2,478 research outputs found
Advances in Functional Decomposition: Theory and Applications
Functional decomposition aims at finding efficient representations for Boolean functions. It is used in many applications, including multi-level logic synthesis, formal verification, and testing.
This dissertation presents novel heuristic algorithms for functional decomposition. These algorithms take advantage of suitable representations of the Boolean functions in order to be efficient.
The first two algorithms compute simple-disjoint and disjoint-support decompositions. They are based on representing the target function by a Reduced Ordered Binary Decision Diagram (BDD). Unlike other BDD-based algorithms, the presented ones can deal with larger target functions and produce more decompositions without requiring expensive manipulations of the representation, particularly BDD reordering.
The third algorithm also finds disjoint-support decompositions, but it is based on a technique which integrates circuit graph analysis and BDD-based decomposition. The combination of the two approaches results in an algorithm which is more robust than a purely BDD-based one, and that improves both the quality of the results and the running time.
The fourth algorithm uses circuit graph analysis to obtain non-disjoint decompositions. We show that the problem of computing non-disjoint decompositions can be reduced to the problem of computing multiple-vertex dominators. We also prove that multiple-vertex dominators can be found in polynomial time. This result is important because there is no known polynomial time algorithm for computing all non-disjoint decompositions of a Boolean function.
The fifth algorithm provides an efficient means to decompose a function at the circuit graph level, by using information derived from a BDD representation. This is done without the expensive circuit re-synthesis normally associated with BDD-based decomposition approaches.
Finally we present two publications that resulted from the many detours we have taken along the winding path of our research
Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays
The goal of this paper is to introduce a new method in computer-aided
geometry of solid modeling. We put forth a novel algebraic technique to
evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with
regularized operators of union, intersection, and difference, i.e., any CSG
tree. The result is obtained in three steps: first, by computing an independent
set of generators for the d-space partition induced by the input; then, by
reducing the solid expression to an equivalent logical formula between Boolean
terms made by zeros and ones; and, finally, by evaluating this expression using
bitwise operators. This method is implemented in Julia using sparse arrays. The
computational evaluation of every possible solid expression, usually denoted as
CSG (Constructive Solid Geometry), is reduced to an equivalent logical
expression of a finite set algebra over the cells of a space partition, and
solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig
Minimization of Quantum Circuits using Quantum Operator Forms
In this paper we present a method for minimizing reversible quantum circuits
using the Quantum Operator Form (QOF); a new representation of quantum circuit
and of quantum-realized reversible circuits based on the CNOT, CV and
CV quantum gates. The proposed form is a quantum extension to the
well known Reed-Muller but unlike the Reed-Muller form, the QOF allows the
usage of different quantum gates. Therefore QOF permits minimization of quantum
circuits by using properties of different gates than only the multi-control
Toffoli gates. We introduce a set of minimization rules and a pseudo-algorithm
that can be used to design circuits with the CNOT, CV and CV quantum
gates. We show how the QOF can be used to minimize reversible quantum circuits
and how the rules allow to obtain exact realizations using the above mentioned
quantum gates.Comment: 11 pages, 14 figures, Proceedings of the ULSI Workshop 2012 (@ISMVL
2012
On the complexity of computing with zero-dimensional triangular sets
We study the complexity of some fundamental operations for triangular sets in
dimension zero. Using Las-Vegas algorithms, we prove that one can perform such
operations as change of order, equiprojectable decomposition, or quasi-inverse
computation with a cost that is essentially that of modular composition. Over
an abstract field, this leads to a subquadratic cost (with respect to the
degree of the underlying algebraic set). Over a finite field, in a boolean RAM
model, we obtain a quasi-linear running time using Kedlaya and Umans' algorithm
for modular composition. Conversely, we also show how to reduce the problem of
modular composition to change of order for triangular sets, so that all these
problems are essentially equivalent. Our algorithms are implemented in Maple;
we present some experimental results
Support-reducing decomposition for FPGA mapping
Decomposition is a technology-independent process, in which a large complex function is broken into smaller, less complex functions. The costs of two-level or factored-form representations (cubes and literals) are used in most decomposition methods, as they have a high correlation with the area of cell-based designs. However, this correlation is weaker for field-programmable gate arrays (FPGAs) based on look-up tables. Furthermore, local optimizations have limited power due to the structural bias of the circuit descriptions. This paper tries to reduce the structural biasing by remapping the LUT network and decomposing the derived functions using the support as cost function. The proposed method improves the FPGA mapping results of a commercial tool for the 20 largest MCNC benchmarks, with gains of 28% in delay plus 18% in area when targeting delay, and a reduction of 28% in area plus 14% in delay with area as cost function. Results with 23% less area and 6% less delay are obtained after physical synthesis (post place-and-route). Moreover, 12 of the best known results for delay (and 3 for area) of the EPFL benchmarks are improved.Peer ReviewedPostprint (author's final draft
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