A critical aspect in the problem of inductive inference is the number of examples needed to accurately infer a boolean function from positive and negative examples. In this paper, we develop an approach for deriving a...
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A critical aspect in the problem of inductive inference is the number of examples needed to accurately infer a boolean function from positive and negative examples. In this paper, we develop an approach for deriving a sequence of examples for this problem. Some computer experiments indicate that, on the average, examples derived according to the proposed approach lead to the inference of the correct function considerably faster than when examples are derived in a random order.
We study a procedure for estimating an upper bound of an unknown noise factor in the frequency domain. A learning algorithm using a Fourier transformation method was originally given by Linial, Mansour and Nisan. Whil...
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We study a procedure for estimating an upper bound of an unknown noise factor in the frequency domain. A learning algorithm using a Fourier transformation method was originally given by Linial, Mansour and Nisan. While Linial, Mansour and Nisan assumed that the learning algorithm estimates Fourier coefficients from noiseless data, Bshouty, Jackson, and Tamon, and also Ohtsuki and Tomita extended the algorithm to ones that are robust for noisy data. The noise process that we consider is as follows: for an example < x, f (x)>, where x is an element of {0, 1}(n), f(x) is an element of {-1, 1}, each bit of x and f (x) gets flipped independently with probability eta during a learning process. The previous learning algorithms for noisy data all assume that the noise factor eta or an upper bound of eta is known in advance. The learning algorithm proposed in this paper works without this assumption. We estimate an upper bound of the noise factor by evaluating a noisy power spectrum in the frequency domain and by using a sampling trick. Combining this procedure with Ohtsuki and Tomita's algorithm, we obtain a quasi-polynomial-time learning algorithm that can cope with noise without knowing any information about the noise in advance. (C) 2011 Elsevier B.V. All rights reserved.
We characterise the aperiodic autocorrelation for a boolean function, f, and define the Aperiodic Propagation Criteria (APC) of degree l and order q. We establish the strong similarity between APC and the Extended Pro...
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We characterise the aperiodic autocorrelation for a boolean function, f, and define the Aperiodic Propagation Criteria (APC) of degree l and order q. We establish the strong similarity between APC and the Extended Propagation Criteria as defined by Preneel et al. in 1991, although the criteria are not identical. We also show how aperiodic autocorrelation can be related to the first derivative off. We further propose the metric APC distance and show that quantum error correcting codes are natural candidates for boolean functions with favourable APC distance. (C) 2006 Elsevier Inc. All rights reserved.
This paper proposes a new type of expression for boolean functions called lexicographical expressions. The basic idea is to impose an input ordering for factoring logical expressions. Several algebraic properties are ...
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This paper proposes a new type of expression for boolean functions called lexicographical expressions. The basic idea is to impose an input ordering for factoring logical expressions. Several algebraic properties are presented and relations with classical algebraic theory are established. The main result is that all elementary factorizations defined by (Cokernel, Kernel) pairs ''compatible'' with an input order are all ''algebraically compatible,'' i.e., are all parts of a single factorization of the function. Thus for a given input order a unique factorization is defined. This leads to fast division procedures. Basic techniques for obtaining lexicographical factorizations are presented. First, a precedence matrix and an updating procedure are defined and used later to select an input order and a corresponding compatible factorization. Second, a factorization technique respecting a fixed order is detailed. This method is then applied to multi-level synthesis using standard cells which was the original motivation of this work. The goal is to reduce wiring complexity. A lexicographical factorization leads to a wiring area reduction due to the structuring of the logic into layers in which the inputs enter the layout in the order given by the factorization. Experimental results comparing this approach to classical ones are given. These results include routing ratio measurements, routing structure observation, global area measurement and critical path estimation. All these results are analyzed after place and route, using an industrial tool (COMPASS Design Automation tool).
A new framework concerning the construction of small-order resilient boolean functions whose nonlinearity is strictly greater than 2(n-1) -2(left perpendicularn/2right perpendicular) is given. First, a generalized Mai...
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A new framework concerning the construction of small-order resilient boolean functions whose nonlinearity is strictly greater than 2(n-1) -2(left perpendicularn/2right perpendicular) is given. First, a generalized Maiorana-McFarland construction technique is described, which extends the current approaches by combining the usage of affine and nonlinear functions in a controllable manner. It is shown that for any given m, this technique can be used to construct a large class of n-variable (n both even and odd) m-resilient degree-optimized boolean functions with currently best known nonlinearity. This class may also provide functions with excellent algebraic properties, measured through the resistance to (fast) algebraic attacks, if the number of n/2-variable affine subfunctions used in the construction is relatively low. Due to a potentially low hardware implementation cost, along with overall good cryptographic properties, this class of functions is an attractive candidate for the use in certain stream cipher schemes.
We give a new lower bound to the covering radius of the first order Reed-Muller code RM(1, n), where n is an element of {9, 11, 13}. Equivalently, we present the n-variable boolean functions for n is an element of {9,...
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We give a new lower bound to the covering radius of the first order Reed-Muller code RM(1, n), where n is an element of {9, 11, 13}. Equivalently, we present the n-variable boolean functions for n is an element of {9,11,13} with maximum nonlinearity found till now. In 2006, 9-variable boolean functions having nonlinearity 241, which is strictly greater than the bent concatenation bound of 240, have been discovered in the class of Rotation Symmetric boolean functions (RSBFs) by Kavut, Maitra and Yucel. To improve this nonlinearity result, we have firstly defined some subsets of the n-variable boolean functions as the generalized classes of "k-RSBFs and k-DSBFs (k-Dihedral Symmetric boolean functions)", where k is a positive integer dividing n. Secondly, utilizing a steepest-descent like iterative heuristic search algorithm, we have found 9-variable boolean functions with nonlinearity 242 within the classes of both 3-RSBFs and 3-DSBFs. Thirdly, motivated by the fact that RSBFs are invariant under a special permutation of the input vector, we have classified all possible permutations up to the linear equivalence of boolean functions that are invariant under those permutations. (C) 2009 Elsevier Inc. All rights reserved.
We investigate the structure of "worst-case" quasi reduced ordered decision diagrams and boolean functions whose truth tables are associated to: we suggest different ways to count and enumerate them. We, the...
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We investigate the structure of "worst-case" quasi reduced ordered decision diagrams and boolean functions whose truth tables are associated to: we suggest different ways to count and enumerate them. We, then, introduce a notion of complexity which leads to the concept of "hard" boolean functions as functions whose QROBDD are "worst-case" ones. So we exhibit the relation between hard functions and the Storage Access function ( also known as Multiplexer).
We propose a quantum algorithm to estimate the Gowers U2 norm of a boolean function, and extend it into a second algorithm to distinguish between linear boolean functions and boolean functions that are E-far from the ...
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We propose a quantum algorithm to estimate the Gowers U2 norm of a boolean function, and extend it into a second algorithm to distinguish between linear boolean functions and boolean functions that are E-far from the set of linear boolean functions, which seems to perform better than the classical BLR algorithm. Finally, we outline an algorithm to estimate Gowers U3 norms of boolean functions.
Classes of set functions defined by the positivity or negativity of the higher-order derivatives of their pseudo-boolean polynomial representations generalize those of monotone, supermodular, and submodular functions....
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Classes of set functions defined by the positivity or negativity of the higher-order derivatives of their pseudo-boolean polynomial representations generalize those of monotone, supermodular, and submodular functions. In this paper, these classes are characterized by functional inequalities and are shown to be closed both under algebraic closure conditions and a local closure criterion. It is shown that for every m : 1, in addition to the class of all set functions, there are only three other classes satisfying these algebraic and local closure conditions: those having positive, respectively negative, mth-order derivatives, and those having a polynomial representation of degree less than m.
A boolean function has an inverse when every output is the result of one and only one input. There are 2 n ! boolean functions of n variables which have an inverse. Equivalence classes of these functions are sets of e...
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A boolean function has an inverse when every output is the result of one and only one input. There are 2 n ! boolean functions of n variables which have an inverse. Equivalence classes of these functions are sets of equivalent functions in the sense that they are identical under a group operation on the input and output variables. This paper counts through five variables the number of equivalence classes of invertible boolean functions under the group operation of complementation, permutation, and complementation and permutation, linear transformations and affine transformations. Lower bounds are given which experimentally give an asymptotic approximation. A representative function is given of each of the 52 classes of invertible boolean functions of three variables under complementation and permutation. These are divided into three types of classes, 21 self-inverting functions, three functions have an inverse in the same class and 14 pairs of functions, each function of the pair in a different class. The four representative functions under the affine transformation are self-invertible.
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