Abstract: Quantum query model is a crucial model for quantum computing, where one query to some input variable of a Boolean function f defined on {0,1}n returns the variable value. The exact query complexity, denoted as QE(f), is defined to be the minimum number of queries required to determine the function value. An important problem in this area is to give a succinct characterization of a k-query exact quantum algorithm for an arbitrary k. To date, the cases k=1 and k=n are already solved and the case k=2 remains unknown. Our result is that there are 27 nondegenerate Boolean functions up to isomorphism with QE(f) being two, among which only two functions can be solved by a 2-query classical algorithm. The input bit number n of the above 27 functions ranges from 2 to 6, where the case n≤3 is already proved and the case n=4 is already found by numerically solving semidefinite programming, which is a complete characterization of quantum query algorithm. Assuming the correctness of the numerical result for n=4, we prove that there are four functions in the case n=5, one in the case n=6 and none in the case n≥7. We further show that the 25 functions for which quantum algorithm has advantage over classical algorithm contain essentially only four different structures.

Abstract: Boolean networks (BNs) have been extensively used to model gene regulatory networks (GRNs). The dynamics of BNs depend on the network architecture and regulatory logic rules (Boolean functions (BFs)) associated with nodes. Nested canalyzing functions (NCFs) have been shown to be enriched among the BFs in the large-scale studies of reconstructed Boolean models. The central question we address here is whether that enrichment is due to certain sub-types of NCFs. We build on one sub-type of NCFs, the chain functions (or chain-0 functions) proposed by Gat-Viks and Shamir. First, we propose two other sub-types of NCFs, namely, the class of chain-1 functions and generalized chain functions, the union of the chain-0 and chain-1 types. Next, we find that the fraction of NCFs that are chain-0 (also holds for chain-1) functions decreases exponentially with the number of inputs. We provide analytical treatment for this and other observations on BFs. Then, by analyzing three different datasets of reconstructed Boolean models we find that generalized chain functions are significantly enriched within the NCFs. Lastly we illustrate that upon imposing the constraints of generalized chain functions on three different GRNs we are able to obtain biologically viable Boolean models.

Abstract: Abstract The main aim of this work is to study important local Banach space constants for Boolean cube function spaces. Specifically, we focus on $\mathcal{B}_{\mathcal{S}}^{N}$, the finite-dimensional Banach space of all real-valued functions defined on the $N$-dimensional Boolean cube $\{-1, +1\}^{N}$ that have Fourier–Walsh expansions supported on a fixed family $\mathcal{S}$ of subsets of $\{1, \ldots , N\}$. Our investigation centers on the projection, Sidon, and Gordon–Lewis constants of this function space. We combine tools from different areas to derive exact formulas and asymptotic estimates of these parameters for special types of families $\mathcal{S}$ depending on the dimension $N$ of the Boolean cube and other complexity characteristics of the support set $\mathcal{S}$. Using local Banach space theory, we establish the intimate relationship among these three important constants.

Abstract: Can every n-bit boolean function with deterministic query complexity k≪ n be restricted to O(k) variables such that the query complexity remains Ω(k)? That is, can query complexity be condensed via restriction? We study such hardness condensation questions in both query and communication complexity, proving two main results. Negative: Query complexity cannot be condensed in general: There is a function f with query complexity k such that any restriction of f to O(k) variables has query complexity O(k3/4). Positive: Randomised communication complexity can be condensed for the sink-of-xor function. This yields a quantitatively improved counterexample to the log-approximate-rank conjecture, achieving parameters conjectured by Chattopadhyay, Garg, and Sherif (2021). Along the way we show the existence of Shearer extractors — a new type of seeded extractor whose output bits satisfy prescribed dependencies across distinct seeds.

Abstract: Abstract Fungal automata are a variation of the two-dimensional sandpile automaton of Bak et al. (Phys Rev Lett 59(4):381–384, 1987. https://doi.org/10.1103/PhysRevLett.59.381 ). In each step toppling cells emit grains only to some of their neighbors chosen according to a specific update sequence. We show how to embed any Boolean circuit into the initial configuration of a fungal automaton with update sequence HV . In particular we give a constructor that, given the description B of a circuit, computes the states of all cells in the finite support of the embedding configuration in $$O(\log \left| {B}\right| )$$ O ( log B ) space. As a consequence the prediction problem for fungal automata with update sequence HV is $$\textsf {P}$$ P -complete. This solves an open problem of Goles et al. (Phys Lett A 384(22):126541, 2020. https://doi.org/10.1016/j.physleta.2020.126541 ).

Abstract: This study presents a novel approach to cryptographic algorithm design that harnesses the power of recurrent neural networks. Unlike traditional mathematical-based methods, neural networks offer nonlinear models that excel at capturing chaotic behavior within systems. We employ a recurrent neural network trained on Monte Carlo estimation to predict future states and generate confusion components. The resulting highly nonlinear substitution boxes exhibit exceptional characteristics, with a maximum nonlinearity of 114 and low linear and differential probabilities. To evaluate the efficacy of our methodology, we employ a comprehensive range of traditional and advanced metrics for assessing randomness and cryptanalytics. Comparative analysis against state-of-the-art methods demonstrates that our developed nonlinear confusion component offers remarkable efficiency for block-cipher applications.

Abstract: The border, or the approximative, model of algebraic computation (VP) is quite popular due to the Geometric Complexity Theory (GCT) approach to P≠NP conjecture, and its complex analytic origins. On the flip side, the definition of the border is inherently existential in the field constants that the model employs. In particular, a poly-size border circuit C(ε, x) cannot be compactly presented in reality, as the limit parameter ε may require exponential precision. In this work we resolve this issue by giving a constructive, or a presentable, version of border circuits and state its applications. We make border presentable by restricting the circuit C to use only those constants, in the function field Fq(ε), that it can generate by the ring operations on {ε}∪Fq, and their division, within poly-size circuit. This model is more expressive than VP as it affords exponential-degree in ε; and analogous to the usual border, we define new border classes called VPε and VNPε. We prove that both these (now called presentable border) classes lie in VNP. Such a 'debordering' result is not known for the classical border classes VP and respectively for VNP. We pose VPε=VP as a new conjecture to study the border. The heart of our technique is a newly formulated exponential interpolation over a finite field, to bound the Boolean complexity of the coefficients before deducing the algebraic complexity. It attacks two factorization problems which were open before. We make progress on (Conj.8.3 in Bürgisser 2000, FOCS 2001) and solve (Conj.2.1 in Bürgisser 2000; Chou,Kumar,Solomon CCC 2018) over all finite fields: 1. Each poly-degree irreducible factor, with multiplicity coprime to field characteristic, of a poly-size circuit (of possibly exponential-degree), is in VNP. 2. For all finite fields, and all factors, VNP is closed under factoring. Consequently, factors of VP are always in VNP. The prime characteristic cases were open before due to the inseparability obstruction (i.e. ‍when the multiplicity is not coprime to q).

Abstract: In this paper, we study the dynamics of synchronous Boolean networks and extend previously obtained results for binary Boolean networks to networks with state variables in a general Boolean algebra of 2p elements, with p>1. The method to do this is based on the Stone representation theorem and the relation of such systems on general Boolean algebras with those with binary-state values. Specifically, we deal with the main periodic orbit problems and predecessor problems (existence, coexistence, uniqueness, and number of them), which allows us to determine the periodic structure and the attractor cycles of the system. These results open opportunities to explore novel applications by means of such general systems.

Abstract: Shapley values, originating in game theory and increasingly prominent in explainable AI, have been proposed to assess the contribution of facts in query answering over databases, along with other similar power indices such as Banzhaf values. In this work we adapt these Shapley-like scores to probabilistic settings, the objective being to compute their expected value. We show that the computations of expected Shapley values and of the expected values of Boolean functions are interreducible in polynomial time, thus obtaining the same tractability landscape. We investigate the specific tractable case where Boolean functions are represented as deterministic decomposable circuits, designing a polynomial-time algorithm for this setting. We present applications to probabilistic databases through database provenance, and an effective implementation of this algorithm within the ProvSQL system, which experimentally validates its feasibility over a standard benchmark.

Abstract: We consider the following basic, and very broad, statistical problem: Given a known high-dimensional distribution D over ℝn and a collection of data points in ℝn, distinguish between the two possibilities that (i) the data was drawn from D, versus (ii) the data was drawn from D|S, i.e. from D subject to truncation by an unknown truncation set S ⊆ ℝn. We study this problem in the setting where D is a high-dimensional i.i.d. product distribution and S is an unknown degree-d polynomial threshold function (one of the most well-studied types of Boolean-valued function over ℝn). Our main results are an efficient algorithm when D is a hypercontractive distribution, and a matching lower bound: 1. For any constant d, we give a polynomial-time algorithm which successfully distinguishes D from D|S using O(nd/2) samples (subject to mild technical conditions on D and S); 2. Even for the simplest case of D being the uniform distribution over {±1}n, we show that for any constant d, any distinguishing algorithm for degree-d polynomial threshold functions must use Ω(nd/2) samples.

Abstract: Block cipher is used as an important technology to protect data confidentiality and user privacy in many fields such as machine learning and cloud storage. Vectorial Boolean functions often serve as the core nonlinear components in block ciphers. When designing cryptographic algorithms, it is crucial to maximize the nonlinearity of vectorial Boolean functions and minimize their autocorrelation to resist relevant cryptographic attacks, to ensure the security of the cryptographic system. In this paper, based on modified Maiorana-McFarland bent functions, we construct a balanced vectorial Boolean functions. Furthermore, we utilize optimal single-weight linear codes to provide specific constructions, and proved that they have high nonlinearity and good differential-linear uniformity.

Abstract: In-memory computing is, in current literature, the most common paradigm used to counteract the Von-Neumann bottleneck, proposing the use of memory elements to define complex input-output relations of the computing kernels. While in classical CMOS computing a similar paradigm can be implemented with look-up tables (LUT), this solution is power and area-hungry. This paper presents the use of Quaternary Content-Addressable Memories (QCAMs), a generalization of the Ternary Content-Addressable Memories (TCAMs), for implementing boolean functions. Content-Addressable Memories can be used as a building block for in-memory processing, using the states of the cells to define a ${\mathbb{B}^{\text{N}}} \to {\mathbb{B}^{\text{M}}}$ function which projects the search word into a new string of bits. The quaternary alphabet allows to represent a more complex function space with respect to the TCAMs while using the same number of cells, enhancing area, power consumption and latency performances achieved when representing arbitrary functions with the CAM hardware. For comparison, it can be demonstrated that QCAMs represent arbitrary Boolean functions with half the number of cells than that would be needed in a standard TCAM implementation, and a ×10 smaller area with respect to SRAM-based LUTs. Along with the table of states and a toy example where the QCAM states are used to define the product among two 2-bit precision real values, this paper presents multiple circuit schemes and encoding schemes for memristor-based QCAMs.

Abstract: Analyzing the relations between Boolean functions has many applications in many fields, such as database systems, cryptography, and collision problems. This paper proposes four quantum algorithms that use amplitude amplification techniques to perform set operations, including Intersection, Difference, and Union, on two Boolean functions in O(N) time complexity. The proposed algorithms employ two quantum amplitude amplification techniques divided into two stages. The first stage uses the Younes et al. algorithm for quantum searching via entanglement and partial diffusion to prepare incomplete superpositions of the truth set of the first Boolean function. In the second stage, a modified version of Arima's algorithm, along with an oracle that represent the second Boolean function, is employed to handle the set operations. The proposed algorithms have a higher probability of success in more general and comprehensive applications when compared with relevant techniques in literature.

Abstract: A crucial system for studying gene regulatory networks is Boolean networks (BNs). Our study employs the Lyapunov function approach to explore the global stability of BN systems under the function perturbations. More specifically, given that the original BN system is globally stable in an ideal environment, a criterion is presented to confirm the robustness of the Lyapunov function of the original system under function perturbations. Finally, an illustrative example of perturbed finite non-cooperative games is provided.

Abstract: A PBN is well known as a mathematical model of complex network systems such as gene regulatory networks. In Boolean networks, interactions between nodes (e.g., genes) are modeled by Boolean functions. In PBNs, Boolean functions are switched probabilistically. In this paper, for a PBN, a simplified representation that is effective in analysis and control is proposed. First, after a polynomial representation of a PBN is briefly explained, a simplified representation is derived. Here, the steady-state value of the expected value of the state is focused, and is characterized by a minimum feedback vertex set of an interaction graph expressing interactions between nodes. Next, using this representation, input selection and stabilization are discussed. Finally, the proposed method is demonstrated by a biological example.

Abstract: Gene Regulatory Networks (GRNs) play a fundamental role in orchestrating the expression of our genes through complex interactions between DNA, RNA, proteins, and other molecules. Accurately reconstructing such networks from gene expression data is a critical yet challenging task in Systems Biology due to their intricate nature and limited data availability. In this work, we introduce a novel Forest-based Evolutionary Algorithm (FP) designed for reconstructing Boolean GRNs from time series data of gene expressions. Unlike traditional methods that struggle with scalability and accurate representation of regulatory interactions, FP utilizes a forest structure where each tree represents the logical relationships between genes, enhancing the model’s capacity to depict complex networks efficiently. Our comprehensive testing indicates that FP rapidly converges towards potential solutions within a limited number of generations, although a higher fitness score does not always equate to a more accurate GRN representation. Implementing mini-batching techniques, inspired by their effectiveness in gradient descent optimization, shows promise in improving computational efficiency without sacrificing performance. A comparative analysis against the main state-of-the-art approaches reveals FP’s tendency towards conservative predictions, emphasizing precision over recall, making it particularly suitable for contexts where the cost of false positives is high. These initial results suggest that FP is a robust and efficient tool for GRN inference.

Abstract: Abstract We present an investigation on threshold circuits and other discretized neural networks in terms of the following four computational resources—size (the number of gates), depth (the number of layers), weight (weight resolution), and energy—where the energy is a complexity measure inspired by sparse coding and is defined as the maximum number of gates outputting nonzero values, taken over all the input assignments. As our main result, we prove that if a threshold circuit C of size s, depth d, energy e, and weight w computes a Boolean function f (i.e., a classification task) of n variables, it holds that log( rk (f))≤ed(logs+logw+logn) regardless of the algorithm employed by C to compute f, where rk (f) is a parameter solely determined by a scale of f and defined as the maximum rank of a communication matrix with regard to f taken over all the possible partitions of the n input variables. For example, given a Boolean function CD n(ξ) =⋁i=1n/2ξi∧ξn/2+i, we can prove that n/2≤ed( log s+logw+logn) holds for any circuit C computing CD n. While its left-hand side is linear in n, its right-hand side is bounded by the product of the logarithmic factors of s,w,n and the linear factors of d,e. If we view the logarithmic terms as having a negligible impact on the bound, our result implies a trade-off between depth and energy: n/2 needs to be smaller than the product of e and d. For other neural network models, such as discretized ReLU circuits and discretized sigmoid circuits, we also prove that a similar trade-off holds. Thus, our results indicate that increasing depth linearly enhances the capability of neural networks to acquire sparse representations when there are hardware constraints on the number of neurons and weight resolution.

Abstract: This article investigates the set stability of switched Boolean networks (SBNs) with function perturbation under arbitrary switching signal. Firstly, the system is converted into algebraic form utilizing semi-tensor product (STP) of matrices. Secondly, based on algebraic expression, four cases are considered. And then, two theorems are obtained, which can be used to determine whether an SBN is set stability after function perturbation. In addition, according to the theorems, two corresponding corollaries are derived. Finally, an example is given to verify the correctness of the conclusion.

Abstract: The main reason for query model’s prominence in complexity theory and quantum computing is the presence of concrete lower bounding techniques: polynomial and adversary method. There have been considerable efforts to give lower bounds using these methods, and to compare/relate them with other measures based on the decision tree. We explore the value of these lower bounds on quantum query complexity and their relation with other decision tree based complexity measures for the class of symmetric functions, arguably one of the most natural and basic sets of Boolean functions. We show an explicit construction for the dual of the positive adversary method and also of the square root of private coin certificate game complexity for any total symmetric function. This shows that the two values cannot be distinguished for any symmetric function. Additionally, we show that the recently introduced measure of spectral sensitivity gives the same value as both positive adversary and approximate degree for every total symmetric Boolean function. Further, we look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity.

Abstract: This paper examines the criteria for the satisfiability and monotonicity of a Boolean function in terms of its polylinear continuation, namely, firstly, a necessary and sufficient condition for the monotonicity of a Boolean function in terms of its polylinear continuation is found, and secondly, proved that by only once calculating the value of the polylinear continuation 𝑝𝐷(𝑥1, 𝑥2, ..., 𝑥𝑛) of the Boolean function 𝑝(𝑥1, 𝑥2, ..., 𝑥𝑛) at any interior point of the unit 𝑛-dimensional cube [0, 1]𝑛, we can determine the satisfiability of the Boolean function 𝑝(𝑥1, 𝑥2, ..., 𝑥𝑛) and, in the central at the cube point [0, 1]𝑛, we can find the number of solutions to the Boolean equation 𝑝(𝑥1, 𝑥2, ..., 𝑥𝑛) = 𝑏.