SYNTHESIS OF FINITE STATE MACHINES: LOGIC OPTIMIZATION

Transcription

1 SYNTHESIS OF FINITE STATE MACHINES: LOGIC OPTIMIZATION

2 SYNTHESIS OF FINITE STATE MACHINES: LOGIC OPTIMIZATION Tiziano Villa University of California/Berkeley Timothy Kam Intel Corporation Robert K. Brayton University of California/Berkeley Alberto Sangiovanni-Vincentelli University of California/Berkeley ~. " SPRINGER-SCIENCE+BUSINESS MEDIA, LLC

3 ISBN ISBN (ebook) DOI / Library of Congress Cataloging-in-Publication Data A C.I.P. Catalogue record for this book is available from the Library of Congress by Springer Science+Business Media New York Originally published by Kluwer Academic Publishers in 1997 Softcover reprint of the hardcover 1 st edition 1997 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo-copying, recording, or otherwise, without the prior written permis sion of the publisher, Springer-Science+Business Media, LLC. Printed on acid-free paper.

4 To Alle mie scorte: Franco e Marta, Maria, Marta e Letizia; Professor and Mrs. Wai-Kee Kam, and Mrs. Kate Kam for unfailing love and support; Ruth for continuous support over many years; To Marcolino, whose laughters and joy of life are bringing so much happiness around him.

5 CONTENTS PREFACE xi Part I FROM SYMBOLIC TO LOGIC REPRESENTATIONS 1 1 INTRODUCTION Logic Synthesis of Sequential Behaviors The Encoding Problem: From Symbolic to Boolean Domain Overview of the Book 9 2 DEFINITIONS Sequential Functions and Their Representation Finite State Machines Taxonomy of Encoding Problems Behavior vs. Structure in Encoding Problems Boolean Algebras and Boolean Functions Discrete Functions as Boolean Functions Two-level Minimization of Multi-Valued Functions Multi-level Multi-Valued Functions Multi-Valued Relations Binary Decision Diagrams Hypercubes 37 3 COMPLEXITY ISSUES Computational Complexity Counting State Assignments 48 Vll

6 viiisvnthesis OF FINITE STATE MACHINES: LOGIC OPTIMIZATION 4 ENCODING FOR SYNTHESIS Algorithms for Optimal Encoding Combinational Encoding Problems State Assignment and FSM Decomposition State Assignment for Testability Sequential Optimization at the Gate Level 130 Part II CONSTRAINED ENCODING SYMBOLIC MINIMIZATION Introduction Encoding for Two-level Implementations A New Symbolic Minimization Algorithm Symbolic Reduction Symbolic Oring Ordering of Symbolic Minimization Symbolic Minimization by Example Experimental Results Conclusions ENCODING CONSTRAINTS Introduction Statement and Complexity of the Encoding Problems Definitions Abstraction of the Problem Input Constraint Satisfaction Input and Output Constraint Satisfaction Bounded Length Encoding Other Applications Results Conclusions 218 Part III GENERALIZED PRIME IMPLICANTS GENERALIZED PRIME IMPLICANTS Introduction 221

7 Contents ix 7.2 Definition of GPIs GPIs by Consensus Operation Encodeability of GPIs Sufficiency of GPIs GPIs by Computation of MV Primes A Procedure for Analysis of GPIs MINIMIZATION OF GPIS Reduction of GPI Minimization to Unate Covering Reduction of GPI Minimization to Binate Covering GPIs and Non-Determinism ENCODEABILITY OF GPIS Introduction Efficient Encodeability Check of GPIs Encoding of a Set of Encodeable GPIs Updating Sets and Raising Graphs Choice of a Branching Column Computation of a Lower Bound 295 Part IV IMPLICIT TECHNIQUES FOR ENCODING IMPLICIT FORMULATION OF UNATE COVERING Introduction Implicit Representations and Manipulations A Fully Implicit Unate Covering Solver Quantifier-Free Reduction of Unate Tables An Application to Encoding Dichotomies IMPLICIT MINIMIZATION OF GPIS Introduction Useful Relations Implicit Generation of GPIs and Minterms Implicit Selection of GPIs 328

8 xsvnthesis OF FINITE STATE MACHINES: LOGIC OPTIMIZATION 11.5 An Example of Implicit GPI Minimization 11.6 Verification of Correctness 11.7 Implementation Issues 11.8 Experiments 11.9 Conclusions Part V CONCL US IONS CONCLUSIONS REFERENCES INDEX

9 PREFACE This book is the second of a set of two monographs devoted to the synthesis of Finite State Machines (FSMs). The first volume addresses functional optimization, whereas this one addresses logic optimization. The result of functional optimization is a symbolic description of an FSM which represents a sequential function chosen from a collection of permissible candidates. Logic optimization is the body of techniques to convert a symbolic description of an FSM into an hardware implementation. The mapping of a given symbolic representation into a two-valued logic implementation is called state encoding (or state assignment) and it impacts heavily area, speed, testability and power consumption of the realized circuit. Here we present a quite extensive account of the subject matter, describing contributions due to the authors and to many other researchers. In the exposition we have emphasized modern approaches pursued since the 80s, but also the previous literature has been integrated into a comprehensive picture. The exposition is self-contained, but some familiarity with Boolean algebra and automata theory will be helpful. Key themes running throughout the book are the minimization of symbolic logic, the satisfaction of encoding constraints and the representation of combinatorial problems of FSM synthesis with binary decision diagrams (BDDs). The first part of the book introduces the relevant background, presents results previously scattered in the literature on the computational complexity of encoding problems, and finally surveys in depth old and new approaches to encoding in logic synthesis. Many recent techniques for encoding can be described as symbolic minimization, i.e., given a function, the generation of an optimal symbolic representation that is implement able in the two-valued domain, provided that some conditions on the binary codes of the states are satisfiable (encoding constraints). The second part of the book presents two main results about symbolic minimization: a new procedure to find minimal two-level symbolic covers, under face, domixi

10 xiisynthesis OF FINITE STATE MACHINES: LOGIC OPTIMIZATION nance and disjunctive constraints, and a unified frame to check encodeability of encoding constraints and find codes of minimum length that satisfy them. This frame has been used for various types of encoding constraints arising in problems that range from encoding for minimum multi-level representation to race-free encoding of asynchronous FSMs. Experiments for different applications are reported. The third part of the book introduces generalized prime implicants (GPIs), which are the counterpart, in symbolic minimization of two-level logic, to prime implicants in two-valued two-level minimization. GPIs enable the design of an exact procedure for two-level symbolic minimization, based on a covering step which is complicated by the need to guarantee encodeability of the final cover. We present a new efficient algorithm to verify encodeability of a selected cover. If a cover is not encodeable, it is shown how to augment it minimally until an encodeable superset of GPIs is determined. To handle encodeability we have extended the frame to satisfy encoding constraints presented in the second part. The covering problems generated in the minimization of GPIs tend to be very large. Recently large covering problems have been attacked successfully by representing the covering table with BDDs. In the fourth part of book we introduce such techniques and extend them to the case of the implicit minimization of GPIs, where also the encodeability and augmentation steps are performed implicitly. In all these years we benefited from close associations with various researchers operating in the field, especially our colleagues at the University of California at Berkeley. We owe gratitude to all of them. A special mention goes to Alex Saldanha with whom we developed the research presented in the second part of this volume. We acknowledge very helpful discussions with present or former colleagues in the Berkeley CAD group: Pranav Ashar, Mark Beardslee, Luca Carloni, Srinivas Devadas, Stephen Edwards, Eugene Goldberg, Sunil Khatri, Yuji Kukimoto, William Lam, Luciano Lavagno, Sharad Malik, Rick McGeer, Rajeev Murgai, Arlindo Oliveira, Tom Shiple, Gitanjali Swamy, Huey-Yih Wang and Yosinori Watanabe. Luciano Lavagno, Sharad Malik and Rajeev Murgai helped in countless ways through discussions and sharing material. Gitanjali Swamy and Rick McGeer contributed an efficient software package for the implicit computation of multivalued prime implicants. Huey-Yih Wang kindly shared some drawings from his dissertation. Stephen Edwards gave advice in matters of style and contributed a software package to visualize FSMs.

11 Preface xiii Ney Calazans, Giovanni De Micheli, Zafar Hasan, Bruce Holmer, Kurt Keutzer, David Johnson and Fabio Somenzi were kindly available to discuss technical questions and share literature. In particular we thank Kurt Keutzer for sending a draft of a paper on the computational complexity of problems in logic synthesis. But as usual the greatest source of sustenance have been our family members to whom we dedicate our work. The funding for this research was provided by DARPA, NSF, SRC and industrial grants from Bell Northern, Digital, Fujitsu, Intel, MICRO, Motorola, and we gratefully acknowledge it. We thank Carl Harris of Kluwer Academic Publishers for his patience and continuous encouragement to persevere in the task of writing the book(s).