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Kdb+ is a database built on top of an interpreter for a programming language, q, itself built on top of another programming language, k. Kdb+, q, and k were created by the Canadian computer scientist Arthur Whitney (b. 1957) with the aim of addressing the inability of traditional relational databases to keep up with increasing volumes of data and speed requirements.

K and q differ in many respects from other programming languages, such as C++, Java, C#, and Python. Whereas these other programming languages are intellectual descendants (for the most part) of C and Simula, k and q are derived from another, very different programming language, APL, designed by the Canadian computer scientist Kenneth (Ken) Eugene Iverson (1920–2004).

Kenneth Iverson

Kenneth Iverson
Kenneth Iverson

Kenneth Iverson was born on 17 December 1920 near Camrose, a town in central Alberta, Canada. His parents were farmers who came to Alberta from North Dakota; his ancestors came from Trondheim, Norway.

Iverson's friend and colleague Roger Kwok Wah Hui (b. 1953) recalls:

Ken dropped out of school after Grade 9 because it was the height of the Depression and there was work to do on the family farm, and because he thought further schooling only led to becoming a schoolteacher and he had no desire to become one.

During World War II, Ken first served in the Canadian army and then in the R.C.A.F. as a flight engineer in PBY Catalina flying patrol boats. He finally learnt about universities from his Air Force mates, many of whom planned to return to university, thanks to government support for servicemen. While in the service he completed high school courses by correspondence. After the war he graduated summa cum laude with a B.A. from Queen's University and the M.A. and Ph.D. degrees from Harvard. His doctoral dissertation, "Machine Solutions of Linear Differential Equations: Applications to a Dynamic Economic Model" (1954), was jointly supervised by Professors Howard Aiken and Wassily Leontief.


Howard Aiken
Howard Aiken

I. Bernard Cohen, in his book Howard Aiken: Portrait of a Computer Pioneer, describes Iverson's recollections of his PhD studies:

Kenneth Iverson has recalled graduate study under Aiken as "like an apprenticeship" in which the student "learned the tools of the scholarship trade". Every topic was "used more as a focus for the development of skills such as clarity of thought and expression than as an end in itself." Once admitted to the program, a graduate student underwent a rite of "adoption into the fold". He was given a desk (or a share of a desk) among a group of other graduate students, the permanent staff, or visiting scholars, "most of whom were engaged in some aspect of the design and building of computers". A student was thus "made to feel part of a scholarly enterprise" and was provided, "often for the first time, with easy and intimate access to others more experienced in his chosen field".


Aiken was especially concerned with clarity of exposition. "Every thesis student," Iverson recalls, "knew and dreaded the fact that Aiken would carefully read and discuss successive chapters of the thesis, insisting upon clear exposition and offering both general and detailed advice." One effect of Aiken's emphasis on writing style was that "budding authors soon learned to call upon their fellow graduate students and other colleagues to review their work." Once a student had a chapter analyzed by Aiken, he would never submit another without the revision necessary to put it into the best form possible.

Iverson served as Assistant Professor of Applied Mathematics at Harvard from 1955 to 1960. He implemented the world's first graduate programme in "automatic data processing":

Many people think that Aiken was interested only in scientific computers. This was simply not so. During one coffee hour, Aiken turned to Ken Iverson, who had just finished his Ph.D., and said: "These machines are going to be immensely important for business, and I want you to prepare and teach a course in business data processing next fall." There had never been such a course anywhere in the world. Ken was qualified only because he was a mathematician. I was so excited by the prospect that I immediately volunteered to be Ken's graduate teaching assistant.

Frederick Brooks Jr. (b. 1931), Aiken and the Harvard "Comp Lab", in I. Bernard Cohen and Gregory W. Welch, editors, Makin' Numbers, MIT Press, 1999, page 141.

Iverson recalls:

Schools of education prepare teachers for elementary schools with a great emphasis on the techniques of teaching. At college level, on the contrary, no preparation for teaching is (or at least was) provided, and graduates, full of their subjects, were unleashed on students. I was fortunate in having as a Teaching Fellow a graduate student who had been born at a podium—Fred Brooks. Moreover, Fred and I were sufficiently close friends that he could, and did, tell me bluntly of my bad teaching techniques in post-mortems held after each lecture. For example, he once said "That was a very interesting point you made about such-and-such—too bad you made it to the blackboard, and could not be heard past the first row."

—Kenneth E. Iverson's Autobiography (with Donald B. McIntyre).

Later Fred Brooks managed the development of IBM's System/360 family of computers and the OS/360 software support package and wrote a seminal book on software development, The Mythical Man-Month.

It was in this period that Iverson developed notation for describing and analyzing various topics in data processing, for teaching classes, and for writing (with Brooks) Automatic Data Processing.

I was appalled to find that the mathematical notation on which I had been raised failed to fill the needs of the courses I was assigned, and I began work on extensions to notation that might serve. In particular, I adopted the matrix algebra used in my thesis work, the systematic use of matrices and higher-dimensional arrays (almost) learned in a course in Tensor Analysis rashly taken in my third year at Queen's, and (eventually) the notion of Operators in the sense introduced by Heaviside in his treatment of Maxwell's equations.

—Kenneth E. Iverson's Autobiography (with Donald B. McIntyre).


IBM Research headquarters in Yorktown Heights, New York
IBM Research headquarters in Yorktown Heights, New York

Iverson explains how he ended up in the Research Division of the International Business Machines Corp. in 1960:

Harvard had a benign rule of five years to tenure or out, benign because it prevented fierce Bostonophiles from clinging forever in the hope of eventual tenure. It was particularly benign in those postwar days of the establishment of commercial research centers, where anyone from the science faculty of a respectable university could expect to double his salary—as I did when I joined the Research Division of IBM.


I did not join IBM until 1960, at which time I was just finishing up my A Programming Language, published in 1962. It included a chapter (called Microprogramming) on the formal description of the IBM 7090 machine.

I left Harvard with offers from IBM Research and Bell Laboratories. Fred Brooks (who had joined IBM somewhat earlier) advised me to choose IBM because computers were their primary concern. After I joined the IBM Research Division, Fred advised that I stick to whatever I really wanted to do, because management was so starved for ideas that anything not clearly crazy would find support. In particular, I was allowed to finish and publish a book on my notation, (A Programming Language, Wiley, 1962), as well as Automatic Data Processing, with co-author Fred Brooks (Wiley, 1963).

Adin Falkoff
Adin Falkoff

I soon met Adin D. Falkoff (1921–2010), and we worked jointly on the notation for some twenty years. It was Adin who suggested the name APL (from the title of my 1962 book). In the 1962 book, I had used APL to describe an IBM computer, and Adin and I (with Ed Sussenguth) used it to produce a formal description of the IBM System 360 computers then under design.

IBM System/360 computer
IBM System/360 computer

An important consequence of this work (published by John Lawrence in the IBM Systems Journal) was that it attracted important contributors to APL, notably Larry Breed (a Stanford student, who joined us at IBM and chose to undertake the crucial work of implementing it as a programming language), and Donald McIntyre, head of Geology at Pomona College (who had acquired a 360 Model 40 for Pomona, and used the formal description to become more expert than the IBM Systems Engineer assigned to Pomona).

—Kenneth E. Iverson's Autobiography (with Donald B. McIntyre).


IBM 2741 APL keyboard
IBM 2741 APL keyboard

In the preface to A Programming Language, Iverson defines a programming language as follows:

Applied mathematics is largely concerned with the design and analysis of explicit procedures for calculating the exact or approximate values of various functions. Such explicit procedures are called algorithms or programs. Because an effective notation for the description of programs exhibits considerable syntactic structure, it is called a programming language.

APL2 keyboard
APL2 keyboard

In A Personal History of APL, Michael S. Montalbano describes the "two key ideas" behind it:

H. G. Wells described human history as a race between education and catastrophe. This observation is more pertinent today than it ever has been in the past. But, in the specific case of the proliferation of stored-program digital computers, the education-catastrophe race, in my opinion, takes a particular form: it is a race between technology and methodology, between gadgets and ideas.

We are developing gadgets at an explosive, accelerating, self-fueling rate. We shall be swamped by these gadgets if we don't hasten to develop and apply the ideas we need to control them.

To me, this need defines the importance and the mission of APL. APL provides the best set of gadget-understanding and gadget-controlling ideas currently available. Looking to the future, it provides the best base for the methodology we must develop if we are ever to bring our gadget-based technology under control.

This opinion is based on experience. I was working with computers when they were merely gleams in the eyes of the early designers: I was working with APL when it was nothing but a collection of incomprehensible characters scattered through publications with catchy titles like: The Description of Finite Sequential Processes. In other words, I've been working with both computers and APL since their very early days. And, in looking back on this experience of just over 34 years, (which is my own personal experience of computing), I find that it can best be summarized in terms of two key ideas:

  • The stored-program idea: the idea that a procedure or algorithm can be stored as a collection of switch settings in exactly the same way as the data on which the procedure is to work and, as a consequence, that executing such a stored procedure consists of starting it and letting it flip switches until it is done. This was apparently first stated explicitly in a paper drafted by John von Neumann in 1945.
  • The efficient-notation idea: the idea that these vast collections of switches, changing their settings in thousandths, millionths, billionths, trillionths, ... of a second, would be pretty hard to manage if we didn't develop a good way to describe them and think about them. I don't know whether this idea was ever presented anywhere in just these terms. Instead, it seemed to be implicit in the activities and writings of many people. But, in my opinion, it received its most effective and fruitful expression in the writings of Kenneth Iverson and others who followed his lead. The most important publication expounding this idea is the book from whose title the letters APL derive their significance.

The stored-program idea provided, and continues to provide, the basis for our current runaway technology. The efficient-notation idea, if we take it seriously and do a lot of thinking and hard work, will help us curb the runaway and direct it into fruitful and productive channels.

Iverson's notation was effective as it was simple. It relied on simple rules of precedence based on right-to-left evaluation. The fundamental data structure in APL is a multidimensional array. Languages such as APL and its progenitors are sometimes referred to as array, vector, or multidimensional programming languages because they implicitly generalize scalar operations to higher-dimensional objects.

APL developers
(L to R) Dick Lathwell, Ken Iverson, Roger Moore, Adin Falkoff, Phil Abrams, and Larry Breed. On the extreme left in the background: Jon McGrew. Taken in the I. P. Sharp Associates hospitality suite during the 1978 APL Users Meeting in Toronto, Ontario.

According to Wikipedia,

With the completion of the formal description Falkoff and Iverson turned their attention to implementation. This work was brought to rapid fruition in 1965 when Larry Breed and Phil Abrams joined the project. They produced a FORTRAN-based implementation on the 7090 called IVSYS (for Iverson system) by autumn 1965, first in batch mode and later, in early 1966, in time-shared interactive mode. Subsequently, Breed, Dick Lathwell (ex University of Alberta), and Roger Moore (of I. P. Sharp Associates) produced the System/360 implementation; the three received the Grace Murray Hopper Award in 1973 "for their work in the design and implementation of APL\360, setting new standards in simplicity, efficiency, reliability and response time for interactive systems." While the 360 implementation work was underway "Iverson notation" was renamed "APL", by Falkoff. The workspace "1 cleanspace" was saved at 1966-11-27 22.53.57 UTC. APL\360 service began within IBM several weeks before that and outside IBM in 1968. Additional information on the implementation of APL\360 can be found in the Acknowledgements of the APL\360 User's Manual and in "Appendix. Chronology of APL development" of The Design of APL.

In 1970 Iverson was named an IBM Fellow in honour of his contribution to the development of APL.

Turing award

The 1979 ACM Turing Award was presented to Iverson by Walter Carlson, Chairman of the Awards Committee, at the ACM Annual Conference in Detroit, Michigan, October 29, 1979:

In making its selection, the General Technical Achievement Award Committee cited Iverson for his pioneering effort in programming languages and mathematical notation resulting in what the computing field now knows as APL. Iverson's contributions to the implementation of interactive systems, to the educational uses of APL, and to programming language theory and practice were also noted.

In 1979 he received the A.M. Turing Award for his work on APL. Iverson opened his Turing Award Lecture, entitled Notation as a Tool of Thought with the following observations:

The importance of nomenclature, notation, and language as tools of thought has long been recognized. In chemistry and in botany, for example, the establishment of systems of nomenclature by Lavoisier and Linnaeus did much to stimulate and to channel later investigation. Concerning language, George Boole in his Laws of Thought asserted "That language is an instrument of human reason, and not merely a medium for the expression of thought, is a truth generally admitted."

Mathematical notation provides perhaps the best-known and best-developed example of language used consciously as a tool of thought. Recognition of the important role of notation in mathematics is clear from the quotations from mathematicians given in Cajori's A History of Mathematical Notations. They are well worth reading in full, but the following excerpts suggest the tone:

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, an in effect increases the mental power of the race.

A. N. Whitehead

The quantity of meaning compressed into small space by algebraic signs, is another circumstance that facilitates the reasonings we are accutomed to carry on by their aid.

Charles Babbage

Nevertheless, mathematical notation has serious deficiencies. In particular, it lacks universality, and must be interpreted differently according to the topic, according to the author, and even according to the immediate context. Programming languages, because they were designed for the purpose of directing computers, offer important advantages as tools of thought. Not only are they universal (general-purpose), but they are also executable and unambiguous. Executability makes it possible to use computers to perform extensive experiments on ideas expressed in a programming language, and the lack of ambiguity makes possible precise thought experiments. In other respects, however, most programming languages are decidedly inferior to mathematical notation and are little used as tools of thought in ways that would be considered significant by, say, an applied mathematician.

The thesis of the present paper is that the advantages of executability and universality found in programming languages can be effectively combined, in a single coherent language, with the advantages offered by mathematical notation.


I. P. Sharp Associates (IPSA) was the vendor of SHARP APL and a major contributor to the APL community in the 1970s and 1980s. It served as a centre of APL research, hiring notable interpreter designers and implementers from IBM and starting the careers of many other notable implementers. It also sponsored the IPSA conferences.

IPSA was founded in 1964 by an eight-person team including president Ian Sharp and Roger Moore.

Iverson recalls in his Autobiography the time that he spent at IPSA (from 1980 to 1987):

I. P. Sharp Associates of Toronto were a pioneer user of APL, establishing it as the base of a time-sharing service widely used in Canada, U.S., and Europe. This came about because Larry Breed had proposed that we hire Roger Moore as a consultant to aid in his implementation of APL. Roger was his classmate at Stanford, and a senior member of I. P. Sharp—he was impressed enough by APL to propose that his firm adopt it.

Having effectively lost control of APL to the Palo Alto Scientific Center (under Horace Flatt), I left IBM and joined Sharp. It was exhilarating to work for a company seriously devoted to APL. Moreover, the seven years to my next retirement (at 67) saw many enhancements to the APL system, largely under the direction of my son Eric.

His group was largely made up of young men who had come to Canada from U.S. to escape service in Vietnam. Their beards and otherwise scruffy appearance led a visiting businessman to say "This place is a zoo". Rather than resent the implied insult, the group proudly adopted the name, and were known thereafter as "The Zoo".

IPSA served an important role as an incubator for entrepreneurs and technologists, including:

  • Bob Bernecky—founded Snake Island Research;
  • Clarke Bruce—founded Soliton Inc.;
  • Gitte Christensen—founded Insight Systems ApS; CEO of Dyalog Ltd.;
  • Lib Gibson—launched Sympatico ISP; CEO of Raft Mobile Systems; CEO of Bell Globemedia Interactive;
  • Leslie Goldsmith—founded Affinity Systems;
  • Rob Hodgkinson—founded Financial Market Software Consultants;
  • Roger Hui—developed the programming language J with Ken Iverson;
  • Hugh Hyndman—founded Affinity Systems;
  • Eric Iverson—founded Jsoftware Inc.;
  • Rohan Jayasekera—founded the Sympatico Internet service;
  • Morten Kromberg—founded Insight Systems ApS; CTO and CXO of Dyalog Ltd.;
  • W.H. Lidington—founded Beyond 20/20 Inc.;
  • Jane Minett—founded the Sympatico Internet service;
  • Roland Pesch—principal at Cygnus Solutions;
  • Richard Proctor—founded APL Borealis Inc.;
  • Scott Remborg—founded the Sympatico Internet service;
  • Morgan Smyth—founded Braegen Inc.;
  • Stephen Taylor—writer, editor, poet, founder of the Iverson College conferences;
  • Arthur Whitney—founded Kx Systems; developed the programming languages A+, k, and q.

Reuters purchased IPSA in 1987.

Roger Hui

Roger Hui
Roger Hui at Dyalog '18

Roger Hui was born in 1953 in Hong Kong. In 1966 he immigrated to Canada with his whole family. In 1973, Hui entered the University of Alberta. In his second year he took a course on probability and statistics in which students were expected to learn the programming language APL with little or no formal instruction. He used all the time he could muster on a heavily burdened computer, and benefited from the APL\360 User's Manual (the book APL Language was not published until March 1975). Because the manual was written by Falkoff and Iverson, Hui thinks it reasonable to say he learned APL from them.

As a summer student in 1975 and 1976, Hui worked at IPSA in Calgary on workspaces for statistical and probability calculations. The major attraction of the job was the unrestricted computer time with access to APL.

After receiving a BSc degree with first class honours in computer science in 1977, Hui worked for two years as a full-time programmer and analyst in the new Edmonton office of IPSA, where his main duty was to support clients in their use of APL time-sharing. He attended the APL79 conference in Rochester, New York, where Iverson gave two papers: "The Role of Operators in APL" and "The Derivative Operator".

Hui recalls:

In my APL career, I have been privileged to be allowed to do my best on programming problems large and small, including a personal list of benchmark problems. They are benchmarks not only of my own understanding but also on advancements in the APL language. One such problem is generating all the permutations of ⍳n.


      p←perm n;i;ind;t
     ⍝ all permutations of ⍳n
      t←perm n-1


      perm←{0=⍵:1 0⍴0 ⋄ ,[⍳2](⊂0,1+∇ ¯1+⍵)⌷⍤1⍒⍤1∘.=⍨⍳⍵}

This last version is not the last word. The details are in the Dyalog blog entry Permutations.

Arthur Whitney

Arthur Whitney
Arthur Whitney

Arthur Whitney was born on October 20, 1957 in Canada. He studied pure mathematics at the graduate level at the University of Toronto in the early 1980s. He then worked at Stanford University.

He recalls his first encounters with APL:

BRYAN CANTRILL You are a bit of a rarity in software engineering in that you have been writing software on a daily basis for decades. Your first introduction to computing was APL with the master, Ken Iverson. What was that like?

ARTHUR WHITNEY In 1969, I was 11, and Ken Iverson was at IBM Research in Yorktown. He had been a friend of my dad's at Harvard in the '40s. We lived in Alberta, but we were driving around the continent and went to visit him. He showed me programming on a terminal in his house in Mount Kisco. This was in the '60s, and already it was interactive, and it was very quick to write programs and get results.

BC You must have been the only 11-year-old on the planet getting that kind of demonstration of programming in 1969.

AW Of course, I had no idea about that, and I didn't really pay much attention. He showed me some stuff, and I thought it was cool. In '74 when I went to a university and took a computer class, they were using punch card, which made no sense because five years earlier I had already seen interactive programming.

BC Did you start working on APL at Waterloo?

AW No, at Waterloo I just did some APL for a week. It was a math major and I wasn't interested in computers because I just wanted to do pure math. So I really missed a big opportunity.

BC Well, I'm not sure if you missed it or if you just found the opportunity a different way.

AW It took me a long time. For the next 10 years I did a little bit of APL in the summers as a consultant, but it wasn't until about 1980 when I was working with Ken at a Canadian company called I. P. Sharp that I really began using it regularly. Ken had retired from IBM after 20 years and was working at I. P. Sharp in Toronto.

I. P. Sharp was an amazing company. It had its own worldwide network that had nothing to do with DARPA (Defense Advanced Research Projects Agency). We were sending e-mails and instant messages to Australia and Singapore. The whole company was APL.

Thus Iverson, Hui, and Whitney all ended up at IPSA. The three of them continued to work on APL, adding new ideas to the programming language.

Morgan Stanley and A+

Morgan Stanley headquarters, New York
Morgan Stanley headquarters, New York

In 1988, Whitney joined Morgan Stanley. He recalls:

My big break was in 1988 when I joined Morgan Stanley. There the motivation was a terabyte of TIC (Treasury International Capital) data, and back then there were a few million transactions a day being processed by realtime trading systems. I think we had one of the biggest trading operations in the world. We had a portfolio that was a billion dollars: half a billion long, half a billion short. We were trading every second electronically. The data set was a terabyte, but we compressed it down. It was pairs trading, and I wrote an APL to do all of that—the big database and the realtime trading—so our entire department was using the language.

I much preferred implementing and coding in LISP, but once I was dealing with big data sets and then having to do fairly simple calculations, APL just seemed to have the better vocabulary.

It had to come up one level. Common LISP even then had about 2,000 primitives. I didn't like that. What I liked was the original LISP, which had car, cdr, cons, and cond, but that was too little. Common LISP was way too big, but a stripped-down version of APL was in the middle with about 50 operations. It's about the same size as C. But the thing about the languages that I implement is that there are no libraries: those 50 operations are it. Everybody builds from there, and the resulting programs are extremely short.

Thus Whitney created A and A+. The A+ website tells the story of the A+ programming language:

A+ is a descendent of the language "A" created in 1988 by Arthur Whitney at Morgan Stanley. At the time, various departments had a significant investment in APL applications and talent, APL being a language well-suited to the manipulation of large arrays of numbers. As technology was moving from the mainframe to distributed systems, there was a search for a suitable APL implementation to run on SunOS, the distributed platform of the period. Not happy with the systems evaluated, Arthur, motivated by management, wrote one geared to the business: large capacity, high performance. He was joined in his efforts as the language took on graphics, systems' interfaces, utility support, and an ever-widening user community. Over the course of the next few years, as the business began to reap tangible value from the efforts, the pieces were shaped into a consistent whole and became A+. The "+" referred to the electric graphical user interface. An A+ development group was formally created in 1992.

A+ soon became the language of choice for development of Fixed Income applications. It offered familiarity to the APL programmers, the advantages of an interpreter in a fast-paced development arena and admirable floating-point performance. A significant driver was that many of Morgan Stanley's best and brightest were the developers and supporters of the language. Through their practical application of technical values, they instilled fervent enthusiasm in talented programmers, regardless of their programming language backgrounds.

A+ was optimized for fast processing of large volumes of time series data. Unlike APL, A+ allowed functions to have up to nine formal parameters, used semicolons to separate statements (so a single statement could be split into multiple lines), used the last statement of a function as its result, and introduced the so-called dependencies which functioned as global variables.

A+ is now available online, Programmers can also download the kapl font, which includes the special characters used by APL and A+.

Jsoftware and J

In one of the essays on the J website Roger Hui tells the story of the J programming language.

One summer weekend in 1989, Arthur Whitney visited Ken Iverson at Kiln Farm and produced—on one page and in one afternoon—an interpreter fragment on the AT&T 3B1 computer. I studied this interpreter for about a week for its organization and programming style; and on Sunday, August 27, 1989, at about four o'clock in the afternoon, wrote the first line of code that became the implementation described in this document.

The essay then goes on to quote Whitney's one-page interpreter fragment:

typedef char C;typedef long I;
typedef struct a{I t,r,d[3],p[2];}*A;
#define P printf
#define R return
#define V1(f) A f(w)A w;
#define V2(f) A f(a,w)A a,w;
#define DO(n,x) {I i=0,_n=(n);for(;i<_n;++i){x;}}
I *ma(n){R(I*)malloc(n*4);}mv(d,s,n)I *d,*s;{DO(n,d[i]=s[i]);}
tr(r,d)I *d;{I z=1;DO(r,z=z*d[i]);R z;}
A ga(t,r,d)I *d;{A z=(A)ma(5+tr(r,d));z->t=t,z->r=r,mv(z->d,d,r);
 R z;}
V1(iota){I n=*w->p;A z=ga(0,1,&n);DO(n,z->p[i]=i);R z;}
V2(plus){I r=w->r,*d=w->d,n=tr(r,d);A z=ga(0,r,d);
 DO(n,z->p[i]=a->p[i]+w->p[i]);R z;}
V2(from){I r=w->r-1,*d=w->d+1,n=tr(r,d);
 A z=ga(w->t,r,d);mv(z->p,w->p+(n**a->p),n);R z;}
V1(box){A z=ga(1,0,0);*z->p=(I)w;R z;}
V2(cat){I an=tr(a->r,a->d),wn=tr(w->r,w->d),n=an+wn;
 A z=ga(w->t,1,&n);mv(z->p,a->p,an);mv(z->p+an,w->p,wn);R z;}
V2(rsh){I r=a->r?*a->d:1,n=tr(r,a->p),wn=tr(w->r,w->d);
 A z=ga(w->t,r,a->p);mv(z->p,w->p,wn=n>wn?wn:n);
 if(n-=wn)mv(z->p+wn,z->p,n);R z;}
V1(sha){A z=ga(0,1,&w->r);mv(z->p,w->d,w->r);R z;}
V1(id){R w;}V1(size){A z=ga(0,0,0);*z->p=w->r?*w->d:1;R z;}
pi(i){P("%d ",i);}nl(){P("\n");}
pr(w)A w;{I r=w->r,*d=w->d,n=tr(r,d);DO(r,pi(d[i]));nl();
 if(w->t)DO(n,P("< ");pr(w->p[i]))else DO(n,pi(w->p[i]));nl();}

C vt[]="+{~<#,";
I st[26]; qp(a){R  a>='a'&&a<='z';}qv(a){R a<'a';}
A ex(e)I *e;{I a=*e;
 if(qp(a)){if(e[1]=='=')R st[a-'a']=ex(e+2);a= st[ a-'a'];}
 R qv(a)?(*vm[a])(ex(e+1)):e[1]?(*vd[e[1]])(a,ex(e+2)):(A)a;}
noun(c){A z;if(c<'0'||c>'9')R 0;z=ga(0,0,0);*z->p=c-'0';R z;}
verb(c){I i=0;for(;vt[i];)if(vt[i++]==c)R i;R 0;}
I *wd(s)C *s;{I a,n=strlen(s),*e=ma(n+1);C c;
 DO(n,e[i]=(a=noun(c=s[i]))?a:(a=verb(c))?a:c);e[n]=0;R e;}

main(){C s[99];while(gets(s))pr(ex(wd(s)));}

On the basis of this fragment Hui developed an interpreter for another APL variant, J.

Unlike APL and A+, J used the ASCII character set. It included advanced features, such as support for parallel MIMD operations. Other ideas by Whitney found their way into J: orienting primitives on the leading axis, using prefix rather than suffix for agreement, and total array ordering.

Ken Iverson, his son, Eric Iverson, and Hui all ended up working in a company called Jsoftware in the 1990s–2000s.

Kx Systems and kdb

UBS headquarters, New York
UBS headquarters, New York

Whitney left Morgan Stanley in 1993 and co-founded Kx Systems with Janet Lustgarten, where he developed another APL variant, called k.

Kx Systems was under an exclusive agreement with Union Bank of Switzerland (UBS). For the next four years Whitney developed various financial and trading applications using k for UBS.

The contract ended in 1997 when UBS merged with Swiss Bank.

In 1998, Kx Systems released kdb, a columnar in-memory database built on k by Arthur Whitney. Kdb included ksql, a query language with an SQL-like syntax, including constructs such as select.


kdb+ was released in June 2003. This was more or less a total rewrite of kdb for 64-bit systems based on the 4th version of k and q, a macro language layer (or a query language, hence the name) on top of k, defined in terms of k. Both q and k compile to the same byte code that is executed in the kdb+ byte interpreter. For example, type in q is the equivalent of @: in k. Q is much more readable than k and most kdb+ developers write their code in q, not k. The q programming language contains its own table query syntax called q-sql, which in some ways resembles the traditional SQL.

Steph Johnson tells the story of kdb+ release in Vector, the journal of the British APL Association:

The single largest annual event for IT executives in the securities industry, the SIA Technology Management Conference & Exhibit takes place in New York City each June and draws an international audience. Kx is traditionally a sponsor of and exhibitor at the event. This year the conference coincided with the announcement of the company’s new 64-bit kdb+ database, which Kx demonstrated on two networked Itanium 2 servers running Linux on the show floor.

Kdb+ has been in beta test at several leading firms including Lehman Brothers. The new database is more than simply a version of kdb for 64-bit architectures, although the need for trading, risk management and other real-time business applications to capture and analyze more transactions and events in memory was the impetus behind its development.

Arthur Whitney, developer of A+, K and all Kx Systems software, designed kdb+ to handle more data types than kdb and to provide data types that map to Java and .net. In addition, kdb+ is several orders of magnitude faster than kdb on operations such as inserts and updates, and it offers greatly reduced latency to speed up transaction processing. Kdb+ also merges the K language with ksql to provide a single language for programming and analytics.

Kdb+ is not a replacement for kdb, which continues to receive growing acceptance in financial and other industries. Kx will continue to develop and sell both platforms. According to Whitney, "After 10 years of experience with customers using K and kdb, we learned a lot, and it was time to incorporate that learning into a next-generation product."

Initially, kdb+ is expected to appeal to firms whose intraday data requirements exceed the capacity of 32-bit architectures. "Many of our customers want to deal with 50 million or more transactions per day." says Whitney. "64-bit makes this easy."

Kx customers are seeing trade data volumes that nearly double each year, and Whitney expects that the financial industry will be one of the first movers to 64-bit technology.

KX Systems party
Michael O'Neill of First Derivatives (centre) acknowledges Arthur Whitney and Janet Lustgarten at a Kx Systems party in New York in June.

The release of kdb+ coincided with a 10-year celebration for Kx Systems. Ibid.:

Founded in 1993, Kx celebrated its first 10 years with a party at mid-town Manhattan's stylish restaurant Noche ("night" in Spanish). The restaurant's Latin theme was complemented by the live music of a 4-piece Brazilian band, and nearly 150 customers, partners, prospects and friends of Kx attended.

Towards the end of the evening, the party took a surprise twist, as employees of Kx and its strategic partner First Derivatives presented Whitney and CEO Janet Lustgarten with a commemorative Steuben glass sculpture. The plaque read:

Kx Systems—10 years of success—2003. Several long-time customers and friends, including Joel Kaplan, who first hired Whitney at Morgan Stanley, were on hand to express their congratulations.

For Whitney, the party and SIA show helped him officially mark the completion of kdb+, which has been in development for several years. "Usually I am too busy programming or thinking about how to program something to pay much attention to product announcements. But this time, being surrounded by so many people who were celebrating the announcement and the company, I had to stop and appreciate what Kx has become over the years."

First Derivatives

On 2 July 2018, First Derivatives plc announced that it was to buy out minority Kx Systems shareholders:

FD (AIM:FDP.L, ESM:DFP.I) announces that it has reached agreement with the minority shareholders of Kx Systems, Inc. ("Kx Systems"), a subsidiary of the Group, regarding the acquisition by FD of their entire remaining shareholding (the "Transaction"). Upon completion of the Transaction, which is expected to take place on or before 29 June 2019, FD will own 100% of the issued share capital of Kx Systems.

Terms of the Transaction

FD has agreed to acquire the remaining 600,022 Kx Systems shares that it doesn't already own from the minority shareholders, namely Arthur Whitney and Janet Lustgarten, who are co-founders and current directors of Kx Systems, and their associated persons. The aggregate consideration is $53.8m in cash, to be financed from FD's available facilities. The terms of the transaction are in line with those agreed between FD and the minority shareholders in October 2014, and include a payment of $12.0m in lieu of anticipated dividends to the minority shareholders for the period up to 31 October 2021.

Brian Conlon, Chief Executive Officer of FD, commented: "Since we acquired a controlling interest in Kx Systems in October 2014 we have invested heavily across our business to target multiple new industries. The agreement to acquire 100% of Kx Systems provides certainty for the Group and its shareholders as we seek to realise the considerable potential we see for further growth."


On 27 November 2018 Sarah Butcher wrote an article in eFinancialCareers entitled The new data platform from the reclusive genius of banking IT:

If you work in financial technology, you should know the name Arthur Whitney. He's the Canadian computer scientist who invented the A+ programming language at Morgan Stanley in the late 1980s. He's also the guy that founded Kx Systems, the data analysis company now owned by First Derivatives, which runs the Kdb databases that underpin most algorithmic trading systems today. Now Whitney—who has been described as having "programming powers beyond the ken of mere mortals"—is back with a newer and better data platform. But he isn't talking to anyone about it.

Released last month, Whitney's new creation is called Shakti (after the name for primordial cosmic energy in Hinduism). According to Whitney, Shakti boasts a level of sophistication that will make it, "a new standard for data storage and analysis within financial services and beyond."

Whitney is reclusive these days and doesn't talk to the press, so Shakti's New York-based director of engineering (and ex-Nomura and Barclays kdb specialist) Fintan Quill is flying the flag in public. "High-data capture rates, combined with low-latency query response times make Shakti the perfect platform for trade signal calculation, pre/post-trade risk, real-time surveillance, back-testing among other things," enthuses Quill.

The new platform is also optimal for cloud computing: "The small memory footprint allows for fast deployment and processing of distributed elastic workloads." It can work with all kinds of datasets, including numerical, temporal and text data, whether structured or not.

Shakti's secret is parallelism, which enables it to work faster than predecessors. Parallelism is built into most of its primitive functions and can expand to multiple jobs and machines via what Quill describes as, "a custom-built interprocess communication protocol as well as a native fork-execution model."

These allow it to function in highly distributed high performance computing (HPC) workloads, while single instruction, multiple data (SIMD) technology allows Shakti to function within the 256 or 512-bit register widths of most modern computers. As a result (says Quill), the platform boasts a combination of low cost and very high performance.

If you think you might want to work with Shakti, it will help to be proficient at programming in Python—The system comes with a Python interface that merges those two interpreters into a single process. "This allows Python developers access to the extremely fast data-handling capabilities of Shakti," says Quill, who says the interface means there's "zero-copy" between the two systems, which reduces processing time.

Shakti is very new and it's not clear who—if anyone—has bought into it already. But after a display of its abilities in London and New York in the past six months, the platform may yet get some takers. Last week, it added a COO, Abby Gruen, who joined from Kx.

Quill suggests that any system involving Arthur Whitney should not go ignored. Morgan Stanley, after all, is still using A+ years after Whitney left. "The simple yet intelligent ideas underlying the APL-derived languages inspired by Ken Iverson have shown the robustness to pass the test of time," says Quill. Shakti is only just starting out, but Whitney's history suggests it might be around for a while to come.