: This paper presents the result of research on heuristic methods for assessment of fault-trees (boolean formulae) using Binary Decision Diagrams (BDDs for short). BDDs are the state-of-the-art compact representation of boolean formulae. The fault trees under consideration are given in the sum-of-products or products-of-sums form. We propose a new heuristic that is both robust and discriminating on these particular formulae. The sizes of the BDD obtained are much smaller than those obtained using standard heuristics. However from the theoretical point of view the size of the corresponding BDD remains exponentially large in the worst case. Nevertheless, in practice we manage to deal with complex formulae by means of a new rewriting method which we propose in this article. 1 INTRODUCTION Faced with the intractability of most of the interesting properties of boolean formulae --- assessing a fault tree, finding satisfying solutions, counting the number of solutions---it becomes necessary...

Constructing a small BDD from a given boolean formula depends on finding a good variable ordering. Finding a good order is NP-complete. In the past, methods based on heuristic analysis and formula (circuit) rewriting have shown to be useful for specific problem domains. We show that these methods need to be integrated: heuristic analysis can drive rewriting, which in turn simplifies analysis. We support this claim with experimental results, and describe Almana, an integrated tool for exploring the combination of analysis and rewriting on large boolean formulae.

This thesis investigates practical and theoretical concerns for the use of Binary Decision Diagrams (BDDs) for qualitative and quantitative risk assessments of complex systems. Boolean models describing failure relationships between components, and fault trees in particular, are boolean formulas whose variables are individual component failures; assessment of these models can be performed by analysis of the boolean function induced by the formula. Resource consumption for BDD computations, which is determined by the form of the boolean formula and the order imposed on its variables, is in many cases exponentially smaller than the truth table for the function. The use of Binary Decision Diagrams has made possible orders-of-magnitude increases in the complexity of systems that can be assessed efficiently. Nonetheless, the practical limits of straightforward use of BDDs for reliability analysis are often surpassed by real-world systems. Understanding why this happens is the first subject...

this report we present experimental results on the size of OBDD representation of a class of boolean functions that are typical of what we can expect to find in a fault tree. These functions do not admit BDDs of small size: they either have an exponential size or polynomial size with a coefficient that is unfortunately much too big from the practical point of view. The experiments were carried on a Sparc Ultra-1 processor with 512 Mb of core memory using a BDD tool Aralia [4]. A 2-level reduncuncies function is is defined as follows.

We present a taxonomy of the heuristic analyses of boolean formulas that are used in BDD-based verifications of circuits and fault trees. This systematic characterization of known analyses gives greater insight into the nature of the problem and clearly indicates several opportunities for investigation. It also provides a formal basis for a novel software architecture for useful and extensible optimization tools for boolean formulas. 1 Introduction Building efficient BDD-based tools for circuit verification, reliability analysis, and model-checking critically depends on techniques for finding good variable orders: poor orders lead to intractably large BDD that are impossible to compute. Between a good and a poor variable order can lie an exponential factor in BDD size[Bry86, Bry91]. Efficiently building efficient tools requires intelligent heuristic analyses that work consistently for a given problem domain. Boolean formulas from different domains have different structural prop...