Binary Decision Diagrams Research Articles

Minimizing Binary Decision Diagrams for Systems of Incompletely Defined Boolean Functions Using Algebraic Cofactor Expansions

Minimizing Binary Decision Diagrams for Systems of Incompletely Defined Boolean Functions Using Algebraic Cofactor Expansions

Exponential separation between quantum and classical ordered binary decision diagrams, reordering method and hierarchies

In this paper, we study quantum Ordered Binary Decision Diagrams(\(\mathrm {OBDD}\)) model; it is a restricted version of read-once quantum branching programs, with respect to “width” complexity. It is known that the maximal gap between deterministic and quantum complexities is exponential. But there are few examples of functions with such a gap. We present a new technique (“reordering”) for proving lower bounds and upper bounds for OBDD with an arbitrary order of input variables if we have similar bounds for the natural order. Using this transformation, we construct a total function \(\mathrm {REQ}\) such that the deterministic \(\mathrm {OBDD}\) complexity of it is at least \(2^{\varOmega (n / \log n)}\), and the quantum \(\mathrm {OBDD}\) complexity of it is at most \(O(n^2/\log n)\). It is the biggest known gap for explicit functions not representable by \(\mathrm {OBDD}\)s of a linear width. Another function(shifted equality function) allows us to obtain a gap \(2^{\varOmega (n)}\) vs \(O(n^2)\). Moreover, we prove the bounded error quantum and probabilistic \(\mathrm {OBDD}\) width hierarchies for complexity classes of Boolean functions. Additionally, using “reordering” method we extend a hierarchy for read-k-times Ordered Binary Decision Diagrams (\({\textit{k}}\text {-}\mathrm {OBDD}\)) of polynomial width, for \(k = o(n / \log ^3 n)\). We prove a similar hierarchy for bounded error probabilistic \({\textit{k}}\text {-}\mathrm {OBDD}\)s of polynomial, superpolynomial and subexponential width. The extended abstract of this work was presented on International Computer Science Symposium in Russia, CSR 2017, Kazan, Russia, June 8 – 12, 2017 Khadiev and Khadieva (2017)

Quadratic Lower Bounds for Algebraic Branching Programs and Formulas

Quadratic Lower Bounds for Algebraic Branching Programs and Formulas

Extracting concepts from triadic contexts using Binary Decision Diagram

Due to the high complexity of real problems, a considerable amount of research that deals with high volumes of information has emerged. The literature has considered new applications of data analysis for high dimensional environments in order to manage the difficulty in extracting knowledge from a database, especially with the increase in social and professional networks. Tri- adic Concept Analysis (TCA) is a technique used in the applied mathematical area of data analysis. Its main purpose is to enable knowledge extraction from a context that contains objects, attributes, and conditions in a hierarchical and systematized representation. There are several algorithms that can extract concepts, but they are inefficient when applied to large datasets because the compu- tational costs are exponential. The objective of this paper is to add a new data structure, binary decision diagrams (BDD), in the TRIAS algorithm and retrieve triadic concepts for high dimen- sional contexts. BDD was used to characterize formal contexts, objects, attributes, and conditions. Moreover, to reduce the computational resources needed to manipulate a high-volume of data, the usage of BDD was implemented to simplify and represent data. The results show that this method has a considerably better speedup when compared to the original algorithm. Also, our approach discovered concepts that were previously unachievable when addressing high dimensional contexts.

Optimizing Binary Decision Diagrams with MaxSAT for Classification

The growing interest in explainable artificial intelligence(XAI) for critical decision making motivates the need for interpretable machine learning (ML) models. In fact, due to their structure (especially with small sizes), these models are inherently understandable by humans. Recently, several exact methods for computing such models are proposed to overcome weaknesses of traditional heuristic methods by providing more compact models or better prediction quality. Despite their compressed representation of Boolean functions, Binary decision diagrams (BDDs) did not gain enough interest as other interpretable ML models. In this paper, we first propose SAT-based models for learning optimal BDDs (in terms of the number of features) that classify all input examples. Then, we lift the encoding to a MaxSAT model to learn optimal BDDs in limited depths, that maximize the number of examples correctly classified. Finally, we tackle the fragmentation problem by introducing a method to merge compatible subtrees for the BDDs found via the MaxSAT model. Our empirical study shows clear benefits of the proposed approach in terms of prediction quality and interpretability (i.e., lighter size) compared to the state-of-the-art approaches.

Editorial

Dear Readers, It gives me great pleasure to announce the sixth regular issue of 2022. First of all, I would like to welcome all the new members of our editorial board and also express my sincere gratitude to all the members who have recently retired. It is only thanks to the expertise of our editorial board and volunteer guest reviewers, as well as the generous support of the J.UCS consortium and the novel contributions of our authors, that we are able to maintain this high standard and even improve the impact factor over the years. In this issue, various topical aspects of computer science are covered by 12 authors from 5 countries in 4 articles. Stefano Masneri, Ana Domínguez, Mikel Zorrilla, Mikel Larrañaga and Ana Arruarte from Italy present a systematic review of the literature on the use of augmented reality applications in primary and secondary schools, with a specific focus on collaborative, multi-user, and interactive applications. In their article, Julio Cesar Vale Neves, Luiz Enrique Zarate and Mark Alan Junho Song from Brazil describe an approach to add a new data structure, binary decision diagrams (BDD), to the TRIAS algorithm and retrieve triadic concepts for high dimensional contexts. In a collaborative research between Croatia and Ukraine, Igor Tomicic, Markus Schatten, and Vadym Shkarupylo propose an open ontology for self-sustainable human settlements in an effort to find a common language for modeling self-sustainable systems and address issues of heterogeneity of physical devices, protocols, software components, data and message formats, and other relevant factors for the implementation of smart systems. Last but not least, Derya Yiltas-Kaplan from Turkey presents her research on traffic optimization with software-defined network controllers.

Operator-Potential Heuristics for Symbolic Search

Symbolic search, using Binary Decision Diagrams (BDDs) to represent sets of states, is a competitive approach to optimal planning. Yet heuristic search in this context remains challenging. The many advances on admissible planning heuristics are not directly applicable, as they evaluate one state at a time. Indeed, progress using heuristic functions in symbolic search has been limited and even very informed heuristics have been shown to be detrimental. Here we show how this connection can be made stronger for LP-based potential heuristics. Our key observation is that, for this family of heuristic functions, the change of heuristic value induced by each operator can be precomputed. This facilitates their smooth integration into symbolic search. Our experiments show that this can pay off significantly: we establish a new state of the art in optimal symbolic planning.

Second-Order Finite Automata

Traditionally, finite automata theory has been used as a framework for the representation of possibly infinite sets of strings. In this work, we introduce the notion of second-order finite automata, a formalism that combines finite automata with ordered decision diagrams, with the aim of representing possibly infinite sets of sets of strings. Our main result states that second-order finite automata can be canonized with respect to the second-order languages they represent. Using this canonization result, we show that sets of sets of strings represented by second-order finite automata are closed under the usual Boolean operations, such as union, intersection, difference and even under a suitable notion of complementation. Additionally, emptiness of intersection and inclusion are decidable. We provide two algorithmic applications for second-order automata. First, we show that several width/size minimization problems for deterministic and nondeterministic ODDs are solvable in fixed-parameter tractable time when parameterized by the width of the input ODD. In particular, our results imply FPT algorithms for corresponding width/size minimization problems for ordered binary decision diagrams (OBDDs) with a fixed variable ordering. Previously, only algorithms that take exponential time in the size of the input OBDD were known for width minimization, even for OBDDs of constant width. Second, we show that for each k and w one can count the number of distinct functions computable by ODDs of width at most w and length k in time h(|Σ|,w) ⋅ kO(1), for a suitable h:mathbb {N}times mathbb {N}rightarrow mathbb {N}. This improves exponentially on the time necessary to explicitly enumerate all such functions, which is exponential in both the width parameter w and in the length k of the ODDs.

Operator-Potentials in Symbolic Search: From Forward to Bi-directional Search

Symbolic search using binary decision diagrams is a state-of-the-art technique for cost-optimal planning. Heuristic search in this context has been problematic as even a very informative heuristic can be detrimental in case it induces difficult-to-represent state partitionings. It was recently shown that operator-potential heuristics can address this issue in forward search by computing a numeric potential for each operator corresponding to the change of the heuristic value induced by that operator. Forward search is, however, not the best known variant of symbolic search. Here we investigate the integration with backward and bi-directional search instead. We prove that forward search (distance-to-goal) operator-potential heuristics can be turned into backward search (distance-to-initial-state) heuristics elegantly in this context, by summing the backward search path operator-potentials with the initial state goal-distance estimate. We run exhaustive experiments on IPC benchmarks, showing that significant performance improvements can be obtained over symbolic forward search and other state-of-the-art techniques.

Discovering the Roots: Uniform Closure Results for Algebraic Classes Under Factoring

Newton iteration is an almost 350-year-old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all roots simultaneously. In this form, the process yields better circuit complexity in the case when the number of rootsris small but the multiplicities are exponentially large. Our method sets up a linear system inrunknowns and iteratively builds the roots as formal power series. For an algebraic circuit\( f(x_1,\ldots ,x_n) \)of sizes, we prove that each factor has size at most a polynomial insand the degree of the squarefree part off. Consequently, if\( f_1 \)is a\( 2^{\Omega (n)} \)-hard polynomial, then any nonzero multiple\( \prod _{i} f_i^{e_i} \)is equally hard for arbitrary positive\( e_i \)’s, assuming that\( \sum _i\deg (f_i) \)is at most\( 2^{O(n)} \).It is an old open question whether the class of poly(n) size formulas (respectively, algebraic branching programs) is closed under factoring. We show that given a polynomialfof degree\( n^{O(1)} \)and formula (respectively, algebraic branching program) size\( n^{O(\log n)} \), we can find a similar-size formula (respectively, algebraic branching program) factor in randomized poly(\( n^{\log n} \)) time. Consequently, if the determinant requires an\( n^{\Omega (\log n)} \)size formula, then the same can be said about any of its nonzero multiples.In all of our proofs, we exploit the following property of multivariate polynomial factorization. Under a random linear transformation\( \tau \), the polynomial\( f(\tau \overline{x}) \)completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. Therefore, with the help of the strong mathematical characterizations and the ‘allRootsNI’ technique, we make significant progress towards the old open problems; supplementing the vast body of classical results and concepts in algebraic circuit factorization (e.g., [17,51,54,111]).

Predicting reliability of software in industrial systems using a Petri net based approach: A case study on a safety system used in nuclear power plant

Software reliability prediction in the early stages of development can be propitious in many ways. The combinatorial models used to predict reliability using architectures such as fault trees, binary decision diagrams, etc. have limitations in modeling complex system behavior. On the other hand, state-based models such as Markov chains suffer from the state-space explosion problem, and they need transition probability among different system states to measure reliability. These probabilities are usually assumed or are obtained from the operational profile for which the system should be used in the field.

Incremental Witness Generation for Branching-Time Logic CTL*

We propose a tester-based symbolic algorithm to incrementally generate tree-like witnesses (the dual concept of counterexamples) for branching-time logic CTL*. The CTL* witnesses are state transition directed graphs of tree-like structure in which all strongly connected components are cycles, and all branches are lasso paths with shortest prefixes. The states and transitions in these graphs may be respectively attached with satisfied state and path subformulas as annotations for intelligibility. The correctness of the lasso path generation algorithm is proved. We have developed an ordered binary decision diagrams based symbolic model checker MCTK2 for CTL* in which the proposed witness generation algorithm has been implemented. To make the generated witnesses graphs more succinct, several optimization strategies are applied to reduce the number of annotations and avoid constructing branching paths as much as possible. The interactive visualization interface of MCTK2 enables users to conveniently inspect counterexamples and expand branches of interest such that they can incrementally analyze larger and more complex counterexamples.

State space and binary decision diagram models for discrete standby systems with multistate components

A state space model for exact analysis of discrete time heterogeneous general standby systems applicable for hot, warm, and cold backups in any combination of them is proposed. The systems have multistate components whose lifetimes follow independent discrete phase-type distributions. The approach is by incorporating a deceleration matrix into the survival matrix of a component while as a backup and proving that such an approach results in a discrete phase-type representation of the system lifetime distribution. The method is applicable for dynamic reliability analyses of general structure systems having combinations of series, parallel, and standby structures. A binary decision diagram model for the same systems is also proposed by representing a multistate component as a single node. The performance of the two models in generating system reliability measures are compared numerically and qualitatively. Applications for a real-world system and finding optimal backup orderings are given.

Reliability analysis for command‐and‐control systems based on edge extension diagram and binary decision diagram

AbstractAs the central nervous system of modern information warfare, the command‐and‐control network becomes the brain of command and the strategic target of the enemy's primary attack during the operation. In this paper, a method based on edge extension diagram (EED) and binary decision diagram (BDD) is proposed to evaluate the reliability of connectivity within the basic command post of command‐and‐control network, which is solving the problem that the conventional methods failed to model failure probabilities of node links and with low calculation efficiency. First, graph simplification of network topology, extension of nodes under reliable state, construction of path functions, replacement of associated edges, variable sorting, construction of binary decision graphs, and recursive calculation to evaluate the reliability of the connection between the two ends of the command‐and‐control network. Then, further analyze and sort the variable importance of the command‐and‐control network and analyze the sensitivity of the network topology. This paper contributes to the reliability assessment of command‐and‐control network system, the identification and maintenance of key network nodes, and the design of network topology structure.

The Use of the Methodology for Determining the Levels of Organisational and Structural System in the Study of the Efficiency of Foreign Students' Social and Pedagogical Support (on the Example of the US HEIs)

The article examines the problem of foreign students’ social pedagogical support in the context of the effectiveness of preparatory systems in the educational institutions, which determines the relevance of the scientific research – the definition of an effective methodology for analysing the level of structure of the adaptation system of foreign students’ training in the educational institution. Due to the relevance of the research, the purpose of the article is to investigate the system of social and pedagogical support for foreign students in higher education institutions of the United States of America using the method of determining the levels of the organisational and structural system of elements of the acculturation triad. That’s why a method of determining the levels of the organisational and structural system has been proposed, which provides the distribution of the analysed branching program into four levels. In addition, the research required the use of a systematic method (to determine the integrity and structure of the process of socio-pedagogical support), the method of classification (which allowed to distribute the preparatory programs by the elements of the acculturation triad) and the abstraction (aimed at a visual representation of the levels of organisational and structural system). It is emphasised that the support consists of elements of the acculturation triad, such as: educational and cultural, psychological, and linguistic. Each of these elements should be presented as a programme branch, coordinated by the preparatory centres. It was found that the number of levels at which the programmes of the analysed unit or educational institution are presented is directly related to the effectiveness of the process of foreign students’ training. As an example of the analysis with the use of this method, the higher educational institutions representing the United States of America were selected (since this country (according to UNESCO) has the largest number of foreign students). As a result, the following HEIs were selected: University of Texas at Austin, Florida Institute of Technology, Ohio State University, and Yale University. The performed analysis allowed saying that the most branched organisational and structural system is presented (among the selected HEI) in the linguistic orientation of Yale University and is characterised by four levels of program branching. The lowest level (only highest) of branching was found in the educational and cultural direction of the Florida Institute of Technology. The presented methodology can be used in theoretical researches and practical analysis of the effectiveness of adaptation centres for foreign students

Comparison of the Representational Power of Random Forests, Binary Decision Diagrams, and Neural Networks.

In this letter, we compare the representational power of random forests, binary decision diagrams (BDDs), and neural networks in terms of the number of nodes. We assume that an axis-aligned function on a single variable is assigned to each edge in random forests and BDDs, and the activation functions of neural networks are sigmoid, rectified linear unit, or similar functions. Based on existing studies, we show that for any random forest, there exists an equivalent depth-3 neural network with a linear number of nodes. We also show that for any BDD with balanced width, there exists an equivalent shallow depth neural network with a polynomial number of nodes. These results suggest that even shallow neural networks have the same or higher representation power than deep random forests and deep BDDs. We also show that in some cases, an exponential number of nodes are required to express a given random forest by a random forest with a much fewer number of trees, which suggests that many trees are required for random forests to represent some specific knowledge efficiently.

Importance measure-based phased mission reliability and UAV number optimization for swarm

Nowadays, the missions of the unmanned aerial vehicle (UAV) swarm are very common. However, it is still difficult to calculate the phased mission reliability of UAV swarms. Meanwhile, there is no method to measure the effect of changes in the number of UAVs on reliability. Furthermore, how to determine the number of UAVs in swarm to maximize mission reliability is also a challenging issue. Hence, the reliability of a single UAV is analyzed based on phased binary decision diagram. Then the calculation method of mission reliability is given by modeling the swarm mission as a k-out-of-n system with phased requirements. What's more, some importance measures are proposed to describe the effect of changes in the number of UAVs on mission reliability. In addition, an optimization model is established to maximize mission reliability by reasonably arranging the number of different UAVs in the swarm. The importance measure and marginal optimization algorithm are combined to solve the optimization model. Finally, the change law of mission reliability is analyzed and the advantages of the proposed solving method are demonstrated.

All-Optical Parallel Programmable Logic Element Based on Binary Decision Diagram Mapping in Micro-Ring Resonators

All-Optical Parallel Programmable Logic Element Based on Binary Decision Diagram Mapping in Micro-Ring Resonators

Insights gained from applying negate-down during quantification for seismic probabilistic safety assessment

Approximations such as the delete-term approximation, rare event approximation, and minimal cutset upper bound (MCUB) need to be prudently applied for the quantification of a seismic probabilistic safety assessment (PSA) model. Important characteristics of seismic PSA models indicate that preserving the success branches in a primary seismic event tree is necessary. Based on the authors' experience in modeling and quantifying plant-level seismic PSA models, the effects of applying negate-down to the success branches in primary seismic event trees on the quantification results are summarized along with the following three insights gained: (1) there are two competing effects on the MCUB-based quantification results: one tending to increase and the other tending to decrease; (2) the binary decision diagram does not always provide exact quantification results; and (3) it is identified when the exact results will be obtained, and which combination provides more conservative results compared to the others. Complicated interactions occur in Boolean variable manipulation, approximation, and the quantification of a seismic PSA model. The insights presented herein can assist PSA analysts to better understand the important theoretical principles associated with the quantification of seismic PSA models.

BDD-Based Algorithm for SCC Decomposition of Edge-Coloured Graphs

Edge-coloured directed graphs provide an essential structure for modelling and analysis of complex systems arising in many scientific disciplines (e.g. feature-oriented systems, gene regulatory networks, etc.). One of the fundamental problems for edge-coloured graphs is the detection of strongly connected components, or SCCs. The size of edge-coloured graphs appearing in practice can be enormous both in the number of vertices and colours. The large number of vertices prevents us from analysing such graphs using explicit SCC detection algorithms, such as Tarjan's, which motivates the use of a symbolic approach. However, the large number of colours also renders existing symbolic SCC detection algorithms impractical. This paper proposes a novel algorithm that symbolically computes all the monochromatic strongly connected components of an edge-coloured graph. In the worst case, the algorithm performs $O(p \cdot n \cdot log~n)$ symbolic steps, where $p$ is the number of colours and $n$ is the number of vertices. We evaluate the algorithm using an experimental implementation based on binary decision diagrams (BDDs). Specifically, we use our implementation to explore the SCCs of a large collection of coloured graphs (up to $2^{48}$) obtained from Boolean networks -- a modelling framework commonly appearing in systems biology.