Analysis and synthesis of weighted-sum functions

A weighted-sum (WS) function computes the sum of selected integers. This paper considers a design method for WS functions by look-up table (LUT) cascades. In particular, it derives upper bounds on the column multiplicities of decomposition charts for WS functions. From these, the size of LUT cascades that realize WS functions can be estimated. The arithmetic decomposition of a WS function is also shown. With this method, a WS function can be implemented with cascades and adders

Worst and best irredundant sum-of-products expressions

In an irredundant sum-of-products expression (ISOP), each product is a prime implicant (Pl) and no product can be deleted without changing the function. Among the ISOPs for some function f, a worst ISOP (WSOP) is an ISOP with the largest number of Pls and a minimum ISOP (MSOP) is one with the smallest number. We show a class of functions for which the Minato-Morreale ISOP algorithm produces WSOPs. Since the ratio of the size of the WSOP to the size of the MSOP is arbitrarily large when it, the number of variables, is unbounded, the Minato-Morreale algorithm can produce results that are very far from minimum. We present a class of multiple-output functions whose WSOP size is also much larger than its MSOP size. For a set of benchmark functions, we show the distribution of ISOPs to the number of Pls. Among this set are functions where the MSOPs have almost as many Pls as do the WSOPs. These functions are known to be easy to minimize. Also, there are benchmark functions where the fraction of ISOPs that are MSOPs is small and MSOPs have many fewer Pls than the WSOPs. Such functions are known to be hard to minimize. For one class of functions, we show that the fraction of ISOPs that are MSOPs approaches 0 as n approaches infinity, suggesting that such functions are hard to minimiz

A discussion on the history of research in arithmetic andReed-Muller expressions

This paper discusses early work by Komamiya in Reed-Muller and arithmetic expressions for switching function

Comparison of the Worst and Best Sum-of-Products Expressions for Multiple-Valued Functions

Because most practical logic design algorithms produce irredundant sum-of-products (ISOP) expressions, the understanding of ISOPs is crucial. We show a class of functions for which Morreale-Minato's ISOP generation algorithm produces worst ISOPs (WSOP), ISOPs with the most product terms. We show this class has the property that the ratio of the number of products in the WSOP to the number in the minimum ISOP (MSOP) is arbitrarily large when the number of variables is unbounded. The ramifications of this are significant; care must be exercised in designing algorithms that produce ISOPs. We also show that 2/sup n-1/ is a firm upper bound on the number of product terms in any ISOP for switching functions on n variables, answering a question that has been open for 30 years. We show experimental data and extend our results to functions of multiple-valued variables

On the Complexity of Classification Functions

A classification function is a multiple-valued input func-tion specified by a set of rules, where each rule is a conjunc-tion of range functions. The function is useful for packet classification for internet, network intrusion detection sys-tem, etc. This paper considers the complexity of range functions and classification functions represented by sum-of-products expressions of binary variables. It gives tighter upper bounds on the number of products for range func-tions. 1

Multiple-Valued Index Generation Functions: Reduction of Variables by Linear Transformation

We consider incompletely specified multiple-valued input index generation functions f : D → {1, 2, . . . , k}, where D ⊆ P n and P = {0, 1, 2, . . . , p − 1}. In such functions, the number of variables to represent f can be often reduced. Let k be the number of elements in D. We show that most functions can be represented with 2 log p (k + 1) or fewer variables, when k is sufficiently smaller than p n . Also, to further reduce the number of variables, we use linear transformations. To find good linear transformations, we introduce the imbalance measure and the ambiguity measure. A heuristic algorithm to reduce the number of variables by linear transformation is presented. Experimental results using randomly generated functions and lists of English words are shown

EVMDD-based analysis and diagnosis methods of multi-state systems with multi-state components

A multi-state system with multi-state components is a model of systems, where performance, capacity, or reliability levels of the systems are represented as states. It usually has more than two states, and thus can be considered as a multi-valued function, called a structure function. Since many structure functions are monotone increasing, their multi-state systems can be represented compactly by edge-valued multi-valued decision diagrams (EVMDDs). This paper presents an analysis method of multi-state systems with multi-state components using EVMDDs. Experimental results show that, by using EVMDDs, structure functions can be represented more compactly than existing methods using ordinary MDDs. Further, EVMDDs yield comparable computation time for system analysis. This paper also proposes a new diagnosis method using EVMDDs, and shows that the proposed method can infer the most probable causes for system failures more efficiently than conventional methods based on Bayesian networks.Japan Society for the Promotion of ScienceMinistry of Education, Culture, Sports, Science and Technology (MEXT)Hiroshima City UniversityGrant-in Aid No. 2500050 (MEXT)Grant no. 0206 (HCU)Grant in Aid for Scientific Research (JSPS

GUEST EDITORIAL

The Reed-Muller Workshop has been held biennially since 1993, and since 2007 has been co-located with the IEEE International Symposium on Multiple-valued Logic and supported by the IEEE Computer Society Technical Committee on Multiple-valued Logic. Papers presented at the Workshop are provided informally to attendees but workshop proceedings are not formally published

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