Proceedings

of the

March 6 and 7, 1978

Computer Science and Statistics:

Eleventh Annual Symposium on theINTERFACE

Held at

North Carolina State University

Raleigh, N.C.

Edited byA. Ronald GallantThomas M. Gerig

Institute of Statistics

North Carolina State University

Copyright 1978by the Institute of Statistics

COMPUTER SCIENCE AND STATISTICS

ELEVENTH ANNUAL SYMPOSIUM ON THE INTERFACE

was supported by grants from

National Science Foundation

Army Research Office, Durham

Office of Naval Research

International Business Machines Corporation

Division of Research Resources, N. I. H.

Division Of Computer Research and Technology, N. Ie H.

Fogerty International Center, N. I. H.

and held in cooperation with the

Statistical Computing Section of the American Statistical Association

Association for Computing Machinery

North Carolina Chapter of the American Statistical Association

Central Carolina Chapter of the Association for Computing Machinery

ii

.ANNOUNCEMENTS

Computer Science and Statistics: Twelfth Annual Symposium on the Interface will

be held at the University of Waterloo, Waterloo, Ontario, Canada, on May 10-11,

1979. For information contact Jane F. Gentleman, Department of Statistics, Univer-

sity of Waterloo, Waterloo, Ontario, Canada N2L 3Gl.

Additional copies of these proceedings can be obtained from The Institute of

Statistics, North Carolina State University, P. O. Box 5457, Raleigh, NC 27650.

The Proceedings of the Tenth Symposium can be obtained from David Hogben, Statis-

tical Engineering Laboratory, Applied Mathematics Division, National Bureau of

Standards, U. S. Department of Commerce, Washington, D. C. 20234.

Proceedings of the Ninth Symposium may be obtained from Prindle, Weber and Schmidt,

Inc., 20 Newbury Street, Boston, Massachusetts 02116.

The Proceedings of the Eighth Interface Symposia can be obtained from Health Sci-

iences Computing Facility, AV-lll, Center for Health Sciences, University of ~

California, Los Angeles, California 90024.

The Proceedings of the Seventh Interface Symposia can be obtained from Statistical

Numerical Analysis and Data Processing Section, 117 Snedecor Hall, Iowa State Uni-

versity, Ames, Iowa 50010.

The Proceedings of the Fourth, Fifth, and Sixth Symposia can be obtained from

Western Periodicals Company, 13000 Raymer Street, North Holly"wood, California

91605.

iii

CONTENTS

PREFACE . . . . . . . . . . . . xKEniOTE ADDRESS 1

Eve.ry"thillg You Always Wanted to KnOl'1 About the History of ComputerScience and Statistics: Annual Symposia on the Interface--and More

Nancy R. Maon ~

WORRSHOP 1 - COMPUTING LANGUAGES

Rights and Responsibillties of Statistical Language UsersLanguage Standards for Statistical Computillg

John. Brocle . . . . . . . . . . . . . . . . . . . . .

Language Design and Statistical SystemsJohn M. Chambers . .

What is a Lang'.lage for Statistical Computing'?Richard L. Wexelblat . . .

~iORRSHOP 2 - COMPUTING METHODS FOR V.ARIANCE COMPONENTS

. . .

6

7

21

25

38

A Simple 'Synthesis '-Based Method of Variance Component EstimationH. O. Har.tley, J. N. Ie. Rao and Lynn. LaMotte

Mixed Model Algorithms for Estimating Non-Homogeneous VariancesWilliam J. Hemmerle and Brian W. Downs ......

Some Problems Faced in Making a Variance Com:gonent Algorithm. intoa General Mixed Model Program

Robert I. Jennrich and Paul F. Sampson . . . . . . . .

A Summary of Recently Developed Methods of Estimating VarianceComponents

S. R. Searle ..... 0

vlORKSHOP 3 - COMPUTING FOR ECONOMETRICS

Computation of the Exact Likelihood for an Ar.ma ProcessCraig F. .A.ns ley . . . . . . . . . . . . . . . . . .

Software for Rank DegeneracyGene Golub and Virginia Klema

39 e48

64

70

71

79

Test Problems and Test Procedures for Least Squares Algori thInsRoy H. Wampler . . 84

WORKSHOP 4 - GRAPHICS . . . . . . . . . . . . . . . . . . . . 91Portable Graphical Software for Data Analysis

Richard A. Becker . .

v

. . . . . . . . . . . . . 92

Visual and Computa.tional Considerations in Smoothing Scatterplotsby Robust Locally Weighted Regression

William S. Cleveland

Camputer-GeneratedMovies as an Analytic ToolRaymond L. Elliott . . . . . . 101

Computer Graphics Standards and Statistical Data PlottingJam.es D. Foley . . . . . . . . . . . . . . . . . . .

Using a Low-Cost Digitizer to Capture Questionnaire ResponsesJ. Michael Hewitt ..

Computer Graphics for Extracting Information from DataRonald K. Lohrding, Myrle M. Johnson and David E. Whiteman

104

110

114

An Interactive Graphical Data An~sis SystemRichard L. Phillips . . . . . . . . . . . 125

Data Structures for Cartographic Analysis and Displayw. R. Tobler . . . . . . . . . . . . . . . . .' . .

WORKSHOP 5 - STATISTICAL COMPUTING IN ADVERSARY PROCEEDINGS

The Lawyer and the Statistical--Computer ExpertAlfred A. Porro, Jr. . . .

. . . .

. . . . . . . .134

140

141

WORKSHOP 6 -' COMPUTING FOR BAYESIAN METHODS . . . . . . . . .. 159

Approximating Marginals From Non Analytically Integrable JointDensities

Marcel G. Dagenais Et Cong Liem Tran . . .

Matrix Weighted Averages: Computation and Presenta~ionHerman B. Leonard . . . . . . . .

Modularity, Readability, and Transportability of the Computer-Assis ted Data Analysi s (CADA) Monitor

Melvin R. Novick, Gerald L. Isaacs and Dennis F. DeKeyrel

160

164

176

A Survey of Numerical IntegrationArthur H. StrOUd . .

Bayesian Analysis and ComputingArnold Zellner . .

WORKSHOP 7 - NUMERICAL ALGORITEMS

. . . . . . . . . . . . . . . . . . . 181

195

203

A Computer Program for Model Building with Stepwise LogisticRegression

L. Engelm:Sl1 . . . . .. 204

Methods in BMDP for Dealing with Ill-Conditioned Data--Multicollinearityand Multivariate Outliers

James W. Fra.n.e . .- .. . . .. . .. .. . . . .. . . .. .. .. .. . .- .. ... 209

vi

Best Subsets Regression Under the Minimax CriterionJames E. Gentle and W. J. Kennedy

The Sweep Operator: its Importance in Statistical ComputingJ. H. Good.r1ight . . . . . . . . . .

215

218

Rejection Using Piece-Wise Linear Majorizing Functions in RandomVariate Generation

Bruce W. Schmeiser and Mohamed A. Shalaby . . 230

Numerical Determination of the Distributions of Stopping VariablesAssociated with Sequential Procedures for Detecting Epochs ofShift in Distributions of Discrete Random Variables

S. Zacks . . . . . . . . . . . . . . . . . . . .

Implementation of Harmonic Data Analysis ProceduresMichael E. Tarter

Finding Influential Subsets of Data in Regression ModelsRoy E. Welsch and Stephen C. Peters .

WORBSHOP 8 - HIGH DIMENSIONAL FILES: LARGE OR COMPLEX . . . .

234

240

245

250

Sir, A Working System for Managing Large and Complex Research DataGary D. Anderson, Eli Cohen, Wally Gazdzik and Barry Robinson 251

Data Management in SAS and Interfaces to Other SystemsA. J . Barr . . .

Effects of Larger Files and Mor~ Varied Usage on the Developmentof the P-Stat System

Roald Buhler and Shirrell Buhler . . . .

. . . . . . . . . . . .Management of Complex Data Structures in a Clinical ResearchEnvironme.p.t

Michael A. Fox 270

The Data Interchange File: Progress Toward Design and ImplementationRichard C. Roistacher . . . . 274

WORKSHOP 9 - COMBINATORIAL COMPUTING IN STATISTICS . . . . . 285

The Whys and Wherefores of Algori tbm DesignJon Louis Bentley .

Algorithms and Data Structures for Range QueriesJon Louis Bentley and Jerome H. Friedman

. . . . . . . .' . . . 286

297

Space-Efficient On-Line Selection AlgorithmsBruce W. Weide

CONTRIBUTIONS BY PAPER

vii

. . . . . . . . .308

313 e

Statistical Methods in Computer Performance Evaluation: A BinomialApproach to the Comparison Problem

Paul D. Amer and Sandra A. Mamrak . . . . . . . .

Gauss-Jordan vs. CholeskiKenneth N. Berk . .

Numerical Solutions of the Beta DistributionHubert Bouver and Rolf E. Bargmann .

. . . . . . . . . . . . .

314

321

325

An Algorithm to Derive Mnemonics for Computer UsageJohn Brode, Jeffrey Stamen and Robert Wallace

Statistical Distance Measures and Test Site Selection: SomeConsiderations

Charles D. Cowan and Randall K. Spoeri . . . . . . .

Smooth Curve Fitting with Shape Preservation Using OsculatoryQuadratic Splines

L. E. Deimel, D. F. McAllister and J. A. Roulier ...

Fosol: A Language for Statistical Computing, Matrix Algebra andTop-to-Bottom Structured Programming

D. Anton Floria.n . . . . . . . . . . . . . . . . . . .

Variance-Component Estimation for the Unbalanced Two-Way RandomClassification

F. Giesbrecht and Lynn Dix . . . . . . . . . . . . . . .

337

338

348

355

Analysis of Data From a Research AircraftJ. E. Grimes . . . . . . . . . . . .

Variance Estimation Using Half-Sample ReplicationsRobert S. Jewett . . . . . . . . .

. . . . . . . . ... 361

Calculation of Chi-SquareClifford J. Maloney . . . . . . . . . 373

The Role of Pseudo-Random Number Generators in the Hope-FilledVariance-Reducing Efforts

G. Arthur Mihram . . . . . . . . . .

Software for Iteratively Reweighted Least Squares ComputationsStephen C. Peters and Virginia C. Ia..ema ~d Paul Holland.

376

380

Survey Software--A Method for Handling the Variance ProblemClark Readler . . . . . . . . . . . . . . . . . . . . . . . 385

Least Squares Spline for Incorrectly-Posed Problems: InformationTheoreti c Approach

Rilni 0 TSllabe . . . . . . 388

A Short Algorithm to Transform Dissimilarities into DistancesHoang M. Thu . . . . . . . . . . . . . . . . . . .. 392

viii

395 e400. . . . . .

. . .A Notation for Specifying Contingent Effects in Analysis-of-Variance Designs

Ervin H. Young

CONTRIBUTIONS BY ABSTRACT

Edit and Imputation Procedures at Statistics CanadaLen Baniuk, J. H. Johnson and G. Sande . . . .

A Fast Algorithm for Estimating the Number of PolyneutronsDetected in the Presence of Poisson Noise

W. A. Beyer and C. Qualls 401

Rummage - A Data Analysis SystemG. Rex Bryce, Del T. Scott and Melvin W. Carter 402

A Preview of P-Stat 78Roald Buhler and Shirrell Buhler 402

Graphics for Seasonal Time Series AnalysisWilliam S. Cleveland, Douglas M. Dunn and Irma J. Terpenning 403

ModUlar Programming in Numerical and Statistical AnalysisW. H. Connelly.. . . . . . . . . . . . . . . . . . . . . . . . . 404

Confidentiality Procedures in Statistical AgenciesL. H. Cox and G. Sande . 404

,URWAS - Universi ty of Rochester Weighted Anova System

Michael L. Davidson and Jerome D. Toporek . G 405

Estimating Latent Score Regressions when Measurement Error Variancesare Known

Noel Dunivant 405

Patient Profiling Systems Through a MinicomputerTurkan K. Gardenier . . . . . . 4

PREFACE

This symposium was the eleventh in a continuing series of symposia organized for

those interested in the intersection of the disciplines and professions of com-

puter science and statistics. This series of symposia has, by now, achieved the

status of a major national meeting within this community.

The customary format of parallel workshops following a plenary session with

keynote address was continued. Each workshop was organized by its chairman and

consists of invited speakers presenting papers on the workshop topic followed by

open discussion of the paper by all participants. Some workshop chairman have

inclUded designated discussants as well. Contributed papers were presented in

poster sessions continuing the successful innovation introduced in the ninth

symposium.

The major departure from previous symposia is the publication of the proceedings

in advance for use by participants during the symposium. Several benefits were

anticipated from pre-publication. Much of the distraction of distribution of

handouts and note taking during workshop presentations is eliminated. Workshop

participants, having read the paper, can eliminate much of the discussion related

to clarification of the invited speakers ideas and can focus their comments on

the ideas themselves. Parallel workshops unavoidably require the participants

to be unable to attend some workshops but, having the papers from these work-

shops in hand, they may read them and recapture many of the benefits of attendance

by discussing the papers with speakers and attendees of the missed workshop during

the session breaks, at lunch, and in the evening. Incidentally, one is also

assured that invited speakers have adequately prepared their presentations.

Pre-publication entails a rigid and compressed pUblication schedule. The task

of arranging so many papers into a camera ready manuscript in such a short time

appeared impossible. Mrs. Margaret Rice single-handedly accomplished the impossi-

ble. There are no words adequate to express our debt to her.

e It is fitting at the end of a decade to review and record the history of thesymposium series. Thus, we were fortunate to have Nancy Mann who is with the

x

Science Center at Rockwell International as our keynote speaker. She is well-

known in the profession for her many professional accomplishments and among them

is that she was one of the founders of this series of symposia. In her keynote

address she recounts the growth of her progeny. There were nine workshops:

2.

3.

4.

5

6.

7

Statistical Computing Languages. Organized by Lawrence C. Rafsky. Dr. Rafsky

is statistician, Chase Manhattan Bank, and is teaching in the Department of

Computer Science, Stevens Institute of Technology. His teaching is in numeri-

cal analysis and his primary research interests are in the interface.

Computing Methods for Variance Components. Organized by William J. Hemmerle.

Professor Hemmerle is Chairman of the Department of Computer Science and

Experimental Statistics at the University of Rhode Island. He is a recognized

authority on statistical computing for linear models (including variance com-

ponents) and has published extensively on the topic.

Computing for Econometrics. Organized by Warren Dent. Professor Dent is

wi th the Institute for Economic Research, The University of Iowa. He is pre-

sently editing a special volume on computing in econometrics for the Journal

of Econometrics.Graphics. Organized by Ronald K. Lohrding. Dr. Lohrding, a statistician, is

the Group Leader of the Energy Systems and Statistics Group in the Energy

Division at the Los Alamos Scientific Laboratory in Los Alamos, New Mexico.

Dr. Lohrding and his staff have been doing extensive research into graphic

displays of statistical information and data.

Statistical Computing in Adversary Proceedings. Organized by John S. deCani.

Professor deCani is Chairman, Department of Statistics, The Wharton School,

University of Pennsylvania. He has much experience in the preparation of

statistical testimony for judicial proceedings.

Computing for Bayesian Statistical Methods. Organized by Joseph B. Kadane.Professor Kadane is Head of the Department of Statistics at Carnegie-Mellon

Universi ty. He is well-known for his research in Bayesian statistical

theory and methods and is a major participant in the development of a

Bayesian regression procedure in the NBER TROLL system.

Numerical Algori throB. Organized by William J. Kennedy. Professor Kennedy is

Head of the Numerical Analysis Section of the Statistical Laboratory at Iowa

State University and Chairman of the Committee for Evaluation of Numerical

Algorithms in the Statistical Computing Section of the American Statistical

Association. He was general chairman for the seventh symposium.

xi

8. Large Data Files. Organized by Frank M. Stitt. Dr. Stitt is Director ofClinical Sciences, ALZA Research, and is associated with the Health Sciences

Computing Facility, University of California at Los Angeles. He has much

experience in data base management and is responsible for new therapeutic

clinical trials activity at ALZA. His research interest is in the develop-

ment of information systems in cli.rrlcal research.

9. Combinatorial Computing in Statistics. Organized by Joseph. B. Kadane.Professor Kadane's professional activities are summarized above.

A. Ronald Gallant

Thomas M. Gerig

Symposium Chairmen

./

xii

KEYNOTE ADDRESS

EVERYTHING YOU ALWAYS WANTED TO mOW.ABOUT TEE HISTORYOF COMPUTER SCIENCE AND STATISTICS: ANNUAL SYMPOSIA

ON THE INTERFACE--AND MORE

NANCY R. MANN

Science Center, Rockwell InternationalThousand Oaks, California 91360

1

KEYNOTE ADDRESS

EVERYTHING YOU ALWAYS WANTED TO KNOW ABOUT THE HISTORYOF COMPUTER SCIENCE AND STATISTICS: ANNUAL SYMPOSIA

ON THE INTERFACE--AND MORE

Nancy R. Mann

Science Center, Rockwell InternationalThousand Oaks, California 91360

INTRODUCTION

This morning I'm going to talk about the history ofComputer Science and Statietias: Symposia on theInterface and related events applying particularly tothe American Statistical Association. This is somethingthat I haven't spoken about before in front of a 1aroeaudience and, in fact. so~ethinQ that I hadn't thouo~tabout for a good number of years at the time Sob r1onroephoned and invited Me to speak. I'm assumin~, since~e are beginning what would be the second decade ofInterface symposia, except for aoproximate1y half-yearslippages betl'leen the third and fourth and the sevl!nthand eighth meetings, that the conference orQanizersfelt this is an appropriate time to look back to seehow it all began and what has happened since itsbeginning.

I wish that on this occasion I could tell you thatreflection upon this history has enabled me to gleansome universal truths to PASS on to you, such as, forexample, those in the three-pronged ecological maxim:evel'1!,:hing is tJonnect:ed to ..vezndthir-{1 eZee, evezn,/thingis dyr.amia(l ZZy changing and there's 1".0 such ';;!~i1!f' ':IS afreeZunch. Unfortunately I haven't been able to findany.

Before I started writing. I read over the keynoteaddresses of Frank ftnscombe, Dick HamminQ, H. O. Hartley,rlartin \'li Ik, Edwin Kuh, John Rice, and Tonv Ralstonappearing in the Proceedings of previous Interfacesymposia. I felt much like Charlie Brown on theoccasion when Lucy commented to Charlie ann to Linusthat she had read that if one stared at clouds. it waspossible to see various familiar objects in them."\4hat do you see in those clouds overhead, Linus?" sheasked. Linus resoonded. "1 see in the configurationof that cloud over there the outline of BritishHonduras. And I see in that second cloud a sculPturemade by the r.;ost outstanding sculptor in contemporaryAmerica. The aesthetics are indesc:oibable and verymoving! And then in that third cloud I see Saul whobecame the great Apostle Paul, standing watchina thosewho were stoning to death the first ehdstian martyr,Stephen."

"':low, Linus." said Lucy, "that's fantastic! "Ihat doyou see, Charl ie Brown?"

And Charlie Brown said "~Iell, I was qOlno to say I seea horsey and a ducky. but ! changed my mi nd. "

Since it's a bit too late now to change my mind aboutdelivering this keynote address, I'll beain DY notinothat i was surprised to discover during my reflectionthat I was present at a substantial number of significantevents in the early part of the history of I..hich I shallspeak. Let me !Jegin at the beginninQ.

2

THE BIRTH OF INTERFft.CE I

The person whom we oldtimers all know to be the oriainalguru of the Interface is Arnie r,ooaman, known to hisparents as Arnold Frank r,oodman. Arnie spent his ~ndergraduate years here at North f4rolina State where heworked two surrmers with. "l':urlv" Lucas. Followino hisgraduation he ~v~d ~est to Stanford, where, in 1961 hereceived his PhD in Statistics under the direction ofHerman Chernoff. He then settlea down in SouthernCalifornia to work in industry, where he soon becameinterested in co~puters and comouter systems.

By the time I met Arnie in 1~63. he had become convincedthat mankind (personkind?) was huncerina. thouoh oer-haps unconsciously, for a syrr.nosium on the interface ofcomputer science and statistics. Sometime in late 195after he took office as President-Elect of theSouthern California ASA chaoter, ~e seized upon theopportunity afforded by this office to assemble asymposium committee, and he attempted to galvanize usinto action.

The committee included representatives frOM the ASAChapter, from a loosely-knit Southern Californiaorganization called SPEC (Statistical Proaram Evalua-tion Committee), to which Arnie had been deliveringhis Interface propaQanda, and from the Los AngelesChaoter of ft~l. Sol Pollack, the Chairman of the Ar.Mchapter, \~rked near Arnie at the Space ~ivision ofRockwell. and he had been converted early on toInterface consciousness.

The cOrmtittee activities got off to a slow startbecause we initially made contact with officials (Ifa Business School of a local university who seemedto view this as an opportunity for a large prooortionof their staff to participate and to receivehonoraria. Since our working budget was what mighteuphemistically be called "zero based" and since I'lewanted more freedom in choosing our format andsoeakers, we declined the honor and looked elsewhere.Committee member Mitchell Locks contacted his secondcousin (or maybe it was his wife's second cousin), aman named Larry Emanuel, who was associated withUniversity of California Extension at UCLA. Larrythought the Interface premise was a good one forUC Extension sponsorship and was more than ~illin9to let us have a free hand in planning the nrogram.So we were off and running, and it turned out to bea pleasant alliance. 8y the time ~(e got toqet~erwith UC Extension, Arnie was looking fon1ar~ tobeing President of the ASA Chapter and decided thatit would be inapprooriate for hi~ to continue tochair the symoosium committee. For reasons knownonly to him and his maker, he saw me as his mostlikely successor and managed to convince me thatInterface chairmanship was my destiny.

Following the initiation of My chairmanshio, tl'iecommittee met and set down tne pur,ose of the Sym-posium, the essence of which can be found inMike Tarter's Preface in the Proceedings of thesixth meeting:

To faaii-ita

The second Interface symposium was held January 25-26. 1968at the International Hotel. near the Los Angeles airport.The third. Chaired by Ed Robison (then of TRW Systemsand now at Union College) was held January 30-31, 1969at the same spot. For both meetings the format remainedthe same as for Interface I, except for a time-frameexpansion to two days. This is in contrast to writteninformation you may have seen describing all of the firstthree symposia as one-day meetings. We added an addi-tional sponsor, the Los Angeles Chapter of IEEE SystemsScience and Cybernetics Group. for the third conference.Otherwise. sponsorshlp remained unchanged.

The topics treated at the second and third meetingswere varied. ranging from SociaZ .4ppEaations andI!/r?ZicatiOM'lationaZ IrtlO1'rrr:r:t:ion Systems, Jw-I-metricsand ComputeI' Aids to Statistical ~,zucation to .'1andom,'lurr;,.bel' Genezoa.tion. ComputeI' ':echnoZog',j. and 11'!lOmationS~ence. Two well-known personalities who are no longerwith us participated in these two symposia.George Forsythe of Stanford spoke about r.umezoiaaZ. aZco-zoithms at the second. and "Curly" Lucas of ~.r.. Statediscussed "The Role of Statistics and the ComputQr inBiomathematics" at the third. -

The tradition of scintillating luncheon addresses con-tinUed. At the second. Ross Adey of the UCLA BrainResearch Institute told us on the first day about "TheUse of Computers in the Search for Memory Traces inthe Living Brain"; and Frank Proschan. then still atBoeing Research Labs. regaled us on the second with"Some Effects Computer Science Has Had on Statistics."If the latter topic seems to you an unlikely one toevoke Chuckles. it's only because you haven't heardFrank Proschan at his best. This was the first ofwhat has turned out to be many times I've introducedFrank. It's a privilege not meant for the faint-hearted.

At lunchtime on the first day of the third meeting.Jack roloshman gave us a "Retrospective View of ProjectedElection Returns: 1968 Version." and Ed Davis. therecently-retired Los Angeles Chief or' POlice. who isbeginning the process of running for Governor ofCalifornia. told us on the second day about "Law.Order, Data and Computers." You may be surpri sed.if you've read about.Ed Davis lately, to learn thathe. has been a part-tlme facul tv member at theUnlversity of Southern California.

The fourth symposium. which was chaired byMitchell Locks and held at U.C. Irvine, marked adeparture from the original format. The most radi-cal change, aside from tne use of campus facilities,was in the organization of the meeting into ~orkshops.a structure that has been continued since. The work-shops for the fourth symposium dealt ~ith ~edicaZ.statistics and computation. secondazood education, hard-Y:aZOelaoft;-.xczoe assign and evaLuation by statisr:icaLr:-:ethocis, and aomputer Zanguages [01' statistia-:.ans.For the first time there was financial support from agovernment agency, the Army Research Office. Durnam.and for the first time a Proceedings was written anddistributed to attendees in microfiche. form. Forsubsequent symposia there have been hard copy ?rc-ceedings.

The fourth symposium also saw the demise of the luncheontalk and the birth of the keynote address. There were,in fact, two keynote addresses at that meeting.Frank Anscombe spoke of the need for flexibility instatistical computing and DicK Hamming discussed tneresponse of statisticians to the computer and computerscience and the res~unse of computer scientists tostatistics. 80th responses were discussed in terms ofwhat had been done contrasted with what Hamming believeds~ould kave "een Aone. Tt "as 1:ki~ tal~ tw qa"""iM 1:"atintroduced the concect of ccmoumetrias. the measurementof computi ng systems, to the Interface. A si gni fi cantconsequence of this presentation was the formation ofSICr1ETRICS, a special-interest group on Compurnetricswithin the Los Angeles Chapter of ACM.

Hamming gave versions of his talK to a meeting of theSouthern California chapter of A.S.A and as the GeneralMethodology Lecture at the 1971 Annual Meeting of ASA

4

in Fort Collins, Colorado. The part of it that r recallmost Vividly from the various cresentations I heardpointed aut the waste of talent involved in "fillinc1 inchinks" by extendin!! or general izing al ready croventheorems in statistics. "If you do this. he said, "youwi 11 read the. paper, and the referee wi 11 read it." Heobserved that the probability that anyone else will wadethrough it is small. He also noted that one way tobecome a famous statistician is to find a nethod ofattacking and solVing an i~por1:ant r~alworld statis-tical probl~. The solution need not be elegant,only useful and easy to apply. He discussed severalsuch problems in computer science.

THE INTERFACE AND ASA

Hamming's talk at the ASA Annual Meeting tOOk placeduring the birth of the ASA Statistical ComoutingSection, following submittal to the Board of Direc-tors and Council of a petition by members of theAssociation.

You may not recall that ASA's first advertisedadvocacy of tne interface of computer science andstatistics took place at its Annual meeting inDecember of 1967 in Washington. D.C. And I'mfairly certain that you're unaware that Arnie Goodmanwas behind it from the beginning.

It happened as it did because I made a trio towasnington. in the early summer of 1977. Shortlybefore the event, Arnie wrote to Don Riley, thenASA Executive Director, that I was planning to bein Washington for a few days and would be availablefor a lunch-time d1scussion on two of those days.The implication seemed to be that it was an oppor-tunity not to be missed.

I had been a member of ASA only a short time in1967. was not at all active and had never heard ofDon Riley. I'm fairly certain that, likewise,Don Riley had never heard of me. Nevertheless.~n wrote me a letter graciously inviting me to meethlm and other ASA representatives inside the ladies'entrance of the Cosmos Club during one of my avail-able lunch times. Since my consciousness of malechauvinism was languishing comfortably, yet to beraised. I entered the ladies' entrance of theCosmos Club at the appoi~ted time with no more thanmild amusement. There I met Don, Ed ~isgyer, thenBusiness Manager and now Managing Director of ASA.and Joe Daly. then Chief Mathematical and Statis-tical Advisor for tne ~ureau of tne Census.

The main remembrance I have of the early part of ourluncheon is that Don and I discovered we were ?e110wOhioans and consequently hit it off extremely well.This was not so long a time before Don passed away,and it turned out to be tne only time I ever saw orinteracted "lith him. On this occasion, hOI'lever, hewas enthusiastic and ebuliient. Arnie had aiven mean Interface message to deliver and before wefinished our coffee. I managed to sur.mon it to mind.I can remember now very little of the details of themessage or even the point of it all. I can onlyrecall that as I listened to myself I was terriblvimpressed by my eloquence, a quaiity I had neverfelt I'd exhibited on a single previous occasion.On the contrary, comments about my oral presentationsusually indicated a discernable lack of eloquence.I do remember that the finish of this impassionedspeech concerned the need for a seSSlon on computerscience and statistics at future national ASAmeetings.

Don. while perhaos not so awed as I bv the e10auenceof my remarks. was sufficiently moved to ooundonthe table a couple of times and declare that therewould be such a session at the next Annual Meetingand that I should organize and chair it. This wouldentail Ed Bisgyer's contacting Carl Bennett. theProgram Chairman for 1967, and having him provide atimeslot.

While organizing the session seemed appropriate enoughto me, I wasn't sure that l shoula be the person tochair it. I remember checking with Ed because he

seemd not to be partlcular1y carried away by the emo-tion of the moment. But he assured me that it wouldbe most proper. I can't remember Joe Daly's reactionto the proceedings, but presumably he was in generalagreement.

Once everything was cleared with Carl Bennett, I con-tacted John Tukey to be a speaker for the session(which my title consultant and I called "ComputerScience and Statistics: A VieW of the Bridge").Thus, even though the session began at 8:30 a.m.and the room we were assigned held about 500 people,we had a capacity audience. The next year at the ASAAnnual Meeting in Pittsburgh, there were four sessionson the Interface. A COM'Ili ttee on Comr;uters inStatistics had also been organized, with membershipinclUding Joe Daly and Will Dixon.

In 1972, just after the StatistiaaZ Ccmputi~~ S~ctionreplaced the ASA Comrr.'it7:se on Ccrrrcut;el's in St;at;istias,Will Dixon, its first Chairman, phoned and aSked me todr.aft its Charter. I thought this sounded about asonerous a task to ask anyone to perform as I couldimagine, about as much fun as watching paint dry--andsai d so. I found, however, tnatit l'lasn' t too di ffi -cult to accomplish by simply selecting pieces I likedfrom existing Section charters. And, of course, themajor part of the Scope of the Section was borrowedfrom the purpose of the Interface Symposia that we hadset down in 1966. You might note that the name Sectionon StatistiaaZ ComDUting really doesn't reflect allaspects of this Scope. However, the name was cnangedfrom Sea1;icn on Statistias c:nd Computers, gi ven in myoriginal draft.

THE MOVE TOl~ARO INTERFACE INDEPENDENCE

In 1971 the Interface conference was held outside ofSouthern California for the first time. The ExecutiveCommittee of the Southern California Chapter of ASAvoted to allow Mitchell Locks to organize the fifthannual Interface symposium at Oklahoma State University,where he had become affiliated upon leaving SouthernCalifornia himself. The next year, Mike Tarter, whohad acted as an Associate Chairman when the symposiumwas held at U.C. Irvine asked the Chapter for permissionto host the meeting at U.C. Berkeley, where he hadmoved from U.C. Irvine. Before the organization ofthe meeting 1n Berkeley got well underway, i1ike, Arnieand I met with l'li 11 Oi xon, in 01 s offi ce, and puttogether a veraal agreement that has given the InterfaceSymposium an identity of its own, independent of potentialsponsors. ?'e decided, with later blessings from the ,4$."Chapter and all pioneer Interfacers, that there "~Uld ~ea committee made up of ex-chairMen of Interface symposiaand headed by the most recent Chairman. This committeewould have the power to decide who would be given theprivilege of organizing a next symposium afte~ the onecurrently being organized. The agreement extends nowto people who may nave oeen unaware of the Interfacesjo'l:lposia at the time it 'lias made. Ultimately, it couldextend to people who weren't yet born when it was made.

DIRECTION OF THE INTERFACE

Since the time of our meeting in Yill Dixon's office.the Seation en Sta7:istiaaZ C~~u"ina has oeen listedamong the sponsors of the Interface"symposia. One cannote, too, a large overlao of recent officers of theSection and recent organizers of the Interface symposia.Following 11ike Tarter's sixth symposium at Berkeley cameSill Kennedy's Iowa State meeting and .Jim Frane's atUCLA (back in Southern California), David Hoaglin andRoy !'/elsch chaired the ninth at Harvard and '~IT, andDavi d Hogben and Denni s Fi fe organi zed the tenth atthe Bureau of Standards.

It is notable that the Statistical Computing Sectionhas seemed to remain true to its name and to havebecome a section devoted almost entirely to statisti-cal computing. The Interface symposia, on the otherhand, nave genera11y continueo to treat a11 threeaspects of computer science and statistics set downin 1966 in the origlnal purpose. The sixth, seventh,eighth, ninth, and tenth meetings have offered work-shops or papers on, for example, aomputer system ,er-

5

fozwrrna'3 evaZuation, robust so.f"::t.re, ra:r.dom numbergeneration, eva~uation of a management info!"!lat:ionsystem, high-reveZ program l.a:nguages and statistias,and pattel'n reaognition. And Tony Ralston's keynoteaddress for the tenth dealt with the mathematizingof computer science.

One could, I believe, describe the published programfor the meeting we're presently attending as one fora symposium on s'tatisticaZ aomputing. The topicsare many and varied, and some sound fascinating tome. Still, it seems evident that the organizers ofthis meeting were not brainwashed in the early tradi-tions we oldtimers were taught to hold near anddear. The chairman of symposium number twelve,Jane Gentleman, can feel free to consider thiscomment an implicit suggestion carried from the guru.Arnold Goodman, by me, Nancy (just plain) Mann.

WORKSHOP l

COMPUTING LANGUAGES

Chair: Lawrence C. Rafsky, Chase Manhattan Bank

6

RIGHTS AND RESPONSIBILITIES OF STATISTICAL LANGUAGE USERSLANGUAGE STANDARDS FOR STATISTICAL COMPUTING

by

John Brode, Cambridge, MA

The fundamental concept presented in this paper is that of the non-procedurallanguage. The us er should only be concerned wi th what needs to be done; the computershould work out the how. Recent advances in computer science (extension of the pro-perties of context free languages to subsets of context sensi tive languages, Petrinets, etc.) can be joined w~th recent work in mathematics (fuzzy sub-sets, theory ofcategories, etc.) leading to useful new ideas for statistical computing languages.Standards are proposed for statistical computing languages.

A)B)

C)

IntroductionJunction of recent advances in linguistic theory and mathematicsa) Recent advances in linguistic theoryb) Recent advances in mathematicsc) Applications of the junction

1) Non-procedural programming; 2) Backtracking; 3) Petri nets;4) Fuzzy set theory; 5) Theory of categories

d) SuromaryLang-llage standards for statistical computing

A) IntroductionIt is the right of every user of a sta-

tistical language to ask that such a languagebe as useful as the current state of com~uterscience will allow. To get this right, 1tmust be ~~e responsibility of each user toknow enough about computer science to dis-tinguish what is known or possible from theindiVidual opinions of practitioners 0:programming. .

ThE! interface between computer science/linguistics and statistics has of~en beenlacking in the past. Far too often, thisinterface has been between a highly trainedstatistician and a programmer who is, per-haps, experienced in the writing of somecomputer language but who is seldom trainedin computer science. Too many statisticalcomputing languages seem to have been im-posed on statisticians by programmers ofvery narrow acquaintance with computerscience.

The purpose of this paper will notbe to present a tutorial in any language,be that language implemented or just pro-posed. (See the Bibliography for a listof references to more than thirty languagesof interest.) Ra~her, I shall try to pre-sent in this paper an outline of what isneeded by statis~ical computing and whatcan be supplied by computer science.

The fundamental concept containedin this paper is that of a non-proceduralprogram or language. RA non-proceduralprogram is a prescription for solving aproblem without regard to details of ~it is solved. R (Leavenworth & Sammet('74) p. 2)

7

one wants to deal in so far aspossible with the behavior whichthe program and its parts are toexhibit--and to suppress the dE!tailof just how this is accom~lisheddemanding-of course that the how'be consistent with (that is, cor-rectly implement) the desired be-havior. a (Cheatham & To~~ley ('75)p. 4)

This is what Sammet ('i2) has called aproblem defining language. (See alsoKassels (177) and Klinkhamer ('72).)

The :undamental a~Droach taken in':his paper is that of ia: precise algebraiclanguage (see Loos ('74)). This is theapproach of APL (Iverson ('62)) but While~~is lanquage is not the answer for evervproblem, especially those tending to the-non-numeric:

NThere is an important lesson tobe learned from APL, however, andthat is the importance of structuralsimplicity and a.'1. excellent-smoothand comfor~able--system surroundingthe lan~~age.ft (Cheatham & Townley('75) p. 10)

I shall try in this paper to extend thissimplicity to handle a wider range ofproblems that may not be deterministic.To do this, we shall need programs thatcan by themselves deduce the how of solvinga problem.

B. Junction of recent advances in lincuistictheorv and mathe~aticsa. Recent advances in linguistic theory

Modern linguistic theory starts withthe work of Chomsky ('56 & '59). He defineda series of language types of which type 1(context sensitive) and type 2 (contextfree) are of interest here. If there is agrammar rule that changes a token 'A' intoa grammatical structure 'w', then these twotypes may be defined as follows:

type 1: t A(/)2~ w

Linguistic work done on type 2 languageshas been more rigorous, since they are moreamenable to precise mathematical treatment.Central to this work is the emptiness pro-blem. Essentially, this problem pertains tothe proof that the application of the rulesof a grammar a finite number of times toan input stream of tokens from the languagegenerated by th~ grammar, will not producean empty res~. (See Aho & Ullman ('73)for a somewhat general survey; Hopcroft &Ullman ('69), Hoare & Lauer ('74), orSalomaa ('73) for more precise surveys.)

Unfortunately, too much attention hasbeen given to the neatness of these proofs.Chomsky, in his original work, finds thatneither type 1 nor type 2 languages are~exactly what is required for the completereconstr~ction of immediate constituentanalysis. I. (Chomsky (I S9) p. 148}

Furthermore, in some cases, inadequateattention has been paid to the rigorousmeaning (in the mathematical sense) ofwhat has been proven. The emptiness problemis solvable fer context free grammars(type 2), but is not solvable for contextsensitive grammars (type 1). (Th. 14.2Hopcroft & Ullman ('69) p. 219) Such a proofassures us that the application of the rulesof the grammar by an automatic algorit~mWill, in a finite number of steps, reducethe input stream to a meaningful statement.This theorem, h.::wever, does not prove that~o context sensitive (and not context free)grammar exists for which the emptinessproblem is solvable.

In particular, starting with the workof Knuth ('65), a form or parsing ( areduction of the input stream accordingto grammatical rules) called LR(k) has beendevelo~ed Which is coterminus with theset ofdeter~inistic languages (essentiallylan~Jages for which the grammatical rulesare unambiguous (Th. 12.1 & Prop. 12.1Salomaa ('73) p 224 & P 2Z8). The set ofdet.er:.u..... S1:l.C lanq..:agos, ilowever, is netcontained in the set of context free lang-uages nor does it Contain the set of con-text free languages.

It can be shown, that some contextsensitive languages are deterministic andthat, therefore, the LR(k) approa::;l isapplicable. 1 (see Walters ('7l)) Thus it

~he 'k' in LRlk} refers basically to the

8

number of symbols that, at most, are tobe examined on either side of the currentsymbol before a decision can be m~de asto the interpretation of tha~ sy:wol. It a.can be shown, however, that for any de- . ~terministic language, there exists an LR(O)grammar. (See Th. 12.9 Hopcroft & Ullman('69) p. 185 for the precise statement)For various reasons, LR(O) grammars aremost readily handled as LR(l) grammars.(see Aho & Johnson (174

is not true that LR(l) parsers need beconfined to context free languages.

BNF (Backus-Naur Form,2 see Naur

ZBNFis often given as Backus Normal Forminstead of Backus-Naur Form. I prefer thelatter, reserving the term "normal form"for the mOst elementary formal structuresuch as the Chomsky Normal Form or theGreibach Normal Form. BNF is more properlya meta-language-Le., a language to des-cribe languages.

('63 & '60) and Backus ('59)} is oftentouted as the best way to write out theformal description of a grammar. BNF iscoterminus with the context free grammars.(see Aho & Ullman ('73 But since it candescribe ambiguous3 grammars, not all BNF

3A grammar is ambiguous if the applicationof its rules toa word in the language itgenerates can give two different meanings.(see Salomaa ('73) p. 54 for a more precisedefinition)

describable grammars are LR(l} parsable.On the contrary, some grammars not des-cribable in BNF are LR(l} parsable. (seeAho Johnson & Ullman (17~) for a statementon the inade~acy of BNF. ote. that Aho &Johnson l'74) seem to be mistaken in assertingthat LR(l) parsing is useful only for con-text free languages)

It is not possible to snow whetheran arbitrary context free grammar isambiguous or not. (Th. 14.7 Hopcroft &Ullman ('69) p. 222) Some context f=eegrammars (and, therefore, their BNF des-criptions) can be made ambiguous. (seeEarley ('75 Some recent work has beendone en the parsing of lan~~ages generatedbv such ambiguous grammars. (see Aho,Johnson, & Ullman ('75); Aho & Johnsen('74)' Earley ('7S); Sheil ('76); andWhart~n ('76}) The approach taken in theseworks is to apply "disambiguating rules"whenever there is more than one possibleinterpretation of an input constr~c~.

An example can be taken from operatorprecedence. Standard FORTRAN precedenceis for the operator '*' (multiplication)to take orecedence over '+' (addition).In B~, this is:

4.rhese two lines are to be read as: "anarbitrary expression is defined either asan arbitrary term or as an arbitrary ex-pression 'plus' an arbitrary term" and "anarbitrary term is defined either as anarbitrary identifying symbol or as anarbitrary term 'times' an arbitrary identi-fying symbol."

This could be written, ambiguously, as:

::=!+

I*Disambiguation would then follow accordingto a table of operator precedence.

A language can be deterministic evenif it has an ambiguous grammar providedthat such ambiguity is resolved (from out-side the grammar) in a deterministicfashion. 5

ssee Sheil ('76) for a more rigorous ex-position of this point. Sheil limits am-biguity away from "direct ambiguity" (i.e.,ambiguity that is preserved through reduction).

The disambiguating rules discussed above(essentially a table lookup) are dete~inistic. Therefore, such a grammar can beLR(l) parsed. (see de Remer ('71

The importance of ambiguity lies in thevirtual impossibility of avoiding it. Thelanguages generated by unambiguous grammarstend to be uninteresting and inflexible.(see Hamlet ('77 ALGOL 60, probably themost rigorous high level language written(and for that reason replaced in actualuse by the less rigorous ALGOL 68) is knownto be ambiguous (Wharton ('76 and, ap-parently, cannot be precisely e~ressed inEnglish. (Hamlet ('77) p. 87-8)

6ALGOL.iS not even a context free languagedue largely to :'ts "semantic" rules forhandling declarations and the like. (seeArbib ('69) p. 172) Nor does it have a"phrase structure grammar" be that contextsensitive or context free. (see Floy ('62

Of equal importance, is the practicalresult that oarsers constructed for certainambiguous grammars are, in some sense, moreefficient thpn those for unambiguous grammars.(Aho & Johnson ('74 Sheil ('76) shows that"boundedly ambiguous grammars" can have atmost a polynomial bound on the number ofsteps needed :'n parsing an input stream ifuse is made of a "well formed substringtable" for disambiguating.

In summa~l, I have tried to show thatrecent work on parsing favors deterministiclanguages and that such languages arenot necessarily generated by context freegrammars. It has been shown that such alanguage can be made compati~le with anambiguous grammar. Such ambiguous grammarsare certainly more interesting than unam-biguous ones. Finally, it would appearthat certain ambiguous grammars can bemore easily parsed than unambiguous ones.

9

We should return now to considera-tion of whether the emptiness problem canbe solved for some subset of context sensitivegrammars. A partial answer is prOVided bythe nature of LR(l) grammars which may becontext sensitive. For such grammars, theemptiness problem can be solved. (seeWalters ('71

Some recent works have shown the empti-ness problem to be solvable for certain sub-sets of context sensitive grammars. Thetechnique used in these proofs is to extendthe class of context free grammars to coverspecial subsets of the context sensitivegrammars. Cannon ('76) derives context sen-sitive grammars by the use of "stategenerators"

interest. Schuler ('75) warns, however,of the enormous complexity of contextsensitivity. He would prefer that gram-mars stuck to the weakly context sensitive.The extensions above are all in this set ofthe weakly context sensitive.

The most promising extension of thecontext free grammars is through the conceptof what are called Petri Nets. This conceptoriginated with the work of Petri in theearly 1970 t s. (see Petri (t 73 & '75A Petri Net is a network relating "places"to each other by means of "transitions"over arcs. Movement from one "place" toanother is referred to as "firing" the arc.Such a network can be diagrammed as follows:

t:.....w4 r ~ /I'"1'l.l.ce

.....c pl. ....

Defining a node as either a "place"or a "transition", a Petri Net languageis the labeling of the nodes of a PetriNet. Such languages can be classified asfollows with regard to the labeling of thenodes (from Peterson ('77:

1) free i.e., no two nodes have thesame label and no node iswithout a label.

2) A-free i.e., no node is without alabel but node labels arenot' necessarily unicp e.

3) ).. i.e., some nodes may nothave a label and not allnodes have unique labels.

Of these, A-free is the most interestingsub-set. The set of AP.~ri Net languagescorresponds to the A-free set with theunlabeled nodes telescoced into labelednodes (which could be considered as blackboxes from a category as aefined below).The set of free Petri Net languages istrivial since every path through a netwould be unique and, therefore, no~amenable to generalization within thenetwork.

Petri Net languages can be furtherdefined as regards the labels that constitutethe terminal state or final markings of thenetwork. (see Peterson ('77 Of mostinterest are those Petri Net languagesthat have labeled terminal states thatare, essentially, distinct from the tran-sition states.

It has been shown that all ~freePetri Net languages are ccntext sensitive.(Hack (176) or Peterson ('77 Hack provesthat the emptiness problem for terminal PetriNet languages (both A-free and free) isrecursively equivalent to the reaChabilityor finiteness problems. (Th. 9.4 & 9.6Hack ('76) pp. 150 & 151) He furtherproves that it is undecidable whether theaddition, removal, or changing of any

10

place, transition, arc, or label modifiesa Petri net language. (Th. 10.3 Hack ('76)p. 159) In Short, it will not be possibleto generalize from a specific Petri netlanguage to any subset of these languagesdefined as an extension of the former. S

SHowever, Berthelot & Roucairol ('76) haveshown that a Petri net can be reduced in afinite number of steps to an irreduciblenet with specific properties (the Church-Rosser property, see b,low). Such a reduc-tion could be used to prove correctnessof the nets resulting from a Petri netlanguage. (see ~autenbach & Schmid ('74

The reachability problem 'is definedfor Petri nets as the question of whether,given an initial set of conditions (ormarkings) on the Petri net, a terminalstate will be reached. The finiteness prob-lem is defined as the question of whethera Petri net will come to a halt after afinite number of firings. Peterson showsthat a general algorithm exists for provingreachability in a Petri net language.(PetersoR ('77 As shown in Hack, thiswill be a recursive problem. (Hack ('76)pp. lSO-l)

Peterson briefly touches on generalizedPetri nets in which there may be multiple 'arcs or inhibitor arcs. To wit:

The inhibitor arc 'i.a.' prevents 'B' fromfiring unless 'A' is set. Such generaliza-tions, while highly desirable for thedescription of complex networkS, are asyet theoretically unmanageable. Neverthe-less, as a practical matter, they may beusable. (They relate to both the theoryof categories and the theo~I of fuzzysub-sets ~entioned below.)

The development and extension of PetriNet languages seems capable of releasingthe full power of the non-procedural ap-proach to programming. They have thepotential of permitting the clear develop-ment of a total approach to a problem.In the introduction to the SIGPI..~ Sym-posium on Very High Level ~anguages,~eavenworth & Sammet define non-proceduralto mean that any sequencing needed is donelogically by the computer. (~eavenworth& Sammet('74 The logic is set by theenvironment defined in a manner that is,in fact, equivalent to a A-free PetriNet language. I expect the full develop-ment of non-procedural languages to comewith that of the Petri Net languages ortheir equivalent.

a class lei of objects A,B,C, for each ordered pair of objects(A,B) of a (possibly empty) set[A,BJe called the set of morphismsfrom A to EI,

iii) for each ordered triple (A,EI,C)of objects in e, a mao

(B,cle X [A,B]e~ [A,CJecalled composition of morphisms."

(Schubert ('72) p. 1)

applicable to multi-linear processingwhere each process is executed with aprobability that is less than one.

Starting in the early 1940's, Eilen-bery & MacLane ('42) & ('45) proposedwhat has come to be called the theory ofcategories. Categories are, essentially,classes of objects that are related to 9each other through some kind of process.

9A precise definition is: ."A Category econsists of the follOWingdata:

i)ii)

Machines, languages, systems can all be viewedas categories. (A good introduction isMacLane ('72); more rigorous works are Herr-lich & Strecher ('73) and Schubert ('72)who is especially clear although qUi~eprecise; Makkai & Reyes ('77) cover cate-gorical logic in a strictly mathematicalway; Arbib & Manes ('74 & '75) and Goguen('73 & '76) have done introductory wOTk aswell. The reader should be aware, however,that the simpler introductions often losein clarity what hey have gained in simplicity.)

B)c) Aoolication of the iunction:1) Non-procedural Proarammina:

Leavenworth & Sammet ('74) write of fivemajor features of non~procedural programminglanguages:

1) associative referencing--L e., much ofwhat is referred to as r.iational database management but extended to auto-matic referencing (retrieval) of data.

2) aggregate operators--i.e., iteratorsover sets, matrices etc.

3) elimination of arbitrary sequencing--i.e., mainly thatthe user suppliedinput sequence is passed over in favorof logical sequencing.

4) non-deterministic programming andparallel processing.

5) pattern directed structures--i.e.,recognition of similar patterns andtheir replasement by a standardexpression or procedure.

(see Kessels ('77) for further discussionof non-procedural programming)

Earley ('74) outlines various basicdata types that he feels should be handledin non-procedural programming:

1) tuples--i.e., relational data (e.g.,entities, cases, observations, or thelike)

2) sequences-- i e., such things astimes series and the like that areordered by their position m the set.

As will be made clearer below, a non-procedural approach best suits those problemsthat require a multi-lateral or parallelattack on the solution. Petri Nets allowa clear definition of both the non-linearityand the multi-linearity of a problem to bemade. The computer can then determine theextent of parallel processing that can beused. (see Lautenbach & Schmid ('74

There remains one further approach tobe examined: namely that of pruning deadends or impossible branches from a network.This involves backtracking from the currentnode to a prior junction and recursivelydescending alternate branches. This willbe discussed below.

B)b) Recent Advances in Mathematics:

It is not my intent to deal with theadvances in mathematics in any detail.The interested reader can pursue thesesUbjects in the references that I shallcite below.

. Starting with the work of Zadeh('65), a great deal has been done onwhat have come to be called "fuzzy" setsor sub-sets. In somewhat loose terms, afuzzy sub-set, F, is a set of orderedpairs:

F == {} for x. U & O~f.4.(x. )6:1~. J. ~ .J.

where xi is a me~~er of some universalset, U, andPr(xi) is the measure of member-ship of xi in the SUb-set F. (see Gaines('77) for an introduction from the stand-point of logic; Ragade & Gupta ('77) for amoderate mathematical introduction; butespecially Kaufmann ('75) for a mathematicaltext; Gaines & Kohout ('7i) is an extensive,annotated bibliography)

The greater part of the work on fuzzi-ness has been algebraic. Topics coveredinclude graphs, logic, netwo~ks, as wellas extensions to lattices and categories.However, some recent studies have dealtwith fuzzy measure theory, thus introducingintegration. (Sugeno & Terano J'77 Inparticular, this can be extended to feedbacksystems through the use of convolutions.(Jain (, 77

FU3ziness has been introduced into therJ.gor of rna~hematics so as to open newfrontiers in our ability to understand theessentially fuzzy world. In the words ofZadeh (forward to Kaufmann S'75) p. ix):

"We have been s low in l::9ming to therealization that muc~, p.l:!rhaps most,of human cognition and interactionwith the outside world. involvesconstraints which are not sets inthe classical sense, but rather"fuzzy sets" (or sUbse.ts), that is,classes with unsharp boundaries inwhich the transition from membershipto non-membership is gradual ratherthan a::>rupt."

A practical orientation oredominates in thestudies of fuzziness as ~ill be seen below.In computer science, fuzziness is most

II

3) sets--i.e., variables, attributes, etc4) relations--i.e., sets of tuples ( a

relational data base).S) functions--i.e., transformations of

data or expressions.

Fagin ('76) shows the equivalence of logicalimplications derived from data and the database structure of the data. Codd ('72) andMcLeod ('77) discuss the language implicationsof data bases. In summary, it seems impor-tant to treat data types and the relationsbetween them with the care and precisionof the appropriate'algebra.

Earley (, 76 & '74) discusses iteratorsat some length. Iterators can be classifiedas operators that determine the domain withinwhich an eXtlressiori or command is to operate.Most commoniy, this will be inclusion orexclusion from the set of data to be analysedor otherwise treated. I~'other words,iterators are functionals that relate onedata base to another or one type of data toanother (e.g. tuples to sequences). (seeSChwarz ('75, "74, & '73) for implementationsof iterators in the set oriented languageSETL' and Loos ('76) for their implementationin AiDES) This is what Tennent ('76 & '77)refers to as "referential transparancy" bywhich he means that the program itselfderives the functionals necessary to executethe user's co~~ands. Such functionals willdetermine the environment (domain of action),its content, and its continuation.

Much of what has been said above canbe implemented in algebra. A property ofalgebraic languages is that they are com-plete. 10 This implies that expressions

lOI.e. that ~~ev have the extended Church-Rosser'property by which is meant that "anysequence of reductions of an expression 'e'will convercre to the same constant '~e'provided 'e

('77) have proven that a fUzzy lan~~age isrecognized by an automaton appropriate tothe type of languagell if and only if a

IlFor a context sensitive language, thiswill be a linear bounded automaton; for acontext free language this will be a push-down autOmaton.

cutoff representation of it on a lattice canbe recognized by the same automaton. 12 They

12(see Honda, Nasu, & Hirose ('77) pp. 154-5) A cutoff representation is defined asfollows: '"

1) an L-fuzzy lan~age, f, over L is amapping from L* to Lwhere L is a lattice with minimal

element 0,L is a finite set of symbols,[- is the set of all strings

and symbols in '[.uA,A is the null stnng. .

and f(x) for x [* is the measure ofmembership.

2) a A-cutoff representation is th~ mappingonto L of the set {xcP I f(x) ~A}

3) A~s an isolated cutpo~nt in the sensethat every measure f(x) of membershipsatisfies tpe following: !f(x)-Al> f >0.

(see Radecki ('77) who argues on practicalgrounds for the use of A-level fuzzy sets forwhich the measure of membershio is set to 0if it is less than A,)

have further shown essentially, that any'rational probabilistic event' is recognizedby a deterministic linear bounded automaton.(Honda, Nasu, & Hirose ('77) p. 163) A'rational probabilistic event' is definedas an event realized by some rationalprobabilistic automaton. 13 This is a further

13 5ee an equivalent result in the work ofChen & Hu ('77) mentioned above. Their cut-off point is 0.5 but is used only to showthat a language exists that is contextsensitive and not context free that isaccepted by a finite state probabilisticautomaton. Thus, the results of Chen &Hu are not as general as that of Honda,Nasu, & Hirose. Note that the 'rationalprobabilistic event' above is essentiallyequivalent to a context sensitive languagein the sense that both the interpretationof a context sensitive lan~~age and theresult of a 'rational probabilistic event'depend on finitely close elements.

proof of the computability of dependentconstructs such as context sensitive lang-uages.

Several applications using fuzzy setshave been made in the planning and controlof industrial processes. (see Mamdani('76) & ('77) for general surveys; and King& Mamdani ('77) for a specific application)Kling ('74) describes the language, PLANNER,e which has been modified to deal with fUZZy

13

concepts such as high, low, etc. Todate, these applications deal only withthose concepts in logical expressions ofthe form: If the water is low, turn thefurnace off.

5) Theory of Cateaories:

Benson ('75) uses category theory toanalyse syntax categories and parsing. Heproves that minimal machines14 can parse

l~inimal here means, essentially, that thereis, at least, a path of a fini te number ofsteps that will accomplish the task.

context free languages and, which is new,that these machines are simple error detect-ing. Burstall & Thatcher deal with a formof parallel processing using category theory.Bainbridge ('76) distinguishes linearstructures which use linear flowcharts andmulti-linear structures which use netWorksHe applies category theory to these latter,prOVing that feedback systems have aminimal representation. Arbib & Manes ('75)show the same for non-deteIrli.nistic machines.

B)d) Summary:

At one level of linguistic research,there is a movement towards semantics ormeaning as opposed to the strJcture or formof languages. (see Mayne ('75 Meaningcan be fuzzy, categorized, or faulty butneeds to be dealt with. (see Hecht ('76It has been specula~ed (Hobbs ('77) thatvery high level programming languagesshould follow the lead of natural languages.By this, Hobbs means redundancy, contextualinterpretations, inter-sentence relations,and the like. In short, a context sensitivegrammar at the least.

At another level, there is a pushtowards a realization of a more formallogic. "Semantics and realization areaspects of the same situation: semanticsis" the problem~of system analysis; whilerealization is the problem of system svn-thesis." (Goguen (, 75) p. 1511 For-Goguen, category theory is the answer sinceit deals with the formal structure ofrealization. He would have us work toachieve artificial intelligence definedas "the scientific study (both experi-mental and theoretical) of the represen-tation, manipUlation, utilization, andacqUisition of concepts and conceptualsystems." (Goguen (, 74) p. 547)

Cherniavsky ('78) in a very recentpaper would remind us at this point thatalgorithms and humans can be distinguishedby their handling of truth. Humans arealways able to adapt to G~del's "rule ofthe excluded third". Or, if I may soconclUde, to cite Heidegger:

"Wahrhei t ist nicht ursprJnglichim Satz beheimatet."

M. Heidegger, Vom Wesen . derWahrheit. p. 12.

(Truth is by nature foreign to thesentence. )

C) Lanquacre Standards for StatisticalComoutinqa) Introduction:

What are the standards that, by right,the statistician should ask for in statis-tical computing languages? We have by nowcovered a considerable amount of ground.What we have seen is, I think, typical ofthe cutting edge of any discipline--farmore questions have been raised thananswers returned.

Theory is clearly and naturally fur-ther along than the practice of the art.It is precisely into a~as where theoryhas broken new ground and presented newapproaches which are of interest to sta-tisticians, that we should insist thatprogrammers venture. Such experimentalwork will prepare us to match the abilitiesof computer hardware developments.

I shall outline below a number ofstandards that, I feel, statisticiansshould insist be encoded in statisticalcomputing programs. Some of the standardsare quite general but some relate to mypersonal preference for algebra. Withoutsuccumbing to the pervasive use of AFL,I feel that since written work on thetheory of statistics is invariably castin algebraic notation, a statisticalcomputing language should do likewise.It is my contention that the rigor accom-panying such an approach will make the useof such programs in fact easier for thenovice than would languages closer to ana~ural language. (see Francis et ale('76) for an orientation more towards anEnglish like syntax; Salton ('75) warnsthat natural language needs to be refinedfor effective computer use by enhancingthe level of disambiguity (sharpen themeaning) ).

Following Tucker ('75), I shall firstoutline the qualities a very high levellanguage should havel5 :, 5 Statistical computing languages are veryhigh level; FORTRAN is high level; andassembly language is low level.

1) easy to learn by a non-programmer,2} functionally extensible,3} minimal writing (i.e., abbreviations),4) permit algorithmic specification,5) natural to use,6} portable (to another machine),7) run-time efficient,8) capabilities for both batch and

interactive,9) modular,

10) pe~.it access to system functions(see also Balzar et ale ('76) Point 4 corres-ponds, more or less, to my interest in analgebraic approach. This conflicts tosome extent with point S.

A further extension of very high levellanguage is to be seen in the work ofKessels ('77) following on that of Klink-hamer ('72). They stress non-procedural-ness and contrast this with the presently

14

more prevalent sequential programming.Kessals' non-procedural language deals withblocks of commands. The components of these ~blocks are then exectued in an order deter- ~mined by the computer--according to needor logic.

Earley ('74) and Goldsmith ('74) intro-duce automatic operators over sets (iterators)Datasets would be referenced by associationwith a name or function. The domain of theset (number of elements, their nature orform, etc.) would be determined by the com-puter. Operations would then be performedover all of the elements (or some subsetthereof) as determined by the environmentof the problem. .

Loos ('74) shows that an algebraiclanguage can be extended to higher levelsof language. Kanoui ('76) describes animplementation of such a higher levelalgebraic language that is, basically,non-procedural. This language, PROLOG,uses predicate logic over a non-deter-ministic tree to integrate the symbolicrepresentation of an algebraic expression.(see Sundblad ('74) and Moses ('71) forsurveys of algebraic computer languaqes.)

In the outline below, I shall attemptto apply the four overall tendenciesdescribed above in this paper--Non-proce-duralness, extended algebra (Petri nets,categories, etc.,), fuzziness, and arti-ficial intelligence. The four are notreally separable. Realizations of Petrinets and categories are not likely to be ~obvious. Therefore, they can not be ~expressed at once in the proper sequentialorder (or only inconveniently so) butrequire that the computer think of ~to proceed.C)b) Theobvious--or whv do we out uo wit~ill

l} Nothing would be fixed.There is no need to require that

starting in column 16 means something, asis the case in SPSS, for instance.

2) The user should not have to countfor the computer.

The computer is capable of countingthe number of cases in a dataset or of fig-uring out the number of variables in aregression.

3) No contrived conventions.E.g., in IBM JCL, the '=' is used to

terminate and begin keyword fields sinceblanks are used in a very special way(separation of certain major fields). Inthis case, the computer is not capableof recogniZing phrases and needs thecontrived convention II keyword

keyword " to recognize one.4) All programs should be interactive,

at least.A system that runs on batch alone

is somewhat comparable to the Model T.

C)c) The not so obvious:

1) Capability for user defined semantics, ~syntax, and lexics. ~

All of these should be table driven(~.e., defined by matching with a table)so that it is trivial for a user to changethe set of terminal symbols, add a newabbreviation, define a composite operator,or the like.

2) Capability for user defined algorithms.The program should be sufficiently

modularized so that a user could rearrangethese modules to create a new algorithm.The user should be required only to specifythe new arrangement. The actual rearrange-ment of machine control should be handledinternally by the program.

3) Capability for continuation by means ofsome other program.

The user should be able to transfercontrol, data, and whatever else to anotherprogram. Again, the actual transferingshould be done internally.

4) Abbreviations should be derived al-gorithmically from the "natural" name.

Thus an user need remember only thefUll, normal name plus the algorithm,rather than a long series of not alwaysobvious character strings.

C)d) The immediate future or what shouldbe done in the next version:

1) Capability to' handle sets and matrices.Databases should be definable and

usable as such (e.g., the database auto-matically defines the number of entitiesor attributes to be considered). Matricesshould be automatically recognized once

~ defined and handled properly.,., 2) Capability to operate over sets.

The user should be able to unite orintersect sets; to sub-set by criteriaof arbitrary compleXity; to relate datasets.

3) Capability to operate over sequences ofordered sets. .

The program should distinguish un-ordered and ordered attributes (such astime series or convergent series) andhandle each appropriately. This canrange from the trivial level of firstdifferences to the not so trivial abilityto loop (iterate) until an arbitraryseries converges to some value.

4) Capability to define the environmentor to set state descriptors.

The user should be able to set thedomain of operations, the extent or formof interaction, output etc. The user shouldbe able to do this on both the global leveland on the level of the single operation.

5) Capability to imbed procedure3.Any process that handles information

can be treated as a function that may returna single value or multiple values. As such,the user should be able to nest them withinother processes or expressions of processes.

6) Capability to recognize patterns.The program Should be able to recognize

patterns such as a description of a processand substitute a call to that process forthe extended description. The user shouldbe able to specify patterns and to use themas appropriate.

e

15

7) Surroundability.The user should be able to imbed the

entire program or just parts of it into someother program or system.C)e) On the horizon: (all of these conceptsare treated more extensively in the text ofthis paper)

1) Non-proceduralness.The order of execution of the component

parts of a problem should be determined bythe computer. This should include thecapability of parallel processing whereappropriate.

2) Capability of extended algebra.The user should be able to describe

complex multi-linear problems (Petri sets,categorical struc"tures and the like) andexpect the program to be able to manipulateand transform them as needed.

3) Inclusion of fuzziness.The ability to deal with fuzzy attri-

butes as well as fuzzy concepts or networks.4) Artificial intelligence.

This covers automatic error correction;interactive probing of errors or inconsisten-cies; determination of how to solve a problem;the ability to use predicate logic; deter-mination of reachability; construction anduse of traces for backtracking; etC.

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