Binary Decision Diagrams Research Articles

EnR: extend and reduce methodology to enable formal verification of truncated adders

Abstract Truncated adders are widely used in applications where using lower bit-width is enough to generate the desired outputs. Truncated adders give benefits in area, power, and delay as compared to their non-truncated counterparts. While prior works deal with the formal verification of n-bit adders, there is no prior work dealing with the formal verification of unverified m-bit truncated adders using verified n-bit adder designs, where m < ${< } $ n. In this work, we propose a methodology that enables the verification for any truncated adder designs, irrespective of the architecture, called Extend and Reduce (EnR). EnR uses a three-step methodology to formally verify the truncated adders, which are considered as Design Under Verification (DUV). Firstly, it extends the bit-width of the DUV (truncated adder) to match with the nearest higher 2 n -bit adder, called an Extended Truncated Adder (ETA). Secondly, it reduces off-the-shelf formally verified golden reference adder design by forcing 0’s at the input bits to match it to the extended DUV design and re-synthesizing it to generate a Reduced Golden Adder (RGA). Lastly, a combinational equivalence check is then performed between the ETA and the RGA for formal verification. If the truncated adder is correct, the ETA and RGA are equivalent and vice versa. We evaluate a number and variety of adder designs to show the efficacy of the EnR methodology. We performed the formal verification using Binary Decision Diagrams (BDDs) and And-Inverter Graphs (AIGs). Lastly, we show the scalability of EnR methodology using adders up to a bit-width of 512 bits. ACM CCS Hardware → Arithmetic and datapath circuits; Combinational circuits; Equivalence checking; Functional verification; Electronic design automation.

Application of Binary Decision Diagrams in Time-Dependent Reliability Analysis

Binary Decision Diagrams (BDDs) are often used in specific types of reliability analysis known as topological (or structure) analysis and time-independent analysis. The previous focuses only on the analysis of system topology, which is defined by structure function. The latter takes into account the structure function together with the time-independent reliabilities of components that the system is composed of. However, the most interesting type of reliability analysis is time-dependent analysis in which reliabilities of the components are time-dependent functions. In this paper, we first present the development of a mathematical model of a non-repairable system composed of independent non-repairable components and explain the properties of this model from the point of view of time-dependent, time-independent, and topological reliability analysis. In the second part of the paper, we present and experimentally compare two methods for time-dependent reliability analysis of the considered mathematical model. The first method is based on the direct application of BDDs and we label it as a basic approach. The second, symbolic approach, combines BDDs with expression trees. The experimental comparison implemented using opensource C++ libraries TeDDy and GiNaC shows that the first method based on the basic approach is much faster than the second method using expression trees.

Rapid computation of survival signature for dynamic fault tree based on sequential binary decision diagram and multidimensional array

Many practical safety-critical systems typically exhibit sequence-dependent failure behaviors, limiting the efficiency of analyzing these systems. Although the survival signature-based method can address this problem to a certain extent, the dependence on Boolean states constrains its application to large systems. In this study, we present a novel method that leverages the sequential binary decision diagram (SBDD) and multidimensional array to rapidly compute survival signatures for dynamic fault trees (DFTs) of these systems. These dynamic nodes in the SBDD are represented through multidimensional arrays, which are then utilized as inputs for the subsequent computations. Ultimately, survival signatures are obtained by iteratively computing the multidimensional arrays. Additionally, two practical engineering cases are examined to highlight the superiority of the proposed methods over other methods. Compared with Boolean state vector-based methods, the proposed method achieves a 689-fold and 209-fold increase in efficiency for calculating survival signatures in their respective cases. Compared with the Monte Carlo (MC) simulation, the simulation efficiency for the reliability results improve by 60-fold and 201-fold in their respective cases.

Perspective on complexity measures targeting read-once branching programs

A model of computation for which reasonable yet still incomplete lower bounds are known is the read-once branching program. Here variants of complexity measures successful in the study of read-once branching programs are defined and studied. Some new or simpler proofs of known bounds are uncovered. Branching program resources and the new measures are compared extensively. The new variants are developed in part in the hope of tackling read-k branching programs for the tree evaluation problem [8]. Other computation problems are studied as well. In particular, a common view of a function studied by Gál [11] and a function studied by Bollig and Wegener [3] leads to the general combinatorics of blocking sets. Technical combinatorial results of independent interest are obtained. New leads towards further progress are discussed. An exponential lower bound for non-deterministic read-k branching programs for the GEN function [17] is also derived, independently from the new measures.

Reliability of a consecutive k-out-of-n: G system with protection blocks

This paper studies the reliability of consecutive k-out-of-n: G systems with protection blocks. The system consists of n components, and it functions if at least k consecutive components are operational. Protection blocks can be used to increase system reliability by reducing the failure rates of exactly k consecutive components. However, the protection blocks also have their own failure rates. System reliability is assessed by calculating the probability of successful operation over a specified time period, taking into account the failure dynamics of both components and protection blocks. Binary decision diagram (BDD) is used to identify combinations of component states that lead to system success. The probability for each combination is then evaluated, considering the states of both the components and the protection blocks. By integrating these probabilities, the overall system reliability is determined. Numerical examples are provided to illustrate the application of this method.

Reliability analysis of dynamic fault trees with Priority-AND gates using conditional binary decision diagrams

The conditional binary decision diagram (CBDD) is an extension of the BDD by introducing the conditioning event node, which has been applied to analyze the reliability of the dynamic fault tree (DFT). However, the previous CBDD-based method is limited to the DFT with spare gates since the Boolean conditioning event only describes the replacement state in spare gates. Hence, it cannot be available for the DFT with Priority-AND (PAND) gates. To address this issue, a novel Boolean conditioning event is proposed to describe the sequence-dependence failure in the dynamic system modeled by the DFT with PAND gates. A DFT with complex PAND gates can be converted into a conditional fault tree (CFT) by recursively applying derived rules that are based on the proposed conditioning event. As a result, the minimal cut set replaces the minimal cut sequence for qualitative analysis, which can reduce the average space complexity in a general case. Then, the CBDD can be generated based on the CFT and it can be used for evaluation. Finally, case studies demonstrate an extension of the cold spare gate and the advantages of our method.

Multivariate to bivariate reduction for noncommutative polynomial factorization

Based on Bergman's theorem, we show that multivariate noncommutative polynomial factorization is deterministic polynomial-time reducible to the factorization of bivariate noncommutative polynomials. More precisely,1.Given an n-variate noncommutative polynomial f∈F〈X〉 over a field F as an arithmetic circuit, computing a complete factorization of f into irreducible factors is deterministic polynomial-time reducible to factorization of a noncommutative bivariate polynomial g∈F〈x,y〉; the reduction transforms f into a circuit for g, and given a complete factorization of g, the reduction recovers a complete factorization of f in polynomial time.The reduction works both in the white-box and the black-box setting.2.We show over the field of rationals that bivariate linear matrix factorization problem for 4×4 matrices is at least as hard as factoring square-free integers and for 3×3 matrices it is in polynomial time.

Hitting Sets for Orbits of Circuit Classes and Polynomial Families

The orbit of an n -variate polynomial \(f({\mathbf {x}})\) over a field \(\mathbb {F}\) is the set \(\text {orb}(f) := \lbrace f(A{\mathbf {x}}+{\mathbf {b}}) : A \in \mathrm{GL}(n,\mathbb {F}) \text{ and } {\mathbf {b}}\in \mathbb {F}^n\rbrace\) . The orbit of a polynomial f is a geometrically interesting subset of the set of affine projections of f . Affine projections of polynomials computable by seemingly weak circuit classes can be quite powerful. For example, the polynomial \(\mathsf {IMM}_{3,d}\) —the \((1,1)\) th entry of a product of d generic \(3 \times 3\) matrices—is computable by a constant-width read-once oblivious algebraic branching program (ROABP), yet every polynomial computable by a size- s general arithmetic formula is an affine projection of \(\mathsf {IMM}_{3,\text {poly}(s)}\) as shown by Ben-or and Cleve [ 12 ]. To our knowledge, no efficient hitting set construction was known for \(\text {orb}(\mathsf {IMM}_{3, d})\) before this work. In this article, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of: Low-individual-degree polynomials computable by commutative ROABPs . This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials and the orbits of multilinear sparse polynomials . Multilinear polynomials computable by constant-width ROABPs . This implies a quasi-polynomial time hitting set for the orbits of the family \(\lbrace \mathsf {IMM}_{3,d}\rbrace _{d \in \mathbb {N}}\) . Polynomials computable by constant-depth, constant-occur formulas . This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in , and also polynomial-time hitting sets for the orbits of the power symmetric polynomials and the sum-product polynomials . Polynomials computable by occur-once formulas . We say a polynomial has low individual degree if the degree of every variable in the polynomial is at most \(\text {poly}(\log n)\) , where n is the number of variables. The first two results are obtained by building upon and strengthening the rank concentration by translation technique of Agrawal, Saha, and Saxena [ 6 ]; the second result additionally uses the merge-and-reduce idea from Forbes and Shpilka [ 30 ] and Forbes, Shpilka, and Saptharishi [ 27 ]. The proof of the third result adapts the algebraic independence-based technique of Agrawal, Saha, Saptharishi, and Saxena [ 5 ] and Beecken, Mittmann, and Saxena [ 11 ] to reduce to the case of constructing hitting sets for the orbits of sparse polynomials. A similar reduction using the Shpilka-Volkovich (SV) generator-based argument in Shpilka and Volkovich [ 90 ] yields the fourth result. The SV generator plays an important role in all four results.

BDD‐based optimized majority/minority gate using electro‐optic Mach‐Zehnder interferometer

AbstractIn this article, an optical majority/minority circuit has been designed using four electro‐optic Mach‐Zehnder interferometers. The binary decision diagram (BDD) is used to optimize the Boolean logical expression. In this design we get two outputs, one performs majority logic and the other performs minority logic simultaneously. Numerical simulation has been done with the obtained data from the beam propagation method. Pseudo‐eye diagram is also plotted to find amplitude modulation (AM) and extinction ratio (ER). AM and ER are found nearly 0.4 and 13 dB, respectively. A new parameter, the optical circuit complexity factor (OCC) factor has been calculated and from the result, it has been shown that the proposed design is more optimized than other majority/minority gate designs so far.

Limitations of the Impagliazzo–Nisan–Wigderson Pseudorandom Generator Against Permutation Branching Programs

AbstractThe classic Impagliazzo–Nisan–Wigderson (INW) pseudorandom generator (PRG) (STOC ‘94) for space-bounded computation uses a seed of length $$O(\log n \cdot \log (nw/\varepsilon )+\log d)$$ O ( log n · log ( n w / ε ) + log d ) to fool ordered branching programs of length n, width w, and alphabet size d to within error $$\varepsilon $$ ε . A series of works have shown that the analysis of the INW generator can be improved for the class of permutation branching programs or the more general regular branching programs, improving the $$O(\log ^2 n)$$ O ( log 2 n ) dependence on the length n to $$O(\log n)$$ O ( log n ) or $${\tilde{O}}(\log n)$$ O ~ ( log n ) . However, when also considering the dependence on the other parameters, these analyses still fall short of the optimal PRG seed length $$O(\log (nwd/\varepsilon ))$$ O ( log ( n w d / ε ) ) . In this paper, we prove that any “spectral analysis” of the INW generator requires seed length $$\begin{aligned} \Omega \left( \log n\cdot \log \log \left( \min \{n,d\}\right) +\log n\cdot \log \left( w/\varepsilon \right) +\log d\right) \end{aligned}$$ Ω log n · log log min { n , d } + log n · log w / ε + log d to fool ordered permutation branching programs of length n, width w, and alphabet size d to within error $$\varepsilon $$ ε . By “spectral analysis” we mean an analysis of the INW generator that relies only on the spectral expansion of the graphs used to construct the generator; this encompasses all prior analyses of the INW generator. Our lower bound matches the upper bound of Braverman–Rao–Raz–Yehudayoff (FOCS 2010, SICOMP 2014) for regular branching programs of alphabet size $$d=2$$ d = 2 except for a gap between their $$O\left( \log n \cdot \log \log n\right) $$ O log n · log log n term and our $$\Omega \left( \log n \cdot \log \log \min \{n,d\}\right) $$ Ω log n · log log min { n , d } term. It also matches the upper bounds of Koucký–Nimbhorkar–Pudlák (STOC 2011), De (CCC 2011), and Steinke (ECCC 2012) for constant-width ($$w=O(1)$$ w = O ( 1 ) ) permutation branching programs of alphabet size $$d=2$$ d = 2 to within a constant factor. To fool permutation branching programs in the measure of spectral norm, we prove that any spectral analysis of the INW generator requires a seed of length $$\Omega \left( \log n\cdot \log \log n+\log n\cdot \log (1/\varepsilon )\right) $$ Ω log n · log log n + log n · log ( 1 / ε ) when the width is at least polynomial in n ($$w=n^{\Omega (1)}$$ w = n Ω ( 1 ) ), matching the recent upper bound of Hoza–Pyne–Vadhan (ITCS 2021) to within a constant factor.

Boosting optimal symbolic planning: Operator-potential heuristics

Heuristic search guides the exploration of states via heuristic functions h estimating remaining cost. Symbolic search instead replaces the exploration of individual states with that of state sets, compactly represented using binary decision diagrams (BDDs). In cost-optimal planning, heuristic explicit search performs best overall, but symbolic search performs best in many individual domains, so both approaches together constitute the state of the art. Yet combinations of the two have so far not been an unqualified success, because (i) h must be applicable to sets of states rather than individual ones, and (ii) the different state partitioning induced by h may be detrimental for BDD size. Many competitive heuristic functions in planning do not qualify for (i), and it has been shown that even extremely informed heuristics can deteriorate search performance due to (ii).Here we show how to achieve (i) for a state-of-the-art family of heuristic functions, namely potential heuristics. These assign a fixed potential value to each state-variable/value pair, ensuring by LP constraints that the sum over these values, for any state, yields an admissible and consistent heuristic function. Our key observation is that we can express potential heuristics through fixed potential values for operators instead, capturing the change of heuristic value induced by each operator. These reformulated heuristics satisfy (i) because we can express the heuristic value change as part of the BDD transition relation in symbolic search steps. We run exhaustive experiments on IPC benchmarks, evaluating several different instantiations of potential heuristics in forward, backward, and bi-directional symbolic search. Our operator-potential heuristics turn out to be highly beneficial, in particular they hardly ever suffer from (ii). Our best configurations soundly beat previous optimal symbolic planning algorithms, bringing them on par with the state of the art in optimal heuristic explicit search planning in overall performance.

Research programs and the progress of theory in sociology: the post-positivist conception of J. Berger, M. Zelditch Jr. and D. Wagner

The article presents an explanation of the post-positivist conception of the progress of sociological theories in research programs, which was developed by J. Berger, M. Zelditch Jr. and D. Wagner. Their conception specified related types of theoretical structure and dynamics of research programs, as well as their functions in sociological theorizing. Structural and dynamic differences of research programs represent different types of conceptual connections between theories and their corresponding basic modes of theoretical growth. In particular, the elaboration of theories of certain explanatory domains by increasing their scope, rigor, precision, or empirical relevance is a mode of progress of linear programs. The proliferation of theories into new explanatory domains and the expansion of the subject range of their applicability is the mode of development of branching programs. The competition of theories in a certain explained domain for the best conceptual representation of its problems is a mode of progress of competing programs. Special types of conceptual connections and modes of theoretical progress may also emerge between theories in linear, branching, and competing research programs: variation of theories as a result of deepening them into their established subject domains due to increased conceptual insight, sophistication and detail; integration of theories. The theoretical development of research programs is provided by their functions, which are analogous to the functions of paradigms in mature sciences: determination of theoretical significance and doability of sociological problems; evaluation and cognitive-based selection of competing theoretical formulations of their solutions; ensuring synergy of the consequences of performing the above-mentioned functions.

CFLOBDDs: Context-Free-Language Ordered Binary Decision Diagrams

This article presents a new compressed representation of Boolean functions, called CFLOBDDs (for Context-Free-Language Ordered Binary Decision Diagrams). They are essentially a plug-compatible alternative to BDDs (Binary Decision Diagrams), and hence are useful for representing certain classes of functions, matrices, graphs, relations, and so forth in a highly compressed fashion. CFLOBDDs share many of the good properties of BDDs, but—in the best case—the CFLOBDD for a Boolean function can be exponentially smaller than any BDD for that function . Compared with the size of the decision tree for a function, a CFLOBDD—again, in the best case—can give a double-exponential reduction in size . They have the potential to permit applications to (i) execute much faster and (ii) handle much larger problem instances than has been possible heretofore. We applied CFLOBDDs in quantum-circuit simulation and found that for several standard problems, the improvement in scalability, compared to BDDs, is quite dramatic. With a 15-minute timeout, the number of qubits that CFLOBDDs can handle are 65,536 for Greenberger-Horne-Zellinger, 524,288 for Bernstein-Vazirani, 4,194,304 for Deutsch-Jozsa, and 4,096 for Grover’s algorithm, besting BDDs by factors of 128×, 1,024×, 8,192×, and 128×, respectively.

A hybrid approach of partially applying BDD for seismic PSA quantification

The binary decision diagram (BDD) method provides a means to calculate the exact probability of a fault tree, but cannot solve large fault trees for nuclear power plant PSAs. However, the BDD method can be used as a supplementary means to increase the accuracy of PSA quantification, especially for a seismic PSA which includes many non-rare events.Since it is difficult to solve a large PSA model using BDD method, several approaches have been developed that partially apply the BDD technique to small-sized branches of a large PSA model. This paper proposes an approach of partially applying BDD technique to large-sized branches. The results from the pilot application to a seismic PSA model shows that the partial BDD approach of this paper provides a relatively accurate way to quantify each sequence as well as the overall core damage in a seismic PSA model.

Efficient non-isomorphic graph enumeration algorithms for several intersection graph classes

Intersection graphs are well-studied in the area of graph algorithms. Some intersection graph classes are known to have algorithms enumerating all unlabeled graphs by reverse search. Since these algorithms output graphs one by one and the numbers of graphs in these classes are vast, they work only for a small number of vertices. Binary decision diagrams (BDDs) are compact data structures for various types of data and useful for solving optimization and enumeration problems. This study proposes enumeration algorithms for five intersection graph classes, which admit O(n)-bit string representations for their member graphs. Our algorithm for each class enumerates all unlabeled graphs with n vertices over BDDs representing the binary strings in time polynomial in n. Moreover, our algorithms are extended so that it enumerates those with constraints on the maximum (bi)clique size and/or the number of edges.

An Experimental Research of the Algorithm for Extracting of a Boolean Function Subsystems for Joint Multilevel Optimization

The results of an experimental comparison of programs for technologically independent minimization of complexity of multilevel representations of systems of fully defined functions based on the Shannon expansion are described. The main attention is paid to the effectiveness of using, when synthesizing logical circuits, a program that implements an algorithm for solving the problem of extracting from a system of functions such subsystems for which it is advisable to carry out joint minimization of multi-level representations of subsystems in the form of Binary Decision Diagrams, called BDD-representations, and in the form of Boolean net. The complexity of the representation is estimated by the total number of literals of Boolean variables in a set of interrelated logical equations that define a system of Boolean functions. After extracting the subsystems and their joint minimization, the synthesis of logic circuits is carried out in the design library of custom digital CMOS VLSIs, the results are compared in terms of chip area and speed (time delay) on a stream of 39 industrial examples of circuits. It was found that for 20 examples, the subsystem selection algorithm allows one to obtain better solutions than joint or separate minimization of multi-level representations based on Shannon expansion (13 examples). The three best solutions are obtained by applying the well-known Espresso program for minimizing functions in the DNF class, and the three solutions are obtained by the synthesizer using the original (non-optimized) matrix representations of DNF systems of Boolean functions.

A new ordering method of basic events in dynamic fault tree for sequential binary decision diagram

AbstractDynamic fault tree (DFT) is a reliable method for modeling complex systems with dynamic characteristics. However, evaluating the reliability of such systems can be complex. The sequential binary decision diagram (SBDD) method simplifies this process by reducing computational complexity. This study investigates factors influencing SBDD size and proposes a new method, the complete priority ordering method, for ordering basic events in SBDDs. Results demonstrate the method's effectiveness in comparison to existing approaches.

BDD efficiency: Survey of BDD edge ordering algorithms in network reliability

AbstractNetwork reliability analysis is vital for ensuring efficient and error‐free communication within networking and communication applications. Binary Decision Diagrams (BDDs) have emerged as a powerful tool for analyzing and optimizing complex network infrastructures. The objective of this research paper is to conduct a comparative analysis of edge‐ordering algorithms for network reliability using BDDs the study aims to evaluate and compare existing algorithms, providing valuable insights for selecting suitable edge‐ordering algorithms that enhance network reliability. The paper concludes that snooker is outperforming among selected algorithms.

Case studies for binary decision diagram for 5G mobile networks optimization

In this paper, we study the binary decision diagrams and mobile network optimization. As we see about the binary decision diagrams, Lee and Akers first introduced the general concept of binary decision diagrams but later spread by Bryant. It turns out that, according to the European Interactive Digital Advertising Alliance (EDAA) order and reduction rules, binary decision diagrams are a standard form of representation. After that, we study the spanning trees and binary decision diagrams. An advantage of binary decision diagrams-based methods is that binary decision diagrams provide a concise and separate description of system success or failure. The effectiveness of binary decision diagram methods is measured by the size of them, which depends heavily on the selected series.

A General Theoretical Framework for Learning Smallest Interpretable Models

We develop a general algorithmic framework that allows us to obtain fixed-parameter tractability for computing smallest symbolic models that represent given data. Our framework applies to all ML model types that admit a certain extension property. By showing this extension property for decision trees, decision sets, decision lists, and binary decision diagrams, we obtain that minimizing these fundamental model types is fixed-parameter tractable. Our framework even applies to ensembles, which combine individual models by majority decision.