A Lattice Boltzmann Method for a Binary Miscible Fluid Mixture and Its Application to a Heat-Transfer Problem A lattice Boltzmann method (LBM) for an isothermal binary miscible fluid mixture is proposed. The binary miscible fluid mixture is assumed to The results indicate that the present LBM is useful for the simulation of fluid flows with heat transfer as well as mass transfer. Recently, the lattice Boltzmann method (LBM) [1] [2] [3] [4] has been used for many kinds of simulations of incompressible viscous flows. In particular, LBM has been successfully applied to problems of multiphase and multicomponent fluid flows [5] [6] [7] [8] [9] [10] . Gunstensen et al. [5] developed a multicomponent LBM based on the twocomponent lattice gas model proposed by Rothman and Keller [6] . Holme and Rothman [7] extended the method to reduce the diffusivity in a miscible two-component system. Flekkøy [8] introduced another two-component LBM model composed of two miscible fluids. Shan and Doolen [9, 10] proposed a multicomponent LBM model including interparticle interaction and external forces.. On the other hand, LBM has also been applied to problems of fluid flows including thermal effects. In general, however, the simulation of thermal fluid systems by LBM has not achieved the same success as that of isothermal flows. For example, McNamara et al. [11] developed a three-dimensional multispeed thermal LBM.. While their method gives accurate results, they pointed out that the numerical stability should be improved. Most previous thermal LBM models [11] [12] [13] [14] are based on such a multispeed approach in which additional particle speeds are needed to obtain the energy equation at the macroscopic level.. As an alternative approach, Shan [15] carried out numerical simulations of RayleighBénard convection by using the previously developed two-component LBM. In his method, the density field of the second component is used to simulate the temperature field by taking advantage of the formal analogy between heat and mass transfer. He et al. [16] also developed a novel thermal LBM model based on a similar approach, in which an independent distribution function for internal energy is introduced to simulate the temperature field. These models are numerically more stable than that of the multispeed thermal LBMs. In both models, however, the equilibrium distribution function for the temperature contains the terms of second order in flow velocity. Since the convection-diffusion equation for the temperature has no terms of second order in flow velocity, it is sufficient to take account of the terms up to first order in flow velocity [17] . Thus there is a possibility of simplifying their models.. In the present paper, we first propose an LBM for an isothermal binary miscible fluid mixture. Next we apply the asymptotic theory proposed by Sone [18] [19] [20] [21] to the present LBM model and obtain the convection-diffusion equation for the diffusing component. Then we calculate a diffusion problem to demonstrate the validity of the proposed method. Finally we apply the present LBM to thermal fluid systems and carry out numerical simulations of Rayleigh-Bénard convection. We have proposed an LBM for an isothermal binary miscible fluid mixture. Applying the asymptotic theory proposed by Sone [18] [19] [20] [21] where U is a characteristic flow speed, a reference density ρ A0 , a reference concentration ρ B0 , and a reference temperature T 0 :. where the subscript A and B represent the variables for A-and B-species, respectively. The dimensional variables in Eq. (A.1) are as follows: c i is the particle velocity, x is the coordinates, t is the time, f Ai and f Bi are the particle distribution functions, ρ A is the density, ρ B is the concentration, u A is the flow velocity, p A is the pressure, g is the gravitational acceleration, β is the volumetric expansion coefficient, T is the temperature, ν is the kinematic viscosity, D BA is the mass diffusivity, and α is the thermal diffusivity.

O O O O O O B I I I O O O O O O O O B I I I O O O B I I I O O O B I I I O O O O O O O O O O O O O O O O O O O B I O O O B I O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


The complexity of constructing pseudorandom generators from hard functions Abstract. We study the complexity of constructing pseudorandom generators (PRGs) from hard functions, focussing on constant-depth circuits. We show that, starting from a function f : {0, 1} l → {0, 1} computable in alternating time O(l) with O(1) alternations that is hard on average (i.e. there is a constant > 0 such that every circuit of size 2 l fails to compute f on at least a 1/poly(l) fraction of inputs) we can construct a PRG : {0, 1} O(log n) → {0, 1} n computable by DLOGTIMEuniform constant-depth circuits of size polynomial in n. Such a PRG implies BP · AC 0 = AC 0 under DLOGTIME -uniformity. On the negative side, we prove that starting from a worst-case hard function f : {0, 1} l → {0, 1} (i.e. there is a constant > 0 such that every circuit of size 2 l fails to compute f on some input) for every positive constant δ < 1 there is no black-box construction of a PRG : {0, 1} δn → {0, 1} n computable by constant-depth circuits of size polynomial in n. We also study worst-case hardness amplification, which is the related problem of producing an average-case hard function starting from a worst-case hard one. In particular, we deduce that there is no blackbox worst-case hardness amplification within the polynomial time hierarchy. These negative results are obtained by showing that polynomialsize constant-depth circuits cannot compute good extractors and listdecodable codes. A rigorous notion of pseudorandom generators (PRGs) was introduced in the seminal works of Blum & Micali (1984) and Yao (1982) , and has since found a striking variety of applications in cryptography and complexity theory. A PRG 148 Viola cc 13 (2004) G : {0, 1} u → {0, 1} n is an efficient procedure that stretches u input bits into n u output bits such that the output of the PRG is indistinguishable from random to small circuits. That is, for every circuit A of size n we have Throughout this work the complexity of a PRG is measured in terms of its output length (denoted n above).. While the existence of PRGs is a major open problem, there has been a series of fascinating works constructing PRGs from weaker and weaker assumptions. Nisan & Wigderson (1994) show how to construct PRGs from strong average-case hardness assumptions, namely the existence of a Boolean function f : {0, 1} l → {0, 1} in E := T IM E(2 O(l) ) that is hard on average for circuits. In particular, they show that starting from a function f : {0, 1} l → {0, 1} in E such that (for some constant > 0) every circuit of size 2 l fails to compute f on at least a 1/2 − 1/2 l fraction of inputs, it is possible to construct a PRG with logarithmic input length, i.e. G : {0, 1}. O(log n) → {0, 1} n , computable in time polynomial in n. Efficient PRGs with logarithmic seed length are of particular interest because, as we discuss below, they imply BP · P = P . Note that the PRG G is computable in time polynomial in n even though it is constructed from a function f that is computable in exponential time (i.e. in E). This is possible because G only evaluates f on inputs of length l = O(log n).. Because the average-case hardness assumption in Nisan & Wigderson (1994) seemed very strong, much research was devoted to constructing PRGs with logarithmic seed length under a weaker worst-case hardness assumption, namely the existence of a function f : {0, 1} l → {0, 1} in E such that (for some constant > 0) every circuit of size 2 l fails to compute f on some input (e.g. Babai et al. 1993; Impagliazzo 1995) . This research culminated in Impagliazzo & Wigderson (1997) where it is shown how to amplify the worst-case hardness of functions in E to the average-case hardness required in Nisan & Wigderson (1994) . We survey these results in Section 3. More direct proofs and improved results were obtained in Sudan et al. (2001) ; Impagliazzo et al. (2000) ; Shaltiel & Umans (2001) ; Umans (2002) .. 1.1. The problem we study. In this paper we address the following problem: What is the complexity of constructing a PRG from a hard function? There are at least two reasons for studying this problem. First, we want to understand the computational relationship between two fundamental quantities in theoretical computer science: hardness and randomness. Second, PRGs are a basic tool whose variety of applications justifies the quest for more and more efficient constructions.. Derandomization. An important application that demands more and more efficient PRGs is the high-end derandomization of a complexity class C, that is, proving BP · C = C. For such application we need PRGs with logarithmic seed length, which as we said above can be constructed starting from a function having exponential circuit complexity. For example, Impagliazzo & Wigderson (1997) show that BP · P = P if there is a function f : {0, 1} l → {0, 1} in E that requires circuits of size 2 Ω(l) . This derandomization works as follows. We run the algorithm we want to derandomize using all the possible outputs of the PRG in place of true random bits. Then we decide according to majority vote. Since the seed length is logarithmic, this process is efficient, i.e. only gives a polynomial slow-down.. It is then clear that if we aim to derandomize a probabilistic complexity class BP · C using a PRG, then the PRG must be computable in C. Therefore, the lower the complexity class we want to derandomize, the more efficient the PRG must be. For example, already to derandomize BP · L (where L := SPACE (log n)), one needs a more efficient PRG construction than the one given in Nisan & Wigderson (1994) and used in Impagliazzo & Wigderson (1997) to derandomize BP · P . This problem is solved by Klivans & van Melkebeek (1999) who obtain BP ·L = L under the assumption that there is a function f : {0, 1} l → {0, 1} computable in linear space that requires circuits of size 2 Ω(l) . In this paper we study more efficient PRG constructions that could be used to derandomize probabilistic classes below BP ·L. In particular, we aim to prove BP · AC 0 = AC 0 subject to DLOGTIME -uniformity, where AC 0 denotes the class of functions computable by polynomial-size constant-depth circuits, and DLOGTIME -uniformity is a strong uniformity condition discussed below (at the end of Section 1.2).. Note that the high-end derandomization BP · AC 0 = AC 0 subject to DLOGTIME -uniformity is not known to hold unconditionally. Indeed, it is still open whether the weaker inclusion P -uniform BP · AC 0 ⊆ P is true: The most efficient unconditional derandomization of P -uniform BP · AC 0 is the one obtained by Nisan (1991) that runs in quasipolynomial time. Such de-cc 13 (2004) randomization is based on the fact that the parity function on l bits cannot be computed (on average) by constant-depth circuits of size 2 l (where is a constant depending on the depth of the circuit; see e.g. Håstad 1987 ). On the other hand, proving P -uniform BP · AC 0 ⊆ P through a PRG would require exhibiting a function f : {0, 1} l → {0, 1} in E that (for some constant > 0) cannot be computed by constant-depth circuits of size 2 l . But it is consistent with the current state of knowledge that every function f : {0, 1} l → {0, 1} in E is computable by circuits of size 2 o(l) and depth 3. It should be noted that a result by Valiant (1977) states that if a function f : {0, 1} l → {0, 1} (for some constant > 0) cannot be computed by circuits of size 2 l and depth 3, then it cannot be computed by circuits of linear size and logarithmic depth. Exhibiting a "natural function" that cannot be computed in the latter circuit class (i.e. linear size and logarithmic depth) is a long-standing open problem proposed in Valiant (1977) .. 1.2. Black-box PRG construction. As we explained in Section 1.1 in this work we study the complexity of constructing PRGs from hard functions. We now discuss what we mean by "constructing". An oracle procedure. n is a black-box PRG construction from worst-case hard functions if for every function f : {0, 1} l → {0, 1} and for every function A : {0, 1} n → {0, 1}, if A distinguishes the output of the PRG from random, i.e.. then there is an oracle circuit C of size s that, given oracle access to A, computes f everywhere, i.e. C A (x) = f (x) for every x. The idea is that if we start with a function f such that no circuit of size s · n can compute f everywhere (i.e. f is worst-case hard for size s · n) then G is a PRG. This is because if a circuit A of size n distinguishes the output of G from random then C A is a circuit of size s · n computing f everywhere, and this contradicts the hardness of f .. Note that in a black-box PRG construction we need the input length l of the hard function f to be Ω(log n). Otherwise it can be shown that every function f : {0, 1} l → {0, 1} can be computed by circuits of size 2 l ≤ n, and so our assumption that f is worst-case hard for size s · n is immediately false.. The PRG construction defined above is called black-box because it works for every function f and every potential distinguisher A, regardless of their complexity. While it is conceivable that non-black-box PRG constructions are more powerful than black-box ones, we note that all known PRG constructions are black-box, e.g. Blum & Micali (1984) ; Yao (1982) ; Impagliazzo & Wigderson cc 13 (2004) Complexity of constructing PRGs 151 (1997); Klivans & van Melkebeek (1999) ; Sudan et al. (2001) ; Impagliazzo et al. (2000) ; Shaltiel & Umans (2001) ; Umans (2002) , and all those in this paper. However, non-black-box PRG constructions are known to be more powerful than black-box ones in the uniform setting, i.e. when we only require that the output of the PRG be indistinguishable from random to uniform machines (as opposed to nonuniform circuits, required by our definition). We refer the reader to Trevisan & Vadhan (2002) for a discussion of this issue.. So far we have discussed black-box PRG constructions from worst-case hard functions. Black-box PRG constructions from average-case hard functions are analogous except that we only require that C A computes f on average (as opposed to everywhere). By the same reasoning this suffices to ensure that G f is a PRG whenever f is hard on average.. Black-box PRG constructions in AC 0 . The main technical question addressed in this paper is: is there a black-box PRG construction in AC 0 ? (Recall AC 0 denotes the class of functions computable by polynomial-size constantdepth circuits.) As we explain below (Section 1.3), in this paper we exhibit both positive and negative results on this question. Our negative results apply regardless of the uniformity of AC 0 , and our positive results hold even under a strict uniformity condition, namely DLOGTIME -uniformity, which we now discuss. Informally, a family of circuits is DLOGTIME -uniform if given indices to two gates one can decide their type and whether they are connected in linear time in the length of the indices (which is logarithmic time in the size of the circuit). There is a consensus that this is the "right" uniformity condition for AC 0 , and the evidence for this is that DLOGTIME -uniform AC 0 has several different and elegant characterizations; see Barrington et al. (1990) . A characterization that we will often use in this work is the following: DLOGTIMEuniform AC 0 is equivalent to ATIME (O(1), log n), where ATIME (O(1), log n) denotes alternating time O(log n) with O(1) alternations (cf. Theorem 2.2). The class ATIME (O(1), log n), introduced by Sipser (1983) , is the logarithmic analogue of the polynomial time hierarchy (ATIME (O(1), n O(1) )), and it is strictly contained in SPACE (log n).. 1.3. Our results. Some of our results are summarized and compared to previous ones in Table 1 . Our main finding is that there are black-box PRG constructions G f from (mild) average-case hard functions such that G f is computable in ATIME (O(1), log n) f . But there is no black-box PRG construction G f from worst-case hard functions such that G f is computable by small nonuniform oracle constant-depth circuits. (Recall ATIME (O(1), log n) = DLOGTIME -uniform AC 0 .). Hardness amplification PRG construction Derandomization from strong hardness given by PRG construction

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B O O O O


Distributed fault-tolerant ring embedding and reconfiguration in hypercubes Abstract-To embed a ring in a hypercube is to find a Hamiltonian cycle through every node of the hypercube. It is obvious that no 2 nnode Hamiltonian cycle exists in an n-dimensional faulty hypercube which has at least one faulty node. However, if a hypercube has faulty links only and the number of faulty links is at most n − 2, at least one 2 n -node Hamiltonian cycle can be found. In this paper, we propose a distributed ring-embedding algorithm that can find a Hamiltonian cycle in a fault-free or faulty n-dimensional hypercube (Q n ), and the complexity is O(n) parallel steps. The algorithm is based on the recursion property of the hypercube and the free-link dimension concept. In some cases, even when the number of faulty links is larger than n − 2, Hamiltonian cycles may still exist. We will show that the largest possible number of faulty links that can be tolerated is 2 n−1 − 1.The performance and the constraints of the fault-tolerant algorithm is also analyzed in detail in this paper. Furthermore, a dynamic reconfiguration algorithm for an embedded ring is proposed and discussed. Due to the distributed nature of the algorithms, they are useful for the simulation of ring-based multiprocessors on MIMD hypercube multiprocessors. THE hypercube is a topology with large connectivity. A vertex in a hypercube graph is connected to n adjacent vertices by n edges, where n is the dimension of the hypercube. Therefore, many other topologies can be embedded in a hypercube, such as the linear array, mesh, binary tree, ring, etc. For many applications on multiprocessors, selecting an appropriate and adequate topological structure of multiprocessors is a critical issue. The reason is that most applications have their own intrinsic communication patterns. By means of embedding, applications that are originally suitable for multiprocessors of other topological structures can be executed on hypercube multiprocessors.. It is well known that systems with a large number of components are more prone to failures. Accordingly, it is not uncommon that there are some link and/or node failures in a hypercube system. Under such circumstances, straightforward embedding may not be possible. Thus, fault-tolerant embedding has become an intriguing issue. Many researchers have addressed this topic [1] , [2] , [3] , [4] , [5] , [6] extensively, but most of them considered node failures only. In [3] , centralized algorithms are devised for optimal ring embedding in an n-dimensional hypercube Q n with at most n − 2 faulty links. The goal of the work in [3] is to find a Hamiltonian cycle through every fault-free node of a hypercube Q n . Their algorithms take O(n 2 ) time to compute the characterization of a. Hamiltonian cycle and, then, O(2 n ) time to construct a Hamiltonian cycle, i.e., O(n 2 + 2 n ) time totally. Our goal is the same, but we achieve this goal in a different and more efficient manner. In this paper, we present distributed algorithms for ring embedding and dynamic reconfiguration. Hence, our work is applicable on MIMD hypercube multiprocessors. Our approach is based on the recursion property of the hypercube and the free-link dimension concept. The free-node dimension concept in [7] applies to node failures while we deal with link failures. The free-node dimension is defined in [7] as that a dimension in a hypercube is said to be free if no pair of nodes across any link in that dimension are both faulty. The concept of free-node dimension was used to embed a ring in a Q n with faulty nodes [5] . In [5] , the embedding procedure is briefly described. A six-node or eightnode subring is first embedded onto each partitioned Q 3 and, then, all the embedded subrings are merged in a distributed manner. However, the details about the internode communications are omitted in [5] . Considering link failures, we say that a dimension is free if all links of this dimension are nonfaulty. We use the concept of free-link dimension to construct subrings, and then these subrings are recursively merged to form larger subrings until the desired largest ring is obtained. Although our approach is motivated by the ring embedding method described in [5] , the concentration on faulty links is very different and unique. Furthermore, we utilize the recursion property of the hypercube to merge subrings efficiently.. The organization of this paper is as follows. In Section 2, the basic concepts regarding ring-embedding are given. In Section 3, we propose a distributed fault-tolerant ring-embedding algorithm for hypercubes. Detailed analysis and comparisons are also provided in this section. Section 4 presents a dynamic reconfiguration algorithm for an embedded ring. Conclusions are given in Section 5.. IN the design of networks, one of the most important topics is their reliability. Connectivity and edge connectivity are widely used as measurements. In practice, we are often interested in a collection of multipaths between a pair of nodes. Some new concepts were proposed to study the collections of multipaths in graphs and networks [9] . The classical approach to studying routing in interconnection networks is to try to find the shortest path between the sending station and the receiving station. Whenever some stations are faulty on the path between the sending station and the receiving station, the management protocol has to find a way to bypass those faulty stations and set up a new path between them. Similarly, if this new path is disconnected again, a third path needs to be set up if it is possible (the network is still connected or there still exists a path between the sending station and the receiving station).. The current Internet provides end-to-end routing as above [16] . If any router on the shortest path becomes congested or goes down, the performance will be dramatically decreased due to routing overhead. IP multicasting [4] faces the same problem. The current routing protocols for IP multicasting, such as DVMRP, MOSPF, and PIM, cannot provide better services in a faulty interconnection networks.. One issue is missing in all the previous management protocols in terms of minimizing the lengths of the new paths in such a faulty interconnection network. It is not enough to only try to make the first connection as short as possible. In order to achieve better performance of communications, it is necessary to consider how to optimize the routing of alternative paths while searching for first connections in case the first connection is broken down, and so on.. This reflects one important aspect of the fault tolerance properties of the interconnection networks which has not been studied for most network topologies. The network researchers want to know what the effects of the faulty stations are, such as increasing the maximum delay in a faulty network. For real-time applications, such real-time control, multimedia applications, and image processing, the propagation delay must meet the time constraints. In a faulty system, connectivity only tells us if the system is still connected. But, people want to know what the longest delay in such a faulty system is and how to do efficient routing to meet the deadlines. In this paper, we answered these questions for pyramid networks.. In the following, we will introduce some concepts to study these effects quantitatively, such as w-distances, fault diameters.. Some of the important networks have been studied in terms of these concepts. For example, Hsu and Lyuu showed the exact bounds of the w-distance for directed toruses [10] . Butterly networks and optical passive stars networks were studied by Cao [1] and Cao et al. [2] .. The pyramid network is one of the important structures in parallel and network computing [14] , [18] and image processing [15] . In image processing, pyramid networks are used as both hardware architectures and software structures. In parallel and network computing, a lot of parallel algorithms are efficiently implemented on pyramid networks. For example, some parallel algorithms are implemented in supercomputers like Cray T3D and T3E and each processor acts as a node in the pyramid network. Other parallel algorithms are implemented by involving by several workstations, each workstation acting as a node in the pyramid network.. To support real-time applications, such as network computing and image processing in pyramid networks, communication among the nodes are very essential. The propagation delay must meet the time constraints. This becomes even more important in a faulty pyramid network for the longest delay in such a faulty environment determines if the pyramid network is suitable to support real-time applications. An answer also needs to be found as to how to efficiently do routing between any working nodes in such a faulty pyramid network.. In this paper, we study the fault tolerance properties of pyramid networks quantitatively in terms of the new concepts such as fault diameter and w-wide diameter, which will be defined in the rest of this section.. A pyramid network (also called a pyramid) is a hierarchy structure based on meshes. A mesh, M[a, b], is a set of nodes V(M[a, b] and two nodes, (x 1 , y 1 ) and (x 2 , y 2 ) , are connected by an edge iff |x 1 − x 2 | + |y 1 − y 2 | = 1 . In this paper, we have proposed distributed algorithms for ringembedding and dynamic reconfiguration in hypercubes. Algorithm RingConstruct works in faulty or fault-free hypercubes with O(n) parallel steps. The number of distinct Hamiltonian cycles which can be identified in a healthy hypercube by our approach is larger than that by using GC. If the number of injured dimensions does not exceed n − 2, Algorithm RingConstruct can tolerate up to 2 n − 1 − 1 faulty links. The conditions under which Algorithm RingConstruct works are analyzed and discussed. Therefore, Algorithm RingConstruct can be applied even when a large number of faulty links exist in the hypercube. In addition, comparisons with previous research results are given. Last, but not least, Algorithm RingReconfig takes only two parallel steps to reconfigure a faulty embedded ring and it is thus good for maintaining the integrity of an embedded ring. One of the applications on ring-based multiprocessors is solving long-range problems [11] . Examples are molecular dynamics calculations, Newtonian gravitational n-body simulations, and the product of two large polynomials. In algorithms for solving these problems, computations must be performed on all pairs of objects. Thus, the communication patterns are ring-based. Our algorithms are devised for real application from the beginning. After executing the algorithms, the connections for a ring are formed and each node has the information about who are its ring neighbors. Moreover, even when the fault-tolerant algorithm cannot complete the embedding successfully, the intermediate results can still be utilized for the simulation of ring-based multiprocessors. Therefore, our approach is practical and useful in terms of performance and fault tolerance capability. In this paper, we study the pyramid network, one of the important architectures in parallel computing, network computing, and image processing. We determine the line connectivity and the fault diameters in pyramid networks. We show how to construct a path between two nodes in the faulty pyramid networks in polynomial time. We also give a polynomial-time algorithm for generating the containers in pyramid networks. The above results show that pyramid networks have very good fault tolerance properties. o. Therefore, the total number of faulty g i -links must be smaller than 2. . Although this number is independent of i, yet the distribution of the locations of faulty g i -links is crucial. It may be the case that the total number of faulty g i -links is smaller than 2 n−2 , but two 2 i -node subrings cannot be merged. This occurs if the number of faulty g i -links corresponding to the two 2 i -node subrings is greater than or equal to PROOF. Let the number of injured dimensions be t. Then, the number of elements in stack F is t and the number of elements in stack E is n − t. Hence, dimensions g 0 through g n−t−1 are free dimensions, while dimensions g n−t through g n−1 are injured dimensions. Let f g i be the total number of faulty g i -links. and m be the total number of faulty links. We have:. Equation (2) . However, we will show that this cannot occur. According to (3), it can be easily seen that the maximum value of f g n t The algorithm presented in this section is capable of finding a ring (Hamiltonian cycle) in Q n with faulty links (up to n − 2 faulty links). We assume each node has the locations of all faulty links. This assumption is reasonable because an optimal broadcasting algorithm which takes n + 1 steps in a hypercube with faulty links has been developed [10] . Therefore, it is possible for nodes in a hypercube with faulty links to broadcast the fault information to other nodes. Thus, each node can make a decision based on the information in its own memory instead of performing communications with other nodes. The dimensions are divided into two groups: free dimensions and injured dimensions. The free dimensions are stored in a stack called E. The injured dimensions are sorted by the number of faulty links and kept in a stack called F, with the top of F being the dimension with the smallest number of faulty links. More details will be given later.. The algorithm has n steps numbered from 0 to n − 1. Stack E or F is popped to give a dimension in each step. Since at most n − 2 faulty links are assumed to exist in Q n , at least two dimensions are free. Consequently, in Step 0, 3.. where is the dimension poped in. Step Before further discussion, we introduce the concept of pseudofaulty links. The introduction of pseudofaulty links is to maintain the uniformity of the structures of subrings formed in Step i. That is, all node labels of a formed subring must be the same as those of another subring in bit positions g 0 through g i . This property must be maintained so that we can guarantee that subrings can be recursively merged. If a certain g i -link is faulty, the corresponding g ilinks in all other merging groups are treated as pseudofaulty. That is, a faulty link makes an "image" in each merging group. Algorithm RingConstruct does not care whether a link is faulty or pseudofaulty. Fig. 3 illustrates the concept of pseudofaulty links. By the introduction of the pseudofaulty link concept, the operations in each merging group are uniform.. The condition for a node N that can involve in merging operations in Step i is:. the g i -links connected to node N and at least one of its ring neighbors are nonfaulty or nonpseudofaulty.. For the convenience of presenting Algorithm RingConstruct, this condition is refered as the merging condition. Note that two g ilinks are required for merging operations in Step i because they are used as ring connections in the merged ring. Each node can check easily if itself and other nodes in the subring satisfy the merging condition since it has the information about the locations of all faulty links. The node with minimal label is chosen to initiate the merging operations. As an example, in Fig. 2 , nodes 001, 011, and 010 satisfy the merging condition in the inner subring and nodes 101, 111, and 110 satisfy the merging condition in the outer subring. Node 001 (101) knows that it is the node to initiate the merging operations. And, then, node 011 (111) is informed to delete connection 001-011 (101-111). Finally, connections 001-101 and 011-111 are established. Algorithm RingConstruct is shown in Fig. 4 .. In a ring, each node has two connections to its adjacent nodes. In iterations from i = 2 to n − 1, subrings are recursively merged. To merge two subrings, one connection of each subring must be broken and two new connections are established. A node simply deletes the corresponding connection record to break a connection. 2). Since we sort the injured dimensions, the probability that the algorithm will fail is reduced. , the two subrings are guaranteed to be merged across dimension g. Fig. 5 shows an example that two subrings cannot be merged.

O O O O O B O O O O O O O O B O O O O B I O O O O O B O O O O O O O B I O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Discovering overlapping communities in social networks: A novel game-theoretic approach Abstract. Identifying communities in social networks has been receiving the increasing attentions recently. However, the overlapping concept has received little attentions in the literature, although it is observed in almost all social networks. In this study, we propose a framework based on the game theory and the structural equivalence concept to address the detection of overlapping communities in social networks. Specifically, we consider the underlying graph as a hypothetical social networking website and regard each vertex of this graph as an agent performing in this multiagent environment. Since each agent may belong to several communities simultaneously, we are able to find overlapping community structure of social networks. The rigorous proof of the existence of Nash equilibrium in this game is provided which shows that the method always reaches to the final solution. Experimental results on the benchmark and real world graphs show superiority of our approach over the other state-of-the-art methods. A social network is a complex network demonstrating social interactions between people [36] . Recently, online social networking websites such as Facebook, 1 Twitter, 2 Myspace 3 and etc., have become tremendously popular, because they let people all around the world communicate with their friends, send emails, spread opinions on some issues, etc., in the cyberspace without in person meetings. These online interactions on the Internet are provided by modern information and communication technology (ICT) .. As a potent representation tool, graph and its literature has become a renowned component in studying social networks and their properties since 20th century. Graph vertices and edges are respectively regarded as the paradigms of the entities in social networks (e.g. people and the interactions between them). Nowadays, the emergence of computational resources, extensive data and recent rapid expansion of these networks to million or even billion of vertices have produced a deep change in the way graphs are approached [2, 5, 25] .. Social network analysis (SNA) has been started in 1930s and since then has become one of the most important topics in sociology [34, 36] . Social networks like many other networks, show several interesting properties such as high network transitivity [37] , power-law degree distributions [4] and the existence of repeated local motifs [24] , yet the significant attribute recently under consideration is "community structure" or "clustering"; the appearance of dense connected groups, modules, clusters or communities of vertices in the network graph and sparser connections between them [25] .. The word community itself refers to a social context. People naturally tend to form groups, within their families, work environments and friends. Communities of social networks can be friendship circles, groups of people sharing common interests and/or activities, etc. Furthermore, many other networked systems including biology and computer science, have built-in communities. This property has high applicability and therefore attracts a lot of researchers from different fields. For example, groups within World Wide Web correspond to web pages on the related topics [11] , groups in social Taking inspiration from human-human interactions, we proposed a game-theoretic framework to identify overlapping communities in social networks based on structural equivalence relations.. Considering the graph of underlying social network as a conceptual social networking website, we attributed community detection problem to an iterative game between the vertices which are regarded as rational agents. Due to the very large strategy space, reaching to the global Nash equilibrium of this game is not viable, so we are just satisfied by a local one. This framework consists of two phases: Personal and System phases where in the first phase, each agent personally decides between join, leave, switch and No Action operations and in the second phase, system decides to suggest a friend to an agent or to evict an agent from an existing community. Regarding to the represented theoretic basis of our method in the Appendix and based on the observations from experiments, it has promising results on the benchmark graphs compared to the other similar methods at the expense of high running time and performed actions.. The framework could be easily extended to use in directed, weighted, multi-dimensional and dynamic networks. Future works include embedding other similar operators borrowed from real life or using some other similarity methods instead of Pearson correlation. Table 6 The best community structure of Football network found by Game method some games don't have Nash Equilibrium. To see when a certain game has Nash equilibrium, recall that potential games are a general class of games that permit pure Nash equilibrium [29] . Indeed for any finite game, there exists a potential function Θ defined on the strat- egy profile S of the agents that maps this profile to some real values. This function must validate the following condition:. Equivalently, if the current strategy profile of the game is S and the agent i switches from strategy s i to s i , the potential function exactly mirrors the changes in the agent utility. It is not hard to see that a game has at most one potential function. A game that does possess a potential function is called a potential game. Consequently we have the following theorem. Community detection problem (CDP) has a long history and has appeared in various forms in several disciplines including sociology and computer science. Perhaps, the first analysis of community structure dates back to 1927, where Stuart Rice looked for clusters of people in small political bodies, based on the similarity of their voting patterns [33] . Two decades later, George Homans showed that social groups could be revealed by suitably rearranging the rows and the columns of matrices describing social ties, until they take an approximate block-diagonal form [17] . In 1955, Weiss and Jacobson searched for work groups within a government agency [38] . The authors studied the matrix of working relationships between agency members who were identified by means of private interviews. Work groups were separated by removing the members working with people of different groups who acted as their connectors. This idea of cutting the bridges between groups is at the core of several modern algorithms of community detection.. In general, traditional techniques to find communities in social networks are hierarchical and partitional clustering, where vertices are joined into groups according to their mutual similarity. Indeed, many works have been done in the literature, which can be categorized into two main groups: optimization methods which look for optimizing some measures and methods with no optimization, which search for some predetermined structures. From these works, one can refer to the works done by Girvan and Newman in 2002 and 2004 introducing two important concepts "edge betweenness" [14] and "modularity" [28] , the work done by Brandes and Erlebach coining the term "conductance" [7] and the work done by Palla et al. [31] . In [14] , Girvan and Newman proposed a new algorithm, aiming at the identification of edges lying between communities and their successive removal that after some iterations leads to the isolation of the communities. The inter-community edges are detected according to the values of a centrality measure, the edge betweenness that expresses the importance of the role of the edges in processes where signals are transmitted across the graph following paths of minimal length. That work triggered a big activity in the field, and many new methods have been proposed in recent years (see [12] to study these methods). In particular, physicists entered the game, bringing in their tools and techniques such as spin models, optimization, percolation, random walks, synchronization and etc., which rapidly became the main ingredients of new original algorithms.. In comparison with the previous discussions, few works have been done considering the overlapping concept. One of the earliest works in this area was carried out by Baumes et al. which proposed two efficient heuristics, Iterative Scan (IS) and Rank Removal (RaRe) to optimize a given function related to the edge density of the clusters [6] . Gregory proposed CONGA [15] and CONGO [16] based on "split betweenness" to detect overlapping communities in the networks. In 2005, Palla et al. showed that searching for some predetermined structures like fully connected graphs or kclique in the network can lead to detecting such communities [31] . In another work, Zhang et al. used fuzzy c-means [41] to detect overlapping communities.. Meanwhile very few works are done based on the game theory (see for example [1, 9] ). These works address the problem of community detection by a gametheoretic framework in which nodes of underlying social network graph are considered as rational agents who want to maximize their payoff according to some criterion. The work done in [9] can also support the overlapping concept. In this work, the difference between gain functions and loss functions is used as a utility function for each of the agents and the Nash equilibrium of the game reveals the final devision of the graph. The gain function used in [9] is based on the Modularity concept [27] and the loss function is defined as a simple linear function with respect to the number of membership labels.. In general, the merit of using the game-theoretic methods is that they are grounded with a systematic theory for formation of communities in the networks, As being humans, it is necessary for all of us to make social interactions with each other. None of us like to be alone and we often have incentives to establish friendships with everyone that has a lot in common with us in behavior, ethics, way of life, social interactions, majors, etc. This is the reason that in both real and cyberspace lives, we often belong to communities whose members are very similar to us, including family and friends. This prompted us to consider the community formation problem in social networks as a play of interactions between their constituents, i.e. people.. We have embedded game theory into our work because it provides rigorous mathematical models of strategic interactions between rational, autonomous and intelligent agents. Indeed game theory is a good tool to capture both the behavior of individuals and strategic interactions among them [1] . According to this fact and based on [9] , we attributed community formation as an iterative game preformed in a multiagent environment in which each node of the underlying network graph is a rational agent who decides to maximize its total utility by joining to communities with members most similar to it in two phases. In the first phase, an agent iteratively decides between join, leave, switch and no operation operators while in the second phase, it responds to two systematic operators as well. These two operators include friend suggestion inspired from social networking websites where we receive suggestions from system and eviction borrowed from real life where there are situations that we might be evicted from groups we belong to. Furthermore, instead of using Modularity function [16] , here the utility for each agent is calculated based on Pearson similarity function; the more similar the community members, the more their utilities. This indeed has a logical and sensible interpretation.

O B I O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O B I O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Cooperative Encoding for Secrecy in Interference Channels This paper investigates the fundamental performance limits of the two-user interference channel in the presence of an external eavesdropper. In this setting, we construct an inner bound, to the secrecy capacity region, based on the idea of cooperative encoding in which the two users cooperatively design their randomized codebooks and jointly optimize their channel prefixing distributions. Our achievability scheme also utilizes message-splitting in order to allow for partial decoding of the interference at the nonintended receiver. Outer bounds are then derived and used to establish the optimality of the proposed scheme in certain cases. In the Gaussian case, the previously proposed cooperative jamming and noise-forwarding techniques are shown to be special cases of our proposed approach. Overall, our results provide structural insights on how the interference can be exploited to increase the secrecy capacity of wireless networks.Index Terms-Cooperative encoding, information theoretic security, interference channel. This work considered the two-user interference channel with an (external) eavesdropper. An inner bound on the achievable secrecy rate region was derived using a scheme that combines cooperative randomized codebook design, channel prefixing, message splitting, and time-sharing techniques. More specifically, our achievable scheme allows the two users to cooperatively construct their randomized codebooks and channel prefixing distributions. Outer bounds are then derived and used to establish the optimality of the proposed scheme in some special cases. For the Gaussian scenario, channel prefixing was used to allow the users to transmit independently generated noise samples using a fraction of the available power. Moreover, as a special case of time sharing, we have developed a novel cooperative TDMA scheme, where a user can add structured and unstructured noise to the channel during the allocated slot for the other user. It is shown that this scheme reduces to the noise forwarding scheme proposed earlier for the relay-eavesdropper channel. In the Gaussian multiple-access setting, our cooperative encoding and channel prefixing scheme was shown to enlarge the achievable regions obtained in previous works. The most interesting aspect of our results is, perhaps, the illumination of the role of interference in cooperatively adding randomness to increase the achievable secrecy rates in multi-user networks. APPENDIX A PROOF OF THEOREM 1 1) Probability of Error Analysis: Below we show that the decoding error probability of user averaged over the ensemble can be arbitrarily made small for sufficiently large . This demonstrates the existence of a codebook with the property that , for any given . The analysis follows from similar arguments given in [2] . See also [17] for joint typical decoding error computations. Here, for any given , each receiver can decode corresponding messages given above with an error probability less than as , if the rates satisfy the following equations:. 2) Equivocation Computation: We first write the following:. (28) where we use the set notation to ease the presentation, and the inequalities are due to the fact that conditioning does not increase entropy.. Here Lastly, for any , , ,. , and , we have , for some as . This is due to the Fano's inequality together with the randomized codebook construction: Given all the message (bin) indices of two users, eavesdropper can decode the randomization indices among those bins. Due to joint typicality, this latter argument holds as long as the rates satisfy the following equations:. This follows as given bin indices , , , and , this reduces to MAC probability of error computation among the codewords of those bins. See [17] . Then, we consider the following bound:. where the first inequality is due to Lemma 15 given at the end of this section, the second inequality is due to the Fano's inequality at the receiver 1 with some as , the last equality is due to observations and (see [14, Lemma 7] ).. We define , , and. . Using standard techniques (see, e.g., [17] ), we introduce a random variable , which is uniformly distributed over , and continue as below (39) where the last inequality follows from the fact that , which implies after using the fact that conditioning does not increase entropy, and the last equality follows by using a standard information theoretic argument in which we define random variables for the single-letter expression, e.g., has the same distribution as . Hence, we obtain the bound (40) for some auxiliary random variables that factors as . . where the second equality follows by , and the last inequality follows due to the fact that conditioning does not increase entropy. Second statement follows from a similar observation.

B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


On well-covered triangulations: Part II A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part I, Discrete Appl. Math., 132, 2004, 97-108] that there are no 5-connected planar well-covered triangulations. It is the aim of the present paper to completely determine the 4-connected well-covered triangulations containing two adjacent vertices of degree 4. In a subsequent paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part III (submitted for publication)], we show that every 4-connected well-covered triangulation contains two adjacent vertices of degree 4 and hence complete the task of characterizing all 4-connected well-covered planar triangulations. There turn out to be only four such graphs. This stands in stark contrast to the fact that there are infinitely many 3-connected well-covered planar triangulations. In 1969, the fourth author first proposed the study of graphs in which each maximal independent set of vertices has the same size and suggested that the name well-covered be applied to them [13] . Although it is now well-known that the independent set problem is NP-complete for graphs in general (cf. Karp [9] ), for certain interesting subfamilies of graphs, such as those called claw-free, the problem becomes polynomially solvable (cf. Minty [12] and Sbihi [16] ). Clearly, the independent set problem has a polynomial solution for the class of well-covered graphs, but how does one recognize this class? It was shown independently by Chvátal and Slater [3] and by Sankaranarayana and Stewart [15] that the recognition problem for well-covered graphs is co-NP-complete. In contrast, if the graphs are claw-free, then the recognition problem becomes polynomial. (See Tankus and Tarsi [17, 18] .) For more comprehensive surveys of well-covered graphs, see Plummer [14] and more recently, Hartnell [7] .. A widely studied subclass of planar graphs are those which are maximal planar and which are commonly called (planar) triangulations. Clearly, any triangulation (larger than a single triangle) must have vertex connectivity 3, 4 or 5. Lebesgue [11] , Kotzig [10] , Borodin [1] and Jendrol' [8] have extensively investigated what kind of configurations must always exist in any triangulation. In an earlier paper [4] these results were used to prove that there is no 5-connected planar well-covered triangulation. In the present paper all 4-connected planar well-covered triangulations containing two adjacent vertices of degree 4 are determined. In another paper [5] the authors show that all 4-connected planar well-covered triangulations contain two adjacent vertices of degree 4. Taken together, these two works allow us to completely determine all 4-connected planar triangulations. There are only four such graphs. (See graphs R 6 , R 7 , R 8 and R 12 in Fig. 2.1 .) Interestingly, all of these graphs were known to the authors some years before they were able to accomplish the much more difficult task of showing that they are indeed the only such graphs. The finiteness of this class also contrasts sharply with the fact that there are known to be infinitely many 3-connected well-covered planar triangulations.. As the complete proof of our main result is quite long, let us present an outline of how we shall proceed. Proposition 2.2 deals with properties of induced 4-cycles in general 4-connected planar triangulations. The remaining results in Section 2 all deal with properties of induced 4-cycles in such graphs which are, in addition, well-covered.. In Section 3 we introduce the concept of a BW -configuration and prove a so-called Extension Lemma based on this idea. BW -configurations and the Extension Lemma, although somewhat technical in nature, serve to eliminate a number of cases which arise in Section 4 and also in a subsequent paper [5] . Hence we deal with them in their own section here. We also point out that this concept applies to general (not necessarily planar) well-covered graphs and hence may prove of assistance in future studies of other families of well-covered graphs.. In Section 4, we apply the aforementioned results to first determine those graphs which contain the induced subgraph Q 3 and finally those which contain a second induced subgraph Q 2 . (The subgraphs Q 3 and Q 2 are shown in Fig. 2.1 .) This then quickly leads to Theorem 4.3, the main result of the paper.. We will adopt the following notation and terminology throughout this paper. In this paper all graphs will be finite and simple. If v is a vertex of a graph G, α(G) will denote the cardinality of a maximum independent set in V (G) and N [v] will denote the closed neighborhood of the vertex v; namely, N [v] = N(v) ∪ {v}. For n > 2, a 1 a 2 . . . a n will denote an n-cycle C with vertex set {a 1 , a 2 , . . . , a n } and with edge set {a 1 a 2 , a 2 a 3 , . . . , a n−1 a n , a n a 1 }. As generally accepted, by the term block we will always mean a maximal 2-connected subgraph. Clearly a graph is well-covered if and only if all its components are. Therefore we shall assume all graphs are connected, unless otherwise specified.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Shape from Periodic Texture Using the Eigenvectors of Local Affine Distortion AbstractÐThis paper shows how the local slant and tilt angles of regularly textured curved surfaces can be estimated directly, without the need for iterative numerical optimization. We work in the frequency domain and measure texture distortion using the affine distortion of the pattern of spectral peaks. The key theoretical contribution is to show that the directions of the eigenvectors of the affine distortion matrices can be used to estimate local slant and tilt angles of tangent planes to curved surfaces. In particular, the leading eigenvector points in the tilt direction. Although not as geometrically transparent, the direction of the second eigenvector can be used to estimate the slant direction. The required affine distortion matrices are computed using the correspondences between spectral peaks, established on the basis of their energy ordering. We apply the method to a variety of real-world and synthetic imagery. THE recovery of surface shape using texture information is a process that is grounded in psychophysics [1] . Moreover, it has been identified by Marr [2] as being a potentially useful component of the P I P h sketch. Stated succinctly, the problem is as follows: Given a two-dimensional image of a curved textured surface, how can the three-dimensional shape of the viewed object be recovered? This is clearly an ill-posed problem. In order to be rendered tractable, restrictive simplifications must be made. These frequently hinge on the assumed periodicity, homogeneity [3] , and isotropy of the underlying surface texture. When viewed from the standpoint of shape recovery, there are two distinct areas of activity in the literature. The first of these confines its attention to planar surfaces and focuses on the recovery of perspective geometry from texture gradient or vanishing point location [4] , [5] , [6] , [7] . The second problem is that of interpreting the geometry of curved surfaces [8] , [9] , [10] , [11] , [12] . Without detailed knowledge of the viewing geometry or camera optics, the latter problem involves recovering local surface orientation from texture gradient.. This second problem is perhaps the most interesting from the standpoint of shape perception and is the focus of this paper. The problem can be approached in two ways. The first of these is to use presegmented structural texture primitives and to measure texture gradient using the variation in the size of the primitives. This approach will clearly be sensitive to the process used to segment the required texture primitives. However, the method can be used with aperiodic textures. The second commonly used method is to adopt a frequency domain representation of the underlying texture distribution. The frequency domain approach can be applied more directly to the raw image data and is potentially less sensitive to the segmentation process. However, the applicability of the method is restricted to periodic textures.. Although it is not dependant on the reliable detection of texture primitives or the measurement of texture gradient, the frequency domain approach to curved shape-from-texture can be criticized on a number of grounds. First, in the methods of both Krumm and Shafer [13] and Malik and Rosenholtz [10] , [11] the recovery of local surface orientation is based on either numerical optimization or exhaustive search. This is done to overcome the problems associated with the fact that no fronto-parallel sample of the texture is to hand. In addition, the method of Rosenholtz and Malik [10] , [11] has difficulty in distinguishing between curved and planar surfaces, is sensitive to initial parameter values, and needs curvature to be specified as a parameter.. To overcome these shortcomings, our aim in this paper is to present a closed form method for recovering shape from curved or planar textured surfaces using frequency information. We follow Krumm and Shafer [13] , Super and Bovik [5] , [14] , Malik and Rosenholtz [10] , [11] by measuring the local texture variations due to the perspectivity using the frequency domain affine distortion of the pattern of spectral peaks. Our main contribution is to show that the directions of the eigenvectors of the affine distortion matrix can be used to directly estimate the slant and tilt angles of local tangent planes to curved surfaces. This result applies under the assumption that the underlying surface is painted with a uniform texture and is viewed under perspective projection onto the image plane. The texture is assumed to be homogeneous but not necessarily isotropic. Our main contribution is, therefore, to develop a direct and simple method for estimating slant and tilt angles.. The main advantage of the method presented in this paper over those of Super and Bovik [14] , Krumm and Shafer [15] , [4] , [13] and Malik and Rosenholtz [11] is that it delivers slant and tilt angles by computing the eigenvectors of the affine distortion matrix for spectral peaks. These related methods, differ from our proposed method in the way in which the affine distortion of the texture spectrum is used to measure the local surface orientation of curved surfaces. Specifically, previous work in this area concentrates on using numerical optimization or exhaustive search to recover the local slant and tilt angles. By contrast, our method recovers the parameters in closed form. To provide some specific examples, Krumm and Shafer [13] numerically adjust the slant and tilt angles so as to minimize a ªfrontalizationº error between the backprojected texture spectra. Although less direct than our method of orientation estimation, this backprojection method involves a feedback loop. The lack of feedback between the measured texture spectra and the estimated orientation field may be viewed as a weakness of our method. Super and Bovik [14] , [5] , on the other hand, invoke the homogeneity assumption and recover the surface orientation parameters which minimize the variance of the backprojected frequency vectors of the spectral peaks. Rosenholz and Malik [10] , [11] have a more sophisticated method which involves using a singular value decomposition method to minimize a 1 P measure of goodness of fit. The recovered parameters are the slant and tilt angles together with local curvature. We have presented a new method for estimating the local orientation of tangent planes to curved textured surfaces. The method commences by finding affine spectral distortions between neighboring points on the image plane. The directions of the eigenvalues of the local distortion matrices can be used to make closed form estimates of the slant and tilt directions. The initial orientation estimates returned by the new method are iteratively refined using a robust smoothing technique to produce a needle map of improved consistency.. The method is demonstrated on both synthetic imagery with known ground-truth and on real-world images of man-made textured surfaces. The method proves useful in the analysis of both planar and curved surfaces. Moreover, the extracted needle maps can be used to make reliable estimates of surface curvature information.. There are a number of ways in which the ideas presented in this paper could be extended and improved. First, our texture measures are relatively crude and could be refined to allow us to analyze textures that are regular but not necessarily periodic. Second, there is scope for improving the quality of the needle map through measuring the back-projection error associated with the smoothed surface normal directions. One possible approach to this problem would be to use a variant of the expectation-maximization algorithm to develop a statistical framework for local orientation estimation and backprojection error assessment. Finally, our current method makes use of a potentially fragile energy ordering heuristic to locate spectral correspondences. A more satisfactory solution to this problem would be to use more than two peaks and to find the set of correspondences which best align the pattern of peaks. We have recently developed a dual-step EM algorithm which can solve this correspondence problem for cluttered pointsets [29] . Our future plans involve using this algorithm for spectral correspondence. We have experimented with both synthetic surfaces with known ground-truth and real-world images. The former are used to assess the accuracy of the method, while we use the latter to demonstrate the practical utility of the method.

O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


An Additivity Theorem for Plain Kolmogorov Complexity We prove the formula C(a, b) = K(a|C(a, b)) + C (b|a, C(a, b)) + O(1) that expresses the plain complexity of a pair in terms of prefix-free and plain conditional complexities of its components.Kolmogorov complexity · Symmetry of information · Additivity of Kolmogorov complexity · Plain complexityThe well known formula from Shannon information theory states that H (ξ, η) = H (ξ) + H (η|ξ). Here ξ, η are random variables and H stands for the Shannon entropy. A similar formula for algorithmic information theory was proven by Kolmogorov and Levin [5] and says thatwhere a and b are binary strings of length at most n and C stands for Kolmogorov complexity (as defined initially by Kolmogorov [4]; now this version is usually called plain Kolmogorov complexity). Informally, C(u) is the minimal length of a program B. Bauwens is supported by Fundaçao para a Ciência e a Tecnologia by grant (SFRH/BPD/75129/2010), and is also partially supported by project CSI 2 (PTDC/EIAC/099951/2008). A. Shen is supported in part by projects NAFIT ANR-08-EMER-008-01 grant and RFBR 09-01-00709a. Author are grateful to their colleagues for interesting discussions and to the anonymous referees for useful comments.B. Bauwens ( )

O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Multi-modal nonmonotonic logics of minimal knowledge In this paper we introduce multi-modal logics of minimal knowledge. Such a family of logics constitutes the first proposal in the field of epistemic nonmonotonic logic in which the three following aspects are simultaneously addressed: (1) the possibility of formalizing multiple agents through multiple modal operators; (2) the possibility of using first-order quantification in the modal language; (3) the possibility of formalizing nonmonotonic reasoning abilities for the agents modeled, based on the principle of minimal knowledge. We illustrate the expressive capabilities of multi-modal logics of minimal knowledge to provide a formal semantics to peer-topeer data integration systems, which constitute one of the most recent and complex architectures for distributed information systems. In this paper we have proposed a first attempt to define a multi-modal, first-order, nonmonotonic family of logics. In particular, the logics S A n presented in this paper generalize recent approaches in epistemic logic and nonmonotonic modal logics in many respects.. We have also illustrated the need for multi-modal nonmonotonic logics in the field of distributed systems. Interestingly, the possibility of modeling knowledge in distributed systems was also the initial motivation behind one of the first nonmonotonic modal logics, i.e., Halpern and Moses' logic of minimal knowledge [18] .. An interesting extension of the present work is towards reasoning in the logics S A n . The first results in this direction appear in [6] , in which an algorithm is presented for reasoning in the restricted fragment of the logic K45 n which is able to logically model information in P2P systems. However, the important issue of defining a general proof system for K45 A n and for other logics in the S A n family is still open. So far, the results in this direction only concern: (1) either single-agent, propositional fragments of S A n , in particular ground nonmonotonic modal logics, for which a tableaux method has been defined in [8] ; or (2) the single-agent propositional fragment of S5 A n , i.e., the logic MKNF, for which reasoning has been studied in [38] .. Finally, it would be very interesting to investigate whether the logics S A n can be characterized by fix-point semantics, in a way analogous to other families of nonmonotonic modal logics [30, 41, 46] .

O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O