Table Of ContentFUNCTIONAL DECOMPOSITION WITH
APPLICATION TO FPGA SYNTHESIS
Functional Decomposition
with Application to
FPGA Synthesis
by
Christoph Scholl
Institute of Computer Science,
Albert-Ludwigs-University,
Freiburg im Breisgau, Germany
......
''
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Cata10gue record for this book is avai1ab1e from the Library of Congress.
ISBN 978-1-4419-4929-5 ISBN 978-1-4757-3393-8 (eBook)
DOI 10.1007/978-1-4757-3393-8
Printed on acid-free paper
AlI Rights Reserved
© 2001 Springer Science+Business Media Dordrecht
Original1y published by Kluwer Academic Publishers in 2001
Softcover reprint ofthe hardcover lst edition 2001
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanica1,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
To Barbara.
Contents
Preface xi
Acknowledgments xiii
Introduction
XV
1. REALIZATIONS OF BOOLEAN FUNCTIONS 1
1.1 Circuits 3
1.2 Boolean Expressions 8
1.3 Binary Decision Diagrams (BDDs) 10
1.3.1 BDDs to represent completely specified function 10
1.3.2 BDDs to represent incompletely specified functions 14
1.4 Field Programmable Gate Arrays (FPGAs) 16
1.5 Complexity of realizations of Boolean functions 20
2. MINIMIZATION OF BDDS 23
2.1 BDD minimization for completely specified functions 23
2.1.1 The role of variable orders 23
2.1.2 ROBDD size and communication complexity 24
2.1.3 Variable reordering 28
2.1.4 Symmetries of completely specified functions 29
2.2 BDD minimization for incompletely specified functions 38
vii
viii FUNCTIONAL DECOMPOSITION APPLIED TO FPGA SYNTHESIS
2.2.1 The 'communication minimization' approach 38
2.2.2 Symmetries of incompletely specified functions 50
2.2.3 Communication minimization and symmetry 61
2.2.4 Don't care minimization and variable reordering 63
2.2.5 Experimental results 64
3. FUNCTIONAL DECOMPOSITION FOR COMPLETELY
SPECIFIED SINGLE-OUTPUT FUNCTIONS 69
3.1 Definitions 70
3.2 Decomposition and Field Programmable Gate Arrays 73
3.3 Minimizing the number of decomposition functions 74
3.4 Nontrivial decomposability-theoretical considerations 81
3.5 Examples of circuits derived by decomposition 88
3.6 Decomposition and ROBDD representations 91
3.6.1 Minimum number of decomposition functions 92
3.6.2 Computing decomposition and composition functions 92
3.6.3 ROBDD size and decomposability 98
3.7 Variable partitioning 101
3.7.1 Partitioning for bound set of fixed size 101
3.7.2 Partitioning for bound set of variable size 103
3.7.3 Other cost functions 105
3.8 The encoding problem 107
3.8.1 Strict decompositions and symmetries 109
3.8.2 Optimizing composition functions by input encoding 113
3.8.3 Optimizing composition functions by look-ahead 114
3.8.4 Support minimization for decomposition functions 117
3.8.5 Common decomposition functions 121
4. FUNCTIONAL DECOMPOSITION FOR COMPLETELY
SPECIFIED MULTI-OUTPUT FUNCTIONS 123
4.1 A motivating example 124
Contents ix
4.2 Output partitioning and input partitioning 129
4.3 Computation of common decomposition functions 134
4.3.1 The problem CDF 139
4.3.2 Applying a solution to CDF 155
4.4 Experiments 162
4.4.1 Example: A binary adder 162
4.4.2 Benchmark results 165
4.5 Alternative methods 166
4.5.1 Interpreting multi-output functions as integer-valued
functions 167
4.5.2 Representing the set of all decomposition functions 170
4.5.3 Decomposition using support minimization 175
5. FUNCTIONAL DECOMPOSITION FOR INCOMPLETELY
SPECIFIED FUNCTIONS 177
5.1 Decomposition and BDD minimization for
incompletely specified functions 178
5.2 One-sided decomposition of
incompletely specified functions 178
5.3 Two-sided decomposition of
incompletely specified functions 180
5.4 Computation of incompletely specified decomposition
and composition functions 181
5.5 Multi-output functions 185
5.5.1 Don't care assignment with respect to logic sharing 187
5.5.2 Compatibility with communication minimization for
single-output functions 193
5.6 Experimental results 194
5.6.1 FPGA synthesis 194
5.6.2 Example: A partial multiplier 196
6. NON-DISJOINT DECOMPOSITIONS 201
6.1 Motivation 202
X FUNCTIONAL DECOMPOSITION APPLIED TO FPGA SYNTHESIS
6.2 Integration of non-disjoint into disjoint decomposition 205
6.2.1 Preselected decomposition functions 205
6.2.2 Variable duplication with don't cares 207
6.3 Computing non-disjoint variable sets for decomposition 208
7. LARGE CIRCUITS 213
Appendices 219
An example for an FPGA device 219
Complexity of CM 223
Characterization of weak and strong symmetry 225
Complexity of MSP 227
Making a function strongly symmetric 229
Compatibility of don't care minimization methods 233
Symmetries and decomposability 235
Complexity of CDF 239
ROBDD based computation of compatibility classes 245
Maximal don't care sets 249
Index
261
Preface
During the last few years Field Programmable Gate Arrays (FPGAs) have be
come increasingly important. Thanks to recent breakthroughs in technology,
FPGAs offer millions of system gates at low cost and considerable speed.
Functional decomposition has emerged as an essential technique in automatic
logic synthesis for FPGAs. Functional decomposition as a technique to find
realizations for Boolean functions was already introduced in the late fifties and
early sixties by Ashenhurst (Ashenhurst, 1959), Curtis (Curtis, 1961), Roth
and Karp (Roth and Karp, 1962; Karp, 1963). In recent years, however, it
has attracted a great deal of renewed attention, for several reasons. First, it is
especially well suited for the synthesis of lookup-table based FPGAs. Also, the
increased capacities of today's computers as well as the development of new
methods have made the method applicable to larger-scale problems. Modern
techniques for functional decomposition profit from the success of Reduced
Ordered Binary Decision Diagrams (ROBDDs) (Bryant, 1986), data structures
that provide compact representations for many Boolean functions occurring in
practical applications. We have now seen the development of algorithms for
functional decomposition which work directly based on ROBDDs, so that the
decomposition algorithm works based on compact representations and not on
function tables or decomposition matrices as in previous approaches.
The book presents, in a consistent manner, a comprehensive presentation of a
multitude of results stemming from the author's as well as various researchers'
work in the field. Apart from the basic method, it also covers functional decom
position for incompletely specified functions, decomposition for multi-output
functions and non-disjoint decomposition.
In essence the exposition is self-contained; a basic knowledge of mathematics
and of some fields of computer science such as complexity theory and graph
theory will be helpful for an understanding.
CHRISTOPH SCHOLL
Freiburg, August 2001
xi
Acknowledgments
I would like to thank all my colleagues and friends both from the University
of Saarland, Saarbriicken, and from the Albert-Ludwigs-University, Freiburg,
who contributed to a pleasant working climate.
Special thanks to Dr. Harry Hengster and Dr. Uwe Sparmann for many helpful
discussions.
I acknowledge the work of Stephan Melchior, who made contributions to the
theory of symmetries of incompletely specified functions.
I would like to express my thanks to Prof. Dr. Paul Molitor for supporting my
research, for cooperating on many subjects in this monograph, for engaging in
helpful discussions, and for encouraging me to write this book.
I am also grateful to Prof. Dr. Gunter Hotz for numerous suggestions, for his
advice, and his confident support.
I would like to thank Prof. Dr. Bernd Becker for giving me the opportunity to
finish this book by sparing me any interruptions and for his continuous support.
I owe gratitude to Prof. Dr. Paul Molitor and Mary Masters, who have read and
improved the draft of this monograph.
Finally, I would like to thank Cindy Lufting, James Finlay, and Mark de Jongh
of Kluwer Academic Publishers for their patience and help in preparing the
final manuscript.
Xlll
