Michael Kearns
Home address: Oce address:
1216 Blo om eld Street AT&T Lab oratories { Research
Hob oken, New Jersey 07030 180 Park Avenue, Ro om A235
Phone: 201 963-5291 Florham Park, New Jersey 07932
Phone: 973 360-8322
Fax: 973 360-8970
Email: [email protected]
Home Page: http://www.research.att.com/~mkearns
Date of Birth Octob er 24, 1962, in California.
Citizenship U.S.A.
Professional In my current p osition as the head of the Arti cial Intelligence Research department
Ob jectives at AT&T Labs, my goals are to build and maintain a world-class research group in AI,
Machine Learning and related disciplines, to develop and oversee pro jects and systems
applying AI to problems of b oth immediate and long-term interest to AT&T, and to
explore applications to new areas of technology and the computing industry. My de-
partment is part of Information Systems and Services ResearchatAT&T Labs, which
has the developmentofnovel internet and web applications and technologies as a central
part of its charter.
My own research examines algorithms and problems in arti cial intelligence and machine
learning and related areas such as data mining, statistics, and neural networks sp eech
recognition, and sp oken dialogue systems. I also have strong interests in cryptography
and security, and theoretical computer science in general.
Education Harvard University, Cambridge, Massachusetts
Ph.D. in computer science, May 1989.
Dissertation : The Computational Complexity of Machine Learning .
Winner of 1989 ACM Distinguished Do ctoral Dissertation Award, published by
MIT Press.
S.M. in computer science, May 1986.
University of California at Berkeley, Berkeley, California
A.B. in mathematics and computer science, June 1985.
Graduated with highest academic honors.
Professional AT&T Laboratories { Research, Florham Park, New Jersey
Exp erience July 1997{Present: Division Manager, Arti cial Intelligence Research Department.
Department recently absorb ed the formerly separate Machine Learning depart-
ment. Resp onsibilities include development and managementofaworld-class re-
search group; pro ject management for developing AI-based systems and applica-
tions; participation in strategy and planning for AT&T Labs research activities;
making visible and lasting research contributions to the elds of arti cial intelli-
gence and machine learning.
AT&T Laboratories { Research, Florham Park, New Jersey
July 1999{November 1999: Division Manager, Secure Systems Research Depart-
ment. Oversaw technical work and p ersonnel in security and cryptography.
AT&T Bel l Laboratories and AT&T Laboratories { Research, Murray Hil l, New Jersey
Septemb er 1991{July 1997: Principal MemberofTechnical Sta , Machine Learning
and Information Retrieval Research Department. Resp onsibilities included making
visible and lasting research contributions to the eld of machine learning and related areas.
Professional International Computer Science Institute, Berkeley, California
Exp erience September 1990{September 1991: Postdo ctoral Asso ciate. Basic research in ma-
continued chine learning and related areas.
Massachusetts Institute of Technology, Cambridge, Massachusetts
May 1989{Septemb er 1990: Postdo ctoral Asso ciate, Lab oratory for Computer Sci-
ence. Basic research in machine learning and related areas.
Teaching Adjunct Professor, University of Pennsylvania, Philadelphia
Exp erience
Winter 1998 { Present. Taught a graduate course in mo dern topics in arti cial
intelligence in 1998.
Lecturer, Columbia University, New York
Fall 1992: Taught a graduate course on machine learning.
Lecturer, University of California, Berkeley
Fall 1992: Taught a graduate course on machine learning.
Grants, Participantina Packard Foundation grant to the Santa Fe Institute for a program on
Honors di erent notions of robustness in the sciences.
and
Awards Awarded a 3-year U.S.-Israel Binational Science Foundation Award grant with Prof.
Haim Somp olinsky of Hebrew University and Dr. H. Sebastian Seung of AT&T Bell
Lab oratories for research on on-line learning algorithms.
Honorable Mention, Best Pap er Award, Eleventh National Conference on Arti cial In-
telligence AAAI '93 , for \Reasoning with Characteristic Mo dels", with H. Kautz and
B. Selman.
1989 ACM-MIT Press Distinguished Dissertation Award see Publications: Bo oks .
1987{1989 A.T. & T. Bell Lab oratories Ph.D. Scholarship.
1985 Award for Academic Distinction, Computer Science Department, U.C. Berkeley.
1985 Klumpke Prize, Mathematics Department, U.C. Berkeley.
1984 Junior year election to Phi Beta Kappa, U.C. Berkeley.
Invited Invited sp eaker, Bernoul li-RIKEN BSI 2000 Symposium on Neural Networks and Learn-
Conference ing,Tokyo, Japan, Octob er 2000.
Lectures
Sole invited lecturer for 1-week EuroCOLT course Reinforcement Learning and Graphical
Models, Cumb erland Lo dge, England, June 2000.
Invited tutorial, Probabilistic Models for Arti cial Intel ligence, 37th Annual Meeting for
Computational Linguistics, June 1999.
Sole invited lecturer for 1-week course Probabilistic Arti cial Intel ligence, Bellairs Re-
search Institute of McGill University,February 1999.
Invited tutorial, Theoretical Issues in Probabilistic Arti cial Intel ligence, 39th Annual
IEEE Symp osium on the Foundations of Computer Science, Novemb er 1998.
Invited sp eaker, Workshop on Probabilistic Graphical Models , September 1997, Isaac
Newton Institute for Mathematical Sciences, Cambridge University, England.
Invited tutorial, Fifteenth National Conference on Arti cial Intel ligence AAAI 1997 ,
August 1997, Providence, Rho de Island.
Invited sp eaker, Fourteenth National ConferenceonArti cial Intel ligence AAAI 1996 ,
August 1996, Portland, Oregon.
Invited sp eaker, Neural Networks: The Statistical Mechanics Perspective,February 1995,
Pohang, Korea.
Invited tutorial, Neural Information Processing Systems NIPS*94 , December 1994,
Denver, Colorado.
Invited sp eaker, European Conference on Machine Learning ECML 1994 , April 1994,
Catania, Italy.
Invited sp eaker, Learning Days in Jerusalem, June 1993, Jerusalem, Israel.
Invited series of four lectures, Vietri Summer School on Learning and Cryptography,
Septemb er 1993, Vietri, Italy.
Invited sp eaker, Maryland Theory Day, March 1993, Baltimore, Maryland.
Invited sp eaker, ThirdAnnual Workshop on Computational Learning Theory and Natu-
ral Learning, August 1992, Madison, Wisconsin.
Invited sp eaker, International Joint Conference on Neural Networks, June 1992, Balti-
more, Maryland.
Invited sp eaker, International Symposium on Arti cial Intel ligence and Mathematics,
January 1990, Ft. Lauderdale, Florida.
Invited sp eaker, Cold Spring Harbor Laboratory Summer Course on Computational Neu-
roscience: Learning and Memory, July 1990, Cold Spring Harb or, New York.
Invited sp eaker, Workshop on Learning, December 1990, Carnegie Mellon University,
Pittsburgh, Pennsylvania.
N.B. In addition, I have given dozens of invited seminars and collo quium lectures at
almost all of the top U.S. universities, and many abroad.
Professional Memb er of the editorial b oard, Journal of the ACM.
Activities
Memb er of the editorial b oard, SIAM Journal on Computing.
Memb er of the editorial b oard, Machine Learning.
Memb er of the editorial b oard, Journal on Arti cial Intel ligenceResearch.
Memb er of the editorial b oard, Adaptive Computation and Machine Learning, book se-
ries, The MIT Press.
Senior Program Committee Memb er, AAAI 2000 .
Program Committee Memb er, UAI 2000.
General chair, 1998 Neural Information Pro cessing Systems Conference NIPS*98 .
Memb er of the program committee, 1998 ACM Symp osium on the Theory of Computa-
tion STOC 1998 .
Memb er of the program committee, 1998 Conference on Uncertainty in Arti cial Intel-
ligence UAI 1998 .
Program chair, 1997 Neural Information Pro cessing Systems Conference NIPS*97 .
Program chair, Ninth Annual ACM Conference on Computational Learning Theory
COLT 1996 .
Theory area chair of the program committee, 1996 Neural Information Pro cessing Sys-
tems Conference NIPS*96 .
Theory area chair of the program committee, 1995 Neural Information Pro cessing Sys-
tems Conference NIPS*95 .
Conference Co-Chair, joint meeting of the Seventh Annual Workshop on Computa-
tional Learning Theory and the Eleventh Annual Conference on Machine Learning, 1994
COLT/ML 1994 .
Memb er of the program committee, Seventh Annual Workshop on Computational Learn-
ing Theory, 1994 COLT 1994 .
Professional Memb er of the program committee, 34th Annual IEEE Symp osium on the Foundations
Activities of Computer Science, 1993 FOCS 1993 .
continued Member of the program committee, 13th International Joint Conference on Arti cial
Intelligence, 1993 IJCAI 1993 .
Organizer, workshop on \Comparison and Uni cation of Algorithms, Loss Functions and
Complexity Measures for Learning", with Esther Levin and Isab elle Guyon, held at Neu-
ral and Information Processing Systems NIPS*92 , Decemb er 1992, Denver, Colorado.
Lo cal arrangements chair, Second Annual Workshop on Computational Learning Theory
and Natural Learning, Septemb er 13{14, 1991, Berkeley, California.
Memb er of the program committee, 32nd Annual IEEE Symp osium on the Foundations
of Computer Science, 1991 FOCS 1991 .
Memb er of the program committee, 10th National Conference on Arti cial Intelligence,
1991 AAAI 1991 .
Memb er of the program committee, Fourth Workshop on Computational Learning The-
ory, 1991 COLT 1991 .
Memb er of the program committee, Seventh International Workshop on Machine Learn-
ing, 1990 ML 1990 .
References Available on request.
Publications: An Intro duction to Computational Learning Theory. With Umesh Vazirani. The
Bo oks MIT Press, Cambridge, Massachusetts, 1994.
The Computational Complexity of Machine Learning. The MIT Press, Cam-
bridge, Massachusetts, 1990. Published in the ACM{MIT Press Distinguished Disser-
tation Series. An earlier version of the dissertation is available as Harvard University
Aiken Computation Lab technical rep ort numb er TR-13-89.
Publications: Cob otDS: A Sp oken Dialogue System for Chat. With C. Isb ell, S. Singh, D.
Articles Litman, J. Howe. Submitted.
A So cial Reinforcement Learning Agent. With C. Shelton, C. Isb ell, S. Singh, P.
Stone. To app ear, Pro ceedings of Fifth International Conference on Autonomous Agents,
2001.
ATTac-2000: An Adaptive Autonomous Bidding Agent. With P. Stone, M.
Littman, S. Singh. To app ear, Pro ceedings of Fifth International Conference on Au-
tonomous Agents, 2001.
An Adaptive Sp oken Dialogue System, and its Empirical Evaluation. With
Satinder Singh, Diane Litman, and Marilyn Walker. In Pro ceedings of the Seventeenth
National Conference on Arti cial Intelligence, AAAI Press/The MIT Press, 2000, pages
645{651.
Cob ot in Lamb daMOO: A So cial Statistics Agent. With Charles Isb ell, David
Kormann, Satinder Singh, and Peter Stone. In Pro ceedings of the Seventeenth National
Conference on Arti cial Intelligence, AAAI Press/The MIT Press, 2000, pages 36{41.
Bias-Variance Error Bounds for Temp oral Di erence Up dates. With S. Singh.
In Pro ceedings of the 13th Annual Conference on Computational Learning Theory, 2000,
pages 142{147.
A Bo osting Approach to Topic Sp otting on Sub dialogues. With Kary Myers,
Satinder Singh, and Marilyn Walker. In Pro ceedings of the 17th International Confer-
ence on Machine Learning, 2000.
Automatic Optimization of Dialogue Management. With Satinder Singh, Diane
Litman, and Marilyn Walker. In Pro ceedings of the 18th International Conference on
Computational Linguistics, 2000.
Fast Planning in Sto chastic Games. With Yishay Mansour and Satinder Singh. In
Pro ceedings of the 16th Conference on Uncertainty in Arti cial Intelligence, 2000.
Nash Convergence of Gradient Dynamics in General-Sum Games. With Satin-
der Singh and Yishay Mansour. In Pro ceedings of the 16th Conference on Uncertainty
in Arti cial Intelligence, 2000.
Reinforcement Learning for Sp oken Dialogue Systems. With Satinder Singh,
Diane Litman, and Marilyn Walker. In Advances in Neural Information Pro cessing Sys-
tems 12, 2000.
Approximate Planning in Large POMDPs via Reusable Tra jectories. With
Yishay Mansour and Andrew Ng. In Advances in Neural Information Pro cessing Sys-
tems 12, 2000.
Automatic Detection of Poor Speech Recognition at the Dialogue Level. With
Diane Litman and Marilyn Walker. In Pro ceedings of the 37th Annual Meeting for Com-
putational Linguistics. 1999, pages 309-316.
Publications: A Sparse Sampling Algorithm for Near-Optimal Planning in Large Markov
Articles Decision Pro cesses. With Yishay Mansour and Andrew Ng. In Pro ceedings of the
continued Sixteenth International Joint Conference on Arti cial Intelligence, Morgan Kaufmann,
1999, pages 1324{1331. To app ear in a sp ecial issue of the journal Machine Learning.
Ecient Reinforcement Learning in Factored MDPs. With Daphne Koller. In
Pro ceedings of the Sixteenth International Joint Conference on Arti cial Intelligence,
Morgan Kaufmann, 1999, pages 740{747.
Finite-Sample Convergence Rates for Q-Learning and Indirect Algorithms.
With Satinder Singh. In Advances in Neural Information Pro cessing Systems 11, M.
Kearns, S. Solla, D. Cohn eds. The MIT Press, 1999, pages 996{1002.
Inference in Multilayer Networks via Large Deviation Bounds. With Lawrence
Saul. In Advances in Neural Information Pro cessing Systems 11, M. Kearns, S. Solla, D.
Cohn eds. The MIT Press, 1999, pages 260{266.
Near-Optimal Reinforcement Learning in Polynomial Time. With Satinder
Singh. In Pro ceedings of the 15th International Conference on Machine Learning, Mor-
gan Kaufmann, 1998, pages 260{268. To app ear in a sp ecial issue of the journal Machine
Learning.
A Fast, Bottom-Up Decision Tree Pruning Algorithm with Near-Optimal
Generalization. With Yishay Mansour. In Pro ceedings of the 15th International Con-
ference on Machine Learning, Morgan Kaufmann, 1998, pages 269{277.
Exact Inference of Hidden Structure from Sample Data in Noisy-OR Net-
works. With Yishay Mansour. In Pro ceedings of the Fourteenth Conference on Uncer-
tainty in Arti cial Intelligence, Morgan Kaufmann, 1998, pages 304{310.
Large Deviation Metho ds for Approximate Probabilistic Inference. With
Lawrence Saul. In Pro ceedings of the Fourteenth Conference on Uncertainty in Arti-
cial Intelligence, Morgan Kaufmann, 1998, pages 311{319.
Testing Problems with Sub-Learning Sample Complexity. With Dana Ron. In
Pro ceedings of the Eleventh Annual Conference on Computational Learning Theory,
ACM Press, 1998. To app ear in the journal Neural Computation.
An Information-Theoretic Analysis of Hard and Soft Assignment Metho ds
for Clustering. With Yishay Mansour and Andrew Ng. In Pro ceedings of the Thir-
teenth Conference on Uncertainty in Arti cial Intelligence, Morgan Kaufmann, 1997,
pages 282{293. Also in Learning in Graphical Mo dels, M.I. Jordan ed., Kluwer Aca-
demic Publishers.
Algorithmic Stability and Sanity-Check Bounds for Leave-One-Out Cross-
Validation. With Dana Ron. Neural Computation 11 6 , pages 1427-1453, 1999. An
earlier version app eared in Pro ceedings of the Tenth Annual Conference on Computa-
tional Learning Theory,ACM Press, 1997, pages 152{162.
Bo osting Theory Towards Practice: Recent Developments in Decision Tree
Induction and the Weak Learning Framework. Pap er accompanying invited talk.
In Pro ceedings of the Thirteenth National Conference on Arti cial Intelligence, AAAI Press/The MIT Press, 1996, pages 1337-1339.
Publications: On the Bo osting Ability of Top-Down Decision Tree Learning Algorithms.
Articles With Yishay Mansour. Journal of Computer and Systems Sciences 58 1 , 1999, pages
continued 109-128. An earlier version app eared in Pro ceedings of the Twenty-Eighth Annual ACM
Symp osium on the Theory of Computing,ACM Press, 1996, pages 459{468.
Applying the Weak Learning Framework to Understand and Improve C4.5.
With Tom Dietterich and Yishay Mansour. In Pro ceedings of the Thirteenth Interna-
tional Conference on Machine Learning, Morgan Kaufmann, 1996, pages 96{104.
A Bound on the Error of Cross Validation, with Applications to the Training-
Test Split. In Neural Computation, 9 5 , 1997, pages 1143{1161. An earlier version
app eared in Advances in Neural Information Pro cessing Systems 8, The MIT Press,
pages 183{189, 1996.
Ecient Algorithms for Learning to Play Rep eated Games Against Compu-
tationally Bounded Adversaries. With YoavFreund, Yishay Mansour, Dana Ron,
Ronitt Rubinfeld, and Rob Schapire. In Pro ceedings of the 36th Annual Symp osium
on the Foundations of Computer Science, IEEE Computer So ciety Press, 1995, pages
332{341.
An Exp erimental and Theoretical Comparison of Mo del Selection Metho ds.
With Yishay Mansour, Andrew Ng, and Dana Ron. Machine Learning 27 1 , 1997,
pages 7{50. An earlier version app eard in Pro ceedings of the Eighth ACM Conference
on Computational Learning Theory,ACM Press, 1995, pages 21{30.
On the Consequences of the Statistical Mechanics Theory of Learning Curves
for the Mo del Selection Problem. In Neural Networks: The Statistical Mechanics
Persp ective,World Scienti c, 1995, pages 277{284.
Rigorous Bounds on Learning Curves from Statistical Physics. With David
Haussler, Sebastian Seung, and Naftali Tishby. Machine Learning 25, 1996, pages 195{
236. An earlier version app eared in Pro ceedings of the Seventh ACM Conference on
Computational Learning Theory,ACM Press, 1994, pages 76{76.
On the Learnability of Discrete Distributions. With Yishay Mansour, Dana Ron,
Ronitt Rubinfeld, Rob ert E. Schapire and Linda Sellie. In Pro ceedings of the 26th An-
nual ACM Symp osium on the Theory of Computing,ACM Press, 1994, pages 273{282.
Weakly Learning DNF and Characterizing Statistical Query Learning Using
Fourier Analysis. With Avrim Blum, MerrickFurst, Je rey Jackson, Yishay Mansour
and Steven Rudich. In Pro ceedings of the 26th Annual ACM Symp osium on the Theory
of Computing,ACM Press, 1994, pages 253{262.
Cryptographic Primitives Based on Hard Learning Problems. With Avrim
Blum, Merrick Furst and Dick Lipton. In Advances in Cryptology, Lecture Notes in
Computer Science, volume 773, Springer-Verlag, 1994, pages 278{291.
Ecient Noise-Tolerant Learning from Statistical Queries. Journal of the ACM
45 6 , 1998, pages 983-1006. An earlier version app ears in Pro ceedings of the 25th An-
nual ACM Symp osium on the Theory of Computing,ACM Press, 1993, pages 392{401.
Publications: Ecient Learning of Typical Finite Automata from Random Walks. With
Articles Yoav Freund, Dana Ron, Ronitt Rubinfeld, Rob ert E. Schapire and Linda Sellie. In-
continued formation and Computation 138 1 , 1997, pages 23-48. An earlier version app eared in
Pro ceedings of the 25th Annual ACM Symp osium on the Theory of Computing,ACM
Press, 1993, pages 315{324.
Learning from a Population of Hyp otheses. With Sebastian Seung. Machine
Learning 18, 1995, pages 255-276. An earlier version app eared in Pro ceedings of the
Sixth Annual Workshop on Computational Learning Theory, ACM Press, 1993, pages
101{110.
Horn Aproximations of Empirical Data. With Henry Kautz and Bart Selman.
Arti cial Intelligence 74 1 , 1995, pages 1129{1145.
Reasoning with Characteristic Mo dels. With Henry Kautz and Bart Selman. In
Pro ceedings of the 11th National Conference on Arti cial Intelligence, AAAI Press/MIT
Press, 1993, pages 34{39.
Toward Ecient Agnostic Learning. With Rob ert E. Schapire and Linda M. Sellie.
Machine Learning 17, 1994, pages 115{141. An earlier version app eared in Pro ceedings
of the Fifth Annual Workshop on Computational Learning Theory, ACM Press, 1992,
pages 341{352.
Oblivious PAC Learning of Concept Hierarchies. In Pro ceedings of the Tenth
National Conference on Arti cial Intelligence, AAAI Press/MIT Press, 1992, pages 215{
222.
Estimating Average-Case Learning Curves Using Bayesian, Statistical
Physics and VC Dimension Metho ds. With David Haussler, Manfred Opp er and
Rob ert E. Schapire. In Advances in Neural Information Pro cessing Systems 4, Morgan
Kaufmann Publishers, 1992, pages 855{862.
Bounds on the Sample Complexityof Bayesian Learning Using Information
Theory and the VC Dimension. With David Haussler and Rob ert E. Schapire. Ma-
chine Learning 14, 1994, pages 83{113. An earlier version app eared in Pro ceedings of
the Fourth Annual Workshop on Computational Learning Theory, Morgan Kaufmann
Publishers, 1991, pages 61{74.
On the Complexity of Teaching. With Sally Goldman. Journal of Computer and
System Sciences 50 1 , 1995, pages 20{31. An earlier version app eared in In Pro ceedings
of the Fourth Annual Workshop on Computational Learning Theory, Morgan Kaufmann
Publishers, 1991, pages 303{314.
Ecient Distribution-free Learning of Probabilistic Concepts. With Rob ert E.
Schapire. Journal of Computer and Systems Sciences 48 3 , 1994, pages 464{497. An
earlier version app eared in Pro ceedings of the 31st Annual IEEE Symp osium on Foun-
dations of Computer Science, IEEE Computer So ciety Press, 1990, pages 382{391.
Exact Identi cation of Circuits using Fixed Points of Ampli cation Func-
tions. With Sally A. Goldman and Rob ert E. Schapire. SIAM Journal on Computing
22 4 , 1993, pages 705{726. An earlier version app eared in Pro ceedings of the 31st An-
nual IEEE Symp osium on Foundations of Computer Science, IEEE Computer So ciety Press, 1990, pages 193{202.
Publications: On the Sample ComplexityofWeak Learning. With Sally A. Goldman and Rob ert
Articles E. Schapire. Information and Computation 117 2 , 1995, pages 276{287. An earlier ver-
continued sion app eared in Pro ceedings of the Third Annual Workshop on Computational Learning
Theory, Morgan Kaufmann Publishers, 1990, pages 217-231.
A Polynomial-time Algorithm for Learning k -variable Pattern Languages
from Examples. With Lenny Pitt. In Pro ceedings of the Second Annual Workshop on
Computational Learning Theory, Morgan Kaufmann Publishers, 1989, pages 57{69.
Cryptographic Limitations on Learning Bo olean Formulae and Finite Au-
tomata. With Leslie G. Valiant. Journal of the ACM 41 1 , 1994, pages 67{95. An
earlier version app eared in Pro ceedings of the 21st ACM Symp osium on the Theory of
Computing,ACM Press, 1989, pages 433{444.
Equivalence of Mo dels for Polynomial Learnability. With David Haussler, Nick
Littlestone, and Manfred Warmuth. Information and Computation 92 2 , 1991, pages
129{161. An earlier version app eared in Pro ceedings of the First Annual Workshop on
Computational Learning Theory, Morgan Kaufmann Publishers, 1988, pages 42{55.
A General Lower Bound on the Number of Examples Needed for Learning.
With Andrzej Ehrenfeucht, David Haussler, and Leslie G. Valiant. Information and
Computation, 82 3 , 1989, pages 247{261. An earlier version app eared in Pro ceedings
of the First Annual Workshop on Computational Learning Theory, Morgan Kaufmann
Publishers, 1988, pages 139{154.
Learning in the Presence of Malicious Errors. With Ming Li. SIAM Journal on
Computing 22 4 , 1993, pages 807{837. An earlier version app eared in Pro ceedings of
the 20th ACM Symp osium on the Theory of Computing,ACM Press, 1988, pages 267{
280.
Recent Results on Bo olean Concept Learning. With Ming Li, Lenny Pitt, and
Leslie G. Valiant. In Pro ceedings of the Fourth International Workshop on Machine
Learning, Morgan Kaufmann Publishers, 1987, pages 337{352.
Learning Bo olean Formulas. With Ming Li and Leslie G. Valiant. Journal of the
ACM 41 6 , 1995, pages 1298{1328.
On the Learnability of Bo olean Formulae. With Ming Li, Lenny Pitt, and Leslie
G. Valiant. In Pro ceedings of the 19th ACM Symp osium on the Theory of Computing,
ACM Press, 1987, pages 285{295.