Boolean Threshold Research Articles

Predicting attractors from spectral properties of stylized gene regulatory networks.

How the architecture of gene regulatory networks shapes gene expression patterns is an open question, which has been approached from a multitude of angles. The dominant strategy has been to identify nonrandom features in these networks and then argue for the function of these features using mechanistic modeling. Here we establish the foundation of an alternative approach by studying the correlation of network eigenvectors with synthetic gene expression data simulated with a basic and popular model of gene expression dynamics: Boolean threshold dynamics in signed directed graphs. We show that eigenvectors of the network adjacency matrix can predict collective states (attractors). However, the overall predictive power depends on details of the network architecture, namely the fraction of positive 3-cycles, in a predictable fashion. Our results are a set of statistical observations, providing a systematic step towards a further theoretical understanding of the role of network eigenvectors in dynamics on graphs.

LogBTF: gene regulatory network inference using Boolean threshold network model from single-cell gene expression data.

From a systematic perspective, it is crucial to infer and analyze gene regulatory network (GRN) from high-throughput single-cell RNA sequencing data. However, most existing GRN inference methods mainly focus on the network topology, only few of them consider how to explicitly describe the updated logic rules of regulation in GRNs to obtain their dynamics. Moreover, some inference methods also fail to deal with the over-fitting problem caused by the noise in time series data. In this article, we propose a novel embedded Boolean threshold network method called LogBTF, which effectively infers GRN by integrating regularized logistic regression and Boolean threshold function. First, the continuous gene expression values are converted into Boolean values and the elastic net regression model is adopted to fit the binarized time series data. Then, the estimated regression coefficients are applied to represent the unknown Boolean threshold function of the candidate Boolean threshold network as the dynamical equations. To overcome the multi-collinearity and over-fitting problems, a new and effective approach is designed to optimize the network topology by adding a perturbation design matrix to the input data and thereafter setting sufficiently small elements of the output coefficient vector to zeros. In addition, the cross-validation procedure is implemented into the Boolean threshold network model framework to strengthen the inference capability. Finally, extensive experiments on one simulated Boolean value dataset, dozens of simulation datasets, and three real single-cell RNA sequencing datasets demonstrate that the LogBTF method can infer GRNs from time series data more accurately than some other alternative methods for GRN inference. The source data and code are available at https://github.com/zpliulab/LogBTF.

On the Compressive Power of Boolean Threshold Autoencoders.

An autoencoder is a layered neural network whose structure can be viewed as consisting of an encoder, which compresses an input vector to a lower dimensional vector, and a decoder, which transforms the low-dimensional vector back to the original input vector (or one that is very similar). In this article, we explore the compressive power of autoencoders that are Boolean threshold networks by studying the numbers of nodes and layers that are required to ensure that each vector in a given set of distinct input binary vectors is transformed back to its original. We show that for any set of n distinct vectors there exists a seven-layer autoencoder with the optimal compression ratio, (i.e., the size of the middle layer is logarithmic in n ), but that there is a set of n vectors for which there is no three-layer autoencoder with a middle layer of logarithmic size. In addition, we present a kind of tradeoff: if the compression ratio is allowed to be considerably larger than the optimal, then there is a five-layer autoencoder. We also study the numbers of nodes and layers required only for encoding, and the results suggest that the decoding part is the bottleneck of autoencoding. For example, there always is a three-layer Boolean threshold encoder that compresses n vectors into a dimension that is twice the logarithm of n .

Predicting cascading extinctions and efficient restoration strategies in plant–pollinator networks via generalized positive feedback loops

The extinction of a species in a plant–pollinator mutualistic community can cause cascading effects and lead to major biodiversity loss. The ecologically important task of predicting the severity of the cascading effects is made challenging by the complex network of interactions among the species. In this work, we analyze an ensemble of models of communities of plant and pollinator species. These models describe the mutualistic inter-species interactions by Boolean threshold functions. We show that identifying generalized positive feedback loops can help pinpoint the species whose extinction leads to catastrophic and substantial damage to the whole community. We compare these results with the damage percentage caused by the loss of species identified as important by previously studied structural measures and show that positive feedback loops and the information gained from them can identify certain crucial species that the other measures fail to find. We also suggest mitigation measures for two specific purposes: (1) prevent the damage to the community by protecting a subset of the species, and (2) restore the community after the damage by restoring a subset of species. Our analyses indicate that the generalized positive feedback loops predict the most efficient strategies to achieve these purposes. The correct identification of species in each category has important implications for conservation efforts and developing community management strategies.

On the Distribution of Successor States in Boolean Threshold Networks.

We study the distribution of successor states in Boolean networks (BNs). The state vector y is called a successor of x if y = F(x) holds, where x, y ∈ {0,1}n are state vectors and F is an ordered set of Boolean functions describing the state transitions. This problem is motivated by analyzing how information propagates via hidden layers in Boolean threshold networks (discrete model of neural networks) and is kept or lost during time evolution in BNs. In this article, we measure the distribution via entropy and study how entropy changes via the transition from x to y , assuming that x is given uniformly at random. We focus on BNs consisting of exclusive OR (XOR) functions, canalyzing functions, and threshold functions. As a main result, we show that there exists a BN consisting of d -ary XOR functions, which preserves the entropy if d is odd and , whereas there does not exist such a BN if d is even. We also show that there exists a specific BN consisting of d -ary threshold functions, which preserves the entropy if [Formula: see text]. Furthermore, we theoretically analyze the upper and lower bounds of the entropy for BNs consisting of canalyzing functions and perform computational experiments using BN models of real biological networks.

Secure approximate pattern matching protocol via Boolean threshold private set intersection

Approximate pattern matching (APM) measures whether the Hamming distance between two strings is less than a threshold value. APM has been widely utilized, such as gene matching and facial recognition. Yet, the genetic data are privacy-sensitive, resulting that the owners are unwilling to share the raw data. This inspires us to explore how to securely perform APM. After revisiting threshold private set intersection (TPSI), we first propose and formalize a functionality named Boolean threshold private set intersection (BTPSI). The new proposed BTPSI primitive returns a Boolean value (0 or 1) to the user, rather than the actual elements in TPSI. We then construct a secure protocol for the BTPSI functionality with semihonest security. Besides, we first combine oblivious transfer and BTPSI to achieve the efficient construction of secure approximate pattern matching (SAPM) protocol in a semihonest model. Furthermore, we implement our SAPM protocol to demonstrate its real practicality. The performance result shows that when the text length is 2 20 ${2}^{20}$ and the pattern length is 2 10 ${2}^{10}$ , the total runtime is less than 3 s.

On Boolean threshold functions with minimum specification number

A set S of Boolean points is a specifying set for a threshold function f if the only threshold function consistent with f on S is f itself. The minimal cardinality of a specifying set for f is the specification number of f and it is never smaller than n+1 for a function with n relevant variables. In the present paper, we develop an inductive approach to describing the set of Boolean threshold functions with minimum specification number by means of operations that allow us to extend functions of n variables in this set to functions of n+1 variables.

Relationships among generalized positive feedback loops determine possible community outcomes in plant-pollinator interaction networks.

Attractors in Boolean network models representing complex systems such as ecological communities correspond to long-term outcomes (e.g., stable communities) in such systems. As a result, identifying efficient methods to find and characterize these attractors allows for a better understanding of the diversity of possible outcomes. Here we analyze networks that model mutualistic communities of plant and pollinator species governed by Boolean threshold functions. We propose a novel attractor identification method based on generalized positive feedback loops and their functional relationships in such networks. We show that these relationships determine the mechanisms by which groups of stable positive feedback loops collectively trap the system in specific regions of the state space and lead to attractors. Put into the ecological context, we show how survival units-small groups of species in which species can maintain a specific survival state-and their relationships determine the final community outcomes in plant-pollinator networks. We find a remarkable diversity of community outcomes: up to an average of 43 attractors possible for networks with 100 species. This diversity is due to the multiplicity of survival units (up to 34) and stable subcommunities (up to 14). The timing of species influx or outflux does not affect the number of attractors, but it may influence their basins of attraction.

Freezing sandpiles and Boolean threshold networks: Equivalence and complexity

The NC versus P-hard classification of the prediction problem for sandpiles on the two dimensional grid with von Neumann neighborhood is a famous open problem. In this paper we make two kinds of progresses, by studying its freezing variant. First, it enables to establish strong connections with other well known prediction problems on networks of threshold Boolean functions such as majority. Second, we can highlight some necessary and sufficient elements to the dynamical complexity of sandpiles, with a surprisingly crucial role of cells with two grains.

Challenge codes for physically unclonable functions with Gaussian delays: A maximum entropy problem

Motivated by a security application on physically unclonable functions, we evaluate the probability distributions and Renyi entropies of signs of scalar products of i.i.d. Gaussian random variables against binary codewords in \begin{document}$ \{\pm1\}^n $\end{document} . The exact distributions are determined for small values of \begin{document}$ n $\end{document} and upper bounds are provided by linking this problem to the study of Boolean threshold functions. Finally, Monte-Carlo simulations are used to approximate entropies up to \begin{document}$ n = 10 $\end{document} .

Separation of Unbounded-Error Models in Multi-Party Communication Complexity

We construct a simple function that has small unbounded-error communication complexity in the $k$-party number-on-forehead (NOF) model but every probabilistic protocol that solves it with subexponential advantage over random guessing has cost essentially $\Omega(\sqrt{n}\,/\,4^k)$ bits. This separates these classes up to $k \leq \delta\log n$ players for any constant $\delta < 1/4$, and gives the largest known separation by an explicit function in this regime of $k$. Our analysis is elementary and self-contained, inspired by the methods of Goldmann, Håstad, and Razborov (Computational Complexity, 1992). After initial publication of our work as a preprint (ECCC, 2016), Sherstov pointed out to us that an alternative proof of an $\Omega((n\,/\,4^k)^{1/7})$ separation is implicit in his prior work (SICOMP, 2016). Furthermore, based on his prior work (SICOMP, 2013 and SICOMP, 2016), Sherstov gave an alternative proof of our constructive $\Omega(\sqrt{n}\,/\,4^k)$ separation and also produced a stronger non-constructive $\Omega(n\,/\,4^k)$ separation. These results are explained in Sherstov's preprint (ECCC, 2016) and in article 14/22 in Theory of Computing. Our result has the following consequence for boolean threshold circuits. Let $\text{THR}$ and $\text{MAJ}$ denote the classes of linear threshold functions that have unbounded weights and polynomially bounded weights, respectively. Further, let $\text{PAR}_k$ ($\text{ANY}_k$) denote the class of functions that are parities of $k$ bits (any $k$-junta, respectively). Denote by $\text{THR} \circ \text{PAR}_k$ the class of depth-2 circuits where the top gate computes a linear threshold function, and the bottom gates compute functions in $\text{PAR}_k$. For every $2 \le k \le \delta \log n$, we show that there exists a function computable by a linear-size $\text{THR} \circ \text{PAR}_k$ circuit, but requires $\exp\left({n^{\Omega(1)}}\right)$-size circuits in the class $\text{MAJ} \circ \text{SYM} \circ \text{ANY}_{k-1}$, where the gates in the middle layer compute symmetric functions. Applying a result of Goldmann et al. (loc. cit.) to the above, similar lower bounds on the size of circuits of the form $\text{MAJ} \circ \text{THR} \circ \text{ANY}_{k-1}$ for computing the function follow. The main technical ingredient of our proof is to exhibit a composed function of the form $f \circ \text{XOR}$ which has exponentially small discrepancy while $f$ has sign-degree just 1. An interesting aspect of our composed function is that the block size of the inner XOR function is fixed to 1, independent of $k$, the number of players. A preliminary version of this paper appeared as ECCC technical report TR 16-095.

Описание некоторых декомпозиций для квадратичных булевых пороговых функций

Описание некоторых декомпозиций для квадратичных булевых пороговых функций

Identifying a Probabilistic Boolean Threshold Network From Samples.

This paper studies the problem of exactly identifying the structure of a probabilistic Boolean network (PBN) from a given set of samples, where PBNs are probabilistic extensions of Boolean networks. Cheng et al. studied the problem while focusing on PBNs consisting of pairs of AND/OR functions. This paper considers PBNs consisting of Boolean threshold functions while focusing on those threshold functions that have unit coefficients. The treatment of Boolean threshold functions, and triplets and -tuplets of such functions, necessitates a deepening of the theoretical analyses. It is shown that wide classes of PBNs with such threshold functions can be exactly identified from samples under reasonable constraints, which include: 1) PBNs in which any number of threshold functions can be assigned provided that all have the same number of input variables and 2) PBNs consisting of pairs of threshold functions with different numbers of input variables. It is also shown that the problem of deciding the equivalence of two Boolean threshold functions is solvable in pseudopolynomial time but remains co-NP complete.

New Enumeration Algorithm for Regular Boolean Functions

This paper introduces a new algorithm for enumerating regular Boolean functions. This algorithm exploits the equivalence between regular Boolean functions and positive threshold functions that can be used to represent instances of the knapsack problem. After proving this equivalence, this paper introduces a novel data structure that may, with further tweaking, enable faster enumeration algorithms for both regular Boolean functions and all-capacities knapsack problem instances.

Circulant discrete dynamical systems with threshold functions of at most three variables

We propose a method for finding sources of discrete dynamical systems of the circulant type with a q-valued arbitrary function at vertices. We find all sources, all fixed points, and some cycles, as well as lengths of some maximal chains outside cycles for the systems with Boolean threshold functions of at most three variables at the vertices.

Некоторые структурные свойства квадратичных булевых пороговых функций

Некоторые структурные свойства квадратичных булевых пороговых функций

Polynomial threshold functions and Boolean threshold circuits

We initiate a comprehensive study of the complexity of computing Boolean functions by polynomial threshold functions (PTFs) on general Boolean domains (as opposed to domains such as {0,1}n or {−1,1}n that enforce multilinearity). A typical example of such a general Boolean domain, for the purpose of our results, is {1,2}n. We are mainly interested in the length (the number of monomials) of PTFs, with their degree and weight being of secondary interest.First we motivate the study of PTFs over the {1,2}n domain by showing their close relation to depth two threshold circuits. In particular we show that PTFs of polynomial length and polynomial degree compute exactly the functions computed by polynomial size THR∘MAJ circuits. We note that known lower bounds for THR∘MAJ circuits extend to the likely strictly stronger model of PTFs (with no degree restriction). We also show that a “max-plus” version of PTFs is related to AC0∘THR circuits.We exploit this connection to gain a better understanding of threshold circuits. In particular, we show that (super-logarithmic) lower bounds for 3-player randomized communication protocols with unbounded error would yield (super-polynomial) size lower bounds for THR∘THR circuits.Finally, having thus motivated the model, we initiate structural studies of PTFs. These include relationships between weight and degree of PTFs, and a degree lower bound for PTFs of constant length.

Total domishold graphs: A generalization of threshold graphs, with connections to threshold hypergraphs

A total dominating set in a graph is a set of vertices such that every vertex of the graph has a neighbor in the set. We introduce and study graphs that admit non-negative real weights associated to their vertices such that a set of vertices is a total dominating set if and only if the sum of the corresponding weights exceeds a certain threshold. We show that these graphs, which we call total domishold graphs, form a non-hereditary class of graphs properly containing the classes of threshold graphs and the complements of domishold graphs, and are closely related to threshold Boolean functions and threshold hypergraphs. We present a polynomial time recognition algorithm of total domishold graphs, and characterize graphs in which the above property holds in a hereditary sense. Our characterization is obtained by studying a new family of hypergraphs, defined similarly as the Sperner hypergraphs, which may be of independent interest.

Dynamic probabilistic threshold networks to infer signaling pathways from time-course perturbation data.

BackgroundNetwork inference deals with the reconstruction of molecular networks from experimental data. Given N molecular species, the challenge is to find the underlying network. Due to data limitations, this typically is an ill-posed problem, and requires the integration of prior biological knowledge or strong regularization. We here focus on the situation when time-resolved measurements of a system’s response after systematic perturbations are available.ResultsWe present a novel method to infer signaling networks from time-course perturbation data. We utilize dynamic Bayesian networks with probabilistic Boolean threshold functions to describe protein activation. The model posterior distribution is analyzed using evolutionary MCMC sampling and subsequent clustering, resulting in probability distributions over alternative networks. We evaluate our method on simulated data, and study its performance with respect to data set size and levels of noise. We then use our method to study EGF-mediated signaling in the ERBB pathway.ConclusionsDynamic Probabilistic Threshold Networks is a new method to infer signaling networks from time-series perturbation data. It exploits the dynamic response of a system after external perturbation for network reconstruction. On simulated data, we show that the approach outperforms current state of the art methods. On the ERBB data, our approach recovers a significant fraction of the known interactions, and predicts novel mechanisms in the ERBB pathway.Electronic supplementary materialThe online version of this article (doi:10.1186/1471-2105-15-250) contains supplementary material, which is available to authorized users.

Lower Bounds Against Weakly-Uniform Threshold Circuits

An ongoing line of research has shown super-polynomial lower bounds for uniform and slightly-non-uniform small-depth threshold and arithmetic circuits (Allender, in Chicago J. Theor. Comput. Sci. 1999(7), 1999; Koiran and Perifel, in Proceedings of the 24th Annual IEEE Conference on Computational Complexity (CCC 2009), pp. 35---40, 2009; Jansen and Santhanam, in Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP 2011), I, pp. 724---735, 2011). We give a unified framework that captures and improves each of the previous results. Our main results are that Permanent does not have threshold circuits of the following kinds.(1) strengthens a result of Jansen and Santhanam (Jansen and Santhanam, in Proceedings of the 38th International Colloquium on Automata, Languages and Programming (ICALP 2011), I, pp. 724---735, 2011), who obtained similar parameters but for arithmetic circuits of constant depth rather than Boolean threshold circuits. (2) and (3) strengthen results of Allender (Allender, in Chicago J. Theor. Comput. Sci. 1999(7), 1999) and Koiran and Perifel (Koiran and Perifel, in Proceedings of the 24th Annual IEEE Conference on Computational Complexity (CCC 2009), pp. 35---40, 2009), respectively, who obtained results with similar parameters but for completely uniform circuits. Our main technical contribution is to simplify and unify earlier proofs in this area, and adapt the proofs to handle some amount of non-uniformity. We also develop a notion of circuits with a small amount of non-uniformity that naturally interpolates between fully uniform and fully non-uniform circuits. We use this notion, which we term weak uniformity, rather than the earlier and essentially equivalent notion of succinctness used by Jansen and Santhanam because the notion of weak uniformity more fully and easily interpolates between full uniformity and non-uniformity of circuits.