Lower bounds for dynamic BDD reordering

Abstract — In this paper we present new lower bounds on BDD size. These lower bounds are derived from more general lower bounds that recently were given in the context of exact BDD minimization. The results presented in this paper are twofold: first, we gain deeper insight by looking at the theory behind the new lower bounds. Examples lead to a better understanding, showing that the new lower bounds are effective in situations where this is not the case for previous lower bounds and vice versa. Following the constraints in practice, we then compromise between runtime and quality of the lower bounds. Finally, a clever combination of old and new lower bounds results in a final lower bound, yielding a significant improvement. Experimental results show the efficiency of our approach. I

Solving Set Constraint Satisfaction Problems using ROBDDs

In this paper we present a new approach to modeling finite set domain constraint problems using Reduced Ordered Binary Decision Diagrams (ROBDDs). We show that it is possible to construct an efficient set domain propagator which compactly represents many set domains and set constraints using ROBDDs. We demonstrate that the ROBDD-based approach provides unprecedented flexibility in modeling constraint satisfaction problems, leading to performance improvements. We also show that the ROBDD-based modeling approach can be extended to the modeling of integer and multiset constraint problems in a straightforward manner. Since domain propagation is not always practical, we also show how to incorporate less strict consistency notions into the ROBDD framework, such as set bounds, cardinality bounds and lexicographic bounds consistency. Finally, we present experimental results that demonstrate the ROBDD-based solver performs better than various more conventional constraint solvers on several standard set constraint problems

Minimization of average path length in BDDs by variable reordering

12th International Workshop on Logic and Synthesis, Laguna Beach, California, USA, May 28-30, 2003, pp.207-213.This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. As such, it is in the public domain, and under the provisions of Title 17, United States Code, Section 105, may not be copyrighted.Minimizing the Average Path Length (APL) in a BDD reduces the time needed to evaluate Boolean functions represented by BDDs. This paper describes an efficient heuristic APL minimization procedure based on BDD variable reordering. The reordering algorithm is similar to classical variable sifting with the cost function equal to the APL rather than the number of BDD nodes. The main contribution of our paper is a fast way of updating the APL during the swap of two adjacent variables. Experimental results show that the proposed algorithm effectively minimizing the APL of large MCNC benchmark functions, achieving reductions of up to 47%. For some benchmarks, minimizing APL also reduces the BDD node count

Selected Topics in Network Optimization: Aligning Binary Decision Diagrams for a Facility Location Problem and a Search Method for Dynamic Shortest Path Interdiction

This work deals with three different combinatorial optimization problems: minimizing the total size of a pair of binary decision diagrams (BDDs) under a certain structural property, a variant of the facility location problem, and a dynamic version of the Shortest-Path Interdiction (DSPI) problem. However, these problems all have the following core idea in common: They all stem from representing an optimization problem as a decision diagram. We begin from cases in which such a diagram representation of reasonable size might exist, but finding a small diagram is difficult to achieve. The first problem develops a heuristic for enforcing a structural property for a collection of BDDs, which allows them to be merged into a single one efficiently. In the second problem, we consider a specific combinatorial problem that allows for a natural representation by a pair of BDDs. We use the previous result and ideas developed earlier in the literature to reformulate this problem as a linear program over a single BDD. This approach enables us to obtain sensitivity information, while often enjoying runtimes comparable to a mixed integer program solved with a commercial solver, after we pay the computational overhead of building the diagram (e.g., when re-solving the problem using different costs, but the same graph topology). In the last part, we examine DSPI, for which building the full decision diagram is generally impractical. We formalize the concept of a game tree for the DSPI and design a heuristic based on the idea of building only selected parts of this exponentially-sized decision diagram (which is not binary any more). We use a Monte Carlo Tree Search framework to establish policies that are near optimal. To mitigate the size of the game tree, we leverage previously derived bounds for the DSPI and employ an alpha–beta pruning technique for minimax optimization. We highlight the practicality of these ideas in a series of numerical experiments

Artificial evolution with Binary Decision Diagrams: a study in evolvability in neutral spaces

This thesis develops a new approach to evolving Binary Decision Diagrams, and uses it to study evolvability issues. For reasons that are not yet fully understood, current approaches to artificial evolution fail to exhibit the evolvability so readily exhibited in nature. To be able to apply evolvability to artificial evolution the field must first understand and characterise it; this will then lead to systems which are much more capable than they are currently. An experimental approach is taken. Carefully crafted, controlled experiments elucidate the mechanisms and properties that facilitate evolvability, focusing on the roles and interplay between neutrality, modularity, gradualism, robustness and diversity. Evolvability is found to emerge under gradual evolution as a biased distribution of functionality within the genotype-phenotype map, which serves to direct phenotypic variation. Neutrality facilitates fitness-conserving exploration, completely alleviating local optima. Population diversity, in conjunction with neutrality, is shown to facilitate the evolution of evolvability. The search is robust, scalable, and insensitive to the absence of initial diversity. The thesis concludes that gradual evolution in a search space that is free of local optima by way of neutrality can be a viable alternative to problematic evolution on multi-modal landscapes

Biconditional-BDD Ordering for Autosymmetric Functions

Autosymmetric functions are particular ``regular'' Boolean functions that are exploited for logic optimization, since it is possible to reduce the number of variables and the number of points of the original autosymmetric function before its synthesis. In this paper we study this regularity in oder to derive a suitable variable ordering for Biconditional Binary Decision Diagrams (BBDDs). BBDDs are a new version of BDD that have EXOR of two variables (instead of a variable) in the nodes. These diagrams are employed for logic synthesis in new technologies such as silicon nanowires and DG-SiNWFETs. We show that it is possible to find a useful variable ordering for these functions and the experimental results validate our approach showing that in the 97% of the cases we get an ordering that gives a number of nodes that is lower or equal to the one obtained with the standard ordering

Early Packet Rejection Using Dynamic Binary Decision Diagram

A firewall is a hardware or software device that performs inspection on a given incoming/outgoing packets and decide whether to allow/deny the packet from entering/leaving the system. Firewall filters the packets by using a set of rules called firewall policies. The policies define what type of packets should be allowed or discarded. These policies describe the field values that the packet header must contain in order to match a policy in the firewall. The decision for any given packet is made by finding the first matching firewall policy, if any. In a traditional firewall, the packet filter goes through each policy in the list until a matching rule is found; the same process is again repeated for every packet that enters the firewall. The sequential lookup that the firewall uses to find the matching rule is time consuming and the total time it takes to perform the lookup increases as the policy in the list increases. Nowadays, a typical enterprise based firewall will have 1000+ firewall policy in it, which is normal. A major threat to network firewalls is specially crafted malicious packets that target the bottom rules of the firewall’s entire set of filtering rules. This attack’s main objective is to overload the firewall by processing a flood of network traffic that is matched against almost all the filtering rules before it gets rejected by a bottom rule. As a consequence of this malicious flooding network traffic, the firewall performance will decrease and the processing time of network traffic may increase significantly The current research work is based on the observation that an alternative method for the firewall policies can provide a faster lookup and hence a better filtering performance. The method proposed in this research relies on a basic fact that the policy c a n be represented as a simple Boolean expression. Thus, Binary Decision Diagrams (BDDs) are used as a basis for the representation of access list in this study. The contribution of this research work is a proposed method for representing firewall Policies using BDDs to improve the performance of packet filtering. The proposed mechanism is called Static Shuffling Binary Decision Diagram (SS-BDD), and is based on restructuring of the Binary Decision Diagram (BDD) by using byte-wise data structure instead of using Field-wise data structure. Real world traffic is used during the simulation phase to prove the performance of packet filtering. The numerical results obtained by the simulation shows that the proposed technique improves the performance for packet filtering significantly on medium to long access lists. Furthermore, using BDDs for representing the firewall policies provides other Useful characteristics that makes this a beneficial approach to in real world

BDD Algortihms and Cache Misses

Within the last few years, CPU speed has greatly overtaken memory speed. For this reason, implementation of symbolic algorithms - with their extensive use of pointers and hashing - must be reexamined. In this paper, we introduce the concept of cache miss complexityas an analytical tool for evaluating algorithms depending on pointer chasing. Such algorithms are typical of symbolic computation found in verification. We show how this measure suggests new data structures and algorithmsfor multi-terminal BDDs. Our ideas have been implemented ina BDD package, which is used in a decision procedure for the Monadic Second-order Logic on strings.Experimental results show that on large examples involving e.g the verification of concurrent programs, our implementation runs 4 to 5 times faster than a widely used BDD implementation.We believe that the method of cache miss complexity is of general interest to any implementor of symbolic algorithms used in verification