Boolean Case Research Articles

A polynomial-time criterion for stability of large-scale switched conjunctive Boolean networks

This paper investigates the global stability of large-scale switched conjunctive Boolean networks (SCBNs) under arbitrary switching signal from the graph-theoretic perspective. With the help of node removal technique, the Thomas’ rule in Boolean case is proved, that is, a Boolean network is globally stable when its network graph is acyclic, and meanwhile the stable equilibrium is constructed. Secondly, the combination of node removal and Thomas’ rule is used to analyse the stability of each mode in SCBNs. Thirdly, when the joint graph is weakly connected and acyclic, a polynomial-time criterion is built for the global stability of large-scale SCBNs under arbitrary switching signal. Finally, the application on the SCBN model in the processing of Toll ligand Spz demonstrates the effectiveness of the criterion.

Avoiding flatness in factoring ordinal data

Factorization of classical, two-valued Boolean data became a widely studied topic in the past decade due to its role in analyzing relational data as well as its significance for other fields. Recently, various extensions to factorization of ordinal data, or data with graded (fuzzy) attributes, have been proposed. We identify and describe a fundamental problem regarding quality of factors, which is non-existent in the Boolean case, but naturally appears in the more general setting of ordinal data. As we demonstrate, the problem gets more significant with growing size of the factorized data. We analyze the problem, propose a method to alleviate it, and evaluate experimentally our solution to the problem. We also provide a discussion regarding ramifications of our findings for the concept of cardinality of fuzzy sets.

Conditional Dichotomy of Boolean Ordered Promise CSPs

Promise Constraint Satisfaction Problems (PCSPs) are a generalization of Constraint Satisfaction Problems (CSPs) where each predicate has a strong and a weak form and given a CSP instance, the objective is to distinguish if the strong form can be satisfied vs. even the weak form cannot be satisfied. Since their formal introduction by Austrin, Guruswami, and H\aa stad, there has been a flurry of works on PCSPs [BBKO19,KO19,WZ20]. The key tool in studying PCSPs is the algebraic framework developed in the context of CSPs where the closure properties of the satisfying solutions known as the polymorphisms are analyzed. The polymorphisms of PCSPs are much richer than CSPs. In the Boolean case, we still do not know if dichotomy for PCSPs exists analogous to Schaefer's dichotomy result for CSPs. In this paper, we study a special case of Boolean PCSPs, namely Boolean Ordered PCSPs where the Boolean PCSPs have the predicate $x \leq y$. In the algebraic framework, this is the special case of Boolean PCSPs when the polymorphisms are monotone functions. We prove that Boolean Ordered PCSPs exhibit a computational dichotomy assuming the Rich 2-to-1 Conjecture [BKM21] which is a perfect completeness surrogate of the Unique Games Conjecture. Assuming the Rich 2-to-1 Conjecture, we prove that a Boolean Ordered PCSP can be solved in polynomial time if for every $\epsilon>0$, it has polymorphisms where each coordinate has Shapley value at most $\epsilon$, else it is NP-hard. The algorithmic part of our dichotomy is based on a structural lemma that Boolean monotone functions with each coordinate having low Shapley value have arbitrarily large threshold functions as minors. The hardness part proceeds by showing that the Shapley value is consistent under a uniformly random 2-to-1 minor. Of independent interest, we show that the Shapley value can be inconsistent under an adversarial 2-to-1 minor.

The algebraic structure of the densification and the sparsification tasks for CSPs

The tractability of certain CSPs for dense or sparse instances is known from the 90s. Recently, the densification and the sparsification of CSPs were formulated as computational tasks and the systematical study of their computational complexity was initiated. We approach this problem by introducing the densification operator, i.e. the closure operator that, given an instance of a CSP, outputs all constraints that are satisfied by all of its solutions. According to the Galois theory of closure operators, any such operator is related to a certain implicational system (or, a functional dependency) Σ. We are specifically interested in those classes of fixed-template CSPs, parameterized by constraint languages Γ, for which there is an implicational system Σ whose size is a polynomial in the number of variables n. We show that in the Boolean case, such implicational systems exist if and only if Γ is of bounded width. For such languages, Σ can be computed in log-space or in a logarithmic time with a polynomial number of processors. Given an implicational system Σ, the densification task is equivalent to the computation of the closure of input constraints. The sparsification task is equivalent to the computation of the minimal key.

Guest Column

Algebraic Natural Proofs is a recent framework which formalizes the type of reasoning used for proving most lower bounds on algebraic computational models. This concept is similar to and inspired by the famous natural proofs notion of Razborov and Rudich [RR97] for boolean circuit lower bounds, but, unlike in the boolean case, it is an open problem whether this constitutes a barrier for proving super-polynomial lower bounds for strong models of algebraic computation. From an algebraic-geometric viewpoint, it is also related to basic questions in Geometric Complexity Theory (GCT), and from a meta-complexity theoretic viewpoint, it can be seen as an algebraic version of the MCSP problem. We survey the recent work around this concept which provides some evidence both for and against the existence of an algebraic natural proofs barrier, with an emphasis on the di erent viewpoints and the connections to other areas.

Forbidden subposet problems in the grid

For posets P and Q, extremal and saturation problems about weak and strong P-free subposets of Q have been studied mostly in the case Q is the Boolean poset Qn, the poset of all subsets of an n-element set ordered by inclusion. In this paper, we study some instances of the problem with Q being the grid, and its connections to the Boolean case and to the forbidden submatrix problem.

On operator-valued infinitesimal Boolean and monotone independence

We introduce the notion of operator-valued infinitesimal (OVI) Boolean independence and OVI monotone independence. Then we show that OVI Boolean (respectively, monotone) independence is equivalent to the operator-valued (OV) Boolean (respectively, monotone) independence over an algebra of [Formula: see text] upper triangular matrices. Moreover, we derive formulas to obtain the OVI Boolean (respectively, monotone) additive convolution by reducing it to the OV case. We also define OVI Boolean and monotone cumulants and study their basic properties. Moreover, for each notion of OVI independence, we construct the corresponding OVI Central Limit Theorem. The relations among free, Boolean and monotone cumulants are extended to this setting. Besides, in the Boolean case we deduce that the vanishing of mixed cumulants is still equivalent to independence, and use this to connect scalar-valued with matrix-valued infinitesimal Boolean independence.

Complexity Assessments for Decidable Fragments of Set Theory. I: A Taxonomy for the Boolean Case*

We report on an investigation aimed at identifying small fragments of set theory (typically, sublanguages of Multi-Level Syllogistic) endowed with polynomial-time satisfiability decision tests, potentially useful for automated proof verification. Leaving out of consideration the membership relator ∈ for the time being, in this paper we provide a complete taxonomy of the polynomial and the NP-complete fragments involving, besides variables intended to range over the von Neumann set-universe, the Boolean operators ∪ ∩ \, the Boolean relators ⊆, ⊈,=, ≠, and the predicates ‘• = Ø’ and ‘Disj(•, •)’, meaning ‘the argument set is empty’ and ‘the arguments are disjoint sets’, along with their opposites ‘• ≠ Ø and ‘¬Disj(•, •)’. We also examine in detail how to test for satisfiability the formulae of six sample fragments: three sample problems are shown to be NP-complete, two to admit quadratic-time decision algorithms, and one to be solvable in linear time.

Representing Fitness Landscapes by Valued Constraints to Understand the Complexity of Local Search

Local search is widely used to solve combinatorial optimisation problems and to model biological evolution, but the performance of local search algorithms on different kinds of fitness landscapes is poorly understood. Here we consider how fitness landscapes can be represented using valued constraints, and investigate what the structure of such representations reveals about the complexity of local search.
 First, we show that for fitness landscapes representable by binary Boolean valued constraints there is a minimal necessary constraint graph that can be easily computed. Second, we consider landscapes as equivalent if they allow the same (improving) local search moves; we show that a minimal constraint graph still exists, but is NP-hard to compute.
 We then develop several techniques to bound the length of any sequence of local search moves. We show that such a bound can be obtained from the numerical values of the constraints in the representation, and show how this bound may be tightened by considering equivalent representations. In the binary Boolean case, we prove that a degree 2 or treestructured constraint graph gives a quadratic bound on the number of improving moves made by any local search; hence, any landscape that can be represented by such a model will be tractable for any form of local search.
 Finally, we build two families of examples to show that the conditions in our tractability results are essential. With domain size three, even just a path of binary constraints can model a landscape with an exponentially long sequence of improving moves. With a treewidth-two constraint graph, even with a maximum degree of three, binary Boolean constraints can model a landscape with an exponentially long sequence of improving moves.

On Design-Theoretic Aspects of Boolean and Vectorial Bent Function

There are two construction methods of designs from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m)$ </tex-math></inline-formula> -bent functions, known as translation and addition designs. In this article we analyze, which equivalence relation for Boolean bent functions, i.e. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,1)$ </tex-math></inline-formula> -bent functions, and vectorial bent functions, i.e. <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m)$ </tex-math></inline-formula> -bent functions with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2\le m\le n/2$ </tex-math></inline-formula> , is coarser: extended-affine equivalence or isomorphism of associated translation and addition designs. First, we observe that similar to the Boolean bent functions, extended-affine equivalence of vectorial <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,m)$ </tex-math></inline-formula> -bent functions and isomorphism of addition designs are the same concepts for all even <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m\le n/2$ </tex-math></inline-formula> . Further, we show that extended-affine inequivalent Boolean bent functions in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> variables, whose translation designs are isomorphic, exist for all <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\ge 6$ </tex-math></inline-formula> . This implies, that isomorphism of translation designs for Boolean bent functions is a coarser equivalence relation than extended-affine equivalence. However, we do not observe the same phenomenon for vectorial bent functions in a small number of variables. We classify and enumerate all vectorial bent functions in six variables and show, that in contrast to the Boolean case, one cannot exhibit isomorphic translation designs from extended-affine inequivalent vectorial <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(6,m)$ </tex-math></inline-formula> -bent functions with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m\in \{ 2,3 \}$ </tex-math></inline-formula> .

Algebraic immunity of vectorial boolean functions and boolean groebner bases

The basic concepts and results related to the Boolean Groebner bases and their application for computing the algebraic immunity of vectorial Boolean functions are considered. This parameter plays an important role for the security evaluation of block ciphers against algebraic attacks. Unlike the available works, the description is carried out at the elementary level using terms of Boolean functions theory. In addition, obtained proofs are shorter than the previous ones. This allows us to achieve significant progress in building the fundamentals of the theory (for the Boolean case) using only elementary methods.The paper can be useful for students and postgraduate students studying cryptology. It may also save time for professionals who want to get familiar with the mathematical techniques used in algebraic attacks on block ciphers.

Spreadability for Quantum Stochastic Processes, with an Application to Boolean Commutation Relations.

In order to manage spreadability for quantum stochastic processes, we study in detail the structure of the involved monoids acting on the index-set of all integers , that is that generated by left and right hand-side partial shifts, the monoid of all strictly increasing maps whose range has finite complement, and finally the collection of all strictly increasing maps of . We show that such three monoids are strictly ordered, and the second-named one is the semidirect product between the first and the action of generated by the one-step shift. Even if the definition of a spreadable stochastic process is provided in terms of the invariance of the finite joint distributions under the natural action of the last monoid on the indices, we see that spreadability can be directly stated in terms of invariance with respect to the action of the first monoid. Concerning the stochastic processes involving the concrete boolean -algebra generated by the annihilators acting on the boolean Fock space (i.e., the concrete -algebra satisfying the boolean commutation relations), we study their spreadability directly in terms of the invariance under the monoid generated by all strictly increasing maps whose range has finite complement because, for this case, such an investigation appears more direct and manageable. Finally, we present the version of the Ryll–Nardzewski theorem for the boolean case, establishing that spreadable, exchangeable and stationary stochastic processes coincide, and describing their common structure.

A constructive proof of a concentration bound for real-valued random variables

Almost 10 years ago, Impagliazzo and Kabanets [5] gave a new combinatorial proof of Chernoff's bound for sums of bounded independent random variables. Unlike previous methods, their proof is constructive. This means that it provides an efficient randomized algorithm for the following task: given a set of Boolean random variables whose sum is not concentrated around its expectation, find a subset of statistically dependent variables. However, the algorithm of Impagliazzo and Kabanets is given only for the Boolean case. On the other hand, the general proof technique works also for real-valued random variables, even though for this case, Impagliazzo and Kabanets obtain a concentration bound that is slightly suboptimal.Herein, we revisit both these issues and show that in fact it is relatively easy to extend the Impagliazzo-Kabanets algorithm to real-valued random variables and to improve the corresponding concentration bound by a constant factor.

Reliability Modeling and Analysis of Generalized Majority Systems by Stochastic Computation

The $k$ -out-of- $n$ : G(F) majority voter consists of n components (or modules) and a number of the components are required to be operating correctly for the overall system to be correct. As per the state discretization of the components, such a system is usually classified as either a binary system or a multi-state system. In practice, the operating conditions of different components may contribute differently to the operation of the entire system. In this manuscript, the $k$ -out-of- $n$ : G(F) majority voter is generalized as a consecutive-weighted- $k$ -out-of- $n$ : G(F) voter with either binary states or multiple states. To overcome the drawbacks of existing approaches, a stochastic analysis is proposed for assessing the system reliability. In the stochastic analysis, the input signal probabilities are encoded into non-Bernoulli sequences with fixed numbers of 0s and 1s for the Boolean case, or randomly permuted sequences for the multi-state scenario. By using stochastic logic, the reliability of a general system consisting of consecutive-weighted- $k$ -out-of- $n$ majority voters is efficiently and accurately predicted. The results are validated by an analysis of several case studies. Although the accuracy of the stochastic analysis is closely related with the employed sequence length, it is shown that a stochastic approach is more efficient than a universal generating function (UGF) method, while still retaining an acceptable accuracy.

SPECIFICATIONS OF APPROXIMATION k-VALUE FUNCTIONS AND THEIR PROPERTIES

This article discribes different characteristics of proximity specifications k -value function. Most of the authors consider characteristics of proximity specifications in boolean case as Hamming distance only, but we can suggest several different approaches to definitions of this terms. The solution to this problem will allow to expand the number of methods for analysis and building information processing and security data system.

On logical bifurcation diagrams

Bifurcation theory provides a powerful framework to analyze the dynamics of differential systems as a function of specific parameters. Abou-Jaoudé et al. (2009) introduced the concept of logical bifurcation diagrams, an analog of bifurcation diagrams for the logical modeling framework. In this work, we propose a formal definition of this concept. Since logical models are inherently discrete, we use the piecewise differential (PWLD) framework to introduce the underlying bifurcation parameters. Given a regulatory graph, a set of PWLD models is mapped to a set of logical models consistent with this graph, thereby linking continuous changes of bifurcation parameters to sequences of valuations of logical parameters. A logical bifurcation diagram corresponds then to a sequence of valuations of the logical parameters (with their associated set of attractors) consistent with at least one bifurcation diagram of the set of PWLD models. Necessary conditions on logical bifurcation diagrams in the general case, as well as a characterization of these diagrams in the Boolean case, exploiting a partial order between the logical parameters, are provided. We also propose a procedure to determine a logical bifurcation diagram of maximum length, starting from an initial valuation of the logical parameters, in the Boolean case. Finally, we apply our methodology to the analysis of a biological model of the p53-Mdm2 network.

BioLQM: A Java Toolkit for the Manipulation and Conversion of Logical Qualitative Models of Biological Networks.

Here we introduce bioLQM, a new Java software toolkit for the conversion, modification, and analysis of Logical Qualitative Models of biological regulatory networks. BioLQM provides core modeling operations as building blocks for the development of integrated modeling software, or for the assembly of heterogeneous analysis workflows involving several complementary tools. Based on the definition of multi-valued logical models, bioLQM implements import and export facilities, notably for the recent SBML qual exchange format, as well as for formats used by several popular tools, facilitating the design of workflows combining these tools. Model modifications enable the definition of various perturbations, as well as model reduction, easing the analysis of large models. Another modification enables the study of multi-valued models with tools limited to the Boolean case. Finally, bioLQM provides a framework for the development of novel analysis tools. The current version implements various updating modes for model simulation (notably synchronous, asynchronous, and random asynchronous), as well as some static analysis features for the identification of attractors. The bioLQM software can be integrated into analysis workflows through command line and scripting interfaces. As a Java library, it further provides core data structures to the GINsim and EpiLog interactive tools, which supply graphical interfaces and additional analysis methods for cellular and multi-cellular qualitative models.

On the Fourier spectrum of functions on Boolean cubes

Let f be a real-valued function of degree d defined on the n-dimensional Boolean cube $$\{ \pm 1\}^{n}$$ , and $$f(x) = \sum _{S \subset \{1,\ldots ,n\}} \widehat{f}(S) \prod _{k \in S} x_k$$ its Fourier-Walsh expansion. The main result states that there is an absolute constant $$C >0$$ such that the $$\ell _{2d/(d+1)}$$ -sum of the Fourier coefficients of $$f:\{ \pm 1\}^{n} \longrightarrow [-1,1]$$ is bounded by $$C^{\sqrt{d \log d}}$$ . It was recently proved that a similar result holds for complex-valued polynomials on the n-dimensional polytorus $$\mathbb {T}^n$$ , but that in contrast to this, a replacement of the n-dimensional torus $$\mathbb {T}^n$$ by the n-dimensional cube $$[-1, 1]^n$$ leads to a substantially weaker estimate. This in the Boolean case forces us to invent novel techniques which differ from the ones used in the complex or real case. We indicate how our result is linked with several questions in quantum information theory.

A circuit-preserving mapping from multilevel to Boolean dynamics

Many discrete models of biological networks rely exclusively on Boolean variables and many tools and theorems are available for analysis of strictly Boolean models. However, multilevel variables are often required to account for threshold effects, in which knowledge of the Boolean case does not generalise straightforwardly. This motivated the development of conversion methods for multilevel to Boolean models. In particular, Van Ham’s method has been shown to yield a one-to-one, neighbour and regulation preserving dynamics, making it the de facto standard approach to the problem. However, Van Ham’s method has several drawbacks: most notably, it introduces vast regions of “non-admissible” states that have no counterpart in the multilevel, original model. This raises special difficulties for the analysis of interaction between variables and circuit functionality, which is believed to be central to the understanding of dynamic properties of logical models. Here, we propose a new multilevel to Boolean conversion method, with software implementation. Contrary to Van Ham’s, our method doesn’t yield a one-to-one transposition of multilevel trajectories; however, it maps each and every Boolean state to a specific multilevel state, thus getting rid of the non-admissible regions and, at the expense of (apparently) more complicated, “parallel” trajectories. One of the prominent features of our method is that it preserves dynamics and interaction of variables in a certain manner. As a demonstration of the usability of our method, we apply it to construct a new Boolean counter-example to the well-known conjecture that a local negative circuit is necessary to generate sustained oscillations. This result illustrates the general relevance of our method for the study of multilevel logical models.

On the Parameterized Complexity of Finding Small Unsatisfiable Subsets of CNF Formulas and CSP Instances

In many practical settings it is useful to find a small unsatisfiable subset of a given unsatisfiable set of constraints. We study this problem from a parameterized complexity perspective, taking the size of the unsatisfiable subset as the natural parameter where the set of constraints is either (i) given a set of clauses, i.e., a formula in conjunctive normal Form (CNF), or (ii) as an instance of the Constraint Satisfaction Problem (CSP). In general, the problem is fixed-parameter in tractable. For an instance of the propositional satisfiability problem (SAT), it was known to be W[1]-complete. We establish A[2]-completeness for CSP instances, where A[2]-hardness prevails already for the Boolean case. With these fixed-parameter intractability results for the general case in mind, we consider various restricted classes of inputs and draw a detailed complexity landscape. It turns out that often Boolean CSP and CNF formulas behave similarly, but we also identify notable exceptions to this rule. The main part of this article is dedicated to classes of inputs that are induced by Boolean constraint languages that Schaefer [1978] identified as the maximal constraint languages with a tractable satisfiability problem. We show that for the CSP setting, the problem of finding small unsatisfiable subsets remains fixed-parameter intractable for all Schaefer languages for which the problem is non-trivial. We show that this is also the case for CNF formulas with the exception of the class of bijunctive (Krom) formulas, which allows for an identification of a small unsatisfiable subset in polynomial time. In addition, we consider various restricted classes of inputs with bounds on the maximum number of times that a variable occurs (the degree), bounds on the arity of constraints, and bounds on the domain size. For the case of CNF formulas, we show that restricting the degree is enough to obtain fixed-parameter tractability, whereas for the case of CSP instances, one needs to restrict the degree, the arity, and the domain size simultaneously to establish fixed-parameter tractability. Finally, we relate the problem of finding small unsatisfiable subsets of a set of constraints to the problem of identifying whether a given variable-value assignment is entailed or forbidden already by a small subset of constraints. Moreover, we use the connection between the two problems to establish similar parameterized complexity results also for the latter problem.