Implicit manipulation of polynomials using zero-suppressed BDDs

Abstract

We present a new technique that broadens the scope of BDD application. It involves manipulating arithmetic polynomials containing higher-degree variables and integer coecients. Our method c an represent large-scale polynomials compactly and uniquely, and it greatly accelerates computation of polynomials. As the polynomial calculus is a basic model in mathematics, our method is very useful in various areas, including formal verication techniques for VLSI design. As our understanding of BDDs has deepened, the range of applications has broadened. Besides Boolean functions, we are often faced with manipulating sets of combinations in many LSI design problems. By mapping a set of combinations into the Boolean space, it can be represented as a characteristic function using a BDD. This method enables us to manipulate a huge number of combinations implicitly, which has never been practical before. Based on implicit set representation, new two-level logic minimization methods have been developed [4][5]. These techniques are also used to solve a kind of covering problem [6].A zero-suppressed BDD (0-sup-BDD) [7] is a new type of BDD adapted the implicit set representation. It can manipulate sets of combinations more eciently than conventional BDDs, especially when dealing with sparse combinations. We have recently studied cube set algebra for manipulating sets of combinations [8], and proposed ecient algorithms for computing cube set operations based on 0-sup-BDDs. This technique is useful for many practical activities related to LSI design, including multi-level logic synthesis [9] and fault simulation.In this paper, we present a new technique that broadens the scope of BDD application. It involves manipulating arithmetic polynomial formulas containing higher-degree variables and integer coecients. Using 0-sup-BDDs, we can represent large-scale polynomials compactly and uniquely. W e developed ecient algorithms for polynomial calculus based on 0-sup-BDD operations. In this method, we can atten arithmetic expressions into canonical forms of polynomials with millions of terms, which have never been represented before. Constructing canonical forms of polynomials immediately leads to equivalence checking of arithmetic expressions. Since polynomial calculus is a basic model in mathematics, our method is expected to be useful for various problems.In this paper, we rst explain 0-sup-BDDs and their operations. We then present a method for representing polynomials with 0-sup-BDDs, and discuss the operation algorithms for polynomial calculus. Finally, w e show implementation of our method and the application for LSI CAD. Eliminate all nodes with the 1-edge pointing to the 0-terminal node. Then connect the edge to the other subgraph directly (Fig. 1).Share all equivalent sub-graphs in the same manner as with conventional BDDs.Notice that, contrary to conventional BDDs, we do not eliminate nodes whose two edges both point to the same node. This reduction rule is asymmetric for the two edges because the nodes remain when their...

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