Archive for the ‘Language design’ Category.

In praise of Knuth and Liskov

In November of 2005, as part of the festivities of its 150-th anniversary, the ETH Zurich bestowed honorary doctorates on Don Knuth and Barbara Liskov. I gave the speech (the “laudatio”). It was published in Informatik Spektrum, the journal of Gesellschaft für Informatik, the German Computer Society, vo. 29, no. 1, February 2006, pages 74-76; I came across it recently and thought others might be interested in this homage to two great computer scientists.  The beginning was in German; I translated it into English. I also replaced a couple of German expressions by their translations: “ETH commencement” for ETH-Tag (the official name of the annual ceremony) and “main building” for Hauptgebäude.

I took this picture of Wirth, Liskov and Knuth (part of my gallery of computer scientists)  later that same day.



 In an institution, Ladies and Gentlement, which so proudly celebrates its hundred-and-fiftieth anniversary, a relatively young disciplines sometimes has cause for envy. We computer scientists are still the babies, or at least the newest kids on the block. Outside of this building, for example, you will see streets bearing such names as Clausius, yet there is neither a Von Neumann Lane nor a a Wirth Square. Youth, however,  also has its advantages; perhaps the most striking is that we still can, in our own lifetime, meet in person some of the very founders of our discipline. No living physicist has seen Newton; no chemist has heard Lavoisier. For us, it works. Today, Ladies and Gentlemen, we have the honor of introducing two of the undisputed pioneers of informatics.

Barbara Liskov

The first of our honorees today is Professor Barbara Liskov. To understand her contributions it is essential to realize the unfair competition in which the so-called Moore’s law pits computer software against computing hardware. To match the astounding progress of computing speed and memory over the past five decades, all that we have on the software side is our own intelligence which, it is safe to say, doesn’t double every eighteen months at constant price. The key to scaling up is abstraction; all advances in programming methodology have relied on new abstraction techniques. Perhaps the most significant is data abstraction, which enables us to organize complex systems on the basis of the types of objects they manipulate, defined in completely abstract terms. This is the notion of abstract data type, a staple component today of every software curriculum, including in the very first programming course here ETH. it was introduced barely thirty years ago in a seemingly modest article in SIGPLAN Notices — the kind of publication that hardly registers a ripple in science indexes — by Barbara Liskov and Stephen Zilles. Few papers have had a more profound impact on the theory and practice of software development than this contribution, “Programming with Abstract Data Types”.

The idea of abstract data types, or ADTs, is one of those Egg of Christopher Columbus moments; a seemingly simple intuition that changes the course of things. An ADT is a class of objects described in terms not of their internal properties, but of the operations applicable to them, and the abstract properties of these operations. Not by what they are, but by what they have. A rather capitalistic view of the world, but well suited to the description of complex systems where each part knows as little as possible about the others to protect itself about their future changes.

An abstraction such as ETH-Commencement could be described in a very concrete way: it happens in a certain place, consists of one event after another, gathers so many people. This is what we computer scientists call an implementation-oriented view, and relying on it means that we can’t change any detail without endangering the consistency of other processes, such as the daily planning of room allocation in the Main Building, which use it. In an ADT view, the abstraction “ETH Commencement” is characterized not by what it is but by what it has: a start, an end, an audience, and operations such as “Schedule the ETH Commencement”, “ Reschedule it”, “Start it”, “End it”. They provide to the rest of the world a clean, precisely specified interface which enables every ADT to use every other based on the minimum properties it requires, thus isolating them from irrelevant internal changes, and providing an irreplaceable weapon in the incessant task of software engineering: battling complexity.

Barbara Liskov didn’t stay with the theoretical concepts but implemented the ideas in the CLU language, one of the most influential of the set of programming languages that in the nineteen-seventies changed our perspective of how to develop good software.

She went on to seminal work on operating systems and distributed computing, introducing several widely applied concepts such as guardians, and always backing her theoretical innovations by building practical systems, from the CLU language and compiler to the Argus and Mercury distributed operating systems. Distributed systems, such as those which banks, airlines and other global enterprises run on multiple machines across multiple networks, raise particularly challenging issues. To quote from the introduction of her article on Argus:

A centralized system is either running or crashed, but a distributed system may be partly running and partly crashed. Distributed programs must cope with failures of the underlying hardware. Both the nodes and the network may fail. The goal of Argus is to provide mechanisms that make it easier for programmers to cope with these problems.

Barbara Liskov’s work introduced seminal concepts to deal with these extremely difficult problems.

Now Ford professor of engineering at MIT, she received not long ago the prestigious John von Neumann award of the IEEE; she has been one of the most influential people in software engineering. We are grateful for how Professor Barbara Liskov has helped shape the field are honored to have her at ETH today.

 Donald Knuth

In computer science and beyond, the name of Donald Knuth carries a unique aura. A professor at Stanford since 1968, now emeritus, he is the only person on record whose job title is the title of his own book: Professor of the Art of Computer Programming. This is for his eponymous multi-volume treatise, which established the discipline of algorithm analysis, and has had more effect than any other computer science publication. The Art of Computer Programming is a marvel of breadth, depth, completeness, mathematical rigor and clarity, not to forget humor. In that legendary book you will find exposed in detail the algorithms and data structures that lie at the basis of all software applications today. A Monte Carlo simulation, as a physicists may use, requires a number sequence that is both very long and very random-looking, even though the computer is a deterministic machine; if the simulation is any good, it almost certainly relies on the devious techniques which The Art of Computer Programming presents for making a perfectly deterministic sequence appear to have no order or other recognizable property. If you are running complex programs on your laptop, and they keep creating millions of software objects without clogging up gigabytes of memory, chances are the author of the garbage collector program is using techniques he learned from Knuth, with such delightful names as “the Buddy System”. If your search engine can at the blink of an eye find a needle of useful information in a haystack of tens of billions of Web pages, it’s most likely because they’ve been indexed using finely tuned data structures, such as hash tables, for which Knuth has been the reference for three decades through volume three, Searching and Sorting.

Knuth is famous for his precision and attention to detail, going so far as to offer a financial reward for every error found in his books, although one suspects this doesn’t cost him too much since people are so proud that instead of cashing the check they have it framed for display. The other immediately striking characteristic of Knuth is how profoundly he is driven by esthetics. This applies to performing arts, as anyone who was in the Fraumünster this morning and found out who the organist was can testify, but even more to his scientific work. The very title “the Art of computer programming” betrays this. Algorithms and data structures for Knuth are never dull codes for computers, but objects of intense esthetic pleasure and friendly discussion. This concern with beauty led to a major turn in his career, which delayed the continuation of the book series by many years but resulted in a development that has affected anyone who publishes scientific text. As he received the page proofs of the second edition of one of the volumes in the late seventies he was so repelled by its physical appearance, resulting from newly introduced computer typesetting technology, that he decided to build a revolutionary font design and text processing system, all by himself, from the ground up. This resulted in a number of publications such as a long and fascinating paper in the Bulletin of the American Mathematical Society entitled “The Letter S”, but even more importantly in widely successful and practical software programs which he wrote himself, TeX and Metafont, which have today become standards for scientific publishing. Here too he has shown the way in quality and rigor, being one of the very few people in the world who promise their software to be free of bugs, and backs that promise by giving a small financial reward for any counter-example.

His numerous other contributions are far too diverse to allow even a partial mention here; they have ranged across wide areas of computer science and mathematics.

To tell the truth, we are a little embarrassed that by bringing Professor Knuth here we are delaying by a bit more the long awaited release of volume 4. But we overcome this embarrassment in time to express our pride for having Donald Erwin Knuth at ETH for this anniversary celebration.

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Reflexivity, and other pillars of civilization

Let me start, dear reader of this blog, by probing your view of equality, and also of assignment. Two questions:  

  • Is a value x always equal to itself? (For the seasoned programmers in the audience: I am talking about a value, as in mathematics, not an expression, as in programming, which could have a side effect.)
  • In programming, if we consider an assignment

       x := y

and v is the value of y before that assignment (again, this little detour is to avoid bothering with side effects), is the value of x always equal to v after the assignment?  

Maybe I should include here one of these Web polls that one finds on newspaper sites, so that you can vote and compare your answer to the Wisdom of Crowds. My own vote is clear: yes to both. Equality is reflexive (every value is equal to itself, at any longitude and temperature, no excuses and no exceptions); and the purpose of assignment is to make the value of the target equal to the value of the source. Such properties are some of the last ramparts of civilization. If they go away, what else is left?  

754 enters the picture

Now come floating-point numbers and the famous IEEE “754” floating-point standard [1]. Because not all floating point operations yield a result usable as a floating-point number, the standard introduces a notion of “NaN”, Not a Number; certain operations involving floating-point numbers may yield a NaN. The term NaN does not denote a single value but a set of values, distinguished by their “payloads”.  

Now assume that the value of x is a NaN. If you use a programming language that supports IEEE 754 (as they all do, I think, today) the test in  

        if x = x then …  

is supposed to yield False. Yes, that is specified in the standard: NaN is never equal to NaN (even with the same payload); nor is it equal to anything else; the result of an equality comparison involving NaN will always be False.  

Assignment behavior is consistent with this: if y (a variable, or an expression with no side effect) has a NaN value, then after  

        x := y  

the test xy will yield False. 

Before commenting further let me note the obvious: I am by no means a numerics expert; I know that IEEE 754 was a tremendous advance, and that it was designed by some of the best minds in the field, headed by Velvel Kahan who received a Turing Award in part for that success. So it is quite possible that I am about to bury myself in piles of ridicule. On the other hand I have also learned that (1) ridicule does not kill, so I am game; and more importantly (2) experts are often right but not always, and it is always proper to question their reasoning. So without fear let me not stop at the arguments that “the committee must have voted on this point and they obviously knew what they were doing” and “it is the standard and implemented on zillions of machines, you cannot change it now”. Both are true enough, but not an excuse to censor questions.  

What are the criteria?

The question is: compatibility with an existing computer standard is great, but what about compatibility with a few hundred years of mathematics? Reflexivity of equality  is something that we expect for any data type, and it seems hard to justify that a value is not equal to itself. As to assignment, what good can it be if it does not make the target equal to the source value?  

The question of assignment is particularly vivid in Eiffel because we express the expected abstract properties of programs in the form of contracts. For example, the following “setter” procedure may have a postcondition (expressed by the ensure clause):  

        set_x (v: REAL)
                        — Set the value of x (an attribute, also of type REAL) the value of v.
                        x := v  
                        x = v

If you call this procedure with a NaN argument for a compiler that applies IEEE 754 semantics, and monitor contracts at run time for testing and debugging, the execution will report a contract violation. This is very difficult for a programmer to accept.  

A typical example arises when you have an assignment to an item of an array of REAL values. Assume you are executing a [i] := x. In an object-oriented view of the world (as in Eiffel), this is considered simplified syntax  for the routine call a.put (x, i). The postcondition is that a [i] = x. It will be violated!  

The experts’ view

I queried a number of experts on the topic. (This is the opportunity to express my gratitude to members of the IFIP working group 2.5 on numerical software [2], some of the world’s top experts in the field, for their willingness to respond quickly and with many insights.) A representative answer, from Stuart Feldman, was:  

If I remember the debate correctly (many moons ago), NaN represents an indefinite value, so there is no reason to believe that the result of one calculation with unclear value should match that of another calculation with unclear value. (Different orders of infinity, different asymptotic approaches toward 0/0, etc.)  

Absolutely correct! Only one little detail, though: this is an argument against using the value True as a result of the test; but it is not an argument for using the value False! The exact same argument can be used to assert that the result should not be False:  

… there is no reason to believe that the result of one calculation with unclear value should not match that of another calculation with unclear value.  

Just as convincing! Both arguments complement each other: there is no compelling reason for demanding that the values be equal; and there is no compelling argument either to demand that they be different. If you ignore one of the two sides, you are biased.  

What do we do then?

The conclusion is not that the result should be False. The rational conclusion is that True and False are both unsatisfactory solutions. The reason is very simple: in a proper theory (I will sketch it below) the result of such a comparison should be some special undefined below; the same way that IEEE 754 extends the set of floating-point numbers with NaN, a satisfactory solution would extend the set of booleans with some NaB (Not a Boolean). But there is no NaB, probably because no one (understandably) wanted to bother, and also because being able to represent a value of type BOOLEAN on a single bit is, if not itself a pillar of civilization, one of the secrets of a happy life.  

If both True and False are unsatisfactory solutions, we should use the one that is the “least” bad, according to some convincing criterion . That is not the attitude that 754 takes; it seems to consider (as illustrated by the justification cited above) that False is somehow less committing than True. But it is not! Stating that something is false is just as much of a commitment as stating that it is true. False is no closer to NaB than True is. A better criterion is: which of the two possibilities is going to be least damaging to time-honored assumptions embedded in mathematics? One of these assumptions is the reflexivity of equality:  come rain or shine, x is equal to itself. Its counterpart for programming is that after an assignment the target will be equal to the original value of the source. This applies to numbers, and it applies to a NaN as well. 

Note that this argument does not address equality between different NaNs. The standard as it is states that a specific NaN, with a specific payload, is not equal to itself.  

What do you think?

A few of us who had to examine the issue recently think that — whatever the standard says at the machine level — a programming language should support the venerable properties that equality is reflexive and that assignment yields equality.

Every programming language should decide this on its own; for Eiffel we think this should be the specification. Do you agree?  

Some theory

For readers who like theory, here is a mathematical restatement of the observations above. There is nothing fundamentally new in this section, so if you do not like strange symbols you can stop here.  

The math helps explain the earlier observation that neither True nor False is more“committing” than the other. A standard technique (coming from denotational semantics) for dealing with undefinedness in basic data types, is to extend every data type into a lattice, with a partial order relation meaning “less defined than”. The lattice includes a bottom element, traditionally written “” (pronounced “Bottom”) and a top element written (“Top”). represents an unknown value (not enough information) and an error value (too much information). Pictorially, the lattice for natural numbers would look like this:  

Integer lattice

The lattice of integers

On basic types, we always use very simple lattices of this form, with three kinds of element: , less than every other element; , larger than all other elements; and in-between all the normal values of the type, which for the partial order of interest are all equal. (No, this is not a new math in which all integers are equal. The order in question simply means “is less defined than”. Every integer is as defined as all other integers, more defined than , and less defined than .) Such lattices are not very exciting, but they serve as a starting point; lattices with more interesting structures are those applying to functions on such spaces — including functions of functions —, which represent programs.  

The modeling of floating-point numbers with NaN involves such a lattice; introducing NaN means introducing a value. (Actually, one might prefer to interpret NaN as , but the reasoning transposes immediately.)  More accurately, since there are many NaN values, the lattice will look more like this:

Float lattice

The lattice of floats in IEEE 754

For simplicity we can ignore the variety of NaNs and consider a single .

Functions on lattices — as implemented by programs — should satisfy a fundamental property: monotonicity. A function f  is monotone (as in ordinary analysis) if, whenever xy, then f (x) ≤ f (y). Monotonicity is a common-sense requirement: we cannot get more information from less information. If we know less about x than about y, we cannot expect that any function (with a computer, any program) f will, out of nowhere, restore the missing information.  

Demanding monotonicity of all floating-point operations reflects this exigency of monotonicity: indeed, in IEEE 754, any arithmetic operation — addition, multiplication … — that has a NaN as one of its arguments must yield a Nan as its result. Great. Now for soundness we should also have such a property for boolean-valued operations, such as equality. If we had a NaB as the  of booleans, just like NaN is the  of floating-point numbers,  then the result of NaN = NaN would be NaB. But the world is not perfect and the IEEE 754 standard does not govern booleans. Somehow (I think) the designers thought of False as somehow less defined than True. But it is not! False is just as defined as True in the very simple lattice of booleans; according to the partial order, True and False are equal for the relevant partial order:

Boolean lattice

The lattice of booleans

Because any solution that cannot use a NaB will violate monotonicity and hence will be imperfect, we must resort to heuristic criteria. A very strong criterion in favor of choosing True is reflexivity — remaining compatible with a fundamental property of equality. I do not know of any argument for choosing False. 

The Spartan approach

There is, by the way, a technique that accepts booleans as we love them (with two values and no NaB) and achieves mathematical rigor. If operations involving NaNs  truly give us pimples, we can make any such operation trigger an exception. In the absence of values,  this is an acceptable programming technique for representing undefinedness. The downside, of course, is that just about everywhere the program must be ready to handle the exception in some way. 

It is unlikely that in practice many people would be comfortable with such a solution. 

Final observations

Let me point out two objections that I have seen. Van Snyder wrote: 

NaN is not part of the set of floating-point numbers, so it doesn’t behave as if “bottom” were added to the set. 

Interesting point, but, in my opinion not applicable: is indeed not part of the mathematical set of floating point numbers, but in the form of NaN it is part of the corresponding type (float in C, REAL in Eiffel); and the operations of the type are applicable to all values, including NaN if, as noted, we have not taken the extreme step of triggering an exception every time an operation uses a NaN as one of its operands. So we cannot free ourselves from the monotonicity concern by just a sleight of hands. 

Another comment, also by Van Snyder (slightly abridged): 

Think of [the status of NaN] as a variety of dynamic run-time typing. With static typing, if  x is an integer variable and y

        x := y 

does not inevitably lead to 

        x = y

 True; and a compelling argument to require that conversions satisfy equality as a postcondition! Such  reasoning — reflexivity again — was essential in the design of the Eiffel conversion mechanism [3], which makes it possible to define conversions between various data types (not just integers and reals and the other classical examples, but also any other user types as long as the conversion does not conflict with inheritance). The same conversion rules apply to assignment and equality, so that yes, whenever the assignment x := y is permitted, including when it involves a conversion, the property x = y  is afterwards always guaranteed to hold.

It is rather dangerous indeed to depart from the fundamental laws of mathematics. 


[1] IEEE floating-point standard, 754-2008; see summary and references in the Wikipedia entry.  

[2] IFIP Working Group 2.5 on numerical software: home page

[3] Eiffel standard (ECMA and ISO), available on the ECMA site.

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A couple of loop examples

(This entry originated as a post on the EiffelStudio user group mailing list.) 

Here are a couple of actual examples of the new loop variants discussed in the blog entry immediately preceding this one. They came out of my current work; I started updating a program to take advantage of the new facility.

As a typical example, I replaced

                eht := item (e)
                Result.extend (eht.key_for_iteration)


        across item (e) as eht loop Result.extend (eht.key) end

 which also gets rid of the local variable declaration. The second form is syntactic sugar for the first, but I find it justified. 

 Another case, involving nested loops: 

— Previously:

                oht := other.item_for_iteration
                e := other.key_for_iteration
                        put (e, oht.item_for_iteration)

— Now:

        across other as o loop
                across o.item as oht loop put (o.key, oht.item) end

here getting rid of two local variable declarations (although I might for efficiency reintroduce the variable e  to compute o.key just once). 

It is important to note that these are not your grandmother’s typical loops: they iterate on complex data structures, specifically hash tables where the keys are lists and the items are themselves hash tables, with lists as both items and keys. 

The mechanism is applicable to all the relevant data structures in EiffelBase (in other words, no need for the programmer to modify anything, just apply the across  loop to any such structure), and can easily extended to any new structure that one wishes to define. In the short term at least, I still plan in my introductory teaching to show the explicit variants first, as it is important for beginners to understand how a loop works. (My hunch based on previous cases is that after a while we will feel comfortable introducing the abstract variants from the start, but it takes some time to learn how to do it right.)

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