Tuples of Disjoint $\mathsf{NP}$ -Sets

Disjoint -pairs are a well studied complexity-theoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint -pairs to disjoint k-tuples of -sets for k≥2. We define subclasses of the class of all disjoint k-tuples of -sets. These subclasses are associated with a propositional proof system and possess complete tuples which are defined from the proof system. In our main result we show that complete disjoint -pairs exist if and only if complete disjoint k-tuples of -sets exist for all k≥2. Further, this is equivalent to the existence of a propositional proof system in which the disjointness of all k-tuples is shortly provable. We also show that a strengthening of this conditions characterizes the existence of optimal proof systems. An extended abstract of this paper appeared in the proceedings of the conference CSR 2006 (Lecture Notes in Computer Science 3967, 80–91, 2006). Supported by DFG grant KO 1053/5-1.     

Topology on the set of R 0 semantics for R 0 algebras

This paper introduces the concepts of R ₀ valuation, R ₀ semantic, countable R ₀ category , R ₀ fuzzy topological category , etc. It is established in a natural way that the fuzzy topology δ and its cut topology on the set Ω _M consisting of all R ₀ valuations of an R ₀ algebra M, and some properties of fuzzy topology δ and its cut topology are investigated carefully. Moreover, the representation theorem for R ₀ algebras by means of fuzzy topology is given, that is to say the category is equivalent to the category . By studying the relation between valuations and filters, the Loomis–Sikorski theorem for R ₀ algebras is obtained. As an application, K-compactness of the R ₀ logic is discussed.     

On the Complexity of 2-Monotone Restarting Automata

The -automaton is the weakest form of the nondeterministic version of the restarting automaton that was introduced by Jančar et al. to model the so-called analysis by reduction. Here it is shown that the class ℒ(R) of languages that are accepted by -automata is incomparable under set inclusion to the class of Church-Rosser languages and to the class of growing context-sensitive languages. In fact this already holds for the class of languages that are accepted by 2-monotone -automata. In addition, we prove that already the latter class contains -complete languages, showing that already the 2-monotone -automaton has a surprisingly large expressive power. The results of this paper have been announced at DLT 2004 in Auckland, New Zealand. This work was mainly carried out while T. Jurdziński was visiting the University of Kassel, supported by a grant from the Deutsche Forschungsgemeinschaft (DFG). F. Mráz and M. Plátek were partially supported by the Grant Agency of the Czech Republic under Grant-No. 201/04/2102 and by the program ‘Information Society’ under project 1ET100300517. F. Mráz was also supported by the Grant Agency of Charles University in Prague under Grant-No. 358/2006/A-INF/MFF.     

A schedulable utilization bound for the multiprocessor \mathsf{EPDF} Pfair algorithm

The earliest-pseudo-deadline-first ( ) Pfair scheduling algorithm is less expensive than some other known Pfair algorithms, but is not optimal for scheduling recurrent real-time tasks on more than two processors. In prior work, sufficient per-task weight (i.e., utilization) restrictions were established for ensuring that tasks either do not miss their deadlines or have bounded tardiness when scheduled under . Implicit in these restrictions is the assumption that the total system utilization may equal the total available processing capacity (i.e., the total number of processors). This paper considers an orthogonal issue, namely, determining a sufficient restriction on the total utilization of a task set for it to be schedulable (i.e., a schedulable utilization bound) under , assuming that there are no per-task weight restrictions. We prove that a task set with total utilization at most is correctly scheduled under on M processors, regardless of how large each task’s weight is. At present, we do not know whether this value represents the worst-case for , but we do provide a counterexample that shows that it cannot be improved to exceed 86% of the total processing capacity. The schedulable utilization bound we derive is expressed in terms of the maximum weight of any task, and hence, if this value is known, may be used to schedule task sets with total utilization greater than .

Algorithms to Separate $\{0,\frac{1}{2}\}$ -Chvátal-Gomory Cuts

Chvátal-Gomory cuts are among the most well-known classes of cutting planes for general integer linear programs (ILPs). In case the constraint multipliers are either 0 or , such cuts are known as -cuts. It has been proven by Caprara and Fischetti (Math. Program. 74:221–235, 1996) that separation of -cuts is -hard. In this paper, we study ways to separate -cuts effectively in practice. We propose a range of preprocessing rules to reduce the size of the separation problem. The core of the preprocessing builds a Gaussian elimination-like procedure. To separate the most violated -cut, we formulate the (reduced) problem as integer linear program. Some simple heuristic separation routines complete the algorithmic framework. Computational experiments on benchmark instances show that the combination of preprocessing with exact and/or heuristic separation is a very vital idea to generate strong generic cutting planes for integer linear programs and to reduce the overall computation times of state-of-the-art ILP-solvers.     

A Novel Class of Symmetric and Nonsymmetric Periodizing Variable Transformations for Numerical Integration

Variable transformations for numerical integration have been used for improving the accuracy of the trapezoidal rule. Specifically, one first transforms the integral via a variable transformation that maps [0,1] to itself, and then approximates the resulting transformed integral by the trapezoidal rule. In this work, we propose a new class of symmetric and nonsymmetric variable transformations which we denote , where r and s are positive scalars assigned by the user. A simple representative of this class is . We show that, in case , or but has algebraic (endpoint) singularities at x = 0 and/or x = 1, the trapezoidal rule on the transformed integral produces exceptionally high accuracies for special values of r and s. In particular, when and we employ , the error in the approximation is (i) O(h ^r ) for arbitrary r and (ii) O(h ^2r ) if r is a positive odd integer at least 3, h being the integration step. We illustrate the use of these transformations and the accompanying theory with numerical examples.      

The <Emphasis Type="Italic">k</Emphasis>-Homotopic Thinning and a Torus-Like Digital Image in <Emphasis Type="Bold">Z</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In order to discuss digital topological properties of a digital image (X,k), many recent papers have used the digital fundamental group and several digital topological invariants such as the k-linking number, the k-topological number, and so forth. Owing to some difficulties of an establishment of the multiplicative property of the digital fundamental group, a k-homotopic thinning method can be essentially used in calculating the digital fundamental group of a digital product with k-adjacency. More precisely, let be a simple closed k _i -curve with l _i elements in . For some k-adjacency of the digital product which is a torus-like set, proceeding with the k-homotopic thinning of , we obtain its k-homotopic thinning set denoted by DT _k . Writing an algorithm for calculating the digital fundamental group of , we investigate the k-fundamental group of by the use of various properties of a digital covering (Z×Z,p ₁×p ₂,DT _k ), a strong k-deformation retract, and algebraic topological tools. Finally, we find the pseudo-multiplicative property (contrary to the multiplicative property) of the digital fundamental group. This property can be used in classifying digital images from the view points of both digital k-homotopy theory and mathematical morphology.

Generalized fuzzy bi-ideals of semigroups

After the introduction of fuzzy sets by Zadeh, there have been a number of generalizations of this fundamental concept. The notion of (∈, ∈ ∨q)-fuzzy subgroups introduced by Bhakat is one among them. In this paper, using the relations between fuzzy points and fuzzy sets, the concept of a fuzzy bi-ideal with thresholds is introduced and some interesting properties are investigated. The acceptable nontrivial concepts obtained in this manner are the (∈, ∈ ∨q)-fuzzy bi-ideals and -fuzzy bi-ideals, which are extension of the concept of a fuzzy bi-ideal in semigroup.     

A semantically complete extension sequence of the system $$\mathcal{L}^* $$

In this paper, the method of well-combined semantics and syntax proposed by Pavelka is applied to the research of the propositional calculus formal system . The partial constant values are taken as formulas, formulas are fuzzified in two manners of semantics and syntax, and inferring processes are fuzzified. A sequence of new extensions { } of the system is proposed, and the completeness of is proved.     

The computational complexity of recognizing permutation functions

Let be a finite field withq elements and a rational function over . No polynomial-time deterministic algorithm is known for the problem of deciding whetherf induces a permutation on . The problem has been shown to be in co-R co-NP, and in this paper we prove that it is inR NP and hence inZPP, and it is deterministic polynomial-time reducible to the problem of factoring univariate polynomials over . Besides the problem of recognizing prime numbers, it seems to be the only natural decision problem inZPP unknown to be inP. A deterministic test and a simple probabilistic test for permutation functions are also presented.     

Asymptotic Fourier Coefficients for a C∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem

In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval , to a function which is periodic on the larger interval . We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval , identically zero for , and varies smoothly in between. Such smoothed “top-hat” functions are “bells” in wavelet theory. Our bell is (for x ≥ 0) where where . By applying steepest descents to approximate the coefficient integrals in the limit of large degree j, we show that when the width L is fixed, the Fourier cosine coefficients a _j of on are proportional to where Λ(j) is an oscillatory factor of degree given in the text. We also show that to minimize error in a Fourier series truncated after the Nth term, the width should be chosen to increase with N as . We derive similar asymptotics for the function f(x)=x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergence     

Generalized fuzzy filters of <Emphasis Type="Italic">R</Emphasis><Subscript>0</Subscript>-algebras

In this paper, we introduce the notions of interval valued -fuzzy filters and interval valued -fuzzy Boolean (implicative) filters in R ₀-algebras and investigate some of their related properties. Some characterization theorems of these generalized fuzzy filters are derived. In particular, we prove that an interval valued fuzzy set F in R ₀-algebras is an interval valued -fuzzy Boolean filter if and only if it is an interval valued -fuzzy implicative filter.     

Ordered fields and {{{\rm \L}\Pi\frac{1}{2}}} -algebras

In this work we further explore the connection between -algebras and ordered fields. We show that any two -chains generate the same variety if and only if they are related to ordered fields that have the same universal theory. This will yield that any -chain generates the whole variety if and only if it contains a subalgebra isomorphic to the -chain of real algebraic numbers, that consequently is the smallest -chain generating the whole variety. We also show that any two different subalgebras of the -chain over the real algebraic numbers generate different varieties. This will be exploited in order to prove that the lattice of subvarieties of -algebras has the cardinality of the continuum. Finally, we will also briefly deal with some model-theoretic properties of -chains related to real closed fields, proving quantifier-elimination and related results.