D.M. Miller

The analysis of one-dimensional linear cellular automata and their aliasing properties

It is shown how to construct a general linear hybrid cellular automaton (CA) such that it has a maximum length cycle, and how the aliasing properties of such automata compare with linear feedback shift registers (LFSRs) when used as signature analyzers. The construction is accomplished by formally demonstrating the isomorphism which binds this kind of CA to the LFSRs. Consequently, these CAs can be analyzed as linear machines. Linear algebraic techniques are then applied appropriately for the transformations, and a useful search algorithm is developed which, given an irreducible characteristic polynomial, finds a corresponding linear hybrid automaton. Such CAs are tabulated for all irreducible and primitive polynomials up to degree 16, plus a selection of others of higher degree. The behavior of a linear hybrid CA and that of its corresponding LFSR are similar-that is, they have the same cycle structure and only relabel the states. The aliasing properties, when they are used as signature analyzers, remain unchanged. >

Toffoli network synthesis with templates

Reversible logic functions can be realized as networks of Toffoli gates. The synthesis of Toffoli networks can be divided into two steps. First, find a network that realizes the desired function. Second, transform the network such that it uses fewer gates, while realizing the same function. This paper addresses the above synthesis approach. We present a basic method and, based on that, a bidirectional synthesis algorithm which produces a network of Toffoli gates realizing a given reversible specification. An asymptotically optimal modification of the basic synthesis algorithm employing generalized mEXOR gates is also presented. Transformations are then applied using template matching. The basis for a template is a network of gates that realizes the identity function. If a sequence of gates in the synthesized network matches a sequence comprised of more than half the gates in a template, then a transformation using the remaining gates in the template can be applied resulting in a reduction in the gate count for the synthesized network. All templates with up to six gates are described in this paper. Experimental results including an exhaustive examination of all 3-variable reversible functions and a collection of benchmark problems are presented. The paper concludes with suggestions for further research.

Multiple-valued logic design tools

A brief overview of past progress in multiple-valued logic design is presented. The methods are considered with respect to the likely development of multiple-valued field programmable gate arrays. Look-up table based arrays are considered in some detail and an algorithm for mapping multiple-valued functions to such an array is presented. This algorithm uses reduced order multiple-valued decision diagrams, an extension of R.E. Bryants (1986) well-studied structure for binary functions. The algorithm works in the functional domain and does not require the synthesis and optimization of a conventional network prior to technology mapping.<<ETX>>

Implementing a multiple-valued decision diagram package

Decision diagrams are the state-of-the-art representation for logic functions, both binary and multiple-valued. Here we consider issues regarding the efficient implementation of a package for the creation and manipulation of multiple-valued decision diagrams (MDDs). In particular we identify issues that differ from binary decision diagram packages. We describe a matrix method for level interchange in MDDs that is essential for implementing variable reordering strategies. In addition, it is the basis for a novel approach to performing logic operations on MDDs, which we also present. Experimental results demonstrate the efficiency of this approach.

On the construction of multiple-valued decision diagrams

Decision diagrams are the state-of-the-art representation for logic functions, both binary and multiple-valued. We consider ways to improve the construction of multiple-valued decision diagrams (MDD). Efficiency is achieved through the use of a simple computed table. We compare the use of recursive MIN and MAX as primitive operations in multiple-valued decision diagram construction to the MV-CASE primitive which is a generalization of the if-then-else (ITE) commonly used in binary DD packages. We also consider the use of cyclic negations and complements as MDD edge operations showing that for certain types of functions this approach can lead to significant reduction in MDD node count. They can also reduce the number of primitives that need to be explicitly implemented. Experimental results showing the efficiency of the proposed approaches are given. The direct implementation of MDDs is briefly compared to representing MDDs using a BDD package.

A synthesis method for MVL reversible logic [multiple value logic]

An r-valued m-variable reversible logic function maps each of the r/sup m/ input patterns to a unique output pattern. The synthesis problem is to realize a reversible function by a cascade of primitive reversible gates. In this paper, we present a simple heuristic algorithm that exploits the bidirectional synthesis possibility inherent in the reversibility of the specification. The primitive reversible gates considered here are one possible extension of the well-known binary Toffoli gates. We present exhaustive results for the 9! 2-variable 3-valued reversible functions, comparing the results of our algorithm to optimal results found by breadth-first search. The approach can be applied to general m-variable, r-valued reversible specifications. Further, we show how the presented technique can be applied to irreversible specifications. The synthesis of a 3-input, 3-valued adder is given as a specific case.

An improved method for computing a generalized spectral coefficient

In 1995, Thornton and Nair studied the computation and application of a general class of spectral coefficients. A method for the computation of individual spectral coefficients using output probabilities computed on ROBDDs was presented. In this paper, that method is improved. Specifically, an alternative method for computing output probabilities is presented which, unlike the previous method, requires no recomputation for shared ROBDD representations and allows the use of edge negations. Based on this method, an alternative approach to computing the value of a spectral coefficient using output probabilities and ROBDDs is given. In contrast to the method described in 1995, one rather than two composition functions need to be considered, and no supplementary ROBDDs need to be constructed.

Augmented sifting of multiple-valued decision diagrams

Discrete functions are now commonly represented by binary (BDD) and multiple-valued (MDD) decision diagrams. Sifting is an effective heuristic technique which applies adjacent variable interchanges to find a good variable ordering to reduce the size of a BDD or MDD. Linear sifting is an extension of BDD sifting where XOR operations involving adjacent variable pairs augment adjacent variable interchange leading to further reduction in the node count. In this paper, we consider the extension of this approach to MDDs. In particular, we show that the XOR operation of linear sifting can be extended to a variety of operations. We term the resulting approach augmented sifting. Experimental results are presented showing sifting and augmented sifting can be quite effective in reducing the size of MDDs for certain types of functions.

Why cellular automata are better than LFSRs as built-in self-test generators for sequential-type faults

This paper presents a combinatorial method of evaluating the effectiveness of linear hybrid cellular automata (LHCA) and linear feedback shift registers (LFSR) as generators for stimulating faults requiring a pair of vectors. We provide a theoretical analysis and empirical comparisons to see why the LHCAs are better than the LFSRs as generators for sequential-type faults in a built-in self-test (BIST) environment. Based on the concept of a partner set, the method derives the number of distinct k-cell substate vectors which have 2/sup 2k/, 1/spl les/k/spl les/[n/2], transition capability for an n-cell LFSR and an n-cell LFSR with maximum length cycles. Simulation studies of the ISCAS85 benchmark circuits provide evidence of the effectiveness of the theoretical metric.<<ETX>>

On disjoint covers and ROBDD size

The relation between the number of nodes in a ROBDD and the number of implicants in the disjoint cover of the function represented by that ROBDD is studied. We identify a class of functions for which there are disjoint covers such that a cover of a larger size can be represented by a ROBDD with a smaller number of nodes. This shows that the size of a ROBDD is not a monotonically increasing function of the size of the disjoint cover.