomorphism. Our general result gives the result of Marker and Miller. In order to apply our general result, we produce a computable enumeration of the types realized in models of $DCF_0$. This also yields the fact that the saturated model of $DCF_0$ has a decidable copy.

Generic Amplification of Recursively Enumerable Sets

Generic amplification is a method that allows algebraically undecidable problems to generate problems undecidable for almost all inputs. It is proved that every simple negligible set is undecidable for almost all inputs, but it cannot be obtained via amplification from any undecidable set. On the other hand, it is shown that every recursively enumerable set with nonzero asymptotic density can be obtained via amplification from a set of natural numbers.

Automated Deduction in Gödel Logic

This article addresses the deduction problem of a formula from a countable theory in the first-order Gödel logic. We generalise the well-known hyperresolution principle for deduction in Gödel logic. Our approach is based on translation of a formula to an equivalent satisfiable finite order clausal theory, consisting of order clauses. We introduce a notion of quantified atom: a formula a is a quantified atom if a = Qx , p ( t 0 , …, t n ), where Q is a quantifier (∀, ∃), p ( t 0 , …, t n ) is an atom, and x is a variable occurring in p ( t 0 , …, t n ); for all i ≤ n , either t i = x or x does not occur in t i . Then an order clause is a finite set of order literals of the form ϵ 1 ⋄ ϵ 2 , where ϵ i is either an atom, or a truth constant ( 0 , 1 ), or a quantified atom, and ⋄ is either a connective ≖, equality, or ≺ strict order. ≖ and ≺ are interpreted by the equality and standard strict linear order on [ 0 , 1 ], respectively. On the basis of the hyperresolution principle, a calculus operating over order clausal theories is devised. The calculus is proved to be refutation sound and complete for the countable case. As an interesting consequence, we get an affirmative solution to the open problem of recursive enumerability of unsatisfiable formulae in Gödel logic.

Degree spectra of real closed fields

Several researchers have recently established that for every Turing degree $\boldsymbol{c}$, the real closed field of all $\boldsymbol{c}$-computable real numbers has spectrum $\{\boldsymbol{d}~:~\boldsymbol{d}'\geq\boldsymbol{c}"\}$. We investigate the spectra of real closed fields further, focusing first on subfields of the field $\mathbb{R}_{\boldsymbol{0}}$ of computable real numbers, then on archimedean real closed fields more generally, and finally on non-archimedean real closed fields. For each noncomputable, computably enumerable set $C$, we produce a real closed $C$-computable subfield of $\mathbb{R}_{\boldsymbol{0}}$ with no computable copy. Then we build an archimedean real closed field with no computable copy but with a computable enumeration of the Dedekind cuts it realizes, and a computably presentable nonarchimedean real closed field whose residue field has no computable presentation.

A mobile app for teaching formal languages and automata

AbstractFormal Languages and Automata (FLA) address mathematical models able to specify and recognize languages, their properties and characteristics. Although solid knowledge of FLA is extremely important for a B.Sc. degree in Computer Science and similar fields, the algorithms and techniques covered in the course are complex and difficult to assimilate. Therefore, this article presents FLApp, a mobile application—which we consider the new way to reach students—for teaching FLA. The application—developed for mobile phones and tablets running Android—provides students not only with answers to problems involving Regular, Context‐free, Context‐Sensitive, and Recursively Enumerable Languages, but also an Educational environment that describes and illustrates each step of the algorithms to support students in the learning process.

On decidability and axiomatizability of some ordered structures

The ordered structures of natural, integer, rational and real numbers are studied here. It is known that the theories of these numbers in the language of order are decidable and finitely axiomatizable. Also, their theories in the language of order and addition are decidable and infinitely axiomatizable. For the language of order and multiplication, it is known that the theories of $\mathbb{N}$ and $\mathbb{Z}$ are not decidable (and so not axiomatizable by any computably enumerable set of sentences). By Tarski's theorem, the multiplicative ordered structure of $\mathbb{R}$ is decidable also; here we prove this result directly and present an axiomatization. The structure of $\mathbb{Q}$ in the language of order and multiplication seems to be missing in the literature; here we show the decidability of its theory by the technique of quantifier elimination and after presenting an infinite axiomatization for this structure we prove that it is not finitely axiomatizable.

On the Problem of Definability of the Computably Enumerable Degrees in the Difference Hierarchy

Questions of definability of computably enumerable degrees in the difference hierarchy (degrees of sets from finite levels of the Ershov difference hierarchy) are studied. Several approaches to the solution of this problem are outlined.

Verification of the Jones unknot conjecture up to 22 crossings

We proved by computer enumeration that the Jones polynomial distinguishes the unknot for knots up to 22 crossings. Following an approach of Yamada, we generated knot diagrams by inserting algebraic tangles into Conway polyhedra, computed their Jones polynomials by a divide-and-conquer method, and tested those with trivial Jones polynomials for unknottedness with the computer program SnapPy. We employed numerous novel strategies for reducing the computation time per knot diagram and the number of knot diagrams to be considered. That made computations up to 21 crossings possible on a single processor desktop computer. We explain these strategies in this paper. We also provide total numbers of algebraic tangles up to 18 crossings and of Conway polyhedra up to 22 vertices. We encountered new unknot diagrams with no crossing-reducing pass moves in our search. We report one such diagram in this paper.

On the Jumps of the Degrees Below a Recursively Enumerable Degree

We consider the set of jumps below a Turing degree, given by JB(a)={x':x≤a}, with a focus on the problem: Which recursively enumerable (r.e.) degrees a are uniquely determined by JB(a)? Initially, this is motivated as a strategy to solve the rigidity problem for the partial order R of r.e. degrees. Namely, we show that if every high2 r.e. degree a is determined by JB(a), then R cannot have a nontrivial automorphism. We then defeat the strategy—at least in the form presented—by constructing pairs a0, a1 of distinct r.e. degrees such that JB(a0)=JB(a1) within any possible jump class {x:x'=c}. We give some extensions of the construction and suggest ways to salvage the attack on rigidity.

The Rogers Semilattices of Generalized Computable Enumerations

We study the cardinality and structural properties of the Rogers semilattice of generalized computable enumerations with arbitrary noncomputable oracles and oracles of hyperimmune Turing degree. We show the infinity of the Rogers semilattice of generalized computable enumerations of an arbitrary nontrivial family with a noncomputable oracle. In the case of oracles of hyperimmune degree we prove that the Rogers semilattice of an arbitrary infinite family includes an ideal without minimal elements and establish that the top, if present, is a limit element under the condition that the family contains the inclusion-least set.

Negative Dense Linear Orders

Considering dense linear orders, we establish their negative representability over every infinite negative equivalence, as well as uniformly computable separability by computable gaps and the productivity of the set of computable sections of their negative representations. We construct an infinite decreasing chain of negative representability degrees of linear orders and prove the computability of locally computable enumerations of the field of rational numbers.

Enumeration of Lattice 3-Polytopes by Their Number of Lattice Points

We develop a procedure for the complete computational enumeration of lattice $3$-polytopes of width larger than one, up to any given number of lattice points. We also implement an algorithm for doing this and enumerate those with at most eleven lattice points (there are 216,453 of them). In order to achieve this we prove that if $P$ is a lattice 3-polytope of width larger than one and with at least seven lattice points then it fits in one of three categories that we call boxed, spiked and merged. Boxed polytopes have at most 11 lattice points; in particular they are finitely many, and we enumerate them completely with computer help. Spiked polytopes are infinitely many but admit a quite precise description (and enumeration). Merged polytopes are computed as a union (merging) of two polytopes of width larger than one and strictly smaller number of lattice points.

Fast Formal Proof of the Erdős–Szekeres Conjecture for Convex Polygons with at Most 6 Points

A conjecture originally made by Klein and Szekeres in 1932 (now commonly known as “Erdős–Szekeres” or “Happy Ending” conjecture) claims that for every $$m \ge 3$$ , every set of $$2^{m-2}+1$$ points in a general position (none three different points are collinear) contains a convex m-gon. The conjecture has been verified for $$m \le 6$$ . The case $$m=6$$ was solved by Szekeres and Peters and required a huge computer enumeration that took “more than 3000 GHz hours”. In this paper we improve the solution in several directions. By changing the problem representation, by employing symmetry-breaking and by using modern SAT solvers, we reduce the proving time to around only a half of an hour on an ordinary PC computer (i.e., our proof requires only around 1 GHz hour). Also, we formalize the proof within the Isabelle/HOL proof assistant, making it significantly more reliable.

Lowness and logical depth

Bennett's concept of logic depth [3] seeks to capture the idea that a language has a lot of useful information. Thus we would expect that neither sufficiently random nor sufficiently computationally trivial sequences are deep. A question of Moser and Stephan [11] explores the boundary of this assertion, asking if there is a low computably enumerable (Bennett) deep language. We answer this question affirmatively by constructing a superlow computably enumerable Bennett deep language.

How many varieties of cylindric algebras are there

Cylindric algebras, or concept algebras in another name, form an interface between algebra, geometry and logic; they were invented by Alfred Tarski around 1947. We prove that there are 2 to the alpha many varieties of geometric (i.e., representable) alpha-dimensional cylindric algebras, this means that 2 to the alpha properties of definable relations of (possibly infinitary) models of first-order logic theories can be expressed by formula schemes using alpha variables, where alpha is infinite. This solves Problem 4.2 in the 1985 Henkin-Monk-Tarski monograph. For solving this problem, we had to devise a new kind of construction, which we then use to solve Problem 2.13 of the 1971 Henkin-Monk-Tarski monograph which concerns the structural description of geometric cylindric algebras. There are fewer varieties generated by locally finite dimensional cylindric algebras, and we get a characterization of these among all the 2 to the alpha varieties. As a by-product, we get a simple, natural recursive enumeration of all the equations true of geometric cylindric algebras, and this can serve as a solution to Problem 4.1 of the 1985 Henkin-Monk-Tarski monograph. All this has logical content and implications concerning ordinary first order logic with a countable number of variables.

Classification of empty lattice 4-simplices of width larger than 2

Combining an upper bound on the volume of empty lattice 4-simplices of large width with a computer enumeration we prove the following conjecture of Haase and Ziegler (2000): Except for 179 classes, of determinant at most 179, all empty 4-simplices have width one or two with respect to some integer functional.