Proper Superset Research Articles

DEDEKIND-FINITE CARDINALS HAVING COUNTABLE PARTITIONS

Abstract We study the possible structures which can be carried by sets which have no countable subset, but which fail to be ‘surjectively Dedekind finite’, in two possible senses, that there is surjection to $\omega $ , or alternatively, that there is a surjection to a proper superset.

Improved Bounds for Induced Poset Saturation

Given a finite poset $\mathcal{P}$, a family $\mathcal{F}$ of elements in the Boolean lattice is induced-$\mathcal{P}$-saturated if $\mathcal{F}$ contains no copy of $\mathcal{P}$ as an induced subposet but every proper superset of $\mathcal{F}$ contains a copy of $\mathcal{P}$ as an induced subposet. The minimum size of an induced-$\mathcal{P}$-saturated family in the $n$-dimensional Boolean lattice, denoted $\mathrm{sat}^*(n,\mathcal{P})$, was first studied by Ferrara et al. (2017).
 Our work focuses on strengthening lower bounds. For the 4-point poset known as the diamond, we prove $\mathrm{sat}^*(n,\Diamond)\geq\sqrt{n}$, improving upon a logarithmic lower bound. For the antichain with $k+1$ elements, we prove $$\mathrm{sat}^*(n,\mathcal{A}_{k+1})\geq \left(1-\frac{1}{\log_2k}\right)\frac{kn}{\log_2 k}$$ for $n$ sufficiently large, improving upon a lower bound of $3n-1$ for $k\geq 3$.

Analog neuron hierarchy

In order to refine the analysis of the computational power of discrete-time recurrent neural networks (NNs) between the binary-state NNs which are equivalent to finite automata (level 3 in the Chomsky hierarchy), and the analog-state NNs with rational weights which are Turing-complete (Chomsky level 0), we study an intermediate model αANN of a binary-state NN that is extended with α≥0 extra analog-state neurons. For rational weights, we establish an analog neuron hierarchy 0ANNs ⊂ 1ANNs ⊂ 2ANNs ⊆ 3ANNs and separate its first two levels. In particular, 0ANNs coincide with the binary-state NNs (Chomsky level 3) being a proper subset of 1ANNs which accept at most context-sensitive languages (Chomsky level 1) including some non-context-free ones (above Chomsky level 2). We prove that the deterministic (context-free) language L#={0n1n∣n≥1} cannot be recognized by any 1ANN even with real weights. In contrast, we show that deterministic pushdown automata accepting deterministic languages can be simulated by 2ANNs with rational weights, which thus constitute a proper superset of 1ANNs. Finally, we prove that the analog neuron hierarchy collapses to 3ANNs by showing that any Turing machine can be simulated by a 3ANN having rational weights, with linear-time overhead.

Unary Watson-Crick automata

We introduce unary Watson-Crick automata and show that the set of all languages accepted by unary Watson-Crick automata is at most equal to the set of languages accepted by one counter automata and is a proper superset of the set of all regular languages.

Reconstructing One-Articulated Networks with Distance Matrices.

Given a distance matrix M that represents evolutionary distances between any two species, an edge-weighted phylogenetic network N is said to satisfy M if between any pair of species, there exists a path in N with a length equal to the corresponding entry in M. In this article, we consider a special class of networks called a one-articulated network, which is a proper superset of galled trees. We show that if the distance matrix M is derived from an ultrametric one-articulated network N (i.e., for any species X and Y, the entry [Formula: see text] is equal to the shortest distance between X and Y in N), we can re-construct a network that satisfies M in [Formula: see text] time, where n denotes the number of species; further, the reconstructed network is guaranteed to be the simplest, in a sense that the number of hybrid nodes is minimized. In addition, one may easily index a one-articulated network N with a minimum number of hybrid nodes in [Formula: see text] space, such that on any given phylogenetic tree T, we can determine whether T is contained in N (i.e., if a spanning subtree [Formula: see text] of N is a subdivision of T) in [Formula: see text] time.

The saturation number of induced subposets of the Boolean lattice

Given a poset P, a family F of elements in the Boolean lattice is said to be P-saturated if (1) F contains no copy of P as a subposet and (2) every proper superset of F contains a copy of P as a subposet. The maximum size of a P-saturated family is denoted by La(n,P), which has been studied for a number of choices of P. The minimum size of a P-saturated family, sat(n,P), was introduced by Gerbner et al. (2013), and parallels the deep literature on the saturation function for graphs.We introduce and study the concept of saturation for induced subposets. As opposed to induced saturation in graphs, the above definition of saturation for posets extends naturally to the induced setting. We give several exact results and a number of bounds on the induced saturation number for several small posets. We also use a transformation to the biclique cover problem to prove a logarithmic lower bound for a rich infinite family of target posets.

Acyclicity in edge-colored graphs

A walk W in edge-colored graphs is called properly colored (PC) if every pair of consecutive edges in W is of different color. We introduce and study five types of PC acyclicity in edge-colored graphs such that graphs of PC acyclicity of type i is a proper superset of graphs of acyclicity of type i+1, i=1,2,3,4. The first three types are equivalent to the absence of PC cycles, PC closed trails, and PC closed walks, respectively. While graphs of types 1, 2 and 3 can be recognized in polynomial time, the problem of recognizing graphs of type 4 is, somewhat surprisingly, NP-hard even for 2-edge-colored graphs (i.e., when only two colors are used). The same problem with respect to type 5 is polynomial-time solvable for all edge-colored graphs. Using the five types, we investigate the border between intractability and tractability for the problems of finding the maximum number of internally vertex-disjoint PC paths between two vertices and the minimum number of vertices to meet all PC paths between two vertices.

Practical Algorithms for Finding Extremal Sets

The minimal sets within a collection of sets are defined as the ones that do not have a proper subset within the collection, and the maximal sets are the ones that do not have a proper superset within the collection. Identifying extremal sets is a fundamental problem with a wide range of applications in SAT solvers, data mining, and social network analysis. In this article, we present two novel improvements of the high-quality extremal set identification algorithm, AMS-Lex , described by Bayardo and Panda. The first technique uses memoization to improve the execution time of the single-threaded variant of the AMS-Lex, while our second improvement uses parallel programming methods. In a subset of the presented experiments, our memoized algorithm executes more than 400 times faster than the highly efficient publicly available implementation of AMS-Lex. Moreover, we show that our modified algorithm's speedup is not bounded above by a constant and that it increases as the length of the common prefixes in successive input itemsets increases. We provide experimental results using both real-world and synthetic datasets, and show our multithreaded variant algorithm outperforming AMS-Lex by 3 to 6 times. We find that on synthetic input datasets, when executed using 16 CPU cores of a 32-core machine, our multithreaded program executes about as fast as the state-of-the-art parallel GPU-based program using an NVIDIA GTX 580 graphics processing unit.

Order compression schemes

Sample compression schemes are schemes for “encoding” a set of examples in a small subset of examples. The long-standing open sample compression conjecture states that, for any concept class C of VC-dimension d, there is a sample compression scheme in which samples for concepts in C are compressed to samples of size at most d.We show that every order over C induces a special type of sample compression scheme for C, which we call order compression scheme. It turns out that order compression schemes can compress to samples of size at most d if C is maximum, intersection-closed, a Dudley class, or of VC-dimension 1 – and thus in most cases for which the sample compression conjecture is known to be true.Since order compression schemes are much simpler than sample compression schemes in general, their study seems to be a promising step towards resolving the sample compression conjecture. We reveal a number of fundamental properties of order compression schemes, which are helpful in such a study. In particular, order compression schemes exhibit interesting graph-theoretic properties as well as connections to the theory of learning from teachers.To obtain small compressed sets, order compression schemes for a concept class C must often use a proper superset H⊃C as a hypothesis space. We thus further compare order compression schemes for C to order compression schemes for such hypothesis spaces, leading to a study of a number of mutually related combinatorial parameters specifying compressibility.

Self-Stabilizing Algorithms for Maximal 2-packing and General k-packing (k ≥ 2) with Safe Convergence in an Arbitrary Graph

In a graph or a network G =( V,E ), a set SâS† V is a 2-packing if ∀ i ∈ V : | N [ i ] ∩S|≤ 1, where N [ i ] denotes the closed neighborhood of node i . A 2-packing is maximal if no proper superset of S is a 2-packing. This paper presents a safely converging self-stabilizing algorithm for maximal 2-packing problem. Under a synchronous daemon, it quickly converges to a 2- packing (a safe state, not necessarily the legitimate state) in three synchronous steps, and then terminates in a maximal one (the legitimate state) in O ( n ) steps without breaking safety during the convergence interval, where n is the number of nodes. Space requirement at each node is O (log n ) bits. This is a significant improvement over the most recent self-stabilizing algorithm for maximal 2-packing that uses O ( n 2 ) synchronous steps with same space complexity and that does not have safe convergence property. We then generalize the technique to design a self- stabilizing algorithm for maximal k -packing, k ≥ 2, with safe convergence that stabilizes in O ( kn 2 ) steps under synchronous daemon; the algorithm has space complexity of O ( kn log n ) bits at each node; existing algorithms for k -packing stabilize in exponential time under a central daemon with O (log n ) space complexity.Â

ON A SUPERCLASS OF A-GRAMMARS

In this paper we consider a superclass of automaton grammars that can be represented in terms of paths on graphs. With this approach, we assume that vertices of graph are labeled by symbols of finite alphabet A . We will call such grammars graph-generated grammars or G-grammars. In contrast to the graph grammars that are used to describe graph structure transformations, G-grammars using a graphs as a means of representing formal languages. We will give an algorithm for constructing G-grammar which generate the language recognized by deterministic finite automaton. Moreover, we will show that the class of languages generated by G-grammars is a proper superset of regular languages.

Reduced Convex Bodies in Finite Dimensional Normed Spaces: A Survey

For a convex body C in a finite dimensional real Banach space M d denote by $${\triangle(C)}$$ its thickness, i.e., its minimal width with respect to the norm. A convex body $${R \subset M^d}$$ is said to be reduced if $${\triangle(C) < \triangle(R)}$$ for each convex body C properly contained in R. The concept of reduced bodies is particularly useful for solving volume-minimizing problems in the area of convexity, and it is also important as extension of basic notions from convexity and functional analysis. Namely, on the one hand the class of reduced bodies is a “dualization” of the concept of complete sets (and, in Euclidean space, that of constant width). On the other hand, it forms a proper superset of the class of complete sets. We present the recent knowledge on this class of convex bodies in finite dimensional real Banach spaces. First we collect general properties of arbitrary reduced bodies. For example, we present constructions supporting our conjecture that in any normed space of dimension larger than 2 there are reduced bodies of unit thickness and diameter at least λ, for every positive number λ. Then we will lay special emphasize on reduced polytopes, and finally on the geometric description of planar reduced bodies. The survey also presents several research problems.

Locality for quantum systems on graphs depends on the number field

Adapting a definition of Aaronson and Ambainis (2005 Theory Comput. 1 47–79), we call a quantum dynamics on a digraph saturated Z-local if the nonzero transition amplitudes specifying the unitary evolution are in exact correspondence with the directed edges (including loops) of the digraph. This idea appears recurrently in a variety of contexts including angular momentum, quantum chaos, and combinatorial matrix theory. Complete characterization of the digraph properties that allow such a process to exist is a long-standing open question that can also be formulated in terms of minimum rank problems. We prove that saturated Z-local dynamics involving complex amplitudes occur on a proper superset of the digraphs that allow restriction to the real numbers or, even further, the rationals. Consequently, among these fields, complex numbers guarantee the largest possible choice of topologies supporting a discrete quantum evolution. A similar construction separates complex numbers from the skew field of quaternions. The result proposes a concrete ground for distinguishing between complex and quaternionic quantum mechanics.

A Polynomial-Time Algorithm for Estimating the Partition Function of the Ferromagnetic Ising Model on a Regular Matroid

We investigate the computational difficulty of approximating the partition function of the ferromagnetic Ising model on a regular matroid. Jerrum and Sinclair have shown that there is a fully polynomial randomised approximation scheme (FPRAS) for the class of graphic matroids. On the other hand, the authors have previously shown, subject to a complexity-theoretic assumption, that there is no FPRAS for the class of binary matroids, which is a proper superset of the class of graphic matroids. In order to map out the region where approximation is feasible, we focus on the class of regular matroids, an important class of matroids which properly includes the class of graphic matroids, and is properly included in the class of binary matroids. Using Seymour's decomposition theorem, we give an FPRAS for the class of regular matroids.

The Parallel versus Branching Recurrences in Computability Logic

This paper shows that the basic logic induced by the parallel recurrence ⅄ \Yup of computability logic (i.e., the one in the signature $\{ ¬, ∧,∨,⅄,\small{𝖸} \}$) is a proper superset of the basic logic induced by the branching recurrence $⫰$ (i.e., the one in the signature $\{ ¬,∧,∨, ⫰, ⫯ \}$). The latter is known to be precisely captured by the cirquent calculus system CL15, conjectured by Japaridze to remain sound—but not complete—with $⅄$ instead of $⫰$. The present result is obtained by positively verifying that conjecture. A secondary result of the paper is showing that $⅄$ is strictly weaker than $⫰$ in the sense that, while $⫰F$ logically implies $⅄F$, the reverse does not hold.

The countable versus uncountable branching recurrences in computability logic

This paper introduces a new simplified version of the countable branching recurrence ▪ of Computability Logic, proves its equivalence to the old one, and shows that the basic logic induced by ▪ (i.e., the one in the signature {¬,∧,∨,▪,▪}) is a proper superset of the basic logic induced by the uncountable branching recurrence ▪ (i.e., the one in the signature {¬,∧,∨,▪,▪}). A further result of this paper is showing that ▪ is strictly weaker than ▪ in the sense that ▪F logically implies ▪F but not vice versa.

Mining frequent cross-graph quasi-cliques

Joint mining of multiple datasets can often discover interesting, novel, and reliable patterns which cannot be obtained solely from any single source. For example, in bioinformatics, jointly mining multiple gene expression datasets obtained by different labs or during various biological processes may overcome the heavy noise in the data. Moreover, by joint mining of gene expression data and protein-protein interaction data, we may discover clusters of genes which show coherent expression patterns and also produce interacting proteins. Such clusters may be potential pathways. In this article, we investigate a novel data mining problem, mining frequent cross-graph quasi-cliques , which is generalized from several interesting applications in bioinformatics, cross-market customer segmentation, social network analysis, and Web mining. In a graph, a set of vertices S is a γ-quasi-clique (0 &lt; γ ≤ 1) if each vertex v in S directly connects to at least γ ⋅ (| S | − 1) other vertices in S . Given a set of graphs G 1 , …, G n and parameter min_sup (0 &lt; min_sup ≤ 1), a set of vertices S is a frequent cross-graph quasi-clique if S is a γ-quasi-clique in at least min_sup ⋅ n graphs, and there does not exist a proper superset of S having the property. We build a general model, show why the complete set of frequent cross-graph quasi-cliques cannot be found by previous data mining methods, and study the complexity of the problem. While the problem is difficult, we develop practical algorithms which exploit several interesting and effective techniques and heuristics to efficaciously mine frequent cross-graph quasi-cliques. A systematic performance study is reported on both synthetic and real data sets. We demonstrate some interesting and meaningful frequent cross-graph quasi-cliques in bioinformatics. The experimental results also show that our algorithms are efficient and scalable.

Hadwiger's conjecture for quasi‐line graphs

AbstractA graph G is a quasi‐line graph if for every vertex v ∈ V(G), the set of neighbors of v in G can be expressed as the union of two cliques. The class of quasi‐line graphs is a proper superset of the class of line graphs. Hadwiger's conjecture states that if a graph G is not t‐colorable then it contains Kt + 1 as a minor. This conjecture has been proved for line graphs by Reed and Seymour. We extend their result to all quasi‐line graphs. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 17–33, 2008

Balanced parentheses strike back

An ordinal tree is an arbitrary rooted tree where the children of each node are ordered. Succinct representations for ordinal trees with efficient query support have been extensively studied. The best previously known result is due to Geary et al. [2004b, pages 1--10]. The number of bits required by their representation for an n -node ordinal tree T is 2 n + o ( n ), whose first-order term is information-theoretically optimal. Their representation supports a large set of O (1)-time queries on T . Based upon a balanced string of 2 n parentheses, we give an improved 2 n + o ( n )-bit representation for T . Our improvement is two-fold: First, the set of O (1)-time queries supported by our representation is a proper superset of that supported by the representation of Geary, Raman, and Raman. Second, it is also much easier for our representation to support new queries by simply adding new auxiliary strings.

WHEN CHURCH-ROSSER BECOMES CONTEXT FREE

We investigate the intersection of Church-Rosser languages and (strongly) context-free languages. The intersection is still a proper superset of the deterministic context-free languages as well as of their reversals, while its membership problem is solvable in linear time. For the problem whether a given Church-Rosser or context-free language belongs to the intersection we show completeness for the second level of the arithmetic hierarchy. The equivalence of Church-Rosser and context-free languages is Π1-complete. It is proved that all considered intersections are pairwise incomparable. Finally, closure properties under several operations are investigated.