布尔矩阵维数的缺断区间(英)

本文证明了n维布尔矩阵维数的一个新的缺断区间为这一结果扩充了一些已知的结论．     

Finding nonnormal bent functions

The question if there exist nonnormal bent functions was an open question for several years. A Boolean function in n variables is called normal if there exists an affine subspace of dimension n/2 on which the function is constant. In this paper we give the first nonnormal bent function and even an example for a nonweakly normal bent function. These examples belong to a class of bent functions found in [J.F. Dillon, H. Dobbertin, New cyclic difference sets with Singer parameters, in: Finite Fields and Applications, to appear], namely the Kasami functions. We furthermore give a construction which extends these examples to higher dimensions. Additionally, we present a very efficient algorithm that was used to verify the nonnormality of these functions.     

Set controllability of Markov jump switching Boolean control networks and its applications

In this article, a novel method is proposed for investigating the set controllability of Markov jump switching Boolean control networks (MJSBCNs). Specifically, the switching signal is described as a discrete-time homogeneous Markov chain. By resorting to the expectation and switching indicator function, an expectation system is constructed. Based on the expectation system, a novel verifiable condition is established for solving the set reachability of MJSBCNs. With the newly obtained results on set reachability, a necessary and sufficient condition is further derived for the set controllability of MJSBCNs. The obtained results are applied to Boolean control networks with Markov jump time delays. Examples are demonstrated to justify the theoretical results.     

The Boolean Prime Ideal Theorem Plus Countable Choice Do Not Imply Dependent Choice

Two Fraenkel-Mostowski models are constructed in which the Boolean Prime Ideal Theorem is true. In both models, AC for countable sets is true, but AC for sets of cardinality 2 and the 2m = m principle are both false. The Principle of Dependent Choices is true in the first model, but false in the second. Mathematics Subject Classification: 03E25, 03E35, 04A25.     

Relevance and truthfulness in information correction and fusion

A general approach to information correction and fusion for belief functions is proposed, where not only may the information items be irrelevant, but sources may lie as well. We introduce a new correction scheme, which takes into account uncertain metaknowledge on the source’s relevance and truthfulness and that generalizes Shafer’s discounting operation. We then show how to reinterpret all connectives of Boolean logic in terms of source behavior assumptions with respect to relevance and truthfulness. We are led to generalize the unnormalized Dempster’s rule to all Boolean connectives, while taking into account the uncertainties pertaining to assumptions concerning the behavior of sources. Eventually, we further extend this approach to an even more general setting, where source behavior assumptions do not have to be restricted to relevance and truthfulness. We also establish the commutativity property between correction and fusion processes, when the behaviors of the sources are independent.     

Linearization of nonlinear filter generators and its application to cryptanalysis of stream ciphers

Nonlinear filter generators are commonly used as keystream generators in stream ciphers. A nonlinear filter generator utilizes a nonlinear filtering function to combine the outputs of a linear feedback shift register (LFSR) to improve the linear complexity of keystream sequences. However, the LFSR-based stream ciphers are still potentially vulnerable to algebraic attacks that recover the key from some keystream bits. Although the known algebraic attacks only require polynomial time complexity of computations, all have their own constraints. This paper uses the linearization of nonlinear filter generators to cryptanalyze LFSR-based stream ciphers. Such a method works for any nonlinear filter generators. Viewing a nonlinear filter generator as a Boolean network that evolves as an automaton through Boolean functions, we first give its linearization representation. Compared to the linearization representation in Limniotis et al. (2008), this representation requires lower spatial complexity of computations in most cases. Based on the representation, the key recoverability is analyzed via the observability of Boolean networks. An algorithm for key recovery is given as well. Compared to the exhaustive search to recover the key, using this linearization representation requires lower time complexity of computations, though it leads to exponential time complexity.     

完备布尔代数商格的注

本文研究了完备布尔代数L的商格Q(L),建立 L上q-集构成的集合Q′(L)与Q(L)之间的同构,得出了完备布尔代数的商格仍是完备布尔代数,并给出了完备布尔代数范畴的某些性质.     

Bool阵的逆阵

We give a characterization of a regular Boolean matrix and prove that AB = I Implies that BA = I, where A and B are Boolean matrices whose elements belong to a Boolean algebra of a set with more than two elements.     

Martin's Axiom is consistent with the existence of nowhere trivial automorphisms

Martin's Axiom does not imply that all automorphisms of are somewhere trivial. An alternate method for obtaining models where every automorphism of is somewhere trivial is explained.