On fuzzy rough sets based on tolerance relations

Dubois and Prade (1990) [1] introduced the notion of fuzzy rough sets as a fuzzy generalization of rough sets, which was originally proposed by Pawlak (1982) [8]. Later, Radzikowska and Kerre introduced the so-called (I,T)-fuzzy rough sets, where I is an implication and T is a triangular norm. In the present paper, by using a pair of implications (I,J), we define the so-called (I,J)-fuzzy rough sets, which generalize the concept of fuzzy rough sets in the sense of Radzikowska and Kerre, and that of Mi and Zhang. Basic properties of (I,J)-fuzzy rough sets are investigated in detail.     

一种基于共轭梯度的LMBP改进学习算法

对神经网络中的LMBP（Levenberg-Marquardt BP）算法的收敛速度慢进行分析,针对矩阵J^TJ+µI求逆过程运算量过大而造成收敛速度慢的缺陷,根据无约束优化理论,提出一种基于共轭梯度方法的改进LMBP网络学习算法,利用求解大规模线性方程组的共轭梯度方法,避免了烦琐的求逆过程,降低了计算复杂度,加快了网络的收敛速度,通过Matlab仿真,比较了算法的收敛速度,证明了方法的有效性。     

Flat families by strongly stable ideals and a generalization of Gröbner bases

Let J be a strongly stable monomial ideal in S=K[x₀,…,x_n] and let Mf(J) be the family of all homogeneous ideals I in S such that the set of all terms outside J is a K-vector basis of the quotient S/I. We show that an ideal I belongs to Mf(J) if and only if it is generated by a special set of polynomials, the J-marked basis of I, that in some sense generalizes the notion of reduced Gröbner basis and its constructive capabilities. Indeed, although not every J-marked basis is a Gröbner basis with respect to some term order, a sort of reduced form modulo I∈Mf(J) can be computed for every homogeneous polynomial, so that a J-marked basis can be characterized by a Buchberger-like criterion. Using J-marked bases, we prove that the family Mf(J) can be endowed, in a very natural way, with a structure of an affine scheme that turns out to be homogeneous with respect to a non-standard grading and flat in the origin (the point corresponding to J), thanks to properties ofJ-marked bases analogous to those of Gröbner bases about syzygies.     

Improving active Mealy machine learning for protocol conformance testing

Using a well-known industrial case study from the verification literature, the bounded retransmission protocol, we show how active learning can be used to establish the correctness of protocol implementation I relative to a given reference implementation R. Using active learning, we learn a model M _R of reference implementation R, which serves as input for a model-based testing tool that checks conformance of implementation I to M _R . In addition, we also explore an alternative approach in which we learn a model M _I of implementation I, which is compared to model M _R using an equivalence checker. Our work uses a unique combination of software tools for model construction (Uppaal), active learning (LearnLib, Tomte), model-based testing (JTorX, TorXakis) and verification (CADP, MRMC). We show how these tools can be used for learning models of and revealing errors in implementations, present the new notion of a conformance oracle, and demonstrate how conformance oracles can be used to speed up conformance checking.     

Approximating the Semantics of Logic Programs by Recurrent Neural Networks

In [1] we have shown how to construct a 3-layered recurrent neural network that computes the fixed point of the meaning function T_P of a given propositional logic program P, which corresponds to the computation of the semantics of P. In this article we consider the first order case. We define a notion of approximation for interpretations and prove that there exists a 3-layered feed forward neural network that approximates the calculation of T_P for a given first order acyclic logic program P with an injective level mapping arbitrarily well. Extending the feed forward network by recurrent connections we obtain a recurrent neural network whose iteration approximates the fixed point of T_P. This result is proven by taking advantage of the fact that for acyclic logic programs the function T_P is a contraction mapping on a complete metric space defined by the interpretations of the program. Mapping this space to the metric space R with Euclidean distance, a real valued function f_P can be defined which corresponds to T_P and is continuous as well as a contraction. Consequently it can be approximated by an appropriately chosen class of feed forward neural networks.     

Honest polynomial degrees and P=? NP

We present a relatively simple proof of a result from Homer (1986) showing that if nonrecursive sets cannot be minimal for honest polynomial-time Turing reducibility (⩽^h_T-minimal), then P ≠ NP. As a corollary to our proof, we strengthen Homer's result by showing, without assuming that P ≠ NP, that there are ⩽^h_T-minimal sets for all tally sets. We also consider the converse of Homer's result, providing some evidence that the nonexistence of ⩽^h_T-minimal sets may folow from P ≠ NP in an interesting way. Finally, we consider structural and/or computability properties of sets that cannot be ⩽^h_T-minimal.     

A measure of information gained through biometric systems

We propose a measure of information gained through biometric matching systems. Firstly, we discuss how the information about the identity of a person is derived from biometric samples through a biometric system, and define the “biometric system entropy” or BSE based on mutual information. We present several theoretical properties and interpretations of the BSE, and show how to design a biometric system which maximizes the BSE. Then we prove that the BSE can be approximated asymptotically by the relative entropy D(f_G(x)∥f_I(x)) where f_G(x) and f_I(x) are probability mass functions of matching scores between samples from individuals and among population. We also discuss how to evaluate the BSE of a biometric system and show experimental evaluation of the BSE of face, fingerprint and multimodal biometric systems.     

On characterization of intuitionistic fuzzy rough sets based on intuitionistic fuzzy implicators

This paper presents a general framework for the study of relation-based (I,T)-intuitionistic fuzzy rough sets by using constructive and axiomatic approaches. In the constructive approach, by employing an intuitionistic fuzzy implicator I and an intuitionistic fuzzy triangle norm T, lower and upper approximations of intuitionistic fuzzy sets with respect to an intuitionistic fuzzy approximation space are first defined. Properties of (I,T)-intuitionistic fuzzy rough approximation operators are examined. The connections between special types of intuitionistic fuzzy relations and properties of intuitionistic fuzzy approximation operators are established. In the axiomatic approach, an operator-oriented characterization of (I,T)-intuitionistic fuzzy rough sets is proposed. Different axiom sets characterizing the essential properties of intuitionistic fuzzy approximation operators associated with various intuitionistic fuzzy relations are explored.     

Connectivity-preserving transformations of binary images

A binary image I is B_a, W_b-connected, where a, b ∈ {4, 8}, if its foreground is a-connected and its background is b-connected. We consider a local modification of a B_a, w_b-connected image I in which a black pixel can be interchanged with an adjacent white pixel provided that this preserves the connectivity of both the foreground and the background of I. We have shown that for any (a, b) ∈ {(4, 8), (8, 4), (8, 8)}, any two B_a, w_b-connected images I and J each with n black pixels differ by a sequence of Θ(n²) interchanges. We have also shown that any two B₄, W₄-connected images I and J each with n black pixels differ by a sequence of O(n⁴) interchanges.     

Preconditioned conjugate gradient methods for the solution of indefinite least squares problems

The conjugate gradient (CG) method is considered for solving the large and sparse indefinite least squares (ILS) problem min  _x (b−Ax) ^T J(b−Ax) where J=diag (I _p ,−I _q ) is a signature matrix. However the rate of convergence becomes slow for ill-conditioned problems. The QR-based preconditioner is found to be effective in accelerating the convergence. Numerical results show that the sparse Householder QR-based preconditioner is superior to the CG method especially for sparse and ill-conditioned problems.     

Necessary and sufficient conditions for existence of J-spectral factorization for para-Hermitian rational matrix functions

For a para-Hermitian rational matrix function G(λ)=J+C(λI_p−A)⁻¹B, where J=J∗ is invertible, and which has no poles and zeros on the imaginary line, we give necessary and sufficient conditions in terms of A,B,C and J for the existence of a J-spectral factorization, as well as an algorithm to obtain the J-spectral factor in case it exists.     

Sublimation enthalpies of terbium and lutetium triiodides and formation enthalpies of their monomer and dimer molecules

The sublimation of terbium and lutetium triiodides was studied by Knudsen effusion mass spectrometry. The temperature dependencies of partial vapor pressures [atm] of the monomer and dimer molecules were determined as: lnp(TbI₃) = – (31.74 ± 0.24)× 10³/T + (21.43 ± 0.29) for 743–906 K; lnp(Tb₂I₆) = – (38.66 ± 0.38)× 10³/T + (25.63 ± 0.46) for 786–906 K; lnp(LuI₃) = – (31.48 ± 0.25)× 10³/T + (22.28 ± 0.30) for 721–951 K; and lnp(Lu₂I₆) = – (36.95 ± 0.37)× 10³/T + (25.21 ± 0.43) for 796–950 K. On the basis of a joint analysis of all literature data, the sublimation enthalpies [kJ mol^–1] at 298.15 K are recommended as 281 ± 3 (TbI₃), 346 ± 30 (Tb₂I₆), 279 ± 12 (LuI₃), and 340 ± 30 (Lu₂I₆). The standard formation enthalpies [kJ mol^–1] of the gaseous species at 298.15 K are –343 ± 4 (TbI₃), –902 ± 30 (Tb₂I₆),–326 ± 12 (LuI₃), and –870 ± 30 (Lu₂I₆).     

Saturation,nonmonotonic reasoning and the closed-world assumption

To deal with the problem of implicit negative information in databases, it is necessary to use a nonmonotonic form of reasoning.The form of reasoning described in this paper uses an ordering relation on interpretations of the type: I is inferior to J if every ‘elementary fact’ true in I is true in J. In simple cases, this rule may be defined by: a set P of formulas implies a formula q if every minimal model of P satisfies q. In more general cases, this definition leads to paradox, if P is consistent and has no minimal model (P must not imply q and its negation).We have:

• 1.(1) carefully formulated the definition of a special type of implication (called ‘sub-implication’) to avoid this type of paradox;
• 2.(2) proved that if P is a set of clauses, then P has a minimal model, and more generally that every model of a set of clauses is minored by one of its minimal models;
• 3.(3) developed (in this last case subject to certain conditions such as the use of a particular set of clauses called groundable clauses or g-clauses) a decidable algorithm to compute whether P implies q, q being practically any formula.

     

Tyrolean termination tool: Techniques and features

The Tyrolean Termination Tool (T_TT for short) is a powerful tool for automatically proving termination of rewrite systems. It incorporates several new refinements of the dependency pair method that are easy to implement, increase the power of the method, result in simpler termination proofs, and make the method more efficient. T_TT employs polynomial interpretations with negative coefficients, like x − 1 for a unary function symbol or x − y for a binary function symbol, which are useful for extending the class of rewrite systems that can be proved terminating automatically. Besides a detailed account of these techniques, we describe the convenient web interface of T_TT and provide some implementation details.     

Anomalous pulse change in gravity-driven microfluidic oscillator and its application to photodiode switching

Electrofluidic analogy is useful because it provides a method to significantly reduce the reliance of microfluidic chips on dynamic off-chip controllers. Among the functions developed by the analogy, conversion from constant to pulsatile pressure is critical and is yet to be studied. Here, unlike its counterpart electrical oscillator generating square pulses more slowly with decreasing the input voltage, we report that a microfluidic oscillator generates sawtooth pressure pulses more rapidly with decreasing the input pressure (P_I) at 1–2 kPa. Further, with decreasing P_I, the oscillator generates square pulses at P_I > 3.4 kPa, but its operation unexpectedly stops at 2.1 < P_I < 3.4 kPa. We analyze its underlying mechanism with a sophisticated model including a dynamic interaction of the oscillator components and reveal the critical role of the dynamic property of oscillator valves. Additionally, we show electrofluidic switching of a photodiode with the oscillator. The understanding obtained in this study would be essential for developing microfluidic circuits using electrofluidic analogy.     

Interactive Restless Multi-armed Bandit Game and Swarm Intelligence Effect

We obtain the conditions for the emergence of the swarm intelligence effect in an interactive game of restless multi-armed bandit (rMAB). A player competes with multiple agents. Each bandit has a payoff that changes with a probability p _c per round. The agents and player choose one of three options: (1) Exploit (a good bandit), (2) Innovate (asocial learning for a good bandit among n _I randomly chosen bandits), and (3) Observe (social learning for a good bandit). Each agent has two parameters (c, p _obs ) to specify the decision: (i) c, the threshold value for Exploit, and (ii) p _obs , the probability for Observe in learning. The parameters (c, p _obs ) are uniformly distributed. We determine the optimal strategies for the player using complete knowledge about the rMAB. We show whether or not social or asocial learning is more optimal in the (p _c , n _I ) space and define the swarm intelligence effect. We conduct a laboratory experiment (67 subjects) and observe the swarm intelligence effect only if (p _c , n _I ) are chosen so that social learning is far more optimal than asocial learning.     

I-Fuzzy equivalence relations and I-fuzzy partitions

A T-fuzzy equivalence relation is a fuzzy binary relation on a set X which is reflexive, symmetric and T-transitive for a t-norm T. Recently, Mesiar et al. [R. Mesiar, B. Reusch, H. Thiele, Fuzzy equivalence relations and fuzzy partitions, J. Multi-Valued Logic Soft Comput. 12 (2006) 167-181] have generalised the t-norm T to any general conjunctor C and investigated the minimal assumptions required on such operations, called duality fitting conjunctors, such that the fuzzification of the equivalence relation admits any value from the unit interval and also the one-one correspondence between the fuzzy equivalence relations and fuzzy partitions is preserved. In this work, we conduct a similar study by employing a related form of C-transitivity, viz., I-transitivity, where I is an implicator. We show that although every I-fuzzy equivalence relation can be shown to be a C-fuzzy equivalence relation, there exist C-fuzzy equivalence relations that are not I-fuzzy equivalence relations and hence these concepts are not equivalent. Most importantly, we show that the class of duality fitting implicators I is much richer than the residuals of the duality fitting conjunctors in the study of Mesiar et al. We also show that the I-fuzzy partitions have a “constant-wise” structure.     

A New Approach to Primary Decomposition

The first purpose of this paper is to describe a new mathematical approach for the computation of an irredundant primary decomposition of a given polynomial ideal I. This presentation will be formed of three parts: a decomposition of the associated radical ideal I to an intersection of prime ideals P_i, then the determination of ideals I_iwhose radical is prime (equal to P_i), and finally, the extraction of the possible embedded components included in I_i. The second is to give an implementation of this algorithm via a new software component, called The Central Control², in which we implemented distributed algorithms performing the basic operations of algebraic geometry.     

Asymptotic behavior and blow up of solutions for semilinear parabolic equations at critical energy level

In this paper we study the initial boundary value problem of semilinear parabolic equations with semilinear term f(u). By using the family of potential wells method we prove that if f(u) satisfies some conditions, J(u₀) ≤ d and I(u₀) > 0, then the solution decays to zero exponentially as t → ∞. On the other hand, if J(u₀) ≤ d, I(u₀) < 0, then the solution blows up in finite time.