Progress in Applications of Boolean Functions

In symmetric cryptology the resistance to attacks depends critically on the nonlinearity properties of the Boolean functions describing cipher components like Substitution boxes (S‐boxes). Some of the most effective methods known to generate functions that satisfy multiple criteria are based on evolutionary heuristics. In this paper, we improve on these algorithms by employing an adaptive strategy. Additionally, using recent improvements in the understanding of these combinatorial structures, we discover essential properties of the graph formed by affine equivalence classes of Boolean functions, which offers several advantages as a conceptual model for multiobjective seeking evolutionary heuristics. Finally, we propose the first major global cooperative effort to discover new bounds for cryptographic properties of Boolean functions.

An asymptotic estimate is established for the number of equivalence classes of Boolean functions of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> variables under transformations of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(x) \rightarrow f(Ax+b)+ L(x)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x=(x_{l}, \cdots ,x_{n}), A</tex> is a nonsingular <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> matrix, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> is a vector, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L(x)</tex> is a linear function.

The simultaneous matching-to-sample procedures that are widely used to study stimulus equivalence in human participants have generally been unsuccessful in animals. However, functional equivalence classes have been demonstrated in pigeons and sea lions using a concurrent repeated reversal discrimination procedure. In this procedure, responding to one set of stimuli is reinforced but responding to a different set is not and the set associated with reinforcement is changed with multiple reversals during the experiment. The experiments reported here were designed to assess whether functional equivalence classes could be demonstrated in rats using similar techniques. Rats were initially trained with two sets of olfactory stimuli (six odors/set). Following many reversals, probe reversal sessions were conducted in which rats were exposed to a subset of the members of each set and, later in the session, the withheld stimuli were introduced. Responding to these delayed probe trials in accord with the reversed contingencies constituted transfer of function. There was some evidence of transfer in Experiment 1, but the effects were relatively weak and variable. Experiment 2 introduced procedural changes and found strong evidence of transfer of function consistent with the formation of functional equivalence classes. These procedures offer a promising strategy to study symbolic behavior in rodents.

In Experiments 1 and 2 a match-to-sample procedure was used to establish the prerequisite conditional discriminations for two 3-member equivalence classes (A1, B1, C1 & A2, B2, C2) where stimuli A1 and A2 were written instructions. In Experiment 1 tests for transitivity confirmed the emergence of derived relations. Following this, an off-the-baseline test was given to verify the existence of functional equivalence classes. During off-the-baseline testing stimuli B and C from both classes controlled responding appropriate to the instructions contained in A1 and A2 respectively. In Experiment 2, the omission of tests for transitivity and the reversal of instructions contained in A1 and A2 did not impede the emergence of functional equivalence classes. In Experiment 3, classes A1, B1, C1 and E1, F1, G1 were linked via D1 to produce one class (A1, B1, C1, D1, E1, F1, G1); another two classes (A2, B2, C2 & E2, F2, G2) were linked with D2 to produce A2, B2, C2, D2, E2, F2, G2. The A and G stimuli in each class contained competing sets of instructions. When transformation of function was examined in the first class, the functions at C1 and E1 were controlled by the rules at A1 and G1 respectively; D1 controlled responding in accordance with the instructions at A1. Results from all experiments are discussed in the context of procedures designed to investigate transfer/transformation of function.

This study extended previous research on equivalence classes that contain more than one function. Initially, separate equivalence classes were established (A1, B1, C1 and A3, B3, C3) using a one-to-many matching-to-sample procedure where A1 and A3 were the sample stimuli. These classes then were transformed into functional equivalence classes by training unique functions at A1 and A3; using modelling clay, a ball was made at A1 and an oblong was made at A3. These two classes then were joined together using another matching-to-sample procedure to establish the class X1-A1-A3. Tests were conducted to see what behaviours occurred in the presence of A1, X1, and A3. Of seven participants, three produced entirely new behaviours at X1, while the others produced the behaviours previously taught at A1 and A3. Results are discussed in the context of variables affecting the generation of novel behaviour.

The present study examined the effects of including stimuli previously trained as members of functional classes or equivalence classes on subsequent equivalence class formation, and isolated the effects of preliminary training from those of the acquired function stimuli. Fifty-six adults were assigned to 1 of 5 conditions. The control group (CONT) received no preliminary training prior to the terminal phase. Participants in the other 4 groups learned two 3-member functional classes and two 3-member equivalence classes during the preliminary phase. The terminal equivalence phase trained two 5-member classes (A → B → C → D → E) comprising abstract forms; the C stimuli in the terminal phase were (a) from the preliminary functional classes for 1 group (ACQ-F), (b) from the preliminary equivalence classes for the second experimental group (ACQ-E), (c) pictures of everyday objects for the picture control group (PIC), and (d) novel, unfamiliar stimuli for the preliminary training control group (PRE-CONT). Class formation yields were 100% in the PIC condition and 11% in the CONT condition; however, low yields in the PRE-CONT, ACQ-F, and ACQ-E conditions were unexpected, suggesting that procedural variables in preliminary training account for more of the subsequent effects on class formation than the stimulus control properties of the acquired function stimuli.

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.

It has been claimed that different types of causes must be considered in biological systems, including top-down as well as same-level and bottom-up causation, thus enabling the top levels to be causally efficacious in their own right. To clarify this issue, the important distinctions between information and signs are introduced here and the concepts of information control and functional equivalence classes in those systems are rigorously defined and used to characterize when top-down causation by feedback control happens, in a way that is testable. The causally significant elements we consider are equivalence classes of lower level processes, realized in biological systems through different operations having the same outcome within the context of information control and networks.

As preparation for a duality theory for saddle programs, a partial conjugacy correspondence is developed among equivalence classes of saddle functions. Three known conjugacy correspondences, including Fenchel’s correspondence among convex functions and Rockafellar’s extension of it to equivalence classes of saddle functions, are shown to be degenerate special cases. Additionally, two new correspondences are brought to light as further special cases. Various questions are answered concerning the lower and upper closures and effective domain of the resulting equivalence class, as well as the effect the correspondence has on the related purely convex function and the subdifferential mapping.

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.

A recursive formula for weights of Boolean rotation symmetric functions

Fuzzy measures or capacities are the most general representation of uncertainty functions. However, this general class has been little explored from the point of view of its information content, when degrees of uncertainty are not supposed to be numerical, and belong to a finite qualitative scale, except in the case of possibility or necessity measures. The thrust of the paper is to define an ordering relation on the set of qualitative capacities expressing the idea that one is more informative than another, in agreement with the possibilistic notion of relative specificity. To this aim, we show that the class of qualitative capacities can be partitioned into equivalence classes of functions containing the same amount of information. They only differ by the underlying epistemic attitude such as pessimism or optimism. A meaningful information ordering between capacities can be defined on the basis of the most pessimistic (resp. optimistic) representatives of their equivalence classes. It is shown that, while qualitative capacities bear strong similarities to belief functions, such an analogy can be misleading when it comes to information content.

A modified version of the original Derived Relational Intrusions Following Training paradigm (DRIFT; Guinther & Dougher, Journal of the Experimental Analysis of Behavior, 93, 329–347; 2010) was used in an attempt to engineer contextually controlled semantic false memories. Participants completed a contextually controlled interrelated conditional discrimination training procedure (match to sample) without tests of symmetry or transitivity. Study list words were nominally assigned to participate in a functional equivalence class with 1 set of non-study words in 1 context but with a different set of non-study words in a different context. The study list words were then presented in 1 of the 2 contexts, between subjects, for free-recall memory testing. Subsequent transfer of the study list words’ remembering function to non-study words (i.e., false recall) was found to be partially dependent upon the context in which the free recall test was administered, thus demonstrating partial contextual control of semantic false memory.

We study counting functions on the free groups F_n and free monoids M_n for n \geq 2 , which we introduce for combinatorial approach to famous Brooks quasimorphisms on free groups. Two counting functions are considered equivalent if they differ by a bounded function. We find the complete set of linear relations between equivalence classes of counting functions and apply this result to construct an explicit basis for the vector space of such equivalence classes. Moreover, we provide a simple graphical algorithm to determine whether two given counting functions are equivalent. In particular, this yields an algorithm to decide whether two linear combinations of Brooks quasimorphisms on F_n represent the same class in bounded cohomology.

UDC 513.88 It the article we study sublinear functionals that are defined on the space of Bochner integrable functions and possess the properties of decomposition and scalar compactness. We give an integral representation for the functionals. Such functionals are used in proving theorems on existence of continuous selections for a family of lower semicontinuous multivalued mappings with decomposable closed nonconvex values in the space of Bochner integrable functions. The article relates to the studies [1-3] and others. 1. We present basic definitions, notation, and a series of auxiliary results. Throughout what follows, X is a separable Banach space with norm H" ]l, T is a Hausdorff compact space with positive Radon measure/z and a-algebra ~ of/,-measurable subsets, and R stands for the real axis. We denote by L1 (T, X) the Banach space of equivalence classes of Bochner integrable functions u: T ---* X, with the norm I[u(.)HL1 = fJ[u(t)Hd#. T By X t and L ~ (T, X) we mean the topological dual spaces of X and L1 (T, X) with the strong (norm) topologies; (x, xl~ and (u(-), u'(.)) are the canonical bilinear forms establishing duality between X and X' and between LI(T, X) and UI(T , X) respectively. For the norm of X ~ we use the same notation as for that of X. The spaces X' and LI(T,X ) endowed with weak a(X~,X) - and a( L~ (T,X), Ll(T,X) )-topologies will be denoted by a-X' and a-L](T,X) (see [4, p. 6831). A function v : T ~ X I is called scalarly measurable if, given x E X, the scalar function (x, v(t)) is measurable. A function v : T ~ X ~ is called weakly measurable if, for every e > 0, there is a closed set Te C T, p(T \ Te) <_ e, such that the restriction of v to Te is a continuous function from T~ into a-X ~. We denote by Loo(T, X') the Banach space of equivalence classes of scalarly measurable, essentially bounded functions v : T --* X', with the norm IIv(.)Hoo = esssup Hv(t)H. Henceforth we identify lET equivalence classes in the spaces L~ (T, X) and L~ (T, X') with their representatives. It is well known (see [4, p. 807]) that U I(T,X) and Loo(T,X') are linearly isometric and that each element x'(.) e L~(T,X) is representable as (x(.),x'(.)) = f(x(t),v(t))d#, x(.) �9 LI(T,X), where T v(.) �9 Loo(T,X'). Therefore, we shall by convention regard Loo(T,X ~) as strongly dual to LI(T, X) and write (x(.),x~(.)) = f(x(t),x~(t))d#, x(.) �9 LI(T,X), considering x~(.) as an element of the space