Trinity algebra and its application ot machine decompositions

General Decomposition and Its Use in Digital Circuit Synthesis

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.

An efficient for the sequential general decomposition of sequential machines

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.

The bit full-decomposition of sequential machines

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.

Simultaneous decompositions of sequential machines

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.

The full-decomposition of sequential machines with the separate realization of the next-state and output functions

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.

Algebraic automata theory

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

Electronic Logic Systems

Abstract: Part 1 Principles of logic systems: combinational logic logic and memory devices combinational logic at different levels of integration synchronous sequential circuits asynchronous sequential circuits arithmetic logic circuits. Part 2 Advanced logic systems: combinational logic techniques partitioning of sequential circuits partition-based design for synchronous sequential circuits partition-based design for asynchronous sequential circuits hybrid design techniques for sequential circuits CAD of logic circuits.

Trinity algebra and full-decompositions of sequential machines

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