Archive for the ‘Algorithms’ Category.

Framing the frame problem (new paper)

Among the open problems of verification, particularly the verification of object-oriented programs, one of the most vexing is framing: how to specify and verify what programs element do not change. Continuing previous work, this article presents a “double frame inference” method, automatic on both sides the specification and verification sides. There is no need to write frame specifications: they will be inferred from routine postconditions. For verification, the method computes the set of actually changed properties through a “change calculus”, itself based on the previously developed alias calculus.

Some verification techniques, such as Hoare-style proofs, require significant annotation effort and potentially yield full functional verification; others, such as model checking and abstract interpretation, have more limited goals but seek full automation. Framing, in my opinion, should be automatic, freeing the programmer-verifier to devote the annotation effort to truly interesting properties.

Reference

[1] Bertrand Meyer: Framing the Frame Problem, in Dependable Software Systems, Proceedings of August 2014 Marktoberdorf summer school, eds. Alexander Pretschner, Manfred Broy and Maximilian Irlbeck, NATO Science for Peace and Security, Series D: Information and Communication Security, Springer, 2015 (to appear), pages 174-185; preprint available here.

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Detecting deadlock automatically? (New paper)

To verify sequential programs, we have to prove that they do the right thing, but also that they do it within our lifetime — that they terminate. The termination problem is considerably harder with concurrent programs, since they add a new form of non-termination: deadlock. A set of concurrent processes or threads will deadlock if they end up each holding a resource that another wants and wanting a resource that another holds.

There is no general solution to the deadlock problem, even a good enough general solution. (“Good enough” is the best we can hope for, since like many important problems deadlock is undecidable.) It is already hard enough to provide run-time deadlock detection, to be able at least to cancel execution when deadlock happens. The research reported in this new paper [1] pursues the harder goal of static detection. It applies to an object-oriented context (specifically the SCOOP model of concurrent OO computation) and relies fundamentally on the alias calculus, a static alias analysis technique developed in previous publications.

The approach is at its inception and considerable work remains to be done. Still, the example handled by the paper is encouraging: analyzing two versions of the dining philosophers problem and proving — manually — that one can deadlock and the other cannot.

References

[1] Bertrand Meyer: An automatic technique for static deadlock prevention, in PSI 2014 (Ershov Informatics Conference), eds. Irina Virbitskaite and Andrei Voronkov, Lecture Notes in Computer Science, Springer, 2015, to appear.; draft available here.

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Lampsort

 

In support of his view of software methodology, Leslie Lamport likes to use the example of non-recursive Quicksort. Independently of the methodological arguments, his version of the algorithm should be better known. In fact, if I were teaching “data structures and algorithms” I would consider introducing it first.

As far as I know he has not written down his version in an article, but he has presented it in lectures; see [1]. His trick is to ask the audience to give a non-recursive version of Quicksort, and of course everyone starts trying to remove the recursion, for example by making the stack explicit or looking for invertible functions in calls. But his point is that recursion is not at all fundamental in Quicksort. The recursive version is a specific implementation of a more general idea.

Lamport’s version — let us call it Lampsort —is easy to express in Eiffel. We may assume the following context:

a: ARRAY [G -> COMPARABLE]        — The array to be sorted.
pivot: INTEGER                                      —  Set by partition.
picked: INTEGER_INTERVAL            — Used by the sorting algorithm, see below.
partition (i, j: INTEGER)
……..require      — i..j is a sub-interval of the array’s legal indexes:
……..……..i < j
……..……..i >= a.lower
……..……..j <= a.upper
……..do
……..……..… Usual implementation of partition
……..ensure     — The expected effect of partition:
……..……..pivot >= i
……..……..pivot < j
……..……..a [i..j] has been reshuffled so that elements in i..pivot are less than
……..……..or equal to those in pivot+1 .. j.
……..end

We do not write the implementation of partition since the point of the present discussion is the overall algorithm. In the usual understanding, that algorithm consists of doing nothing if the array has no more than one element, otherwise performing a partition and then recursively calling itself on the two resulting intervals. The implementation can take advantage of parallelism by forking the recursive calls out to different processors. That presentation, says Lamport, describes only a possible implementation. The true Quicksort is more general. The algorithm works on a set of integer intervals; in Eiffel:

intervals: SET [INTEGER_INTERVAL]

It initializes intervals to contain a single element, the entire interval; at each iteration, it removes an interval from the set, partitions it if that makes sense (i.e. the interval has more than one element), and inserts the resulting two intervals into the set. It ends when the set is empty. Here it is:

……..from                                 — Initialize interval set to contain a single interval, the array’s entire index range:
……..…..create intervals.make_one (a.lower |..| a.upper)….         ..……..
……..invariant
……..…..— See below
……..until
……..…..intervals.is_empty                                                            — Stop when there are no more intervals in set
……..loop
……..…..picked := intervals.item                                                    — Pick an interval from (non-empty) interval set.
……..……if picked.count > 1 then                                                  — (The precondition of partition holds, see below.)
……..……..…..partition (picked.lower, picked.upper)               — Split it, moving small items before and large ones after pivot.
……..……..…..intervals.extend (picked.lower |..| pivot)            — Insert new intervals into interval set: first
……..……....intervals.extend (pivot + 1 |..| picked.upper)     — and second.
……..……end
……..…...intervals.remove (picked)                                               — Remove interval that was just partitioned.
…….end

Eiffel note: the function yielding an integer interval is declared in the library class INTEGER using the operator |..| (rather than just  ..).

The query item from SET, with the precondition not is_empty,  returns an element of the set. It does not matter which element. In accordance with the Command-Query Separation principle, calling item does not modify the set; to remove the element you have to use the command remove. The command extend adds an element to the set.

The abstract idea behind Lampsort, explaining why it works at all, is the following loop invariant (see [2] for a more general discussion of how invariants provide the basis for understanding loop algorithms). We call “slice” of an array a non-empty contiguous sub-array; for adjacent slices we may talk of concatenation; also, for slices s and t s <= t means that every element of s is less than or equal to every element of t. The invariant is:

a is the concatenation of the members of a set slices of disjoint slices, such that:
– The elements of a are a permutation of its original elements.
– The index range of any member  of slices having more than one element is in intervals.
– For any adjacent slices s and t (with s before t), s <= t.

The first condition (conservation of the elements modulo permutation) is a property of partition, the only operation that can modify the array. The rest of the invariant is true after initialization (from clause) with slices made of a single slice, the full array. The loop body maintains it since it either removes a one-element interval from intervals (slices loses the corresponding slice) or performs partition with the effect of partitioning one slice into two adjacent ones satisfying s <= t, whose intervals replace the original one in intervals. On exit, intervals is empty, so slices is a set of one-element slices, each less than or equal to the next, ensuring that the array is sorted.

The invariant also ensures that the call to partition satisfies that routine’s precondition.

The Lampsort algorithm is a simple loop; it does not use recursion, but relies on an interesting data structure, a set of intervals. It is not significantly longer or more difficult to understand than the traditional recursive version

sort (i, j: INTEGER)
……..require
……..……..i <= j
……..……..i >= a.lower
……..……..j <= a.upper
……..do
……..……if j > i then                    — Note that precondition of partition holds.
……..……..…..partition (i, j)         — Split into two slices s and t such that s <= t.
……..……..…..sort (i, pivot)          — Recursively sort first slice.
……..……..…..sort (pivot+1, j)      — Recursively sort second slice.
……..……end……..…..
……..end

Lampsort, in its author’s view, captures the true idea of Quicksort; the recursive version, and its parallelized variants, are only examples of possible implementations.

I wrote at the start that the focus of this article is Lampsort as an algorithm, not issues of methodology. Let me, however, give an idea of the underlying methodological debate. Lamport uses this example to emphasize the difference between algorithms and programs, and to criticize the undue attention being devoted to programming languages. He presents Lampsort in a notation which he considers to be at a higher level than programming languages, and it is for him an algorithm rather than a program. Programs will be specific implementations guided in particular by efficiency considerations. One can derive them from higher-level versions (algorithms) through refinement. A refinement process may in particular remove or restrict non-determinism, present in the above version of Lampsort through the query item (whose only official property is that it returns an element of the set).

The worldview underlying the Eiffel method is almost the reverse: treating the whole process of software development as a continuum; unifying the concepts behind activities such as requirements, specification, design, implementation, verification, maintenance and evolution; and working to resolve the remaining differences, rather than magnifying them. Anyone who has worked in both specification and programming knows how similar the issues are. Formal specification languages look remarkably like programming languages; to be usable for significant applications they must meet the same challenges: defining a coherent type system, supporting abstraction, providing good syntax (clear to human readers and parsable by tools), specifying the semantics, offering modular structures, allowing evolution while ensuring compatibility. The same kinds of ideas, such as an object-oriented structure, help on both sides. Eiffel as a language is the notation that attempts to support this seamless, continuous process, providing tools to express both abstract specifications and detailed implementations. One of the principal arguments for this approach is that it supports change and reuse. If everything could be fixed from the start, maybe it could be acceptable to switch notations between specification and implementation. But in practice specifications change and programs change, and a seamless process relying on a single notation makes it possible to go back and forth between levels of abstraction without having to perform repeated translations between levels. (This problem of change is, in my experience, the biggest obstacle to refinement-based approaches. I have never seen a convincing description of how one can accommodate specification changes in such a framework without repeating the whole process. Inheritance, by the way, addresses this matter much better.)

The example of Lampsort in Eiffel suggests that a good language, equipped with the right abstraction mechanisms, can be effective at describing not only final implementations but also abstract algorithms. It does not hurt, of course, that these abstract descriptions can also be executable, at the possible price of non-optimal performance. The transformation to an optimal version can happen entirely within the same method and language.

Quite apart from these discussions of software engineering methodology, Lamport’s elegant version of Quicksort deserves to be known widely.

References

[1] Lamport video here, segment starting at 0:32:34.
[2] Carlo Furia, Bertrand Meyer and Sergey Velder: Loop invariants: Analysis, Classification and Examples, in ACM Computing Surveys, September 2014, preliminary text here.

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Hungarian rotation

The 2013 Informatics Europe “Best Practices in Education” award was devoted, this year, to initiatives for teaching informatics in schools [1].  It was given out last week at the European Computer Science Summit in Amsterdam [2]. Two teams shared it, one from Poland and the other from Romania. Both teams showed excellent projects, but the second was beyond anything I expected.

The project comes from the Hungarian-speaking Sapientia University in Transylvania and is devoted to teaching algorithms visually and “at the same time enhancing intercultural communication” in the region. It illustrates the classical sorting algorithms through folk dances. Quicksort is Hungarian, selection sort is gypsy, merge sort is “Transylvanian-saxon”. I think my favorite is Shell sort [3]. For more, see their YouTube channel [4].

Now  if only they could act the loop invariants [5].

References

[1] 2013 Best Practices in Education award, see here.

[2] 2013 European Computer Science Summit, here.

[3] “Shell sort with Hungarian (Székely) folk dance”, see here.

[4] YouTube Algorythmics channel, here.

[5] Carlo Furia, Bertrand Meyer and Sergey Velder: Loop invariants: Analysis, Classification and Examples, in ACM Computing Surveys, Septembre 2014, to appear, available here.

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The invariants of key algorithms (new paper)

 

I have mentioned this paper before but as a draft. It has now been accepted by ACM’s Computing Surveys and is scheduled to appear in September 2014; the current text, revised from the previous version, is available [1].

Here is the abstract:

Software verification has emerged as a key concern for ensuring the continued progress of information technology. Full verification generally requires, as a crucial step, equipping each loop with a “loop invariant”. Beyond their role in verification, loop invariants help program understanding by providing fundamental insights into the nature of algorithms. In practice, finding sound and useful invariants remains a challenge. Fortunately, many invariants seem intuitively to exhibit a common flavor. Understanding these fundamental invariant patterns could therefore provide help for understanding and verifying a large variety of programs.

We performed a systematic identification, validation, and classification of loop invariants over a range of fundamental algorithms from diverse areas of computer science. This article analyzes the patterns, as uncovered in this study,governing how invariants are derived from postconditions;it proposes a taxonomy of invariants according to these patterns, and presents its application to the algorithms reviewed. The discussion also shows the need for high-level specifications based on “domain theory”. It describes how the invariants and the corresponding algorithms have been mechanically verified using an automated program prover; the proof source files are available. The contributions also include suggestions for invariant inference and for model-based specification.

Reference

[1] Carlo Furia, Bertrand Meyer and Sergey Velder: Loop invariants: Analysis, Classification and Examples, in ACM Computing Surveys, to appear in September 2014, preliminary text available here.

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Quiz (1): What is this function?

For various reasons there have been no articles in recent weeks; now we are restarting on a regular basis!

A the first topic for this new season, here is a little quiz. I have a function:

  • For 0 it yields 0.
  • For 1 it yields 1.
  • For 2 it yields 4.
  • For 3 it yields 9.
  • For 4 it yields 16.

The question: what is the value for 5?

Answer next Wednesday (at least it will be Wednesday in some time zone).

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Ado About The Resource That Was (Not)

 

After a few weeks of use, Microsoft Outlook tends in my experience to go into a kind of thrashing mode where the user interface no longer quite functions as it should, although to the tool’s credit it does not lose information. Recently I have been getting pop-up warnings such as

 

A required resource was

 

A required resource was what ? The message reminded me of an episode in a long-ago game of Scrabble, in which I proposed ADOABOUT as a word. “Ado about what? ”, the other players asked, and were not placated by my answer.

The message must have been trying to say  that a required resource was missing, or not found, but at the time of getting the final detail Outlook must have run out of UI resources and hence could not summon the needed text string. Not surprising, since running out of resources is precisely what caused the message to appear, in a valiant attempt to tell the user what is going on. (Valiant but not that useful: if you are not a programmer on the Outlook development team but just a customer trying to read email, it is not absolutely obvious how the message, even with the missing part, helps you.) The irony in the example is that the title bar suggests the problem arose in connection with trying to display the “Social Connector” area, a recent Outlook feature which I have never used. (Social connector? Wasn’t the deal about getting into computer science in the first place that for the rest of your life you’d be spared the nuisance of social connections? One can no longer trust anything nowadays.)

We can sympathize with whoever wrote the code. The Case Of The Resource That Was (Not) is an example of a general programming problem which we may call Space Between Your Back And Wall  or SBYBAW:  when you have your back against the wall, there is not much maneuvering space left.

A fairly difficult case of the SBYBAW problem arises in garbage collection, for example for object-oriented languages. A typical mark-and-sweep garbage collector must traverse the entire object structure to remove all the objects that have not been marked as reachable from the stack. The natural way to write a graph traversal algorithm is recursive: visit the roots; then recursively traverse their successors, flagging visited objects in some way to avoid cycling. Yes, but the implementation of a recursive routine relies on a stack of unpredictable size (the longest path length). If we got into  garbage collection, most likely it’s that we ran out of memory, precisely the kind of situation in which we cannot afford room for unpredictable stack growth.

In one of the early Eiffel garbage collectors, someone not aware of better techniques had actually written the traversal recursively; had the mistake not been caught early enough, it would no doubt have inflicted unbearable pain on humankind. Fortunately there is a solution: the Deutsch-Schorr-Waite algorithm [1], which avoids recursion on the program side by perverting the data structure to  replace some of the object links by recursion-control links; when the traversal’s execution proceeds along an edge, it reverses that edge to permit eventual return to the source. Strictly speaking, Deutsch-Schorr-Waite still requires a stack of booleans — to distinguish original edges from perverted ones — but we can avoid a separate stack (even just  a stack of booleans, which can be compactly represented in a few integers) by storing these booleans in the mark field of the objects themselves. The resulting traversal algorithm is a beauty — although it is fairly tricky, presents a challenge for verification tools, and raises new difficulties in a multi-threaded environment.

Deutsch-Schorr-Waite is a good example of “Small Memory Software” as studied in a useful book of the same title [2]. The need for Small Memory Software does not just arise for embedded programs running on small devices, but also in mainstream programming whenever we face the SBYBAW issue.

The SBYBAW lesson for the programmer is tough but simple. The resources we have at our disposal on a computing system may be huge, but they are always finite, and our programs’ appetite for resources will eventually exhaust them. At that stage, we have to deal with the SBYBAW rule, which sounds like a tautology but is an encouragement to look for clever algorithms:  techniques for freeing resources when no resources remain must not request new resources.

References

[1] Deutsch-Schorr-Waite is described in Knuth and also in [2]. Someone should start a Wikipedia entry.

[2] James Noble and Charles Weir: Small Memory Software: Patterns for Systems with Limited Memory, Addison-Wesley, 2001.

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