Learning a Function of r Relevant Variables

Two Boolean functions are affine equivalent if one can be obtained from the other by applying an affine transformation to the input variables. For a long time, there have been efforts to investigate the affine equivalence of Boolean functions. Due to the complexity of the general problem, only affine equivalence under certain groups of permutations is usually considered. Boolean functions which are invariant under the action of cyclic rotation of the input variables are known as rotation symmetric (RS) Boolean functions. Due to their speed of computation and the prospect of being good cryptographic Boolean functions, this class of Boolean functions has received a lot of attention from cryptographic researchers. In this paper, we study affine equivalence for the simplest rotation symmetric Boolean functions, called MRS functions, which are generated by the cyclic permutations of a single monomial. Using Pólya’s enumeration theorem, we compute the number of equivalence classes, under certain large groups of permutations, for these MRS functions in any number

The class of affine Boolean functions is rich enough to express constant bits and dependencies between different bits of different words. For example, the function (x0) ∧ (¬y1) ∧ (x4 ⇔ y7) ∧ (x5 ⇔ ¬y9) is affine and expresses the invariant that the low bit (bit 0) of the variable x is true, that bit 1 of y is false, that the bits 4 and 7 of x and y coincide whereas bits 5 and 9 of x and y differ. This class of Boolean function is amenable to bit-precise reasoning since it satisfies strong chain properties which bound the number of times a system of semantic fixpoint equations need to be reapplied when reasoning about loops. This paper address the key problem of abstracting an arbitrary Boolean function to either a general affine function or a so-called affine function of width 2, when the function is represented as an ROBDD. Novel algorithms are presented for this task: one that manipulates Boolean vectors and another which is inspired by anti-unification. The speed and precision of both algorithms are compared on benchmark circuits, to draw conclusions on the tractability of affine abstraction.

This paper investigates the optimum OBDD representation problem based on two classes of Boolean functions. The first class is defined by OBDDs, in which the number of non-terminal nodes is equal to the number of input variables. We refer to such OBDDs and their corresponding Boolean functions as thin OBDDs and thin Boolean functions. The second class is the thin factored Boolean functions, which is defined by factored forms with each variable appearing exactly once. Many interesting properties of these two classes of functions are presented. Based on which, a revised dynamic shortest cube first OBDD variable ordering algorithm is developed. This algorithm is shown to be optimum for thin factored Boolean functions.

Information processing in dendrites: II. Information theoretic complexity

Several underlying structural and functional factors that determine the fault behavior of a network are not yet well understood. In this paper, we show that there exists a large class of Boolean functions, called root functions, which can never appear as faulty response in an irredundant two-level circuit even when any arbitrary multiple stuck-at faults are injected. Conversely, we show that any other Boolean function can appear as a faulty response in an irredundant realization of some root function under certain stuck-at faults. We characterize this new class of functions and show that for n variables, their number is exactly equal to the number of independent dominating sets (Harary and Livingston, Appl. Math. Lett., 1993) in a Boolean n-cube. Similar properties are observed for multiple-valued logic functions as well. Finally, we discuss its application to logic design and point out some open problems.

Random Boolean networks (RBNs) have been widely explored as models for understanding complex systems, in particular gene regulatory networks (GRNs). Stability (order) and instability (chaos), critical characteristics of any dynamic system, have naturally been a central focus in RBN research. Several measures have previously been introduced to assess stability, e.g., via damage spreading or via statistical properties of attractors such as their number, lengths or basin sizes. Undoubtedly, network topology plays an important role in shaping the dynamics of RBN. Another key factor influencing the dynamics is the Boolean functions (BFs) assigned to each node. In this work, we conduct a systematic investigation to examine the influence of five different classes of BFs on the dynamics of RBNs. By employing various dynamical stability measures, we show that biologically meaningful BFs consistently drive the dynamics toward the stable regime compared to random BFs. This observation holds across networks with varying size, connectivity, and degree distributions. Additionally, we show that the change in the values of our stability measures with increasing connectivity or size of RBNs is different across the BF classes. For most observables, random BFs change the values significantly, pushing the dynamics toward a more chaotic regime with increasing connectivity or network size. In contrast, biologically meaningful BFs show very minimal fluctuations for most stability measures. These findings emphasize the advantage of restricting to biologically meaningful classes in the reconstruction and modeling of biological systems within Boolean framework.

In this paper, we study the nonexistence of bent functions in the class of Boolean functions without monomials of degree less than d in their algebraic normal forms (ANF). We prove that n-variable Boolean functions in such class are not bent when there are not more than n + d - 3 monomials in their ANFs. We also show that an n-variable Boolean function is not bent if it has no monomial of degree less than ⌈3n/8 + 3/4⌉ in its ANF.

Optimal ordered binary decision diagrams for read-once formulas

Rotation Symmetric Boolean Functions –; Count and Cryptographic Properties

We prove that polynomial size discrete synchronous Hopfield networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial space-bounded nonuniform Turing machines. As a corollary to the construction, we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions P/poly.

International audience

The number and probability of canalizing functions

An analysis of root functions—A subclass of the Impossible Class of Faulty Functions (ICFF)

Boolean functions play a central role in security applications because they constitute one of the basic primitives for modern cryptographic services. In the last decades, research on Boolean functions has been boosted due to the importance of security in many diverse public systems relying on such technology. A main focus is to find Boolean functions with specific properties. An open problem in this context is to find a balanced Boolean function with an 8-bit input and nonlinearity 118. Theoretically, such a function has been shown to exist, but it has not been found yet. In this work we focus on specific classes of Boolean functions, and analyze the landscape of results obtained by integrating algebraic and evolutionary computation (EC) based approaches. Results indicate that combinations of these approaches give better results although not reaching 118 nonlinearity.

It is a difficult task to compute the r-th order nonlinearity of a given function with algebraic degree strictly greater than r > 1. Though lower bounds on the second order nonlinearity are known only for a few particular functions, the majority of which are cubic. We investigate lower bounds on the second order nonlinearity of cubic Boolean functions [Formula: see text], where [Formula: see text], dl = 2il + 2jl + 1, m, il and jl are positive integers, n > il > jl. Furthermore, for a class of Boolean functions [Formula: see text] we deduce a tighter lower bound on the second order nonlinearity of the functions, where [Formula: see text], dl = 2ilγ + 2jlγ + 1, il > jl and γ ≠ 1 is a positive integer such that gcd(n,γ) = 1. Lower bounds on the second order nonlinearity of cubic monomial Boolean functions, represented by fμ(x) = Tr(μx2i+2j+1), [Formula: see text], i and j are positive integers such that i > j, were obtained by Gode and Gangopadhvay in 2009. In this paper, we first extend the results of Gode and Gangopadhvay from monomial Boolean functions to Boolean functions with more trace terms. We further generalize and improve the results to a wider range of n. Our bounds are better than those of Gode and Gangopadhvay for monomial functions fμ(x). Especially, our lower bounds on the second order nonlinearity of some Boolean functions F(x) are better than the existing ones.