Abstract: In Boolean networks (BNs), robustness typically refers to the system’s ability to tolerate perturbations in either the state or the rule-based structure, both of which can significantly affect network dynamics. For function perturbation of rule-based structure, most existing studies have focused on analyzing the characteristics of the perturbations or investigating the resulting dynamics through simulations. However, few works have employed external analytical tools, such as Lyapunov-based methods, to evaluate the stability of the perturbed BNs. This paper introduces a Lyapunov-based framework for analysing the robust stability of BNs under one-bit function perturbations. We establish a theoretical result showing that if a perturbed BN admits an adaptively adjustable Lyapunov function to some extent, then it is guaranteed to achieve global finite-time stability at the original equilibrium state. Finally, a numerical example is provided to illustrate the theoretical results.

Abstract: This paper delves into the robust observability and robust (weak) detectability problems of Boolean networks (BNs) subject to function perturbation, which results in alterations to columns in the state transition matrix of BNs. To begin, based on whether transitions of states outside the set of perturbed points are affected, two distinct cases are introduced to analyze the evolution of state transitions at step k following the perturbation. Subsequently, the necessary and sufficient conditions for robust observability are established by examining changes of state transitions, employing a logical equation set. Additionally, a novel criterion to ascertain the (weak) detectability is proposed based on the data form of BNs, from which criteria are proposed to evaluate robust (weak) detectability. Finally, the main conclusions are illustrated with a numerical example and the p53-MDM2 negative feedback gene regulatory network.

Abstract: In this paper, capabilities of a software tool for topological reliability analysis of complex systems are experimentally analysed. This tool is called DDD (Decision Diagram Distributor) and is available with all its source files and experiment scripts for public on GitHub. DDD combines modular decomposition with MPI-based (Message Passing Interface) distributed computation and TeDDy (Templated Decision Diagram library). A user-driven configuration splits the global Boolean structure function into independent submodules, each represented and evaluated as a BDD (Binary Decision Diagram) in the TeDDy. Our main goal is to determine how different computational improvements, including our own, affect the overall runtime of performing complex tasks through various experiments made out of Boolean functions. Just decomposition alone slashes worst-case runtimes from tens of seconds to under one. That is an improvement of over two orders of magnitude on challenging Boolean PLA benchmarks. Parallel execution with 16 processes yields up to 8 times speedup over single process decomposed runs. DDD enables practical reliability evaluation for systems with thousands of components.

Abstract: The Boolean formula satisfiability problem (SAT) plays a fundamental role in many practical applications, but is computationally challenging due to its NP-hard nature. Leveraging cloud computing has become an effective way to address larger-scale instances. This article proposes a secure outsourcing framework for solving Boolean formula SAT, with a focus on preserving the confidentiality of both the Boolean formulas and their solutions. The core innovation of our scheme lies in its robust obfuscation technique, which prevents the original formula from being exposed during outsourcing. This is an essential property when formulas encode sensitive designs, such as advanced digital circuits. Our approach transforms Boolean formulas into systems of equations over the finite field $\mathbf {GF}(2)$, which are then modified through arithmetic operations to generate an obfuscated equivalent. These transformed systems can be solved either by applying linearization techniques in the cloud or by reverting to Boolean form for processing by specialized SAT solvers. Compared to existing methods, our scheme offers enhanced protection by generating obfuscated instances that are indistinguishable from any formula with the same solution space. Moreover, the transformation incurs only linear growth in problem size, ensuring practical efficiency is maintained.

Abstract: Characterizing the minimum, necessary and sufficient components to generate the dynamics of a biological system has always been a priority to understand its functioning. In this sense, the canonical form of biological systems modeled by Boolean networks accurately defines the components in charge of controlling the dynamics of such systems. However, the calculation of the canonical form might be complicated in mathematical terms. In addition, computing the canonical form does not consider the dynamical properties found when using the synchronous and asynchronous update schemes to solve Boolean networks. Here, we analyze both update schemes and their connection with the canonical form of Boolean networks. We found that the synchronous scheme can be expressed by the Chapman-Kolmogorov equation, being a particular case of Markov chains. We also discovered that the canonical form of any Boolean network can be easily obtained by solving this matrix equation. Finally, we found that, the update order of the asynchronous scheme generates a set of functions that, when composed together, produce characteristic properties of this scheme, such as the conservation of fixed-point attractors or the variability in the basins of attraction. We concluded that the canonical form of Boolean networks can only be obtained for systems that use the synchronous update scheme, which opens up new possibilities for study.

Abstract: Boolean networks (BNs) are widely used to model biological regulatory networks. Attractors here hold significant meaning as they represent long-term behaviors such as homeostasis and the results of cell differentiation. As such, computing attractors is of critical importance to guarantee the validity of a model or to assess its stability and robustness. However, this problem is quite challenging when it comes to large real-world models. To overcome the limits of state-of-the-art BDD-based or ASP-based enumeration approaches, we introduce a SAT-based approach to compute fixed points (singleton attractors) of BN and exhibit its merits for counting the number of singleton attractors of large-scale benchmarks well established in the literature.

Abstract: We discuss the second-order differential uniformity of vectorial Boolean functions. The closely related notion of second-order zero differential uniformity has recently been studied in connection to resistance to the boomerang attack. We prove that monomial functions with univariate form $x^d$ where $d=2^{2k}+2^k+1$ and $\gcd(k,n)=1$ have optimal second-order differential uniformity. Computational results suggest that, up to affine equivalence, these might be the only optimal cubic power functions. We begin work towards generalising such conditions to all monomial functions of algebraic degree 3. We also discuss further questions arising from computational results.

Abstract: Bent functions are Boolean functions that are maximally nonlinear. They can be represented as bent squares, i.e., square matrices for which each row and each column is the Walsh spectrum of a Boolean function. Using this representation, it is shown in this note that the number of bent functions in $n$ variables is at least $2^{n \cdot 2^{\frac{n}{2}} \left(1 + O\left(\frac{1}{n}\right)\right)}$ for even integers $n$.

Abstract: All digital devices have components that implement Boolean functions, mapping that component's input to its output. However, any fixed Boolean function can be implemented by an infinite number of circuits, all of which vary in their resource costs. This has given rise to the field of circuit complexity theory, which studies the minimal resource cost to implement a given Boolean function with any circuit. Traditionally, circuit complexity theory has focused on the resource costs of a circuit's size (its number of gates) and depth (the longest path length from the circuit's input to its output). In this paper, we extend circuit complexity theory by investigating the minimal thermodynamic cost of a circuit's operation. We do this by using the mismatch cost of a given circuit that is run multiple times in a row to calculate a lower bound on the entropy production incurred in each such run of the circuit. Specifically, we derive conditions for mismatch cost to be proportional to the size of a circuit, and conditions for them to diverge. We also use our results to compare the thermodynamic costs of different circuit families implementing the same family of Boolean functions. In addition, we analyze how heterogeneity in the underlying physical processes implementing the gates in a circuit influences the minimal thermodynamic cost of the overall circuit. These and other results of ours lay the foundation for extending circuit complexity theory to include mismatch cost as a resource cost.

Abstract: We reduce the problem of proving deterministic and nondeterministic Boolean circuit size lower bounds to the analysis of certain two-dimensional combinatorial cover problems. This is obtained by combining results of Razborov (1989), Karchmer (1993), and Wigderson (1993) in the context of the fusion method for circuit lower bounds with the graph complexity framework of Pudlák, Rödl, and Savický (1988). For convenience, we formalize these ideas in the more general setting of "discrete complexity", i.e., the natural set-theoretic formulation of circuit complexity, variants of communication complexity, graph complexity, and other measures. We show that random graphs have linear graph cover complexity, and that explicit super-logarithmic graph cover complexity lower bounds would have significant consequences in circuit complexity. We then use discrete complexity, the fusion method, and a result of Karchmer and Wigderson (1993) to introduce nondeterministic graph complexity. This allows us to establish a connection between graph complexity and nondeterministic circuit complexity. Finally, complementing these results, we describe an exact characterization of the power of the fusion method in discrete complexity. This is obtained via an adaptation of a result of Nakayama and Maruoka (1995) that connects the fusion method to the complexity of "cyclic" Boolean circuits, which generalize the computation of a circuit by allowing cycles in its specification.

Abstract: This paper studies the important problem of quantum classification of Boolean functions from a entirely novel perspective. Typically, quantum classification algorithms allow us to classify functions with a probability of $1.0$, if we are promised that they meet specific unique properties. The primary objective of this study is to explore whether it is feasible to obtain any insights when the input function deviates from the promised class. For concreteness, we use a recently introduced quantum algorithm that is designed to classify with probability $1.0$ using just a single oracular query a large class of imbalanced Boolean functions. Fist, we establish a completely new concept characterizing ``nearness'' between Boolean function. Utilizing this concept, we show that, as long as the input function is close enough to the promised class, it is still possible to obtain useful information about its behavioral pattern from the classification algorithm. In this regard, the current study is among the first to provide evidence that shows how useful it is to apply quantum classification algorithms to functions outside the promised class in order to get a glimpse of important information.

Abstract: This paper introduces a novel analytic framework for analyzing the effective solvability of certain NP problem instances through spectral and information-theoretic techniques. By applying Boolean Fourier analysis and entropy flow via the SAPZ–MMLS dynamics, the author defines the subclass Peffective⊂NPP_{\mathrm{effective}} \subset \text{NP}Peffective⊂NP, characterized by spectral sparsity and entropy-rigid collapse. A conditional theorem is proven showing that such instances can be approximated in polynomial time via entropy-regularized inference, without contradicting the P ≠ NP conjecture. This work initiates an entropy-based classification theory within NP and opens a new direction in complexity theory. 🔹 Description (한글) 이 논문은 NP 문제 중 일부가 스펙트럼 희소성과 엔트로피 붕괴 조건을 만족할 경우, SAPZ–MMLS 흐름 하에서 사실상 P에 준하는 계산 가능성을 갖는다는 새로운 정보론적 분석 틀을 제시합니다. Boolean Fourier 해석과 엔트로피 흐름을 기반으로 PeffectiveP_{\mathrm{effective}}Peffective라는 조건부 서브클래스를 정의하고, 이에 속하는 문제들이 다항 시간 내 근사적으로 해결 가능함을 조건부 정리로 증명합니다. 본 연구는 P ≠ NP 가설을 유지하면서도 NP 내부의 구조를 엔트로피 기준으로 정량적 분류하는 새로운 이론적 가능성을 제시합니다. 🔹 Author Lee ByoungwooEmail: leeclinic@protonmail.com

Abstract: In 1985, Razborov [Razborov, 1985] proved the first superpolynomial size lower bound for monotone Boolean circuits for the perfect matching the clique functions, and, independently, Andreev [Andreev, 1985] obtained exponential size lower bounds. These breakthroughs were soon followed by further advancements in monotone complexity, including better lower bounds for clique [Alon and Boppana, 1987; Ingo Wegener, 1987], superlogarithmic depth lower bounds for connectivity by Karchmer and Wigderson [Karchmer and Wigderson, 1990], and the separations mon-NC ≠ mon-P and that mon-NC^i ≠ mon-NC^{i+1} by Raz and McKenzie [Ran Raz and Pierre McKenzie, 1999]. Karchmer and Wigderson [Karchmer and Wigderson, 1990] proved their result by establishing a relation between communication complexity and (monotone) circuit depth, and Raz and McKenzie [Ran Raz and Pierre McKenzie, 1999] introduced a new technique, now called lifting theorems, for obtaining communication lower bounds from query complexity lower bounds, In this talk, we will survey recent advancements in monotone complexity driven by query-to-communication lifting theorems. A decade ago, Göös, Pitassi, and Watson [Mika Göös et al., 2018] brought to light the generality of the result of Raz and McKenzie [Ran Raz and Pierre McKenzie, 1999] and reignited this line of work. A notable extension is the lifting theorem [Ankit Garg et al., 2020] for a model of DAG-like communication [Alexander A. Razborov, 1995; Dmitry Sokolov, 2017] that corresponds to circuit size. These powerful theorems, in their different flavours, have been instrumental in addressing many open questions in monotone circuit complexity, including: optimal 2^Ω(n) lower bounds on the size of monotone Boolean formulas computing an explicit function in NP [Toniann Pitassi and Robert Robere, 2017]; a complete picture of the relation between the mon-AC and mon-NC hierarchies [Susanna F. de Rezende et al., 2016]; a near optimal separation between monotone circuit and monotone formula size [Susanna F. de Rezende et al., 2020]; exponential separation between NC^2 and mon-P [Ankit Garg et al., 2020; Mika Göös et al., 2019]; and better lower bounds for clique [de Rezende and Vinyals, 2025; Lovett et al., 2022], improving on [Cavalar et al., 2021]. Very recently, lifting theorems were also used to prove supercritical trade-offs for monotone circuits showing that there are functions computable by small circuits for which any small circuit must have superlinear or even superpolynomial depth [de Rezende et al., 2024; Göös et al., 2024]. We will explore these results and their implications, and conclude by discussing some open problems.

Abstract: Boolean networks, inspired by gene regulatory networks, were developed to understand the complex behaviors observed in biological systems, with network attractors corresponding to biological phenotypes or cell types. In this article, we present a proof for a conjecture by Williadsen, Triesch and Wiles about upper bounds for the stability of basins of attraction in Boolean networks. We further extend this result from a single basin of attraction to the entire network. Specifically, we demonstrate that the asymptotic upper bound for the robustness and the basin entropy of a Boolean network are negatively linearly related.

Abstract: Canalization is a key organizing principle in complex systems, particularly in gene regulatory networks. It describes how certain input variables exert dominant control over a function's output, thereby imposing hierarchical structure and conferring robustness to perturbations. Degeneracy, in contrast, captures redundancy among input variables and reflects the complete dominance of some variables by others. Both properties influence the stability and dynamics of discrete dynamical systems, yet their combinatorial underpinnings remain incompletely understood. Here, we derive recursive formulas for counting Boolean functions with prescribed numbers of essential variables and given canalizing properties. In particular, we determine the number of non-degenerate canalizing Boolean functions -- that is, functions for which all variables are essential and at least one variable is canalizing. Our approach extends earlier enumeration results on canalizing and nested canalizing functions. It provides a rigorous foundation for quantifying how frequently canalization occurs among random Boolean functions and for assessing its pronounced over-representation in biological network models, where it contributes to both robustness and to the emergence of distinct regulatory roles.

Abstract: In this article, we aim to define a Boolean entropy notion parallel to the framework of free entropy proposed by Voiculescu. Motivated by the work of Lenczewski and the work of Cébron & Gillers, we mainly investigated two random matrix models (the Gaussian Symmetric Block model and the Conditioned GUE model), in which asymptotic Boolean independence appears. We showed a large deviation principle for both models. As a result, the two rate functions coincide up to scaling and are minimized by the Rademacher distribution. Therefore, we refer to the logarithmic integral in the rate function as Boolean entropy. Finally, we proved this logarithmic integral is maximized by the Rademacher distribution and monotone along the Boolean Central Limit Theorem.

Abstract: This paper offers a gentle introduction into the realm of monotone span programs and their connection with linear secret sharing schemes and attribute-based encryption while emphasizing the cryptographic importance of finding efficient MSPs for representing complex access structures. We provide a proof that there is no ideal LSSS for Boolean circuits, thus tackling the open problem of finding LSSSes of non-exponential size for Boolean circuits. Moreover, we present an application of our proof to graph access structures and a backtracking approach to finding efficient MSPs for given access structures.