Advanced BDD Optimization by Rudiger Ebendt, Görschwin Fey, Rolf Drechsler

By Rudiger Ebendt, Görschwin Fey, Rolf Drechsler

VLSI CADhas drastically bene?ted from using lowered ordered Binary determination Diagrams (BDDs) and the clausal illustration as an issue of Boolean Satis?ability (SAT), e.g. in common sense synthesis, ver- cation or design-for-testability. In contemporary sensible functions, BDDs are optimized with recognize to new aim capabilities for layout house exploration. the most recent traits express more and more proposals to fuse the options of BDD and SAT. This ebook provides a contemporary presentation of the verified in addition to of modern recommendations. newest leads to BDD optimization are given, c- ering di?erent facets of paths in BDDs and using e?cient decrease bounds in the course of optimization. The awarded algorithms comprise department ? and sure and the prevalent A -algorithm as e?cient innovations to - plore huge seek areas. ? The A -algorithm originates from Arti?cial Intelligence (AI), and the EDA group has been blind to this idea for a very long time. Re- ? cently, the A -algorithm has been brought as a brand new paradigm to discover layout areas in VLSI CAD. in addition to AI seek strategies, the e-book additionally discusses the relation to a different ?eld of job bordered to VLSI CAD and BDD optimization: the clausal illustration as a SAT challenge.

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This equation is given in the next result. 33 For a node v in a BDD, let λ(v) denote the sum of the lengths of all paths starting at v and ending at a terminal node. 7) Proof. In the case of v ∈ {1, 0} there is nothing to show. Now let v be an inner node. Let p = (v, e1 , then(v), . . , t) be a path from v via then(v) onto a terminal node t. Further, let p = (then(v), . . , t) be a path coinciding with p except that we start at then(v) instead. e. the length of every path via then(v) is increased by one if started at v instead.

2 it is explained, how the problem of finding an optimal variable ordering can be expressed in terms of minimum cost path search in a state space. It is outlined, how the generic A∗ -algorithm can be used for this task. To ensure efficiency of A∗ -based approaches, the heuristic function used must have an important property: this is the property of monotonicity. 3, motivation and a formal proof of this property are given for the heuristic function chosen in the new approach. 4. First, two techniques to combine A∗ and B&B are presented.

0 Two BDDs for f = x1 · x2 + x3 · x4 + . . + xn−1 · xn . To describe approaches to minimization of BDD size (as will be done in Chapters 3 and 4), a formalism expressing variable orderings, changes of these orderings and movements of variables is necessary. This section introduces the notation used throughout this book for this purpose. In some cases, when considering the variable ordering of a given BDD, the ordering is not needed explicitly. g. this is the case, when a BDD is given with respect to an initial ordering: considering methods for BDD size minimization, we are often interested in expressing (or restricting) the effect of the changes from one ordering to another.

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