Spectral Methods for Boolean and Multiple-Valued Input Logic Functions

Spectral techniques in digital logic design have been known for more than thirty years. They have been used for Boolean function classification, disjoint decomposition, parallel and serial linear decomposition, spectral translation synthesis (extraction of linear pre- and post-filters), multiplexer synthesis, prime implicant extraction by spectral summation, threshold logic synthesis, estimation of logic complexity, testing, and state assignment. This dissertation resolves many important issues concerning the efficient application of spectral methods used in the computer-aided design of digital circuits. The main obstacles in these applications were, up to now, memory requirements for computer systems and lack of the possibility of calculating spectra directly from Boolean equations. By using the algorithms presented here these obstacles have been overcome. Moreover, the methods presented in this dissertation can be regarded as representatives of a whole family of methods and the approach presented can be easily adapted to other orthogonal transforms used in digital logic design. Algorithms are shown for Adding, Arithmetic, and Reed-Muller transforms. However, the main focus of this dissertation is on the efficient computer calculation of Rademacher-Walsh spectra of Boolean functions, since this particular ordering of Walsh transforms is most frequently used in digital logic design. A theory has been developed to calculate the Rademacher-Walsh transform from a cube array specification of incompletely specified Boolean functions. The importance of representing Boolean functions as arrays of disjoint ON- and DC- cubes has been pointed out, and an efficient new algorithm to generate disjoint cubes from non-disjoint ones has been designed. The transform algorithm makes use of the properties of an array of disjoint cubes and allows the determination of the spectral coefficients in an independent way. By such an approach each spectral coefficient can be calculated separately or all the coefficients can be calculated in parallel. These advantages are absent in the existing methods. The possibility of calculating only some coefficients is very important since there are many spectral methods in digital logic design for which the values of only a few selected coefficients are needed. Most of the current methods used in the spectral domain deal only with completely specified Boolean functions. On the other hand, all of the algorithms introduced here are valid, not only for completely specified Boolean functions, but for functions with don\u27t cares. Don\u27t care minterms are simply represented in the form of disjoint cubes. The links between spectral and classical methods used for designing digital circuits are described. The real meaning of spectral coefficients from Walsh and other orthogonal spectra in classical logic terms is shown. The relations presented here can be used for the calculation of different transforms. The methods are based on direct manipulations on Karnaugh maps. The conversion start with Karnaugh maps and generate the spectral coefficients. The spectral representation of multiple-valued input binary functions is proposed here for the first time. Such a representation is composed of a vector of Walsh transforms each vector is defined for one pair of the input variables of the function. The new representation has the advantage of being real-valued, thus having an easy interpretation. Since two types of codings of values of binary functions are used, two different spectra are introduced. The meaning of each spectral coefficient in classical logic terms is discussed. The mathematical relationships between the number of true, false, and don\u27t care minterms and spectral coefficients are stated. These relationships can be used to calculate the spectral coefficients directly from the graphical representations of binary functions. Similarly to the spectral methods in classical logic design, the new spectral representation of binary functions can find applications in many problems of analysis, synthesis, and testing of circuits described by such functions. A new algorithm is shown that converts the disjoint cube representation of Boolean functions into fixed-polarity Generalized Reed-Muller Expansions (GRME). Since the known fast algorithm that generates the GRME, based on the factorization of the Reed-Muller transform matrix, always starts from the truth table (minterms) of a Boolean function, then the described method has advantages due to a smaller required computer memory. Moreover, for Boolean functions, described by only a few disjoint cubes, the method is much more efficient than the fast algorithm. By investigating a family of elementary second order matrices, new transforms of real vectors are introduced. When used for Boolean function transformations, these transforms are one-to-one mappings in a binary or ternary vector space. The concept of different polarities of the Arithmetic and Adding transforms has been introduced. New operations on matrices: horizontal, vertical, and vertical-horizontal joints (concatenations) are introduced. All previously known transforms, and those introduced in this dissertation can be characterized by two features: ordering and polarity . When a transform exists for all possible polarities then it is said to be generalized . For all of the transforms discussed, procedures are given for generalizing and defining for different orderings. The meaning of each spectral coefficient for a given transform is also presented in terms of standard logic gates. There exist six commonly used orderings of Walsh transforms: Hadamard, Rademacher, Kaczmarz, Paley, Cal-Sal, and X. By investigating the ways in which these known orderings are generated the author noticed that the same operations can be used to create some new orderings. The generation of two new Walsh transforms in Gray code orderings, from the straight binary code is shown. A recursive algorithm for the Gray code ordered Walsh transform is based on the new operator introduced in this presentation under the name of the bi-symmetrical pseudo Kronecker product . The recursive algorithm is the basis for the flow diagram of a constant geometry fast Walsh transform in Gray code ordering. The algorithm is fast (N 10g2N additions/subtractions), computer efficient, and is implemente

A STUDY OF BINARY DECISION DIAGRAM CHARACTERISTICS OF BENT BOOLEAN FUNCTIONS

Bent Boolean functions exist only for an even number of variables, moreover, they are unbalanced. Therefore, they are used in coding theory and in many areas of computer science. General form of bent functions is still unknown. One way of representing Boolean functions is with a reduced ordered binary decision diagram (ROBDD). The strength of ROBDDs is that they can represent Boolean functions data with a high level of redundancy in a compact form, as long as the data is encoded in such a way that the redundancy is exposed. This paper investigates characteristics of bent functions with focus on their ROBDD parameters. Decision diagram experimental framework has been used for implementation of a program for calculation of the ROBDD parameters. The results presented in this paper are intended to be used to create methods for the construction of bent functions using a ROBDD as a data structure from which the bent functions can be discovered

Formal Methods in Quantum Circuit Design

The design and compilation of correct, efficient quantum circuits is integral to the future operation of quantum computers. This thesis makes contributions to the problems of optimizing and verifying quantum circuits, with an emphasis on the development of formal models for such purposes. We also present software implementations of these methods, which together form a full stack of tools for the design of optimized, formally verified quantum oracles. On the optimization side, we study methods for the optimization of Rz and CNOT gates in Clifford+Rz circuits. We develop a general, efficient optimization algorithm called phase folding, which reduces the number of Rz gates without increasing any metrics by computing its phase polynomial. This algorithm can further be combined with synthesis techniques for CNOT-dihedral operators to optimize circuits with respect to particular costs. We then study the optimal synthesis problem for CNOT-dihedral operators from the perspectives of Rz and CNOT gate optimization. In the case of Rz gate optimization, we show that the optimal synthesis problem is polynomial-time equivalent to minimum-distance decoding in certain Reed-Muller codes. For the CNOT optimization problem, we show that the optimal synthesis problem is at least as hard as a combinatorial problem related to Gray codes. In both cases, we develop heuristics for the optimal synthesis problem, which together with phase folding reduces T counts by 42% and CNOT counts by 22% across a suite of real-world benchmarks. From the perspective of formal verification, we make two contributions. The first is the development of a formal model of quantum circuits with ancillary bits based on the Feynman path integral, along with a concrete verification algorithm. The path integral model, with some syntactic sugar, further doubles as a natural specification language for quantum computations. Our experiments show some practical circuits with up to hundreds of qubits can be efficiently verified. Our second contribution is a formally verified, optimizing compiler for reversible circuits. The compiler compiles a classical, irreversible language to reversible circuits, with a formal, machine-checked proof of correctness written in the proof assistant F*. The compiler is structured as a partial evaluator, allowing verification to be carried out significantly faster than previous results

The 1991 3rd NASA Symposium on VLSI Design

Papers from the symposium are presented from the following sessions: (1) featured presentations 1; (2) very large scale integration (VLSI) circuit design; (3) VLSI architecture 1; (4) featured presentations 2; (5) neural networks; (6) VLSI architectures 2; (7) featured presentations 3; (8) verification 1; (9) analog design; (10) verification 2; (11) design innovations 1; (12) asynchronous design; and (13) design innovations 2

[Research Pertaining to Physics, Space Sciences, Computer Systems, Information Processing, and Control Systems]

Research project reports pertaining to physics, space sciences, computer systems, information processing, and control system