The bit full-decomposition of sequential machines

Abstract: Modem microelectronic technology.gives opportunities to build digital circuits of huge complexity and provides a wide diversity of logic building blocks. Although logic designers have been building circuits for many years, they have realized that advances in microelectronic technology are outstripping their abilities to make use of the created opportunities. In this paper, we present the fundamentals of a logic design methodology which meets the requirements of today's complex circuits and modem building blocks. The methodology is based on the theory of general full-decompositions which constitutes the theory of digital circuit structures at the highest abstraction level. The paper explains the theory and shows how it can be used for digital circuit synthesis. The decomposition methodology that is presented ensures “correctness by construction” and enables very effective and efficient post-factum validation. It makes possible extensive examination of the structural features of the required information processing in relation to a given set of objectives and constraints.

Abstract: Sequential machines which derine control and serial processing units of modern digital systems are large and complex and, therefore, difficult to design, implement, optimize and verify. So, methods and CAD-tools that can decompose complex machines have attracted a great deal of interest recently. In this paper, a heuristic method is presented for suboptimal multiple-objective sequential general decomposition of sequential machines into submachines with limited input/output bits, product terms and state variables. The experimental results obtained from the prototypic implementation of the method show that the method is efficient. It produces high quality decompositions using relatively small memory and in an appropriately short time. The method is flexible and after some modifications can be applied to other decomposition problems.

Abstract: During the last decade, many different approaches have been proposed to solve the multiple-level synthesis problem with different minimum functionally complete systems of primitive logic blocks. The most popular of them is the division-based approach. However, modem microelectronic technology provides a large variety of building blocks which considerably differ from those typically considered. The traditional methods are therefore not suitable for synthesis with many modem building blocks. Furthermore, they often fail to find global optima for complex designs and leave unconsidered some important design aspects. Some of their weaknesses can be eliminated without leaving the paradigm they are based on, other ones are more fundamental. A paradigm which enables efficient exploitation of the opportunities created by the microelectronic technology is the general decomposition paradigm. The aim of this paper is to analyze and compare the general decomposition approach and the division-based approach. The most important advantages of the general decomposition approach are its generality (any network of any building blocks can be considered) and totality (all important design aspects can be considered) as well as handling the incompletely specified functions in a natural way. In many cases, the general decomposition approach gives much better results than the traditional approaches.

Abstract: Large sequential machines are difficult to design, to optimize, to implement and to verify. Therefore, methods and CAD tools are needed that can decompose sequential machines. In this paper, we briefly describe the theoretical and practical results that were obtained in the field of simultaneous decompositions which divide the process described by a given sequential machine into a number of interacting parallel partial processes, each implemented by one partial machine.

Abstract: The decomposition theory of sequential machines aims to find answers to the following important practical problem: how to decompose a complex sequential machine into a number of simpler partial machines in order to: simplify the design, implementation and verification process; make it possible to process (to optimize, to implement, to test, ••. ) the separate partial machines al though it may be impossible to process the whole machine with existing tools; make it possible to implement the machine with existing building blocks or inside of a limited silicon area. For many years, decomposition of the internal states of sequential machines has been investigated. Here, decomposition of the states, as well as, the inputs and outputs of sequential machines is considered, i.e. full-decomposition. In [16], classification of full-decompositions is presented and theorems about the existence of different full-decompositions are provided. In this report a special full-decomposition strategy is investigated the full-decomposition of sequential machines with the separate realization of the next-state and output functions. This strategy has several advantages comparing to the case where a sequential machine is considered as a unit. In the report, the results of theoretical investigations are presented; however, the notions and theorems provided here have straightforward practical interpretations and they can be directly used in order to develope programs computing different sorts of decompositions for sequential machines. INDEX TERMS Automata theory, decomposition, logic system design, sequential machines. ACKNOWLEDGEMENTS The author is indebted to Prof. ir. A. Heetman and Prof. ir. M. P.J. Stevens for making it possible to perform this work, to Dr. P.R. Attwood for making corrections to the English text and to mr. C. van de Watering for typing the text.

Abstract: Introduction 1 Semigroups and their relatives 2 Machines and semigroups 3 Decompositions 4 The holonomy decomposition 5 Recognizers 6 Sequential machines and functions Appendix References Index of notation Index

Abstract: Much scientific work today is directed towards understanding complexity — the complexity of numerical algorithms, of the English syntax, of living organisms or ecological systems, to cite only a few examples. The aim of this chapter is to introduce the reader to the theory of discrete information processing systems (automata) and to develop an algebraic framework within which we can talk about their complexity.

Abstract: The object of this paper is to study the realization of a sequential machine from several smaller machines. The basic tools in this investigation are the partitions with the substitution property and the partition pairs. It is shown that to every (loop-free) realization of a sequential machine from n smaller machines corresponds a set of n partitions with the substitution property whose product is the zero partition. Conversely, it is shown that to every such set of n partitions corresponds a realization of the given sequential machine from n smaller machines. The natural ordering of these partitions is reflected in the information flow between the corresponding component machines and the algebraic operations defined between these partitions corresponding to the realization, govern the modifications of this realization. Finally, it is shown how the amount of information flowing between the component machines in a realization can be studied by means of partition pairs.