In this Chapter, we imagine a decision maker (or a group of experts) trying to establish or examine fair procedures to combine opinions about alternatives related to different points of view.

Multiple criteria decision making

A digital combinational logic circuit is reversible if it maps each input pattern to a unique output pattern. Such circuits are of interest in quantum computing, optical computing, nanotechnology and low-power CMOS design. Synthesis approaches are not well developed for reversible circuits even for small numbers of inputs and outputs.In this paper, a transformation based algorithm for the synthesis of such a reversible circuit in terms of n × n Toffoli gates is presented. Initially, a circuit is constructed by a single pass through the specification with minimal look-ahead and no back-tracking. Reduction rules are then applied by simple template matching. The method produces near-optimal results for 3-input circuits and also produces very good results for larger problems.

/pdf/a-transformation-based-algorithm-for-reversible-logic-27g0s28056.pdf

A transformation based algorithm for reversible logic synthesis

Reversible or information-lossless circuits have applications in digital signal processing, communication, computer graphics, and cryptography. They are also a fundamental requirement in the emerging field of quantum computation. We investigate the synthesis of reversible circuits that employ a minimum number of gates and contain no redundant input-output line-pairs (temporary storage channels). We prove constructively that every even permutation can be implemented without temporary storage using NOT, CNOT, and TOFFOLI gates. We describe an algorithm for the synthesis of optimal circuits and study the reversible functions on three wires, reporting the distribution of circuit sizes. We also study canonical circuit decompositions where gates of the same kind are grouped together. Finally, in an application important to quantum computing, we synthesize oracle circuits for Grover's search algorithm, and show a significant improvement over a previously proposed synthesis algorithm.

/pdf/synthesis-of-reversible-logic-circuits-3j3y79pcmo.pdf

Synthesis of reversible logic circuits

Learning techniques used in computer vision for food quality evaluation: a review

: A given switching function of n variables can frequently be decomposed into a composite function of several essentially simpler switching functions. Such decompositions lead to designs of more economical switching circuits to realize the given switching function. Ashenhurst's chart method is generalized to nondisjunctive decompositions by means of the don't care conditions. This extension provides an effective method of constructing all decompositions of switching functions. (Author)

On the decomposition of switching functions

Logic synthesis for reversible logic differs considerably from standard logic synthesis. The gates are multi-output and the unutilized outputs from these gates are called “garbage”. One of the synthesis tasks is to reduce the number of garbage signals. Previous approaches to reversible logic synthesis minimized either only the garbage or (predominantly) the number of gates. Here we present for the first time a method that minimizes concurrently the number of gates, their total delay and the total garbage. Our method adopts for reversible logic many ideas developed previously for standard logic synthesis (such as Ashenhurst/Curtis Decomposition, Dietmeyer’s Composition, non-linear preprocessing for BDDs), methods created in ReedMuller Logic (such as Pseudo-Kronecker Decision Diagrams with Complemented Edges, Pseudo-Kronecker Lattice Diagrams and their generalizations) and introduces also new methods specific to reversible logic.

/pdf/a-general-decomposition-for-reversible-logic-1v3rbti523.pdf

A General Decomposition for Reversible Logic

This paper presents a generalized method for decomposition of multiple-valued functions. The main reason for using the described method is efficient implementation of logic circuits as well as effective representation of data in information systems. In logic synthesis, the method reduces the demand for silicon space required to implement designs. It is shown that the decomposition technique leads to additional compressing capabilities in PLA implementations. Another very promising area of application of decomposition is its effective representation of data in information systems, data bases and in other applications of information storing systems.

Decomposition of multiple-valued functions

We introduce a Reversible Programmable Gate Array (RPGA) based on regular structure to realize binary functions in reversible logic. This structure, called a 2 * 2 Net Structure, allows for more efficient realization of symmetric functions than the methods shown by previous authors. In addition, it realizes many non-symmetric functions even without variable repetition. Our synthesis method to RPGAs allows to realize arbitrary symmetric function in a completely regular structure of reversible gates with smaller “garbage” than the previously presented papers. Because every Boolean function is symmetrizable by repeating input variables, our method is applicable to arbitrary multi-input, multi-output Boolean functions and realizes such arbitrary function in a circuit with a relatively small number of garbage gate outputs. The method can be also used in classical logic. Its advantages in terms of numbers of gates and inputs/outputs are especially seen for symmetric or incompletely specified functions with many outputs.

/pdf/regularity-and-symmetry-as-a-base-for-efficient-realization-3nsw6v9ewu.pdf

Regularity and Symmetry as a Base for Efficient Realization of Reversible Logic Circuits

The analysis of information relationships is of primary importance for the analysis and synthesis of digital information systems. The paper aims to introduce and discuss the fundamental apparatus for the analysis and evaluation of information relationships. It defines and explains various relationships between information, measures for the amount and importance of information, and measures for the strength and importance of the information relationships. It demonstrates importance of the introduced relationships and measures for efficient synthesis, and shows how to apply them in the synthesis process. The analysis apparatus makes operational the famous theory of partitions and set systems of Hartmanis (1966). While partitions and set systems enable one to model information, the relationships and measures enable one to analyze and measure information and information relationships. Both together form a complete information modeling and analysis apparatus that can be applied in logic design, decision system design, pattern recognition, knowledge discovery, machine learning and other areas.

Information relationships and measures: an analysis apparatus for efficient information system synthesis

We present a new theory of non-disjoint serial decomposition We also present our new decomposition tool DEMAIN The decomposition approach implemented in DEMAIN relies on: a partition-based representation of Boolean functions; and an effective balanced decomposition strategy that switches between the parallel and non-disjoint serial decomposition In consequence, we applied the non-disjoint serial decomposition and parallel decomposition for efficient synthesis of FPGA-based circuits directed towards area or delay optimisation

Lech Jozwiak

Papers

A General Decomposition for Reversible Logic

Decomposition of multiple-valued functions

Regularity and Symmetry as a Base for Efficient Realization of Reversible Logic Circuits

Information relationships and measures: an analysis apparatus for efficient information system synthesis

Non-disjoint decomposition of Boolean functions and its application in FPGA-oriented technology mapping