Introduction to linear and nonlinear programming

We describe a method for using abstraction to reduce the complexity of temporal-logic model checking. Using techniques similar to those involved in abstract interpretation, we construct an abstract model of a program without ever examining the corresponding unabstracted model. We show how this abstract model can be used to verify properties of the original program. We have implemented a system based on these techniques, and we demonstrate their practicality using a number of examples, including a program representing a pipelined ALU circuit with over 101300 states.

Model checking and abstraction

Efficient manipulation of Boolean functions is an important component of many computer-aided design tasks This paper describes a package for manipulating Boolean functions based on the reduced, ordered, binary decision diagram (ROBDD) representation The package is based on an efficient implementation of the if-then-else (ITE) operator A hash table is used to maintain a strong canonical form in the ROBDD, and memory use is improved by merging the hash table and the ROBDD into a hybrid data structure A memory function for the recursive ITE algorithm is implemented using a hash-based cache to decrease memory use Memory function efficiency is improved by using rules that detect when equivalent functions are computed The usefulness of the package is enhanced by an automatic and low-cost scheme for recycling memory Experimental results are given to demonstrate why various implementation trade-offs were made These results indicate that the package described here is significantly faster and more memory-efficient than other ROBDD implementations described in the literature

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Efficient implementation of a BDD package

The ordered binary decision diagram (OBDD) has proven useful in many applications as an efficient data structure for representing and manipulating Boolean functions. A serious drawback of OBDD's is the need for application-specific heuristic algorithms to order the variables before processing. Further, for many problem instances in logic synthesis, the heuristic ordering algorithms which have been proposed are insufficient to allow OBDD operations to complete within a limited amount of memory. The paper proposes a solution to these problems based on having the OBDD package itself determine and maintain the variable order. This is done by periodically applying a minimization algorithm to reorder the variables of the OBDD to reduce its size. A new OBDD minimization algorithm, called the sifting algorithm, is proposed and appears especially effective in reducing the size of the OBDD. Experiments with dynamic variable ordering on the problem of forming the OBDD's for the primary outputs of a combinational circuit show that many computations complete using dynamic variable ordering when the same computation fails otherwise.

Dynamic variable ordering for ordered binary decision diagrams

In this paper, we propose Zero-Suppressed BDDs (0-Sup-BDDs), which are BDDs based on a new reduction rule. This data structure brings unique and compact representation of sets which appear in many combinatorial problems. Using 0-Sup-BDDs, we can manipulate such sets more simply and efficiently than using original BDDs. We show the properties of 0-Sup-BDDs, their manipulation algorithms, and good applications for LSI CAD systems.

Zero-Suppressed BDDs for Set Manipulation in Combinatorial Problems

R.E. Bryant proposed a method to handle logic expressions (IEEE Trans. Comp., vol.25, no.8, p.667-91, 1986) which is based on binary decision diagrams (BDD) with restriction; variable ordering ix fixed throughout a diagram. The method is more efficient than other methods proposed so far and depends heavily on variable ordering. A simple but powerful algorithm for variable ordering is developed. The algorithm tries to find a variable ordering which minimizes the number of crosspoints of nets when the circuit diagram is drawn. This is applied to the Boolean comparison of ISCAS benchmark circuits for test pattern generation. The results show that binary decision diagrams (BDD) with the proposed ordering method can verify almost all benchmark circuits in less than several central processor unit (CPU) minutes, which is one hundred times (or more) faster than times reported in the literature. Some techniques for circuit evaluation ordering are also mentioned. >

Evaluation and improvement of Boolean comparison method based on binary decision diagrams

SUMMARY Although intervention practices like mindfulness medita- tion have proven e ﬀ ective in treating psychosis, there is no clarity on the mechanism of information propagation in the brain. In this study, we for- mulated a network optimization problem and searched for the optimal solution using Digital Annealer developed by Fujitsu Ltd. This is inspired by quantum computing and is e ﬀ ective in solving large-scale combinatorial optimization problems to ﬁnd the information propagation pathway in the brain that contributes to the realization of mindfulness. Speciﬁcally, we deﬁned the optimal network state as the state of the brain network that is considered to be associated with the mindfulness state. We formulated the problem into two network optimization problems—the minimum vertex- cover problem and the maximum-ﬂow problem—to search for the information propagation pathway that is important for realizing the state. In the minimum vertex-cover problem, we aimed to identify brain regions that are important for the realization of the mindfulness state, and identiﬁed eight regions, including four that were suggested to be consistent with previous studies. We formulated the problem as a maximum-ﬂow problem to iden- tify the information propagation pathways in the brain that contribute to the activation of these four identiﬁed regions. As a result, approximately 30% of the connections in the brain network structure of this study were identiﬁed, and the pathway with the highest ﬂow rate was considered to charac- terize the bottom-up emotion regulation during mindfulness. The ﬁndings of this study could be useful for more direct interventions in the context of mindfulness, which are being investigated by neurofeedback and other methods. This is because existing studies have not clariﬁed the information propagation pathways that contribute to the realization of the brain network states that characterize mindfulness states. In addition, this approach may be useful as a methodology to identify information propagation pathways in the brain that contribute to the realization of higher-order human cognitive activities, such as mindfulness, within large-scale brain networks.

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H. Fujisawa

Papers

Evaluation and improvement of Boolean comparison method based on binary decision diagrams

Formulation of Mindfulness States as a Network Optimization Problem and an Attempt to Identify Key Brain Pathways Using Digital Annealer