A Unifying Approach to Edge-valued and Arithmetic Transform Decision Diagrams (original) (raw)
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Edge-valued binary decision diagrams
1996
We describe a canonical and compact data structure, called Edge Valued Binary Decision Diagrams (evbdd), for representing and manipulating pseudo Boolean functions (PBF). evbdds are particularly useful when both arithmetic and Boolean operations are required. We describe a general algorithm on evbdds for performing any binary operation that is closed over the integers. Next, we discuss the relation between the probability expression of a Boolean function and its representation as a pseudo Boolean function. Utilizing this, we present algorithms for computing the probability spectrum and the Reed-Muller spectrum of a Boolean function directly on the evbdd. Finally, w e describe an extension of evbdds which associates both an additive a n d a m ultiplicative w eight with the true edges of the function graph.
Decision diagrams for discrete functions: classification and unified interpretation
Proceedings of 1998 Asia and South Pacific Design Automation Conference
This paper classies dierent decision diagrams (DDs) for discrete functions with respect to the domain and range of represented functions. Relationships among dierent DDs and their relations to spectral transforms are also shown. That provides a unied interpretation of DDs, and their further classication with respect to the spectral transforms.
Techniques for formal transformations of binary decision diagrams
2004
Abstract Binary decision diagrams (BDDs), when used for the representation of discrete functions, permit the direct technology mapping into multi-level logic networks. Complexity of a network derived from a BDD is expressed by its number of non-terminal nodes. The paper discusses the problem of reducing the BDDs. It makes two main contributions:(a) the bounds of the potential complexity of the BDD are determined and proven;(b) a formal technique is presented for simplification of Boolean operations on a set of BDDs.
Remarks on Shapes of Decision Diagrams and Classes of Multiple-Valued Functions
The paper studies binary and ternary functions that have decision diagrams of identical shape in the original and spectral (Fourier) domain. These functions are called Fouriersweet functions. This class of functions involves certain classes of bent functions and quadratic forms in both binary and ternary cases. Bent functions and quadratic forms have applications in cryptography and error-correcting codes. Not all bent functions are Fourier-sweet functions. It follows, that Fourier-sweet functions are capable of capturing the differences among the classes of bent functions, and at the same time link them to quadratic forms. Representation by shape invariant decision diagrams in the original and spectral domain might provide some better insight into features of bent functions and quadratic forms. The functions represented by the disjoint quadratic forms in the binary case and diagonal forms in the ternary case are elementary Fourier-sweet functions. In both binary and ternary cases, the application of affine transformations, under certain precisely specified restrictions, to the elementary Fourier-sweet functions produces other Fourier-sweet functions.
Numeric Function Generators Using Decision Diagrams for Discrete Functions
2009
This paper introduces design methods for numeric function generators (NFGs) using decision diagrams. NFGs are hardware accelerators to compute values of numeric functions such as trigonometric, logarithmic, square root, and reciprocal functions. Most existing design methods for NFGs are intended only for a specific class of numeric functions. However, by using decision diagrams for discrete functions (i.e., word-level decision diagrams), we can systematically design fast and compact NFGs for a larger class of functions. This paper shows three design methods for NFGs using 1) multi-terminal binary decision diagrams (MTBDDs), 2) binary moment diagrams (BMDs), and 3) edge-valued binary decision diagrams (EVBDDs).
Proceedings of the 31st annual conference on Design automation conference - DAC '94, 1994
An ecient package for construction of and operation on ordered K r onecker F unctional D e cision Diagrams (OKFDD) is presented. OKFDDs are a generalization o f OBDDs and OFDDs and as such provide a more c ompact representation of the functions than either of the two decision diagrams. In this paper b asic properties of OKFDDs and their ecient representation and manipulation a r e presented. Based on the comparison of the three d e cision diagrams for several b enchmark functions, a 25% improvement in size over OBDDs is observed for OKFDDs.
IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 1999
This paper proposes a method to construct smaller binary decision diagrams for characteristic functions (BDDs for CFs). A BDD for CF represents an n-input m-output function, and evaluates all the outputs in O(n + m) time. We derive an upper bound on the number of nodes of the BDD for CF of n-bit adders (adrn). We also compare complexities of BDDs for CFs with those of shared binary decision diagrams (SBDDs) and multi-terminal binary decision diagrams (MTBDDs). Our experimental results show: 1) BDDs for CFs are usually much smaller than MTBDDs; 2) for adrn and for some benchmark circuits, BDDs for CFs are the smallest among the three types of BDDs; and 3) the proposed method often produces smaller BDDs for CFs than an existing method.
A characterization of binary decision diagrams
IEEE Transactions on Computers, 1993
Binary Decision Diagrams (BDD's) are a representation of Boolean functions. Its use in the synthesis, simulation and testing of Boolean circuits has been proposed by various researchers. In all these applications of BDD's solutions to some fundamental computational problems are needed. We present a characterization of BDD's in terms of the complexity of these computational problems. A tighter bound on the size of an Ordered BDD that can be computed from a given Boolean Circuit is presented. Based on our results we make a case for exploring the use of Repeated BDD's, with a small number of Repeated Variables, and Free BDD's for some applications for which only Ordered BDD's have be used so far.
Mutual conversions between generalised arithmetic expansions and free binary decision diagrams
IEE Proceedings - Circuits, Devices and Systems, 1998
Calculation of generalised arithmetic expansions from free binary decision diagrams of incompletely specified Boolean functions is shown. The way of decomposing generalised arithmetic expansion coefficients in terms of cofactors of Boolean functions is presented. Based on the decomposition a second new algorithm to synthesise quasi-optimal free binary decision diagrams directly from generalised arithmetic expansion of Boolean functions is developed.
Numerical Function Generators Using Edge-Valued Binary Decision Diagrams
2007 Asia and South Pacific Design Automation Conference, 2007
In this paper, we introduce the edge-valued binary decision diagram (EVBDD) to reduce the memory and delay in numerical function generators (NFGs). An NFG realizes a function, such as a trigonometric, logarithmic, square root, or reciprocal function, in hardware. NFGs are important in, for example, digital signal applications, where high speed and accuracy are necessary. We use the EVBDD to produce a fast and compact segment index encoder (SIE) that is a key component in our NFG. We compare our approach with NFG designs based on multi-terminal BDD's (MTBDDs), and show that the EVBDD produces SIEs that have, on average, only 7% of the memory and 40% of the delay of those designed using MTBDDs. Therefore, our NFGs based on EVBDDs have, on average, only 38% of the memory and 59% of the delay of NFGs based on MTBDDs.