Boolean Functions Research Articles

Theoretical understanding of gradients of spike functions as boolean functions

Theoretical understanding of gradients of spike functions as boolean functions

A New Multicommodity Network Flow Model and Branch and Cut for Optimal Quantum Boolean Circuit Synthesis

This study introduces a new optimization model and a branch-and-cut approach for synthesizing optimal quantum circuits for reversible Boolean functions, which are pivotal components in quantum algorithms. Although heuristic algorithms have been extensively explored for quantum circuit synthesis, research on exact counterparts remains relatively limited. However, the need to design quantum circuits with guaranteed optimality is increasing, especially for improving computational fidelity on noisy intermediate-scale quantum devices. This study presents mathematical optimization as a viable option for optimal synthesis, with the potential to accommodate practical considerations arising in fast-evolving quantum technologies. We set a demonstrative problem to implement reversible Boolean functions using high-level gates known as multiple control Toffoli gates while minimizing a technology-based proxy called quantum cost—the number of low-level gates used to realize each high-level gate. To address this problem, we propose a discrete optimization model based on a multicommodity network and discuss potential future variations at an abstract level to incorporate technical considerations. A customized branch and cut is then developed upon different aspects of our model, including polyhedron integrality, surrogate constraints, and variable prioritization. Our experiments demonstrate the robustness of the proposed approach in finding cost-optimal circuits for all benchmark instances within a two-hour time frame. Furthermore, we present interesting intuitions from these experiments and compare our computational results with relevant studies, highlighting newly discovered circuits with the lowest quantum costs reported in this paper. History: Accepted by Giacomo Nannicini, Area Editor for Quantum Computing. This paper has been accepted for the INFORMS Journal on Computing Special Issue on Quantum Computing. Funding: This research was supported by the Ministry of Science and Information and Communication Technology, South Korea [Grants 2017R1E1A1A0307098814 and 2020R1A4A307986411]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0562 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0562 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Generating Bent Functions and Dynamic Filters: A Novel Equivalence-Based Approach

Boolean functions are fundamental building blocks in both discrete mathematics and computer science, with applications spanning from cryptography to coding theory. Bent functions, a subset of Boolean functions with maximal nonlinearity, are particularly valuable in cryptographic applications. This study introduces a novel equivalence relation among all Boolean functions and presents an algorithm to generate bent functions based on this relation. We systematically generated a collection of 10,000 bent functions over eight variables, all originating from the same equivalence class, and analyzed their structural complexity through rank determination. Our findings revealed the presence of at least five distinct affine classes of bent functions within this collection. By employing this construction, we devised an algorithm to generate a filter function capable of combining Boolean functions. This filter function can be dynamically adjusted based on a key, offering potential applications in symmetric cipher design, such as enhancing security or improving efficiency.

A Note on APN Permutations and Their Derivatives

Prior to the discovery of an APN permutation in six dimension it was conjectured that such functions do not exist in even dimension, as none had been found at that time. However, finding APN permutations in even dimension ≥8 remains a significant challenge. Understanding and determining more properties of these functions is a crucial approach to discovering them. In this note, we study the properties of vectorial Boolean functions based on the weights of the first-order and second-order derivatives of their components. We show that a function is an APN permutation if and only if the sum of the squares of the weights of the first-order derivatives of its components is exactly 22n−1(2n−1+1)(2n−1). Additionally, we determined that the sum of the weights of the second-order derivatives of the components of any vectorial Boolean function is at most 22n−1(2n−1)(2n−2). This bound is achieved if and only if a function is APN.

Implementation of a non-standard system for simplifying Peirce-Webb functions

The object of research are models of optimal logic circuits based on universal Peirce-Webb functions. The problem solved is the efficiency of the technique for simplifying the Peirce-Webb functions. The extension of the non-standard system to the simplification of Peirce-Webb functions makes it possible to discover new rules of equivalent transformations of Boolean functions, and to complete the simplification procedure in one step. A feature of the simplification of functions in the Peirce-Webb basis by a non-standard system is fixing the digital project at the level of abstraction, followed by the application of the mechanism of logical synthesis to generate the corresponding equivalent at the level of gates of the logic circuit. The result of the transformation of the terms of the binary matrix in the end is some combinatorial system, metadata that can explain other data, for example, determine the minimum function for another logical basis. The interpretation of the result consists in the use of combinatorial properties of binary structures of functions in the Peirce-Webb basis and binary structures of functions in the basic basis. These properties do not depend on the selected logical basis, which makes it possible to carry out equivalent transformations on binary matrices of Peirce-Webb functions according to the rules of the algebra of the main basis. It has been experimentally confirmed that a non-standard system enables: – to reduce the algorithmic complexity of simplifying the Peirce-Webb functions; – to increase the performance of the simplification of Peirce-Webb functions by 200–300 %; – to demonstrate the visibility of the process of simplifying functions. In terms of application, the non-standard system of simplifying the Peirce-Webb functions could ensure the transfer of innovations to material production: from conducting fundamental research, expanding the capabilities of digital component design technology to organizing serial or mass production of novelties.

Estimation of function's supports under arithmetic constraints

AbstractThe well-known inequality $$\lvert {\rm supp}(f) \rvert \lvert {\rm supp}( \widehat f) \rvert \geq |G|$$ | supp ( f ) | | supp ( f ^ ) | ≥ | G | gives a lower estimation for each support. In this paper we consider the case where there exists a slowly increasing function $$F$$ F such that $$\lvert {\rm supp}(f) \rvert \leq F(\lvert {\rm supp}( \widehat f) \rvert )$$ | supp ( f ) | ≤ F ( | supp ( f ^ ) | ) . We will show that this can be done under some arithmetic constraint. The two links that help us come from additive combinatorics and theoretical computer science. The first is the additive energy which plays a central role in additive combinatorics. The second is the influence of Boolean functions. Our main tool is the spectral analysis of Boolean functions. We prove an uncertainty inequality in which the influence of a function and its Fourier spectrum play a role.

Polynomial quantum computing algorithms for solving the dualization problem for positive Boolean functions

Polynomial quantum computing algorithms for solving the dualization problem for positive Boolean functions

OPTIMIZATION PROBLEM FOR NUMBER OF LOGIC GATES NEEDED TO IMPLEMENT MULTIPLE BOOLEAN FUNCTIONS USING DECODER

Abstract. Background. Computer circuits enable almost every piece of electronics to function. An opportunity for their optimization was found in one of the ways to implement a circuit, which involves a decoder. Decoders are basic circuit components whose behavior makes it easy to implement multiple Boolean functions. Functions are just rules by which informational signals are being processed in the circuit. To implement a function Logic gatEs (LEs) are used. LEs group multiple input signals and process them using binary logic operators like AND or NOR. Purpose. Develop a computer program that, based on provided functions, will find a solution associated with approximately minimal number of logic gates. Relevance. Prevalence of circuits increases the value of small optimizations, since any improvement gets multiplied by millions of circuits produced. Optimization at the logic level is being explored, which will retain value regardless of the physical properties of the circuit. Additionally, the theory of optimizing LE usage for circuits built on decoders is examined, which will alleviate future research. Methods. Development of the theory arose from an understanding that input signals can be grouped using LEs consciously. This way, LE outputs can be used in more than one function, which results in reduction of total number of LEs. It becomes evident that the problem can be restated in terms of sets. Similarity to known NP-hard problems emerges, which prompts usage of existing solution approaches. Among them, the greedy algorithm is chosen for its simplicity in program implementation. The connection between the problem and digital circuitry is weakened by the introduction of special evaluation functions that work with sets. So, the problem is reduced to several mathematical rules, which then are used as the appropriate parts of the greedy algorithm. Development of knowledge necessary to create the target program is thus concluded. Results. The target program is successfully implemented and can be seen in full at https://github.com/Gleb-05/MakeFuncForDC. A test is conducted on 1000 sets of four pseudorandom eight-term functions. It is estimated that the implementations proposed by the program require on average 0.75 of the initial number of LEs. The average running time is estimated at 2 milliseconds. In contrast, brute-force search for the same problem is theorized to run for years. Conclusions. A working program that finds an almost optimal solution is successfully created and tested. Its performance makes it potentially useful in the industrial scale of circuit manufacturing. Since the program essentially offers workload management, its possible applications go beyond the domain of digital electronics. Additionally, the theory of optimizing the use of LEs is presented for the first time. It can be used as the basis for future research on the implementation of multiple Boolean functions using decoder.

Dynamics of ecosystem services along ecological network seascapes

Abstract Human societies depend on services provided by ecosystems, from local needs such as clean water and pest control to global services like the ozone layer and the ocean biological pump. Ecosystem services are linked to the states of the ecosystem, which are, in turn, governed by a web of ecological interactions. These interactions, along with the services they support, are under threat from environmental changes driven by human activities. Therefore, safeguarding these vital services requires an understanding of how the structure and dynamics of ecological interactions are affected by environmental change. A critical step towards this goal is the development of a theoretical framework that can elucidate how ecosystem services are sustained or impaired by interactions within ecosystems in fluctuating environments. Recent years have seen progress in characterizing the organization and dynamics of ecological networks. However, linking temporally varying network structure in fluctuating environments, the seascapes of ecological networks, and their impact on services remains a challenge. We propose an approach based on merging ecological network analysis with Boolean functions and modeling of fluctuating environments to address how services are affected by environmental change. We review aspects of Boolean Network models and illustrate the approach using biologically inspired Boolean rules that involve predator-prey cycles, trophic cascades, and mutualisms formed by plants and their frugivores. This approach aims to contribute to the study of how the organization of ecological interactions affects the persistence of ecosystem services. Specifically, we discuss how this approach can provide new insights into how environmental change affects the relationship between ecological networks and ecosystem services. The combination of information on the natural history of species interactions and ecosystem services, Boolean networks, and models for fluctuating environments may contribute to conservation strategies for preserving biodiversity and ecosystem services in the face of ongoing environmental change.

A Counterexample to a Directed KKL Inequality

We show that the natural directed analogues of the KKL theorem and the Eldan– Gross inequality from the analysis of Boolean functions fail to hold. This is in contrast to several other isoperimetric inequalities on the Boolean hypercube (such as the Poincare inequality, Margulis’s inequality and Talagrand’s inequalit) for which directed strengthenings have recently been established.

Energy Aware Technology Mapping of Genetic Logic Circuits.

Energy and its dissipation are fundamental to all living systems, including cells. Insufficient abundance of energy carriers─as caused by the additional burden of artificial genetic circuits─shifts a cell's priority to survival, also impairing the functionality of the genetic circuit. Moreover, recent works have shown the importance of energy expenditure in information transmission. Despite living organisms being non-equilibrium systems, non-equilibrium models capable of accounting for energy dissipation and non-equilibrium response curves are not yet employed in genetic design automation (GDA) software. To this end, we introduce Energy Aware Technology Mapping, the automated design of genetic logic circuits with respect to energy efficiency and functionality. The basis for this is an energy aware non-equilibrium steady state model of gene expression, capturing characteristics like energy dissipation─which we link to the entropy production rate─and transcriptional bursting, relevant to eukaryotes as well as prokaryotes. Our evaluation shows that a genetic logic circuit's functional performance and energy efficiency are disjoint optimization goals. For our benchmark, energy efficiency improves by 37.2% on average when comparing to functionally optimized variants. We discover a linear increase in energy expenditure and overall protein expression with the circuit size, where Energy Aware Technology Mapping allows for designing genetic logic circuits with the energetic costs of circuits that are one to two gates smaller. Structural variants improve this further, while results show the Pareto dominance among structures of a single Boolean function. By incorporating energy demand into the design, Energy Aware Technology Mapping enables energy efficiency by design. This extends current GDA tools and complements approaches coping with burden in vivo.

Efficient Algorithm for Generating Optimal Inequality Candidates for MILP Modeling of Boolean Functions

Mixed-Integer Linear Programming (MILP) modeling has become an important tool for both the analysis and the design of symmetric cryptographic primitives. The bit-wise modeling of their nonlinear components, especially the S-boxes, is of particular interest since it allows more informative analysis compared to word-oriented models focusing on counting active S-boxes. At the same time, the size of these models, especially in terms of the number of required inequalities, tends to significantly influence and ultimately limit the applicability of this method to real-world ciphers, especially for larger number of rounds. It is therefore of great cryptographic significance to study optimal linear inequality descriptions for Boolean functions. The pioneering works of Abdelkhalek et al. (FSE 2017), Boura and Coggia (FSE 2020) and Li and Sun (FSE 2023) provided various heuristic techniques for this computationally hard problem, decomposing it into two algorithmic steps, coined Problem 1 and Problem 2, with the latter being identical to the well-known NP-hard set cover problem, for which there are many heuristic and exact algorithms in the literature. In this paper, we introduce a novel and efficient branch-and-bound algorithm for generating all minimal, non-redundant candidate inequalities that satisfy a given Boolean function, therefore solving Problem 1 in an optimal manner without relying on heuristics. We furthermore prove that our algorithm correctly computes optimal solutions. Using a number of dedicated optimizations, it provides significantly improved runtimes compared to previous approaches and allows the optimal modeling of the difference distribution tables (DDT) and linear approximation tables (LAT) of many practically used S-boxes. The source code for our algorithm is publicly available as a tool for researchers and practitioners in symmetric cryptography.

Boolean Functions with Two Distinct Nega-Hadamard Coefficients

Boolean Functions with Two Distinct Nega-Hadamard Coefficients

Searching for Continuous n-Clusters with Boolean Reasoning

A bicluster consists of a subset of rows and columns of a given matrix, whose intersection defines the region (bicluster) of values of precisely defined condition. Through the decades, a variety of biclustering techniques have been successfully developed. Recently, it was proved that many possible patterns defined in two-dimensional data could be found with the application of Boolean reasoning. The provided theorems showed that any existing pattern in the data could be unequivocally encoded as an implicant of a proper Boolean function. Moreover, a prime implicant of that function encoded the inclusion-maximal (non-extendable) pattern. On the other hand, the definition of some two-dimensional patterns may be easily extended to three-dimensional patterns (triclusters) as well as to any number of dimensions (n-clusters). This paper presents a new approach for searching for three- and higher-dimensional simple patterns in continuous data with Boolean reasoning. Providing the definition of the Boolean function for this tasks, it is shown that the similar correspondence—implicants encode patterns, and prime implicants encode inclusion-maximal patterns—has a strong mathematical background: the proofs of appropriate theorems are also presented in this paper.

Biotic stress and yield stability in English organic silvoarable agroforestry

In-field trees are thought to buffer arable crops from climate extremes through the creation of microclimates that may reduce the impacts of heat, wind, and cold. Much less is known about how trees and their biotic interactions (e.g. with natural enemies of pests and wild understory plants) impact crop yield stability to biotic stresses such as crop pests and disease. Modelling these interactions using conventional approaches is complex and time consuming, and we take a simplified approach, representing the agroecosystem as a Boolean regulatory network and parameterising Boolean functions using expert opinion. This allies our approach with decision analysis, which is increasingly finding applications in agriculture. Despite the naivety of our model, we demonstrate that it outputs complex and realistic agroecosystem dynamics. It predicts that, in English silvoarable, the biotic interactions of in-field trees boost arable crop yield overall, but they do not increase yield stability to biotic stress. Sensitivity analysis shows that arable crop yield is very sensitive to disease and weeds. We suggest that the focus of studies and debate on ecosystem service provision by English agroforestry needs to shift from natural enemies and pests to these ecosystem components. We discuss how our model can be improved through validation and parameterisation using real field data. Finally, we discuss how our approach can be used to rapidly model systems (agricultural or otherwise) than can be represented as dynamic interaction networks.

Design of concatenative complete complementary codes for CCC-CDMA via specific sequences and extended Boolean functions

A complete complementary code (CCC) consists of M sequence sets with size M. The sum of the auto-correlation functions of each sequence set is an impulse function, and the sum of cross-correlation functions of the different sequence sets is equal to zero. Thanks to their excellent correlation, CCCs received extensive use in engineering. In addition, they are strongly connected to orthogonal matrices. In some application scenarios, additional requirements are made for CCCs, such as recently proposed for concatenative CCC (CCCC) division multiple access (CCC-CDMA) technologies. In fact, CCCCs are a special kind of CCCs which requires that each sequence set in CCC be concatenated to form a zero-correlation-zone (ZCZ) sequence set. However, this requirement is challenging, and the literature is thin since there is only one construction in this context. We propose to go beyond the literature through this contribution to reduce the gap between their interest and our limited knowledge of CCCCs. This paper will employ novel methods for designing CCCCs and precisely derive two constructions of these objects. The first is based on perfect cross Z-complementary pair and Hadamard matrices, and the second relies on extended Boolean functions. Specifically, we highlight that optimal and asymptotic optimal CCCCs could be obtained through the proposed constructions. Besides, we shall present a comparison analysis with former structures in the literature and examples to illustrate our main results.

Energy Aware Technology Mapping of Genetic Logic Circuits.

Energy and its dissipation are fundamental to all living systems, including cells. Insufficient abundance of energy carriers -as caused by the additional burden of artificial genetic circuits- shifts a cell's priority to survival, also impairing the functionality of the genetic circuit. Moreover, recent works have shown the importance of energy expenditure in information transmission. Despite living organisms being non-equilibrium systems, non-equilibrium models capable of accounting for energy dissipation and non-equilibrium response curves are not yet employed in genetic design automation (GDA) software. To this end, we introduce Energy Aware Technology Mapping, the automated design of genetic logic circuits with respect to energy efficiency and functionality. The basis for this is an energy aware non-equilibrium steady state (NESS) model of gene expression, capturing characteristics like energy dissipation -which we link to the entropy production rate- and transcriptional bursting, relevant to eukaryotes as well as prokaryotes. Our evaluation shows that a genetic logic circuit's functional performance and energy efficiency are disjoint optimization goals. For our benchmark, energy efficiency improves by 37.2% on average when comparing to functionally optimized variants. We discover a linear increase in energy expenditure and overall protein expression with the circuit size, where Energy Aware Technology Mapping allows for designing genetic logic circuits with the energy efficiency of circuits that are one to two gates smaller. Structural variants improve this further, while results show the Pareto dominance among structures of a single Boolean function. By incorporating energy demand into the design, Energy Aware Technology Mapping enables energy efficiency by design. This extends current GDA tools and complements approaches coping with burden in vivo.

Complete Characterization of Fairness in Secure Two-Party Computation of Boolean Functions

Complete Characterization of Fairness in Secure Two-Party Computation of Boolean Functions

On complexity of colloid cellular automata

The colloid cellular automata do not imitate the physical structure of colloids but are governed by logical functions derived from them. We analyze the space-time complexity of Boolean circuits derived from the electrical responses of colloids-specifically ZnO (zinc oxide, an inorganic compound also known as calamine or zinc white, which naturally occurs as the mineral zincite), proteinoids (microspheres and crystals of thermal abiotic proteins), and their combinations in response to electrical stimulation. To extract Boolean circuits from colloids, we send all possible configurations of two-, four-, and eight-bit binary strings, encoded as electrical potential values, to the colloids, record their responses, and infer the Boolean functions they implement. We map the discovered functions onto the cell-state transition rules of cellular automata-arrays of binary state machines that update their states synchronously according to the same rule-creating the colloid cellular automata. We then analyze the phenomenology of the space-time configurations of the automata and evaluate their complexity using measures such as compressibility, Shannon entropy, Simpson diversity, and expressivity. A hierarchy of phenomenological and measurable space-time complexity is constructed.

ReSG: A Data Structure for Verification of Majority-based In-memory Computing on ReRAM Crossbars

Recent advancements in the fabrication of Resistive Random Access Memory (ReRAM) devices have led to the development of large-scale crossbar structures. In-memory computing architectures relying on ReRAM crossbars aim to mitigate the processor-memory bottleneck that exists with current complementary metal-oxide semiconductor technology. With this motivation, several synthesis and mapping approaches focusing on the realizations of Boolean functions in the ReRAM crossbars have been proposed earlier. Thus far, the verification of the designs realized on ReRAM crossbars is done either through manual inspection or using simulation-based approaches. Since manual inspections and simulation-based approaches are limited to smaller designs, they cannot be applied to the verification of complex designs on large-scale ReRAM crossbars. Motivated by this, we propose, for the first time, an automatic equivalence checking flow that determines the equivalence between the original function specification (e.g., Majority-inverter Graph ) and the crossbar micro-operations file formats. We consider two crossbar structures, zero-transistor, one-memristor (0T1R) and one-transistor, one-memristor (1T1R) to implement the micro-operations. While the micro-operations file format exists for 0T1R crossbar structures, no representations for micro-operations to be executed in 1T1R crossbars exist yet. In this work, we introduce the micro-operation file format for 1T1R crossbar structures to efficiently represent the micro-operations as ReRAM crossbar netlists. Afterwards, we introduce two intermediate data structures, ReRAM Sequence Graph for 0T1R crossbars (ReSG-0T1R) and for 1T1R crossbars (ReSG-1T1R) , that are derived from the 0T1R and 1T1R crossbar micro-operations file formats, respectively. These ReSGs are then translated into Boolean Satisfiability (SAT) formula, and then the verification is done by checking the generated SAT formulae against the golden functional specification (represented in Verilog) using Z3 Satisfiability solver. Experimental evaluations confirm the effectiveness of the proposed verification methodology on MCNC and ISCAS benchmarks.