Chain conditions on \textit{BL}-algebras

The aim of this paper is to solve the open problem appeared in Motamed and Moghaderi (Soft Comput 2012), about the relation between Noetherian (Artinian) $\textit{BL}$ -algebras in short exact sequences. Also, a better theorem to improve its results is suggested. The relation between Noetherian and Artinian $\textit{BL}$ -algebras is found, the concept of length for a filter in $\textit{BL}$ -algebras is introduced and properties of finite length $\textit{BL}$ -algebras are developed. Finally, it is proved that any $\textit{BL}$ -algebra has finite length if and only if be Noetherian and Artinian.     

Arithmetizing Classes Around \textsf{NC} 1 and \textsf{L}

The parallel complexity class $\textsf{NC}$ ¹ has many equivalent models such as polynomial size formulae and bounded width branching programs. Caussinus et al. (J. Comput. Syst. Sci. 57:200–212, 1992) considered arithmetizations of two of these classes, $\textsf{\#NC}$ ¹ and $\textsf{\#BWBP}$ . We further this study to include arithmetization of other classes. In particular, we show that counting paths in branching programs over visibly pushdown automata is in $\textsf{FLogDCFL}$ , while counting proof-trees in logarithmic width formulae has the same power as $\textsf{\#NC}$ ¹. We also consider polynomial-degree restrictions of $\textsf{SC}$ ⁱ , denoted $\textsf{sSC}$ ⁱ , and show that the Boolean class $\textsf{sSC}$ ¹ is sandwiched between $\textsf{NC}$ ¹ and $\textsf{L}$ , whereas $\textsf{sSC}$ ⁰ equals $\textsf{NC}$ ¹. On the other hand, the arithmetic class $\textsf{\#sSC}$ ⁰ contains $\textsf{\#BWBP}$ and is contained in $\textsf{FL}$ , and $\textsf{\#sSC}$ ¹ contains $\textsf{\#NC}$ ¹ and is in $\textsf{SC}$ ². We also investigate some closure properties of the newly defined arithmetic classes.     

A new set of generators and a physical interpretation for the SU (3) finite subgroup D(9,1,1;2,1,1)

After 100 years of effort, the classification of all the finite subgroups of $SU(3)$ is yet incomplete. The most recently updated list can be found in Ludl (J Phys A Math Theory 44:255204, 2011), where the structure of the series $(C)$ and $(D)$ of $SU(3)$ -subgroups is studied. We provide a minimal set of generators for one of these groups which has order $162$ . These generators appear up to phase as the image of an irreducible unitary braid group representation issued from the Jones–Kauffman version of $SU(2)$ Chern–Simons theory at level $4$ . In light of these new generators, we study the structure of the group in detail and recover the fact that it is isomorphic to the semidirect product $\mathbb Z _9\times \mathbb Z _3\rtimes S_3$ with respect to conjugation.     

\lambda -Statistical convergence of fuzzy numbers and fuzzy functions of order \theta

In this paper, we introduce the concept of $\lambda $ -statistical convergence of order $\theta $ and strong $\lambda $ -summability of order $\theta $ for the sequence of fuzzy numbers. Further the same concept is extended to the sequence of fuzzy functions and introduce the spaces like $S_\lambda ^\theta (\hat{f})$ and $\omega _{\lambda p} ^\theta (\hat{f})$ . Some inclusion relations in those spaces and also the underlying relation between these two spaces are also obtained.     

Improved Approximation of Linear Threshold Functions

We prove two main results on how arbitrary linear threshold functions ${f(x) = {\rm sign}(w \cdot x - \theta)}$ over the n-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every n-variable threshold function f is ${\epsilon}$ -close to a threshold function depending only on ${{\rm Inf}(f)^2 \cdot {\rm poly}(1/\epsilon)}$ many variables, where ${{\rm Inf}(f)}$ denotes the total influence or average sensitivity of f. This is an exponential sharpening of Friedgut’s well-known theorem (Friedgut in Combinatorica 18(1):474–483, 1998), which states that every Boolean function f is ${\epsilon}$ -close to a function depending only on ${2^{O({\rm Inf}(f)/\epsilon)}}$ many variables, for the case of threshold functions. We complement this upper bound by showing that ${\Omega({\rm Inf}(f)^2 + 1/\epsilon^2)}$ many variables are required for ${\epsilon}$ -approximating threshold functions. Our second result is a proof that every n-variable threshold function is ${\epsilon}$ -close to a threshold function with integer weights at most ${{\rm poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2/3})}.}$ This is an improvement, in the dependence on the error parameter ${\epsilon}$ , on an earlier result of Servedio (Comput Complex 16(2):180–209, 2007) which gave a ${{\rm poly}(n) \cdot 2^{\tilde{O}(1/\epsilon^{2})}}$ bound. Our improvement is obtained via a new proof technique that uses strong anti-concentration bounds from probability theory. The new technique also gives a simple and modular proof of the original result of Servedio (Comput Complex 16(2):180–209, 2007) and extends to give low-weight approximators for threshold functions under a range of probability distributions other than the uniform distribution.     

An improved FPTAS for maximizing the weighted number of just-in-time jobs in a two-machine flow shop problem

Recently, Shabtay and Bensoussan (2012) developed an original exact pseudo-polynomial algorithm and an efficient $\upvarepsilon $ -approximation algorithm (FPTAS) for maximizing the weighted number of just-in-time jobs in a two-machine flow shop problem. The complexity of the FPTAS is $O$ (( $n^{4}/\upvarepsilon $ )log( $n$ / $\upvarepsilon $ )), where $n$ is the number of jobs. In this note we suggest another pseudo-polynomial algorithm that can be converted to a new FPTAS which improves Shabtay–Bensoussan’s complexity result and runs in $O(n^{3}/\upvarepsilon )$ time.     

New types of fuzzy ideals of near-rings

In this paper, we consider the $(\in_{\gamma},\in_{\gamma} \vee \; \hbox{q}_{\delta})$ -fuzzy and $(\overline{\in}_{\gamma},\overline{\in}_{\gamma} \vee \; \overline{\hbox{q}}_{\delta})$ -fuzzy subnear-rings (ideals) of a near-ring. Some new characterizations are also given. In particular, we introduce the concepts of (strong) prime $(\in_{\gamma},\in_{\gamma} \vee \; \hbox{q}_{\delta})$ -fuzzy ideals of near-rings and discuss the relationship between strong prime $(\in_{\gamma},\in_{\gamma} \vee \; \hbox{q}_{\delta})$ -fuzzy ideals and prime $(\in_{\gamma},\in_{\gamma} \vee \; \hbox{q}_{\delta})$ -fuzzy ideals of near-rings.     

Computational complexity and approximability of social welfare optimization in multiagent resource allocation

A central task in multiagent resource allocation, which provides mechanisms to allocate (bundles of) resources to agents, is to maximize social welfare. We assume resources to be indivisible and nonshareable and agents to express their utilities over bundles of resources, where utilities can be represented in the bundle form, the $k$ -additive form, and as straight-line programs. We study the computational complexity of social welfare optimization in multiagent resource allocation, where we consider utilitarian and egalitarian social welfare and social welfare by the Nash product. Solving some of the open problems raised by Chevaleyre et al. (2006) and confirming their conjectures, we prove that egalitarian social welfare optimization is $\mathrm{NP}$ -complete for the bundle form, and both exact utilitarian and exact egalitarian social welfare optimization are $\mathrm{DP}$ -complete, each for both the bundle and the $2$ -additive form, where $\mathrm{DP}$ is the second level of the boolean hierarchy over  $\mathrm{NP}$ . In addition, we prove that social welfare optimization by the Nash product is $\mathrm{NP}$ -complete for both the bundle and the $1$ -additive form, and that the exact variants are $\mathrm{DP}$ -complete for the bundle and the $3$ -additive form. For utility functions represented as straight-line programs, we show $\mathrm{NP}$ -completeness for egalitarian social welfare optimization and social welfare optimization by the Nash product. Finally, we show that social welfare optimization by the Nash product in the $1$ -additive form is hard to approximate, yet we also give fully polynomial-time approximation schemes for egalitarian and Nash product social welfare optimization in the $1$ -additive form with a fixed number of agents.     

Pseudorandom generators for CC0[p] and the Fourier spectrum of low-degree polynomials over finite fields

In this paper, we give the first construction of a pseudorandom generator, with seed length O(log n), for CC⁰[p], the class of constant-depth circuits with unbounded fan-in MOD _p gates, for some prime p. More accurately, the seed length of our generator is O(log n) for any constant error ${\epsilon > 0}$ . In fact, we obtain our generator by fooling distributions generated by low-degree polynomials, over ${\mathbb{F}_p}$ , when evaluated on the Boolean cube. This result significantly extends previous constructions that either required a long seed (Luby et al. 1993) or could only fool the distribution generated by linear functions over ${\mathbb{F}_p}$ , when evaluated on the Boolean cube (Lovett et al. 2009; Meka & Zuckerman 2009). En route of constructing our PRG, we prove two structural results for low-degree polynomials over finite fields that can be of independent interest.

1. Let f be an n-variate degree d polynomial over ${\mathbb{F}_p}$ . Then, for every ${\epsilon > 0}$ , there exists a subset ${S \subset [n]}$ , whose size depends only on d and ${\epsilon}$ , such that ${\sum_{\alpha \in \mathbb{F}_p^n: \alpha \ne 0, \alpha_S=0}|\hat{f}(\alpha)|^2 \leq \epsilon}$ . Namely, there is a constant size subset S such that the total weight of the nonzero Fourier coefficients that do not involve any variable from S is small.
2. Let f be an n-variate degree d polynomial over ${\mathbb{F}_p}$ . If the distribution of f when applied to uniform zero–one bits is ${\epsilon}$ -far (in statistical distance) from its distribution when applied to biased bits, then for every ${\delta > 0}$ , f can be approximated over zero–one bits, up to error δ, by a function of a small number (depending only on ${\epsilon,\delta}$ and d) of lower degree polynomials.

     

Distinguishing and relating higher-order and first-order processes by expressiveness

This is a paper on distinguishing and relating two important kinds of calculi through expressiveness, settling some critical but long unanswered questions. The delimitation of higher-order and first-order process calculi is a basic and pivotal topic in the study of process theory. Particularly, expressiveness studies mutual encodability, which helps decide whether process-passing or name-passing is more fundamental, and the way they ought to be used in both theory and practice. In this paper, we contribute to such demarcation with three major results. Firstly $\pi $ (first-order pi-calculus) can faithfully express $\varPi $ (basic higher-order pi-calculus). The calculus $\varPi $ has the elementary operators (input, output, composition and restriction). This actually is a corollary of a more general result, that $\pi $ can encode $\varPi ^r$ ( $\varPi $ enriched with the relabelling operator). Secondly $\varPi $ cannot interpret $\pi $ reasonably. This is of more significance since it separates $\varPi $ and $\pi $ by drawing a well-defined boundary. Thirdly an encoding from $\pi $ to $\varPi ^r$ is revisited and discussed, which not only implies how to make $\varPi $ more useful but also stresses the importance of name-passing in $\pi $ .     

Derandomized Parallel Repetition Theorems for Free Games

Raz’s parallel repetition theorem (SIAM J Comput 27(3):763–803, 1998) together with improvements of Holenstein (STOC, pp 411–419, 2007) shows that for any two-prover one-round game with value at most ${1- \epsilon}$ 1 - ? (for ${\epsilon \leq 1/2}$ ? ≤ 1 / 2 ), the value of the game repeated n times in parallel on independent inputs is at most ${(1- \epsilon)^{\Omega(\frac{\epsilon^2 n}{\ell})}}$ ( 1 - ? ) Ω ( ? 2 n ? ) , where ? is the answer length of the game. For free games (which are games in which the inputs to the two players are uniform and independent), the constant 2 can be replaced with 1 by a result of Barak et al. (APPROX-RANDOM, pp 352–365, 2009). Consequently, ${n=O(\frac{t \ell}{\epsilon})}$ n = O ( t ? ? ) repetitions suffice to reduce the value of a free game from ${1- \epsilon}$ 1 - ? to ${(1- \epsilon)^t}$ ( 1 - ? ) t , and denoting the input length of the game by m, it follows that ${nm=O(\frac{t \ell m}{\epsilon})}$ n m = O ( t ? m ? ) random bits can be used to prepare n independent inputs for the parallel repetition game. In this paper, we prove a derandomized version of the parallel repetition theorem for free games and show that O(t(m + ?)) random bits can be used to generate correlated inputs, such that the value of the parallel repetition game on these inputs has the same behavior. That is, it is possible to reduce the value from ${1- \epsilon}$ 1 - ? to ${(1- \epsilon)^t}$ ( 1 - ? ) t while only multiplying the randomness complexity by O(t) when m = O(?). Our technique uses strong extractors to “derandomize” a lemma of Raz and can be also used to derandomize a parallel repetition theorem of Parnafes et al. (STOC, pp 363–372, 1997) for communication games in the special case that the game is free.     

Combinatorial algorithms for distributed graph coloring

Numerous problems in Theoretical Computer Science can be solved very efficiently using powerful algebraic constructions. Computing shortest paths, constructing expanders, and proving the PCP Theorem, are just few examples of this phenomenon. The quest for combinatorial algorithms that do not use heavy algebraic machinery, but are roughly as efficient, has become a central field of study in this area. Combinatorial algorithms are often simpler than their algebraic counterparts. Moreover, in many cases, combinatorial algorithms and proofs provide additional understanding of studied problems. In this paper we initiate the study of combinatorial algorithms for Distributed Graph Coloring problems. In a distributed setting a communication network is modeled by a graph $G=(V,E)$ of maximum degree $\varDelta $ . The vertices of $G$ host the processors, and communication is performed over the edges of $G$ . The goal of distributed vertex coloring is to color $V$ with $(\varDelta + 1)$ colors such that any two neighbors are colored with distinct colors. Currently, efficient algorithms for vertex coloring that require $O(\varDelta + \log ^* n)$ time are based on the algebraic algorithm of Linial (SIAM J Comput 21(1):193–201, 1992) that employs set-systems. The best currently-known combinatorial set-system free algorithm, due to Goldberg et al. (SIAM J Discret Math 1(4):434–446, 1988), requires $O(\varDelta ^2+\log ^*n)$ time. We significantly improve over this by devising a combinatorial $(\varDelta + 1)$ -coloring algorithm that runs in $O(\varDelta + \log ^* n)$ time. This exactly matches the running time of the best-known algebraic algorithm. In addition, we devise a tradeoff for computing $O(\varDelta \cdot t)$ -coloring in $O(\varDelta /t + \log ^* n)$ time, for almost the entire range $1 < t < \varDelta $ . We also compute a Maximal Independent Set in $O(\varDelta + \log ^* n)$ time on general graphs, and in $O(\log n/ \log \log n)$ time on graphs of bounded arboricity. Prior to our work, these results could be only achieved using algebraic techniques. We believe that our algorithms are more suitable for real-life networks with limited resources, such as sensor networks.     

Error Analysis of a Compact ADI Scheme for the 2D Fractional Subdiffusion Equation

In this paper, a Crank–Nicolson-type compact ADI scheme is proposed for solving two-dimensional fractional subdiffusion equation. The unique solvability, unconditional stability and convergence of the scheme are proved rigorously. Two error estimates are presented. One is $\mathcal{O }(\tau ^{\min \{2-\frac{\gamma }{2},\,2\gamma \}}+h_1^4+h^4_2)$ in standard $H^1$ norm, where $\tau $ is the temporal grid size and $h_1,h_2$ are spatial grid sizes; the other is $\mathcal{O }(\tau ^{2\gamma }+h_1^4+h^4_2)$ in $H^1_{\gamma }$ norm, a generalized norm which is associated with the Riemann–Liouville fractional integral operator. Numerical results are presented to support the theoretical analysis.     

Structuring unreliable radio networks

In this paper we study the problem of building a constant-degree connected dominating set (CCDS), a network structure that can be used as a communication backbone, in the dual graph radio network model (Clementi et al. in J Parallel Distrib Comput 64:89–96, 2004; Kuhn et al. in Proceedings of the international symposium on principles of distributed computing 2009, Distrib Comput 24(3–4):187–206 2011, Proceedings of the international symposium on principles of distributed computing 2010). This model includes two types of links: reliable, which always deliver messages, and unreliable, which sometimes fail to deliver messages. Real networks compensate for this differing quality by deploying low-layer detection protocols to filter unreliable from reliable links. With this in mind, we begin by presenting an algorithm that solves the CCDS problem in the dual graph model under the assumption that every process $u$ is provided with a local link detector set consisting of every neighbor connected to $u$ by a reliable link. The algorithm solves the CCDS problem in $O\left( \frac{\varDelta \log ^2{n}}{b} + \log ^3{n}\right) $ rounds, with high probability, where $\varDelta $ is the maximum degree in the reliable link graph, $n$ is the network size, and $b$ is an upper bound in bits on the message size. The algorithm works by first building a Maximal Independent Set (MIS) in $\log ^3{n}$ time, and then leveraging the local topology knowledge to efficiently connect nearby MIS processes. A natural follow-up question is whether the link detector must be perfectly reliable to solve the CCDS problem. With this in mind, we first describe an algorithm that builds a CCDS in $O(\varDelta $ polylog $(n))$ time under the assumption of $O(1)$ unreliable links included in each link detector set. We then prove this algorithm to be (almost) tight by showing that the possible inclusion of only a single unreliable link in each process’s local link detector set is sufficient to require $\varOmega (\varDelta )$ rounds to solve the CCDS problem, regardless of message size. We conclude by discussing how to apply our algorithm in the setting where the topology of reliable and unreliable links can change over time.     

A Speculative Parallel DFA Membership Test for Multicore,SIMD and Cloud Computing Environments

We present techniques to parallelize membership tests for Deterministic Finite Automata (DFAs). Our method searches arbitrary regular expressions by matching multiple bytes in parallel using speculation. We partition the input string into chunks, match chunks in parallel, and combine the matching results. Our parallel matching algorithm exploits structural DFA properties to minimize the speculative overhead. Unlike previous approaches, our speculation is failure-free, i.e., (1) sequential semantics are maintained, and (2) speed-downs are avoided altogether. On architectures with a SIMD gather-operation for indexed memory loads, our matching operation is fully vectorized. The proposed load-balancing scheme uses an off-line profiling step to determine the matching capacity of each participating processor. Based on matching capacities, DFA matches are load-balanced on inhomogeneous parallel architectures such as cloud computing environments. We evaluated our speculative DFA membership test for a representative set of benchmarks from the Perl-compatible Regular Expression (PCRE) library and the PROSITE protein database. Evaluation was conducted on a 4 CPU (40 cores) shared-memory node of the Intel Academic Program Manycore Testing Lab (Intel MTL), on the Intel AVX2 SDE simulator for 8-way fully vectorized SIMD execution, and on a 20-node (288 cores) cluster on the Amazon EC2 computing cloud. Obtained speedups are on the order of $\mathcal O \left( 1+\frac{|P|-1}{|Q|\cdot \gamma }\right) $ , where $|P|$ denotes the number of processors or SIMD units, $|Q|$ denotes the number of DFA states, and $0<\gamma \le 1$ represents a statically computed DFA property. For all observed cases, we found that $0.02<\gamma <0.47$ . Actual speedups range from 2.3 $\times $ to 38.8 $\times $ for up to 512 DFA states for PCRE, and between 1.3 $\times $ and 19.9 $\times $ for up to 1,288 DFA states for PROSITE on a 40-core MTL node. Speedups on the EC2 computing cloud range from 5.0 $\times $ to 65.8 $\times $ for PCRE, and from 5.0 $\times $ to 138.5 $\times $ for PROSITE. Speedups of our C-based DFA matcher over the Perl-based ScanProsite scan tool range from 559.3 $\times $ to 15079.7 $\times $ on a 40-core MTL node. We show the scalability of our approach for input-sizes of up to 10 GB.     

Quantum discord in spin systems with dipole–dipole interaction

The behavior of total quantum correlations (discord) in dimers consisting of dipolar-coupled spins 1/2 are studied. We found that the discord $Q=0$ at absolute zero temperature. As the temperature $T$ increases, the quantum correlations in the system increase at first from zero to its maximum and then decrease to zero according to the asymptotic law $T^{-2}$ . It is also shown that in absence of external magnetic field $B$ , the classical correlations $C$ at $T\rightarrow 0$ are, vice versa, maximal. Our calculations predict that in crystalline gypsum $\hbox {CaSO}_{4}\cdot \hbox {2H}_{2}{\hbox {O}}$ the value of natural $(B=0)$ quantum discord between nuclear spins of hydrogen atoms is maximal at the temperature of 0.644  $\upmu $ K, and for 1,2-dichloroethane $\hbox {H}_{2}$ ClC– $\hbox {CH}_{2}{\hbox {Cl}}$ the discord achieves the largest value at $T=0.517~\upmu $ K. In both cases, the discord equals $Q\approx 0.083$  bit/dimer what is $8.3\,\%$ of its upper limit in two-qubit systems. We estimate also that for gypsum at room temperature $Q\sim 10^{-18}$  bit/dimer, and for 1,2-dichloroethane at $T=90$  K the discord is $Q\sim 10^{-17}$  bit per a dimer.     

Solving Convection-Diffusion Problems on Curved Domains by Extensions from Subdomains

We present a technique for numerically solving convection-diffusion problems in domains $\varOmega $ with curved boundary. The technique consists in approximating the domain $\varOmega $ by polyhedral subdomains $\mathsf{{D}}_h$ where a finite element method is used to solve for the approximate solution. The approximation is then suitably extended to the remaining part of the domain $\varOmega $ . This approach allows for the use of only polyhedral elements; there is no need of fitting the boundary in order to obtain an accurate approximation of the solution. To achieve this, the boundary condition on the border of $\varOmega $ is transferred to the border of $\mathsf{D }_h$ by using simple line integrals. We apply this technique to the hybridizable discontinuous Galerkin method and provide extensive numerical experiments showing that, whenever the distance of $\mathsf{{D}}_h$ to $\partial \varOmega $ is of order of the meshsize $h$ , the convergence properties of the resulting method are the same as those for the case in which $\varOmega =\mathsf{{D}}_h$ . We also show numerical evidence indicating that the ratio of the $L^2(\varOmega )$ norm of the error in the scalar variable computed with $d>0$ to that of that computed with $d=0$ remains constant (and fairly close to one), whenever the distance $d$ is proportional to $\min \{h,Pe^{-1}\}/(k+1)^2$ , where $Pe$ is the so-called Péclet number.     

A connection between the Cantor–Bendixson derivative and the well-founded semantics of finite logic programs

Results of Schlipf (J Comput Syst Sci 51:64?C86, 1995) and Fitting (Theor Comput Sci 278:25?C51, 2001) show that the well-founded semantics of a finite predicate logic program can be quite complex. In this paper, we show that there is a close connection between the construction of the perfect kernel of a $\Pi^0_1$ class via the iteration of the Cantor?CBendixson derivative through the ordinals and the construction of the well-founded semantics for finite predicate logic programs via Van Gelder??s alternating fixpoint construction. This connection allows us to transfer known complexity results for the perfect kernel of $\Pi^0_1$ classes to give new complexity results for various questions about the well-founded semantics ${\mathit{wfs}}(P)$ of a finite predicate logic program P.     

On weighted first-order logics with discounting

We introduce a linear temporal logic and a first-order logic in the weighted setup of the max-plus semiring with discounting parameters in $[0,1)$ . Furthermore, we define $\omega \hbox {-}d$ -star-free series and counter-free weighted Büchi automata. We show that the classes of series definable in fragments of the weighted linear temporal logic and first-order logic, the class of $\omega \hbox {-}d$ -star-free series, and a subclass of $\omega \hbox {-}d$ -counter-free series coincide. This extends a fundamental result, for first-order logic theory, to series over the max-plus semiring with discounting.