Boolean Systems Research Articles

Improving Communication Networks to Transfer Data in Real Time for Environmental Monitoring and Data Collection

Integrated communication networks (CN) have proven successful in tracking environmental activities, wherein several sensors are installed throughout diverse surroundings to gather data or observe certain events. CNs, comprising several interacting detectors, have proven effective in various applications by transmitting data via diverse transmission methods inside the communication system. The erratic and constantly changing surroundings necessitate conventional CNs to engage in frequent conversations to disseminate the latest data, potentially incurring substantial connection expenses through joint data gathering and dissemination. High-frequency communications are prone to failure due to the extensive distance of data transfer. This research presents a unique methodology for multi-sensor environmental monitoring networks utilizing autonomous systems. The transmission system can mitigate elevated communication costs and Single Point of Failing (SPOF) challenges by employing a decentralized method that facilitates in-network processing. The methodology employs Boolean systems, enabling a straightforward verification process while preserving essential details about the dynamics of the communication system. The methodology further simplifies the data collection process and employs a Reinforcement Learning (RL) technique to forecast future events inside the surroundings by recognizing patterns.

Bayesian Optimization for State and Parameter Estimation of Dynamic Networks with Binary Space.

This paper focuses on joint state and parameter estimation in partially observed Boolean dynamical systems (POBDS), a hidden Markov model tailored for modeling complex networks with binary state variables. The majority of current techniques for parameter estimation rely on computationally expensive gradient-based methods, which become intractable in most practical applications with large size of networks. We propose a gradient-free approach that uses Gaussian processes to model the expensive log-likelihood function and utilizes Bayesian optimization for efficient likelihood search over parameter space. Joint state estimation is also achieved alongside parameter estimation using the Boolean Kalman filter. The performance of the proposed method is demonstrated using gene regulatory networks observed through synthetic gene-expression data. The numerical results demonstrate the scalability and effectiveness of the proposed method in the joint estimation of the model parameters and genes' states.

Lattice structures that parameterize regulatory network dynamics

We consider two types of models of regulatory network dynamics: Boolean maps and systems of switching ordinary differential equations. Our goal is to construct all models in each category that are compatible with the directed signed graph that describe the network interactions. This leads to consideration of lattice of monotone Boolean functions (MBF), poset of non-degenerate MBFs, and a lattice of chains in these sets. We describe explicit inductive construction of these posets where the induction is on the number of inputs in MBF.Our results allow enumeration of potential dynamic behavior of the network for both model types, subject to practical limitation imposed by the size of the lattice of MBFs described by the Dedekind number.

New Attacks on LowMC Using Partial Sets in the Single-Data Setting

The LowMC family of block ciphers was proposed by Albrecht et al. in Eurocrypt 2015, specifically targeting adoption in FHE and MPC applications due to its low multiplicative complexity. The construction operates a 3-bit quadratic S-box as the sole non-linear transformation in the algorithm. In contrast, both the linear layer and round key generation are achieved through multiplications of full rank matrices over GF(2). The cipher is instantiable using a diverse set of default configurations, some of which have partial non-linear layers i.e., in which the S-boxes are not applied over the entire internal state of the cipher. The significance of cryptanalysing LowMC was elevated by its inclusion into the NIST PQC digital signature scheme PICNIC in which a successful key recovery using a single plaintext/ciphertext pair is akin to retrieving the secret signing key. The current state-of-the-art attack in this setting is due to Dinur at Eurocrypt 2021, in which a novel way of enumerating roots of a Boolean system of equation is morphed into a key-recovery procedure that undercuts an ordinary exhaustive search in terms of time complexity for the variants of the cipher up to five rounds. In this work, we demonstrate that this technique can efficiently be enriched with a specific linearization strategy that reduces the algebraic degree of the non-linear layer as put forward by Banik et al. at IACR ToSC 2020(4). This amalgamation yields new attacks on certain instances of LowMC up to seven rounds.

A generalized uniqueness theorem for generalized Boolean dynamical systems

We characterize the canonical diagonal subalgebra of the C⁎-algebra associated with a generalized Boolean dynamical system. We also introduce a particular commutative subalgebra, which we call the abelian core, in our C⁎-algebra. We then establish a uniqueness theorem under the assumptions that B and L are countable, which says that a ⁎-homomorphism of our C⁎-algebra is injective if and only if its restriction to the abelian core is injective.

Technology independent optimization when implementing sparse systems of disjunctive normal forms of Boolean functions in ASIC

Objectives. The problem of choosing the best methods and programs for circuit implementation as part of digital ASIC (Application-Specific Integrated Circuit) sparse systems of disjunctive normal forms (DNF) of completely defined Boolean functions is considered. For matrix forms of sparse DNF systems, the ternary matrix specifying elementary conjunctions contains a large proportion of undefined values corresponding to missing literals of Boolean input variables, and the Boolean matrix specifying the occurrences of conjunctions in DNF functions contains a large proportion of zero values.Methods. It is proposed to investigate various methods of technologically independent logical optimization performed at the first stage of logical synthesis: joint minimization of systems of functions in the DNF class, separate and joint minimization in classes of multilevel representations in the form of Boolean networks and BDD representations using mutually inverse cofactors, as well as the division of a system of functions into subsystems with a limited number of input variables and the method of block cover of DNF systems, focused on minimizing the total area of the blocks forming the cover.Results. When implementing sparse DNF systems of Boolean functions in ASIC, along with traditional methods of joint minimization of systems of functions in the DNF class, methods for optimizing multilevel representations of Boolean function systems based on Shannon expansions can be used for technologically independent optimization, while separate minimization and joint minimization of the entire system as a whole turn out to be less effective compared with block partitions and coatings of the DNF system and subsequent minimization of multilevel representations. Schemes obtained as a result of synthesis using minimized representations of Boolean networks often have a smaller area than schemes obtained using minimized BDD representations.Conclusion. For the design of digital ASIC, the effectiveness of combined approach is shown, when initially the block coverage programs of the DNF system is used, followed by the use of programs to minimize multilevel block representations in the form of Boolean networks minimized based on Shannon expansion.

Bisimulations of nonlinear feedback shift registers

This work investigates how to reduce nonlinear feedback shift registers (NFSRs) in combination with bisimulation method via semi-tensor product (STP). By considering a large-scale NFSR as a smaller-scale Boolean system utilising bisimulation relations, the computing complexity of the system can be decreased. On the one hand, the properties of the bisimulation relation between two different NFSRs are examined, while on the other hand, a necessary and sufficient condition is discovered, as well as a reduction strategy utilising bisimulation. Besides, it is talked about how NFSR properties spread throughout the reduction process. The nonsingularity, driven stability and other aspects of the original NFSR are confirmed to be preserved by the reduced Boolean system.

Kernel-Based Particle Filtering for Scalable Inference in Partially Observed Boolean Dynamical Systems

This paper addresses the inference challenges associated with a class of hidden Markov models with binary state variables, known as partially observed Boolean dynamical systems (POBDS). POBDS have demonstrated remarkable success in modeling the ON and OFF dynamics of genes, microbes, and bacteria in systems biology, as well as in network security to represent the propagation of attacks among interconnected elements. Despite existing optimal and approximate inference solutions for POBDS, scalability remains a significant issue due to the computational cost associated with likelihood evaluations and the exploration of extensive parameter spaces. To overcome these challenges, this paper proposes a kernel-based particle filtering approach for large-scale inference of POBDS. Our method employs a Gaussian process (GP) to efficiently represent the expensive-to-evaluate likelihood function across the parameter space. The likelihood evaluation is approximated using a particle filtering technique, enabling the GP to account for various sources of uncertainty, including limited likelihood evaluations. Leveraging the GP’s predictive behavior, a Bayesian optimization strategy is derived for effectively seeking parameters yielding the highest likelihood, minimizing the overall computational burden while balancing exploration and exploitation. The proposed method’s performance is demonstrated using two biological networks: the mammalian cell-cycle network and the T-cell large granular lymphocyte leukemia network.

Rough set theory applied to finite dimensional vector spaces

In this paper, we study finite dimensional vector spaces using rough set theory (RST) by defining a Boolean information system IB associated with a vector space V for a given basis B. We define an indiscernibility relation on V and investigate different partitions induced on V. We identify that every reduct of the information system is a basis of V and the core is empty in the case of V over field F with |F|≥3. We study the measure of dependency between different subsets of V. Moreover, we define three posets and prove that these posets are isomorphic and complete. We study lower and upper approximations for different subsets of V and determine rough and exact sets. We give the general form of the entries of the discernibility matrix. We determine essential sets and the essential dimension of the information system, and prove that minimal entries of the discernibility matrix coincide with the essential sets. Finally, we demonstrate the use of the developed theory by providing two practical examples.

Dynamical interplay between random Boolean networks and awareness propagation

The stability of random Boolean networks (RBNs) has aroused continuous attention due to its profound applications in genetic regulatory. With extensive research on the joint influence of network topology and update rules, the interaction between RBN stability and other dynamic processes remains to be investigated. In this paper we propose an interactive multiplex model and study the dynamical interplay between RBN stability and awareness propagation. Theoretical analysis reveals that the stability of RBN is enhanced by awareness propagation. Specifically, its transition point between stable and unstable phases is determined by the main eigenvalue of H, which is closely related to the prevalence of awareness propagation and coupling strength between two layers. In addition, we illustrate the existence of a meta-critical point, from which awareness propagates and thus inhibits chaotic behaviors in RBN. Our work sheds light on the interaction between RBN stability and awareness propagation, and provides theoretical guidance to intervention of real Boolean systems.

Joint and Separate Minimization of Multilevel Representations of Boolean Function Systems

The results of experimental studies of the effectiveness of programs for minimizing multilevel representations of systems of Boolean functions performed during the synthesis of combinational circuits in the design library of custom CMOS VLSI are presented. The efficiency of joint and separate minimization programs using Shannon expansions for constructing multilevel representations of systems of fully defined Boolean functions in the form of interconnected systems of logical equations — complete or abbreviated formulas of Shannon expansion or formulas corresponding to Boolean networks is investigated. A comparison is made with the results of experiments performed for various types of joint minimization only. A comparison is also made with the results of solutions obtained by the program of joint and separate minimization of Boolean function systems in the DNF class. It is shown that the use of joint minimization of multi-level representations makes it possible to obtain circuits of a smaller area more often, and the use of a separate one — circuits with a lower delay.

Synthesis of Modular Multipliers

The results of experiments on the circuit implementation of modular multipliers in the design library of ASIC (Application-Specific Integrated Circuits) and FPGA (Field-Programmable Gate Array) are presented. The initial descriptions of modular multiplier projects were given by systems of not fully defined (partial) Boolean functions and algorithmic VHDL descriptions. Logical optimization was carried out in the class of disjunctive normal forms (DNF) and representations of Boolean function systems by BDD (Binary Decision Diagrams). The synthesized circuits were evaluated by area and time delay. It is established that the use of partial Boolean function models and preliminary logi­cal BDD optimization allows one to improve the parameters of synthesized ASIC and FPGA blocks for small module values, however, the best solutions for large module values can be obtained using algorithmic VHDL descriptions of modular multipliers. In the synthesis of modular multiplier circuits as part of an FPGA and the use of ISE and Vivado design systems it is advisable to use synthesized VHDL operations (a*b) mod p.

Limitations of the Macaulay matrix approach for using the HHL algorithm to solve multivariate polynomial systems

Recently Chen and Gao \cite{ChenGao2017} proposed a new quantum algorithm for Boolean polynomial system solving, motivated by the cryptanalysis of some post-quantum cryptosystems. The key idea of their approach is to apply a Quantum Linear System (QLS) algorithm to a Macaulay linear system over C, which is derived from the Boolean polynomial system. The efficiency of their algorithm depends on the condition number of the Macaulay matrix. In this paper, we give a strong lower bound on the condition number as a function of the Hamming weight of the Boolean solution, and show that in many (if not all) cases a Grover-based exhaustive search algorithm outperforms their algorithm. Then, we improve upon Chen and Gao's algorithm by introducing the Boolean Macaulay linear system over C by reducing the original Macaulay linear system. This improved algorithm could potentially significantly outperform the brute-force algorithm, when the Hamming weight of the solution is logarithmic in the number of Boolean variables.Furthermore, we provide a simple and more elementary proof of correctness for our improved algorithm using a reduction employing the Valiant-Vazirani affine hashing method, and also extend the result to polynomial systems over Fq improving on subsequent work by Chen, Gao and Yuan \citeChenGao2018. We also suggest a new approach for extracting the solution of the Boolean polynomial system via a generalization of the quantum coupon collector problem \cite{arunachalam2020QuantumCouponCollector}.

Observability of Boolean control networks with stochastic disturbances

In this paper, the observability problem of Boolean control networks (BCNs) with stochastic disturbances is investigated via two kinds of control schemes: deterministic control and state feedback control. Firstly, based on the proposed indicator matrix, a simplified system of the original augmented Boolean system is constructed. Based on the analysis of the auxiliary system, observability of the original BCN is converted to determine whether an observable set can be reached from another unobservable set. After that, some necessary and sufficient conditions are obtained to judge the observability of BCNs. At the same time, two algorithms are proposed for designing these two types of control sequences. Finally, numerical simulations are also provided to demonstrate feasibility of the theoretical results.

From Boolean networks to linear dynamical systems: a simplified route

ABSTRACT Boolean Networks can be converted to discrete linear dynamical systems on finite spaces by a semi-tensor-product approach. This approach has been used by many to study the dynamics and control of Boolean systems. However, the process of getting the linear representation using the semi-tensor-product method is complicated even for a simple three-node network and requires the help of a computer program. In this work, we show that we can skip the semi-tensor process and obtain the same linear representation with a straightforward mapping. Moreover, our approach produces a large number of isomorphic representations which provides a flexible framework. Importantly, it could simplify the analytical study of networks with unspecified number of nodes that have some structure.

Software JimenaE allows efficient dynamic simulations of Boolean networks, centrality and system state analysis

The signal modelling framework JimenaE simulates dynamically Boolean networks. In contrast to SQUAD, there is systematic and not just heuristic calculation of all system states. These specific features are not present in CellNetAnalyzer and BoolNet. JimenaE is an expert extension of Jimena, with new optimized code, network conversion into different formats, rapid convergence both for system state calculation as well as for all three network centralities. It allows higher accuracy in determining network states and allows to dissect networks and identification of network control type and amount for each protein with high accuracy. Biological examples demonstrate this: (i) High plasticity of mesenchymal stromal cells for differentiation into chondrocytes, osteoblasts and adipocytes and differentiation-specific network control focusses on wnt-, TGF-beta and PPAR-gamma signaling. JimenaE allows to study individual proteins, removal or adding interactions (or autocrine loops) and accurately quantifies effects as well as number of system states. (ii) Dynamical modelling of cell–cell interactions of plant Arapidopsis thaliana against Pseudomonas syringae DC3000: We analyze for the first time the pathogen perspective and its interaction with the host. We next provide a detailed analysis on how plant hormonal regulation stimulates specific proteins and who and which protein has which type and amount of network control including a detailed heatmap of the A.thaliana response distinguishing between two states of the immune response. (iii) In an immune response network of dendritic cells confronted with Aspergillus fumigatus, JimenaE calculates now accurately the specific values for centralities and protein-specific network control including chemokine and pattern recognition receptors.

Preimage attacks on reduced‐round Keccak hash functions by solving algebraic systems

AbstractIn this paper, improved preimage attacks are presented on 3‐round Keccak‐256 and Keccak‐512 and 4‐round Keccak‐256 based on algebraic methods. The authors propose some new properties about the components of Keccak permutation, reconsider the existing preimage attacks, and further refine the linearisation processes of quadratic bits to lower the complexities. For 3‐round Keccak‐256 and Keccak‐512, priority is given to values with higher probability for quadratic bits, such that the guessing complexities decrease from slightly more than 265 and 2440 to 264.79 and 2424, respectively. For preimage attack on 4‐round Keccak‐256, some strategies of saving degrees of freedom are applied to solve Boolean multivariate quadratic systems and reduce the guessing complexity from 2196 to 2188.

Synchronous Boolean Finite Dynamical Systems on Directed Graphs over XOR Functions

In this paper, we investigate the complexity of a number of computational problems defined on a synchronous boolean finite dynamical system, where update functions are chosen from a template set of exclusive-or and its negation. We first show that the reachability and path-intersection problems are solvable in logarithmic space-uniform AC1 if the objects execute permutations, while the reachability problem is known to be in P and the path-intersection problem to be in UP in general. We also explore the case where the reachability or intersection are tested on a subset of objects, and show that this hardens complexity of the problems: both problems become NP-complete, and even \({\Pi }^{p}_{2}\)-complete if we further require universality of the intersection. We next consider the exact cycle length problem, that is, determining whether there exists an initial configuration that yields a cycle in the configuration space having exactly a given length, and show that this problem is NP-complete. Lastly, we consider the t-predecessor and t-Garden of Eden problem, and prove that these are solvable in polynomial time even if the value of t is also given in binary as part of instance, and the two problems are in logarithmic space-uniform NC2 if the value of t is given in unary as part of instance.

Quotients of Probabilistic Boolean Networks

A probabilistic Boolean network (PBN) is a discrete-time system composed of a collection of Boolean networks between which the PBN switches in a stochastic manner. This article focuses on the study of quotients of PBNs. Given a PBN and an equivalence relation on its state set, we consider a probabilistic transition system that is generated by the PBN; the resulting quotient transition system then automatically captures the quotient behavior of this PBN. We therefore describe a method for obtaining a probabilistic Boolean system that generates the transitions of the quotient transition system. Applications of this quotient description are discussed, and it is shown that for PBNs, controller synthesis can be performed easily by first controlling a quotient system and then lifting the control law back to the original network. A biological example is given to show the usefulness of the developed results.

Probabilistic edge weights fine-tune Boolean network dynamics.

Biological systems are noisy by nature. This aspect is reflected in our experimental measurements and should be reflected in the models we build to better understand these systems. Noise can be especially consequential when trying to interpret specific regulatory interactions, i.e. regulatory network edges. In this paper, we propose a method to explicitly encode edge-noise in Boolean dynamical systems by probabilistic edge-weight (PEW) operators. PEW operators have two important features: first, they introduce a form of edge-weight into Boolean models through the noise, second, the noise is dependent on the dynamical state of the system, which enables more biologically meaningful modeling choices. Moreover, we offer a simple-to-use implementation in the already well-established BooleanNet framework. In two application cases, we show how the introduction of just a few PEW operators in Boolean models can fine-tune the emergent dynamics and increase the accuracy of qualitative predictions. This includes fine-tuning interactions which cause non-biological behaviors when switching between asynchronous and synchronous update schemes in dynamical simulations. Moreover, PEW operators also open the way to encode more exotic cellular dynamics, such as cellular learning, and to implementing edge-weights for regulatory networks inferred from omics data.