Fixed Polarity Reed-Muller Research Articles

Algorithm to generate fixed polarity Reed-Muller GF(5) spectra from disjoint cubes

A theory has been developed to calculate fixed polarity Reed-Muller spectra over GF(5) from disjoint cubes representation of 5-valued functions. The algorithm makes use of the properties of disjoint cubes and allows the determination of the spectral coefficients in an independent way. It is simple and can be implemented with small amount of storage space. The algorithm is interactive so the users can manually determine the generation of a part of a spectrum. At the end of the article, experimental results of the algorithm for several 5-valued test files are shown.

Power minimization of FPRM functions based on polarity conversion

For an n-variable Boolean function, there are 2n fixed polarity Reed-Muller (FPRM) forms. In this paper, a frame of power dissipation estimation for FPRM functions is presented and the polarity conversion is introduced to minimize the power for FPRM functions. Based on searching the best polarity for low power dissipation, an optimal algorithm is proposed and implemented in C. The algorithm is tested on seven single output functions from MCNC benchmark circuits. The experimental results are shown in this paper.

Exact minimisation of large multiple output FPRM functions

The properties of the polarity for sum-of-products (SOP) expressions of Boolean functions are formally investigated. A transform matrix S is developed to convert SOP expressions from one polarity to another polarity. It is shown that the effect of SOP polarity is to reorder the on-set minterms of a Boolean function. Furthermore, the transform matrix P for fixed polarity Reed-Muller (FPRM) expressions for the conversion between two different polarities, based on the properties of SOP polarity, is achieved. Comparison of these two matrices shows that the Reed-Muller transform matrix P has a much more complex structure. Additionally, the best polarity of FPRM forms with the least on-set terms corresponds with the polarity of SOP forms with the best ‘order’ of the on-set minterms. Applying these algebraic properties of the transform matrix P, a fast algorithm is presented to obtain the best polarity of FPRM expressions for large multiple output Boolean functions. The computation time is independent of the number of outputs. The developed program is tested on common personal computers and the results for benchmark examples of up to 25 inputs and 29 outputs are presented.

Reed--Muller tree-based minimisation of fixed polarity Reed–Muller expansions

A heuristic method for the determination of optimum or near-optimum fixed polarity Reed-Muller (FPRM) representation of multiple output, completely specified Boolean systems is presented. The Reed-Muller (RM) tree representation forms the conceptual framework for the method, which involves manipulations of arrays of cubes. A coding method that is well adapted to RM tree representation is presented. The minimisation method takes as input a disjoint sum of cubes representation of the Boolean system. Using Karpovsky's complexity estimates as the basis for polarity selection, the method obtains the FPRM expansion by generating in one run an optimum or near optimum Reed-Muller tree representation.

Efficient algorithm to calculate Reed–Muller expansions over GF(4)

A new algorithm to generate the full polarity matrix of fixed polarity Reed-Muller expansions over Galois fields of order 4, GF(4), has been developed. By using directly the truth vector of the original function, a recursive formula is developed to generate the whole polarity matrix. The algorithm uses the properties of the fixed polarity matrix to speed up the calculation and reduce the number of necessary multipliers and adders. The computational complexity of the algorithm is compared with other works. It is shown that, for practical hardware implementations of quaternary functions, the new algorithm is better than all other existing algorithms. The fast flow diagrams for computation of the whole or partial matrix are also presented.

Comments on "Sympathy: fast exact minimization of fixed polarity Reed-Muller expansion for symmetric functions"

The above paper finds an optimal fixed-polarity Reed-Muller expansion of an n-variable totally symmetric function using an OFDD-based algorithm that requires O(n/sup 7/) time and O(n/sup 6/) storage space. However, an algorithm based on Suprun's transient triangles requires only O(n/sup 3/) time and O(n/sup 2/) storage space. An implementation of this algorithm yields computation times lower by several orders of magnitude.

Spectral interpretation of the fast tabular technique for fixed-polarity Reed-Muller expressions

This paper presents a uniform spectral approach to the fast tabular technique for generating fixed polarity Reed–Muller expressions. Basic operations in the tabular technique are described through the discrete dyadic convolution. The presented results can be extended to various polynomial expansions of discrete functions.

Fast conversion algorithm for very large Boolean functions

Fixed polarity Reed-Muller (FPRM) expressions are alternatives to the traditional sum-of-products forms (SOPs) for the representation of Boolean functions. A fast algorithm is proposed to convert from SOPs to FPRM forms without generating disjoint cube covers or functional decision diagrams (FDDs). This procedure is based on the property of input redundancy and is tailored for very large multiple output Boolean functions. Test results for benchmark examples of up to 199 inputs and 99 outputs are given.

Generalised k-variable-mixed-polarity Reed–Muller expansions for system of Boolean functions and their minimisation

A lookup table based method to minimise generalised partially-mixed-polarity Reed–Muller (GPMPRM) expansions with k mixed polarity variables is presented. The developed algorithm can produce solutions based on the desired cost criteria for the systems of completely specified functions. A heuristic approach based on the exclusion rule is adopted to extract the best dual polarity variables from any fixed polarity Reed–Muller (FPRM) expansion. The obtained experimental results compared favourably with the recently published results and outperform those generated by the exact minimal FPRM expansion minimisers.

Boolean Matching Filters Based on Row and Column Weights of Reed–Muller Polarity Coefficient Matrix

In this article, we have shown, by means of the EXOR Ternary Decision Diagram that the number of literals and product terms of the Fixed Polarity Reed–Muller (FPRM) expansions can be used to fully classify all Boolean functions in NP equivalent class and NPN equivalent class, respectively. Efficient graph based algorithms to compute the complete weight vectors have been presented. The proof and computation method has led to the derivation of a set of characteristic signatures that has low probability of aliasing when used as the Boolean matching filters in library mapping.

Genetic algorithm for minimisation of fixed polarity Reed-Muller expressions

A genetic algorithm (GA) is developed to find small or minimal fixed polarity Reed-Muller expressions (FPRMs) for large functions. The authors combine the GA with greedy heuristics, i.e. They use hybrid GAs (HGAs). They show by experiments that results superior to all other approaches for large functions can be obtained using GAs. This mainly results from the use of a problem specific data structure and the incorporation of heuristics in the GA.

Mapping of on-set fixed polarity Reed-Muller coefficients from on-set canonical sum of products coefficients and the minimization of pseudo Reed-Muller expressions

In this paper, a more efficient tabular method for mapping on-set fixed polarity Reed–Muller coefficients for a given polarity vector from on-set canonical sum of products coefficients is developed. Using this mapping technique, a heuristic algorithm for generating optimal pseudo Reed–Muller expressions from canonical sum of products expressions is also developed. Illustrative examples and experimental results are included to show the performance of the developed algorithms.

Efficient polarity conversion for large Boolean functions

The concept of polarity for canonical sum-of-products (SOP) Boolean functions is introduced. This facilitates efficient conversion between SOP and fixed polarity Reed-Muller (FPRM) forms. New algorithms are presented for the bidirectional conversion between the two paradigms. Multiple segment and multiple pointer techniques are employed to achieve fast conversion for large Boolean functions. Experimental results are given using a personal computer with a Cyrix 6x86-166 CPU and 32MB RAM. The results show that the algorithm is very efficient in terms of time and space for large Boolean functions.

Testability of 2-Level AND/EXOR Circuits

It is often stated that AND/EXOR circuits are much easier to test than AND/OR circuits. This statement, however, only holds true for circuits derived from restricted classes of AND/EXOR expressions, like positive polarity Reed-Muller and fixed polarity Reed-Muller expressions. For these two classes of expressions, circuits with good deterministic testability properties are known. In this paper we show that these circuits also have good random pattern testability attributes. An input probability distribution is given that yields a short expected test length for biased random patterns. This is the first time theoretical results on random pattern testability are presented for 2-level AND/EXOR circuit realizations of arbitrary Boolean functions. It turns out that analogous results cannot be expected for less restricted classes of 2-level AND/EXOR circuits. We present experiments demonstrating that generally minimized 2-level AND/OR circuits can be tested as easy (or hard) as minimized 2-level AND/EXOR circuits.

Fast tabular technique for fixed-polarity Reed-Muller logic with inherent parallel processes

We have developed a fast tabular technique for Boolean to fixed-polarity Reed–Muller logic conversion. The idea is derived from the popular tabular technique but the processes of generating new terms and cancelling pairs are vastly different. The feature of bit-by-bit operation in the tabular technique has also been changed. In our technique, an index table is used to capture the number of occurrences of initial and new terms. This instant updating process eliminates the time-consuming step of cancelling pairs and makes the method much more efficient than the tabular technique. In addition, the fast tabular technique possesses inherent parallelism since different minterms can be processed concurrently and added to the index table independently. Our method is applicable to any number of variables and is suitable for manual computations as well as computer programming. Some comparison results are given, and we believe our technique has achieved the best time and space complexities so far.

Mapping of fixed polarity Reed-Muller coefficients from minterms and the minimization of fixed polarity Reed-Muller expressions

In this paper, a relationship between on-set minterms of a logic function, an arbitrary polarity vector of the variables, and the corresponding fixed polarity Reed-Muller (FPRM) coefficients is established. Using this relationship, an efficient and simple algorithm for mapping FPRM coefficients from the on-set minterms of the function for a given polarity vector is presented. Another heuristic algorithm for finding an optimal polarity vector from the on-set minterms that produces the near minimum fixed polarity Reed-Muller expression (FPRME) is also presented. Both these algorithms are developed for single-output fully specified logic functions.

Boolean functions classification via fixed polarity Reed-Muller forms

In this paper, we present a new method to characterize completely specified Boolean functions. The central theme of the classification is the functional equivalence (a.k.a. Boolean matching). Two Boolean functions are equivalent if there exists input permutation, input negation, or output negation that can transform one function to the other. We have derived a method that can efficiently identify equivalence classes of Boolean functions. The well-known canonical Fixed Polarity Reed-Muller (FPRM) forms are used as a powerful analysis tool. The necessary transformations to derive one function from the other are inherent in the FPRM representations. To identify uniquely each equivalence class, a set of well-known characteristics of Boolean functions and their variables (including linearity, symmetry, total symmetry, self-complement, and self-duality) are employed. It is shown that all the equivalence classes of four-variable functions are uniquely identified where majority of the classes have a single FPRM form as their representative. The Boolean matching has applications in technology mapping and in design of standard cell libraries.

Sympathy: fast exact minimization of fixed polarity Reed-Muller expressions for symmetric functions

In this paper, a polynomial time algorithm for the minimization of fixed polarity Reed-Muller expressions (FPRMs) for totally symmetric functions based on ordered functional decision diagrams (OFDDs) is presented. A generalization to partially symmetric functions is investigated. The algorithm has been implemented as the program Sympathy. Experimental results in comparison to previously published methods are given to show the efficiency of the approach.

Relation between OFDDs and FPRMs

Ordered functional decision diagrams (OFDDs) are often used as a data structure for representing fixed polarity Reed-Muller expressions (FPRMs). The authors show that OFDDs are never much larger in size than FPRMs. In contrast there exists Boolean functions for which FPRMS always have exponential size, while the OFDD representation remains small. In this sense the authors show that OFDDs are a good data structure for FPRM representation.

Fast OFDD-based minimization of fixed polarity Reed-Muller expressions

We present methods to minimize fixed polarity Reed-Muller expressions (FPRMs), i.e., two-level fixed polarity AND/EXOR canonical representations of Boolean functions, using ordered functional decision diagrams (OFDDs). We investigate the close relation between both representations and use efficient algorithms on OFDDs for exact and heuristic minimization of FPRMs. In contrast to previously published methods, our algorithm can also handle circuits with several outputs. Experimental results on large benchmarks are given to show the efficiency of our approach.