Representation of Rotation Symmetric Multiple-Valued Functions Using Decision Diagrams

A recursive formula for weights of Boolean rotation symmetric functions

This special issue of the VLSI DESIGN: an International Journal of Custom-Chip Design, Simulation and Testing is devoted to spectral techniques, new types of decision trees and diagrams, their applications, efficient ways of their calculation and mutual relations between spectral techniques and classical decision diagrams. Manipulations and calculations with discrete functions are fundamental tasks in many areas of Computer Science and Engineering. Many problems in digital system design, simulation and testing can be expressed as a sequence of operations on discrete functions. The performance of Computer-Aided Design systems used in solving various problems in this area strongly depends on the efficiency of representations of discrete functions. Decision Diagrams (DDs) have proved very convenient data structure for discrete function representations, permitting manipulations and calculations with large functions efficiently in terms of time and space. In many applications, as for example those involving large matrices, conventional algorithms are significantly improved by using DDs. In logic design, such applications relate to basic problems of the design, verification, simulation and testing logical networks. Besides Boolean functions, some other types of discrete representations, such as multi-valued functions, cube sets, EXOR-based spectral and probabilistic formulas have to be dealt with. Hence the numbers of different decision trees and diagrams have been introduced for these various applications and representations. The spectral or polynomial representation of Boolean function is obtained by expressing it in terms of an orthogonal basis. The most popular basis is formed by discrete Walsh functions. Note that the spectral and the polynomial representation of a Boolean function are equivalent. The discrete orthogonal functions are referred to as the basis functions when the spectral or abstract harmonic terminology is used and as monomials when the polynomial terminology is used. Such polynomial representations have extensive applications in coding theory and cryptography as well as design of nonlinear filters. Spectral techniques based on Walsh, Haar, Arithmetic and Reed–Muller transforms have been used in digital logic design for more than 30 years. These techniques have been used for Boolean function classification, disjoint decomposition, identification of symmetries, parallel and serial linear decomposition, spectral translation synthesis (extraction of linear preand post-filters), multiplexer synthesis, threshold logic synthesis, state assignment and testing, and evaluation of logic complexity. The renewed interest in applications of spectral methods in design of VLSI digital circuits is caused by their excellent design for testability properties and in their efficiency in Boolean mapping problems. Furthermore, the practical applicability of these techniques was greatly improved with the development of efficient methods that allow calculating and operating on spectra of Boolean functions directly from reduced representations of such functions in the form of arrays of cubes or decision diagrams. Following from these recent developments, techniques have been presented for efficient spectral translation to decompose a circuit into a cascade of two sub circuits: linear block composed of EXOR gates fed by the primary input of the overall circuit. Such linear decomposition drastically simplifies the synthesis task. The choice of suitable linear transformation is based on a complexity measure assigned to each Boolean function that is heuristically related to the complexity of the final circuit implementation. To avoid the experimental costs of computing the Walsh transform, Binary Decision Diagrams (BDDs) based techniques are used. Recently the idea of linear transformations is applied to decision diagrams. By combining powerful spectral linear techniques with variable reordering techniques, it is possible to synthesize large Boolean functions with standard Computer-Aided Design tools that fail otherwise. The areas addressed by this special issue span spectral transforms, their applications and efficient calculation methods through decision diagrams. New spectral decision diagrams are also emerging that support larger data structures allowing more complex discrete representations to be implemented that was previously impossible.

Affine equivalence for cubic rotation symmetric Boolean functions with [formula omitted] variables

On the weight and nonlinearity of homogeneous rotation symmetric Boolean functions of degree 2

Bent functions are a class of discrete functions which exhibit the highest degree of nonlinearity. As such bent functions form an essential part of cryptographic systems. Original concept of bent functions defined in GF(2) can be extended to multiple-valued case. Multiple-valued bent functions are defined in therms of properties of their Vilenkin-Chrestenson spectra. Decision diagrams are a method of compact representation of discrete functions. Special types of decision diagrams have been introduced for various types of discrete functions. In this paper we demonstrate how Vilenkin-Chrestenson decision diagrams can be used for efficient representation of multiple-valued bent functions.

In this paper we give a spectral transform interpretation of AND-EXOR representations of switching functions and related decision diagrams in the vector space over GF(2). The consideration is uniformly extended to the Fourier series-like expressions of functions in the complex vector space and the decision diagrams for integer-valued functions. It is shown that the multi-terminal decision diagrams, MTBDDs, and edge-valued decision diagrams, EVBDDs, for integer-valued functions are derived by using the same sets of basic functions already applied for the decision diagrams attached to some AND-EXOR expressions, but considered over the complex field. The algebraic transform decision diagrams, ATDDs, are considered as the integer counterparts of the functional decision diagrams, FDDs, attached to the algebraic transform in the same way as the FDDs are attached to the Reed-Muller expressions. It is shown that the EVBDDs are the ATDDs in different notation.

Loading classical data into quantum registers is one of the most important primitives of quantum computing. While the complexity of preparing a generic quantum state is exponential in the number of qubits, in many practical tasks the state to prepare has a certain structure that allows for faster preparation. In this paper, we consider quantum states that can be efficiently represented by (reduced) decision diagrams, a versatile data structure for the representation and analysis of Boolean functions. We design an algorithm that utilizes the structure of decision diagrams to prepare their associated quantum states. Our algorithm has a circuit complexity that is linear in the number of paths in the decision diagram. Numerical experiments show that our algorithm reduces the circuit complexity by up to 31.85% compared to the state-of-the-art algorithm, when preparing generic $n$-qubit states with ${n}^{3}$ nonzero amplitudes. Additionally, for states with sparse decision diagrams, including the initial state of the quantum Byzantine agreement protocol, our algorithm reduces the number of controlled-nots by 86.61--99.9%.

Minimization of Haar wavelet series and Haar spectral decision diagrams for discrete functions

Logic functions are usually represented by logical expressions or decision diagrams using AND and OR operators. However, some functions have more compact representations with EXOR operators. This chapter surveys representations of logic functions using EXOR operators.

Ordered Kronecker functional decision diagrams (OKFDD's) are a data structure for efficient representation and manipulation of Boolean functions. OKFDD's are a generalization of ordered binary decision diagrams (OBDD)s) and ordered functional decision diagrams and thus combine the advantages of both. In this paper, basic properties of OKFDD's and their efficient representation and manipulation are given. Starting with elementary manipulation algorithms, we present methods for the construction of small OKFDD's. Our approach is based on dynamic variable ordering and decomposition-type choice. For changing the decomposition type, we use an efficient reordering-based method. We briefly discuss the implementation of PUMA, an OKFDD package, which was used in all our experiments. These experiments demonstrate the quality of our methods in comparison to sifting and interleaving for OBDD's.

This paper presents a design method for multiple-output networks using time domain multiplexing (TDM) and shared multi-terminal multiple-valued decision diagrams (SMTMDDs). SMTMDDs represent multiple-valued multiple-output functions, while TDM systems transmit several signals on a single line. This method reduces: 1) hardware; 2) logic levels; and 3) pins. In the logic design, we use two types of decision diagrams (DDs): shared binary decision diagrams (SBDDs) and SMTMDDs. We propose heuristic algorithms to derive shared multiterminal binary decision diagrams (SMTBDDs) from SBDDs, and SMTMDDs from SMTBDDs. Experimental results show the compactness of SMTMDDs over SBDDs in terms of number of non-terminal nodes, where the nodes for output selection variables are not included in the non-terminal nodes. In the network, each non-terminal node of an SBDD and an SMTMDD is realized by a multiplexer (MUX). We also present upper bounds on the number of nonterminal nodes to realize an n-input m-output function by an SBDD and an SMTMDD. In addition, we compare the proposed TDM realization with the conventional one.

The Sentential Decision Diagram (SDD) is a compact and canonical representation of Boolean functions that generalizes the Ordered Binary Decision Diagrams (OBDDs). A variant of SDDs, namely Zero-suppressed Sentential Decision Diagrams (ZSDDs), was proposed recently by using different trimming rules. SDDs are suitable for functions where adjacent input assignments have the same outcome, while ZSDDs are more compact for spare functions. In this paper, we introduce a novel canonical SDD variant, called the Tagged Sentential Decision Diagrams (TSDDs). The key insight of TSDDs is to combine both trimming rules of SDDs and ZSDDs. With both characteristics of SDDs and ZSDDs, the TSDD representation is at least as small as the SDD or ZSDD representation for any Boolean functions. This is also shown in our experimental evaluation.

Symbolic model checking tries to reduce the state explosion problem by implicit construction of the state space. The major limiting factor is the size of the symbolic representation mostly stored in huge binary decision diagrams. A new approach to symbolic model checking of Petri nets and related models of computation is presented, outperforming the conventional one and avoiding some of its drawbacks. Our approach is based on a novel, efficient form of representation for multi-valued functions called interval decision diagram (IDD) and the corresponding image computation technique using interval mapping diagrams (IMDs). IDDs and IMDs are introduced, their properties are described, and the feasibility of the new approach is shown with some experimental results.

Implicit techniques for representing and generating the reachability set of a high-level model have become quite efficient. However, such techniques are usually restricted to models whose events have equal priority. Models containing events with differing classes of priority or complex priority structure, in particular models with immediate events, have thus been required to use explicit reachability set generation techniques. In this paper, we present an efficient implicit technique, based on multi-valued decision diagram representations for sets of states and matrix diagram representations for next-state functions, that can handle models with complex priority structure. If the model contains immediate events, the vanishing states can be eliminated either during generation, by manipulating the matrix diagram, or after generation, by manipulating the multi-valued decision diagram. We apply both techniques to several models and give detailed results.

In this paper, a novel spectral technique based on the transform constructed from a Complex Hadamard matrix is introduced. The technique finds potential applications in various synthesis and optimization processes of logic circuits. Classification of switching function is shown here as one of the possible applications. The technique uses the Complex Hadamard matrix as the spectral transform, manipulates the complex spectral coefficients of the functions and classifies them into the NPN-equivalent and linearly separable functions. The method has proven to be a more efficient classification scheme than the existing classification method using the standard spectral technique based on the Walsh Transform. For efficient manipulation of switching functions, some new decision diagrams to represent Complex Hadamard matrices and their spectra of integervalued functions are proposed. With the intrinsic properties of the new transform in handling complex numbers, the presented decision diagrams lead to a more efficient and compact representation of both switching and complex functions.