The bit full-decomposition of sequential machines
01 Jan 1989-
TL;DR: It is proved, that the bit full- decomposition can be treated as a special case of the symbol full-decomposition; therefore, no new decomposition theory is needed for this case, but the symbolFulldecompositions theory together with the theorems introduced here constitute the theory of bitFullDecomposition.
Abstract: Control units and serial processing units of today's information processing systems must realize complex processes, which are usually described in the form of a sequential machine or a number of cooperating sequential machines. Large machines are difficult to: design, optimize, implement and verify. Therefore, there is a real need for CAD tools, which could decompose a complex sequential machine into a number of smaller and less complicated partial machines. For many years, the decomposition of only the internal states of sequential machines has been studied. However, this sort of decomposition is not a sUfficient solution. The complexity of a circuit implementing a sequential machine is a function not only of machine's internal states but as well of inputs and outputs. Furthermore, the possibility to implement a machine with today's array logic building blocks depends not only on the number of internal states but as well on inputs and outputs. So, there is a real need for decompositions upon the states, inputs and outputs of a sequential machine, i.e. for fulldecompositions. During the full-decomposition process, the input and/or state and/or output symbols (values) can be decomposed or the input and/or state and/or output bits. So, it is possible to perform the symbol fulldecomposition or the bit full-decomposition. This report provides the classification of full-decompositions and describes briefly the theoretical foundations of bit fulldecomposition. Comparing to the symbol full-decomposition,the bit fulldecomposition has the following advantage: input and output decoders are reduced to an appropriate distribution of the primary input and output bits between the partial machines. In the report, definitions of a bit partition and bit partition pairs are introduced and their usefulness to bit full-decompositions is shown. It is proved, that the bit full-decomposition can be treated as a special case of the symbol full-decomposition; therefore, no new decomposition theory is needed for this case, but the symbol fulldecomposition theory together with the theorems introduced here constitute the theory of bit full-decomposition. Finally, a comparison is made between the symbol and the bit fulldecompositions and some practical conclusions and remarks are presented. In the appendix, an example is provided that illustrates the possibility and the practical usefulness of bit full-decomposition. Based on the developed theory, the CAD algorithms calculating different bit full-decompositions have been developed and implemented. Those algorithms and the practical results are presented and estimated in the separate paper [5]. Index Terms Automata theory, decomposition, logic 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.
Citations
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TL;DR: The fundamentals of a logic design methodology which meets the requirements of today's complex circuits and modem building blocks are presented and the decomposition methodology that is presented ensures “correctness by construction” and enables very effective and efficient post-factum validation.
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.
46 citations
Additional excerpts
...A general composition GC of n sequential machines defines the general composition machine M6c(GC) M6c({Mi }, { Coni }) (IGc, SGC, OGC, GC, k6C) with I6c Ii, SGC Si, OGC Oi, GC (SGc’XGc) GC ((S1 Sn), (X1 Xn)) i(Si, (Xi’ Cni(Yl Yn)), hGC(SGc, XGC) hGc((S Sn), (X1 Xn)) hi(si, (Xi’ Cni(Yl Yn))" Formal definitions for compositions TC of various special types T can be introduced in a very similar way, as special cases of the above definition [22]-[25]....
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...9); bitfull-decompositions [25]-[27], where the decod-...
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TL;DR: 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 and shows that it is efficient and flexible.
22 citations
TL;DR: The aim of this paper is to analyze and compare the general decomposition approach and the division-based approach to solve the multiple-level synthesis problem with different minimum functionally complete systems of primitive logic blocks.
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.
19 citations
TL;DR: 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 are described.
18 citations
01 Jan 1989
TL;DR: A special full-decomposition strategy is investigated, which has several advantages comparing to the case where a sequential machine is considered as a unit, and can be directly used in order to develope programs computing different sorts of decompositions for sequential machines.
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.
17 citations
References
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Book•
01 Jan 1966
571 citations
"The bit full-decomposition of seque..." refers methods in this paper
...The concepts of partitions and partition pairs introduced by Hartmanis [9][10][11][12] are useful tools for analyzing the information flow in and between machines; therefore, they were used in this work....
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Book•
01 Jan 1982
TL;DR: In this paper, the holonomy decomposition is used to decompose semigroups and their relatives into machines and semiggroups, and their relation to semigroup decompositions.
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
174 citations
01 Jan 1980
TL;DR: 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 to talk about their complexity.
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.
124 citations
TL;DR: It is shown how the amount of information flowing between the component machines in a realization can be studied by means of partition pairs.
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.
73 citations
"The bit full-decomposition of seque..." refers methods in this paper
...The concepts of partitions and partition pairs introduced by Hartmanis [9][10][11][12] are useful tools for analyzing the information flow in and between machines; therefore, they were used in this work....
[...]
56 citations
"The bit full-decomposition of seque..." refers methods in this paper
...The concepts of partitions and partition pairs introduced by Hartmanis [9][10][11][12] are useful tools for analyzing the information flow in and between machines; therefore, they were used in this work....
[...]