Branching Time Logics \mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha } with Operations Until and Since Based on Bundles of Integer Numbers, Logical Consecutions, Deciding Algorithms

This paper is intended as an attempt to describe logical consequence in branching time logics. We study temporal branching time logics $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ which use the standard operations Until and Next and dual operations Since and Previous (LTL, as standard, uses only Until and Next). Temporal logics $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ are generated by semantics based on Kripke/Hinttikka structures with linear frames of integer numbers $\mathcal {Z}$ with a single node (glued zeros). For $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ , the permissible branching of the node is limited by α (where 1≤α≤ω). We prove that any logic $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ is decidable w.r.t. admissible consecutions (inference rules), i.e. we find an algorithm recognizing consecutions admissible in $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ . As a consequence, it implies that $\mathcal {BTL}^{\mathrm {U,S}}_{\mathrm {N},\mathrm {N}^{-1}}(\mathcal {Z})_{\alpha }$ itself is decidable and solves the satisfiability problem.     

On the Exact Complexity of Evaluating Quantified k -CNF

We relate the exponential complexities 2 ^s(k)n of $\textsc {$k$-sat}$ and the exponential complexity $2^{s(\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf}))n}$ of $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ (the problem of evaluating quantified formulas of the form $\forall\vec{x} \exists\vec{y} \textsc {F}(\vec {x},\vec{y})$ where F is a 3-cnf in $\vec{x}$ variables and $\vec{y}$ variables) and show that s(∞) (the limit of s(k) as k→∞) is at most $s(\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf}))$ . Therefore, if we assume the Strong Exponential-Time Hypothesis, then there is no algorithm for $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ running in time 2 ^cn with c<1. On the other hand, a nontrivial exponential-time algorithm for $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ would provide a $\textsc {$k$-sat}$ solver with better exponent than all current algorithms for sufficiently large k. We also show several syntactic restrictions of the evaluation problem $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ have nontrivial algorithms, and provide strong evidence that the hardest cases of $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ must have a mixture of clauses of two types: one universally quantified literal and two existentially quantified literals, or only existentially quantified literals. Moreover, the hardest cases must have at least n?o(n) universally quantified variables, and hence only o(n) existentially quantified variables. Our proofs involve the construction of efficient minimally unsatisfiable $\textsc {$k$-cnf}$ s and the application of the Sparsification lemma.     

Delayed theory combination vs. Nelson-Oppen for satisfiability modulo theories: a comparative analysis

Most state-of-the-art approaches for Satisfiability Modulo Theories $(SMT(\mathcal{T}))$ rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory $\mathcal{T} (\mathcal{T}{\text {-}}solver)$ . Often $\mathcal{T}$ is the combination $\mathcal{T}_1 \cup \mathcal{T}_2$ of two (or more) simpler theories $(SMT(\mathcal{T}_1 \cup \mathcal{T}_2))$ , s.t. the specific ${\mathcal{T}_i}{\text {-}}solvers$ must be combined. Up to a few years ago, the standard approach to $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ was to integrate the SAT solver with one combined $\mathcal{T}_1 \cup \mathcal{T}_2{\text {-}}solver$ , obtained from two distinct ${\mathcal{T}_i}{\text {-}}solvers$ by means of evolutions of Nelson and Oppen’s (NO) combination procedure, in which the ${\mathcal{T}_i}{\text {-}}solvers$ deduce and exchange interface equalities. Nowadays many state-of-the-art SMT solvers use evolutions of a more recent $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ procedure called Delayed Theory Combination (DTC), in which each ${\mathcal{T}_i}{\text {-}}solver$ interacts directly and only with the SAT solver, in such a way that part or all of the (possibly very expensive) reasoning effort on interface equalities is delegated to the SAT solver itself. In this paper we present a comparative analysis of DTC vs. NO for $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ . On the one hand, we explain the advantages of DTC in exploiting the power of modern SAT solvers to reduce the search. On the other hand, we show that the extra amount of Boolean search required to the SAT solver can be controlled. In fact, we prove two novel theoretical results, for both convex and non-convex theories and for different deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ , which relate the amount of extra Boolean search required to the SAT solver by DTC with the number of deductions and case-splits required to the ${\mathcal{T}_i}{\text {-}}solvers$ by NO in order to perform the same tasks: (i) under the same hypotheses of deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ required by NO, DTC causes no extra Boolean search; (ii) using ${\mathcal{T}_i}{\text {-}}solvers$ with limited or no deduction capabilities, the extra Boolean search required can be reduced down to a negligible amount by controlling the quality of the $\mathcal{T}$ -conflict sets returned by the ${\mathcal{T}_i}{\text {-}}solvers$ .     

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.     

On matrices, automata, and double counting in constraint programming

Matrix models are ubiquitous for constraint problems. Many such problems have a matrix of variables $\mathcal{M}$ , with the same constraint C defined by a finite-state automaton $\mathcal{A}$ on each row of $\mathcal{M}$ and a global cardinality constraint $\mathit{gcc}$ on each column of $\mathcal{M}$ . We give two methods for deriving, by double counting, necessary conditions on the cardinality variables of the $\mathit{gcc}$ constraints from the automaton $\mathcal{A}$ . The first method yields linear necessary conditions and simple arithmetic constraints. The second method introduces the cardinality automaton, which abstracts the overall behaviour of all the row automata and can be encoded by a set of linear constraints. We also provide a domain consistency filtering algorithm for the conjunction of lexicographic ordering constraints between adjacent rows of $\mathcal{M}$ and (possibly different) automaton constraints on the rows. We evaluate the impact of our methods in terms of runtime and search effort on a large set of nurse rostering problem instances.     

Congestion Games with Linearly Independent Paths: Convergence Time and Price of Anarchy

We investigate the effect of linear independence in the strategies of congestion games on the convergence time of best improvement sequences and on the pure Price of Anarchy. In particular, we consider symmetric congestion games on extension-parallel networks, an interesting class of networks with linearly independent paths, and establish two remarkable properties previously known only for parallel-link games. We show that for arbitrary (non-negative and non-decreasing) latency functions, any best improvement sequence reaches a pure Nash equilibrium in at most as many steps as the number of players, and that for latency functions in class $\mathcal{D}$ , the pure Price of Anarchy is at most $\rho(\mathcal{D})$ , i.e. it is bounded by the Price of Anarchy for non-atomic congestion games. As a by-product of our analysis, we obtain that for symmetric network congestion games with latency functions in class $\mathcal{D}$ , the Price of Stability is at most $\rho(\mathcal{D})$ .     

\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.     

Integrated damping parameter and control design in structural systems for \bf{\mathcal{H}^2} and {\mathcal{H}^{\infty}} specifications

The paper presents a linear matrix inequality (LMI)-based approach for the simultaneous optimal design of output feedback control gains and damping parameters in structural systems with collocated actuators and sensors. The proposed integrated design is based on simplified $\mathcal{H}^2$ and $\mathcal{H}^{\infty}$ norm upper bound calculations for collocated structural systems. Using these upper bound results, the combined design of the damping parameters of the structural system and the output feedback controller to satisfy closed-loop $\mathcal{H}^2$ or $\mathcal{H}^{\infty}$ performance specifications is formulated as an LMI optimization problem with respect to the unknown damping coefficients and feedback gains. Numerical examples motivated from structural and aerospace engineering applications demonstrate the advantages and computational efficiency of the proposed technique for integrated structural and control design. The effectiveness of the proposed integrated design becomes apparent, especially in very large scale structural systems where the use of classical methods for solving Lyapunov and Riccati equations associated with $\mathcal{H}^2$ and $\mathcal{H}^{\infty}$ designs are time-consuming or intractable.     

Order optimal information spreading using algebraic gossip

In this paper we study gossip based information spreading with bounded message sizes. We use algebraic gossip to disseminate $k$ distinct messages to all $n$ nodes in a network. For arbitrary networks we provide a new upper bound for uniform algebraic gossip of $O((k+\log n + D)\varDelta )$ rounds with high probability, where $D$ and $\varDelta $ are the diameter and the maximum degree in the network, respectively. For many topologies and selections of $k$ this bound improves previous results, in particular, for graphs with a constant maximum degree it implies that uniform gossip is order optimal and the stopping time is $\varTheta (k + D)$ . To eliminate the factor of $\varDelta $ from the upper bound we propose a non-uniform gossip protocol, TAG, which is based on algebraic gossip and an arbitrary spanning tree protocol $\mathcal{S } $ . The stopping time of TAG is $O(k+\log n +d(\mathcal{S })+t(\mathcal{S }))$ , where $t(\mathcal{S })$ is the stopping time of the spanning tree protocol, and $d(\mathcal{S })$ is the diameter of the spanning tree. We provide two general cases in which this bound leads to an order optimal protocol. The first is for $k=\varOmega (n)$ , where, using a simple gossip broadcast protocol that creates a spanning tree in at most linear time, we show that TAG finishes after $\varTheta (n)$ rounds for any graph. The second uses a sophisticated, recent gossip protocol to build a fast spanning tree on graphs with large weak conductance. In turn, this leads to the optimally of TAG on these graphs for $k=\varOmega (\text{ polylog }(n))$ . The technique used in our proofs relies on queuing theory, which is an interesting approach that can be useful in future gossip analysis.     

Parallel black box \mathcal {H}-LU preconditioning for elliptic boundary value problems

Hierarchical ( $\mathcal {H}$ -) matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an $\mathcal {H}$ -matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In the context of finite element discretisations of elliptic boundary value problems, $\mathcal {H}$ -matrices can be used for the construction of preconditioners such as approximate $\mathcal {H}$ -LU factors. In this paper, we develop a new black box approach to construct the necessary partition. This new approach is based on the matrix graph of the sparse stiffness matrix and no longer requires geometric data associated with the indices like the standard clustering algorithms. The black box clustering and a subsequent $\mathcal {H}$ -LU factorisation have been implemented in parallel, and we provide numerical results in which the resulting black box $\mathcal {H}$ -LU factorisation is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation.     

On (\overline{\in},\overline{\in} \vee \overline{q})-fuzzy ideals of BCI-algebras

The concepts of $(\overline{\in},\overline{\in} \vee \overline{q})$ -fuzzy (p-, q- and a-) ideals of BCI-algebras are introduced and some related properties are investigated. In particular, we describe the relationships among ordinary fuzzy (p-, q- and a-) ideals, (??,?????? q)-fuzzy (p-, q- and a-) ideals and $(\overline{\in},\overline{\in} \vee \overline{q})$ -fuzzy (p-,q- and a-) ideals of BCI-algebras. Moreover, we prove that a fuzzy set??? of a BCI-algebra X is an $(\overline{\in},\overline{\in} \vee \overline{q})$ -fuzzy a-ideal of X if and only if it is both an $(\overline{\in},\overline{\in} \vee \overline{q})$ -fuzzy p-ideal and an $(\overline{\in},\overline{\in} \vee \overline{q})$ -fuzzy q-ideal. Finally, we give some characterizations of three particular cases of BCI-algebras by these generalized fuzzy ideals.     

Near-Linear Approximation Algorithms for Geometric Hitting Sets

Given a range space $(X,\mathcal{R})$ , where $\mathcal{R}\subset2^{X}$ , the hitting set problem is to find a smallest-cardinality subset H?X that intersects each set in $\mathcal{R}$ . We present near-linear-time approximation algorithms for the hitting set problem in the following geometric settings: (i)? $\mathcal{R}$ is a set of planar regions with small union complexity. (ii)? $\mathcal{R}$ is a set of axis-parallel d-dimensional boxes in ? ^d . In both cases X is either the entire ? ^d , or a finite set of points in ? ^d . The approximation factors yielded by the algorithm are small; they are either the same as, or within very small factors off the best factors known to be computable in polynomial time.     

Detecting and predicting privacy violations in online social networks

Online social networks have become an essential part of social and work life. They enable users to share, discuss, and create content together with various others. Obviously, not all content is meant to be seen by all. It is extremely important to ensure that content is only shown to those that are approved by the content’s owner so that the owner’s privacy is preserved. Generally, online social networks are promising to preserve privacy through privacy agreements, but still everyday new privacy leakages are taking place. Ideally, online social networks should be able to manage and maintain their agreements through well-founded methods. However, the dynamic nature of the online social networks is making it difficult to keep private information contained. We have developed $\mathcal{PROTOSS}$ , a run time tool for detecting and predicting $\mathcal{PR}\mathrm{ivacy}\ \mathrm{vi}\mathcal{O}\mathrm{la}\mathcal{T}\mathrm{ions}\ \mathrm{in}\ \mathcal{O}\mathrm{nline}\ \mathcal{S}\mathrm{ocial}\ \mathrm{network}\mathcal{S}$ . $\mathcal{PROTOSS}$ captures relations among users, their privacy agreements with an online social network operator, as well as domain-based semantic information and rules. It uses model checking to detect if relations among the users will result in the violation of privacy agreements. It can further use the semantic information to infer possible violations that have not been specified by the user explicitly. In addition to detection, $\mathcal{PROTOSS}$ can predict possible future violations by feeding in a hypothetical future world state. Through a running example, we show that $\mathcal{PROTOSS}$ can detect and predict subtle leakages, similar to the ones reported in real life examples. We study the performance of our system on the scenario as well as on an existing Facebook dataset.     

A new PAC bound for intersection-closed concept classes

For hyper-rectangles in $\mathbb{R}^{d}$ Auer (1997) proved a PAC bound of $O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$ , where $\varepsilon$ and $\delta$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $d$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$ and $O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$ . For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$ examples to learn some particular maximum intersection-closed concept classes.     

Exact and Parameterized Algorithms for Max Internal Spanning Tree

We consider the $\mathcal{NP}$ -hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form $\mathcal{O}^{*}(c^{n})$ with c≤2. For graphs with bounded degree, c<2. The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of $\mathcal{O}(1.8612^{n})$ when analyzed with respect to the number of vertices. We also show that its running time is $2.1364^{k} n^{\mathcal{O}(1)}$ when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.     

A note on reverse scheduling with maximum lateness objective

The inverse and reverse counterparts of the single-machine scheduling problem $1||L_{\max }$ are studied in [2], in which the complexity classification is provided for various combinations of adjustable parameters (due dates and processing times) and for five different types of norm: $\ell _{1},\ell _{2},\ell _{\infty },\ell _{H}^{\Sigma } $ , and $\ell _{H}^{\max }$ . It appears that the $O(n^{2})$ -time algorithm for the reverse problem with adjustable due dates contains a flaw. In this note, we present the structural properties of the reverse model, establishing a link with the forward scheduling problem with due dates and deadlines. For the four norms $\ell _{1},\ell _{\infty },\ell _{H}^{\Sigma }$ , and $ \ell _{H}^{\max }$ , the complexity results are derived based on the properties of the corresponding forward problems, while the case of the norm $\ell _{2}$ is treated separately. As a by-product, we resolve an open question on the complexity of problem $1||\sum \alpha _{j}T_{j}^{2}$ .     

On the class of λ-statistically convergent difference sequences of fuzzy numbers

In this study, we introduce the sets $\left[ V,\lambda ,p\right] _{\Updelta }^{{\mathcal{F}}},\left[ C,1,p\right] _{\Updelta }^{{\mathcal{F}}}$ and examine their relations with the classes of $ S_{\lambda }\left( \Updelta ,{\mathcal{F}}\right)$ and $ S_{\mu }\left( \Updelta ,{\mathcal{F}}\right)$ of sequences for the sequences $\left( \lambda _{n}\right)$ and $\left( \mu _{n}\right) , 0<p<\infty $ and difference sequences of fuzzy numbers.     

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.     

Some properties of Rényi entropy over countably infinite alphabets

We study certain properties of Rényi entropy functionals $H_\alpha \left( \mathcal{P} \right)$ on the space of probability distributions over ?₊. Primarily, continuity and convergence issues are addressed. Some properties are shown to be parallel to those known in the finite alphabet case, while others illustrate a quite different behavior of the Rényi entropy in the infinite case. In particular, it is shown that for any distribution $\mathcal{P}$ and any r ∈ [0,∞] there exists a sequence of distributions $\mathcal{P}_n$ converging to $\mathcal{P}$ with respect to the total variation distance and such that $\mathop {\lim }\limits_{n \to \infty } \mathop {\lim }\limits_{\alpha \to 1 + } H_\alpha \left( {\mathcal{P}_n } \right) = \mathop {\lim }\limits_{\alpha \to 1 + } \mathop {\lim }\limits_{n \to \infty } H_\alpha \left( {\mathcal{P}_n } \right) + r$ .