Bourbaki

Lance Fortnow recently devoted a post to The Bourbaki Lecture. Reading his post, reminded me that I had in mind of scribbling a couple of lines on the book Bourbaki: A Secret Society of Mathematicians by Maurice Mashaal, Pour la Science - AMS, 2006, 168 pp. I came across that book while looking for another one at the university library, and decided to borrow it. I am glad I did!

For one, I did not know that André Weil (1906-1998) introduced the symbol for the empty set that we now universally use. (A whole issue of the Notices of the AMS is devoted to Weil.) Nor did I know that Boubarki coined the terms injection, surjection and bijection, and invented the notion of filter. Not to mention the striking similarity between André Weil and Eugenio Moggi! (Eugenio, I hope that you don't mind me mentioning this!)

More seriously, I really enjoyed reading that book, and I warmly recommend it. It tells what was good, and what was bad, with Bourbaki and its view of mathematics. It is also full of eminently quotable sentences from the Bourbaki members. Examples?

"You can't do good work in applied mathematics until you can do good work in pure mathematics." (Jean Dieudonné (1906-1992), cited on p. 119)

"[...] logic is the mathematician's hygiene, but it does not provide his meals. The big problems constitute the daily bread on which he lives." (André Weil in The Future of Mathematics, 1948) Shame on you Weil!

The closing paragraph of the book is poignant:

"Despite some mistakes, Bourbaki did add a little to the "honour of the human spirit." In an era when sports and money are such great idols of civilization, this is no small virtue."

Indeed, not to mention that the members of Bourbaki have included some of the greatest mathematical minds of the last century.

Great minds can also make great mistakes (for instance, Bourbaki's disdain of probability theory had a very negative effect on French work in that very important area of mathematics, and Bourbaki was very dogmatic in his view of mathematics), and one of the virtues of the book is that these are not swept under the carpet. Read it, and form your own opinion on this non-existent, and most prolific mathematician.

Change of Dean

At a meeting with the staff members of the School of Science and Engineering held yesterday, the new rector, Svafa Grönfeld, announced that the dean of the school, Bjarki Brynjarsson, is stepping down from his position.

A five-person evaluation committee will be formed by early next month, and its first assignment will be to set down criteria for the selection of the new dean for the School. I expect that an advertisement for the position will be published in early March.

Watch this space if you are interested in the job!

Trying to Influence Research Funding in Iceland

Parliamentary elections are coming up in Iceland, and a bunch of scientists are trying to put funding for research on the political agenda. This group blog is an attempt at keeping the discussion of this issue alive. As you can see, most of the posts are encrypted in Icelandic. However, since my knowledge of Icelandic is not what it should be, I use English for my posts on that blog.

In case anybody is interested, I posted an entry entitled More Funding and More Variety! In that post I sing the praise of some types of funding that is available in countries like The Netherlands and Denmark, but not in Iceland. Comments are most welcome.

Too Many People Have Way Too Much Money

Icelandic business is thriving, and Icelandic businessmen are buying companies, football teams, supermarket chains etc. all over the place. There is a new breed of very wealthy people out here, who are not particularly bothered by the fact that, e.g., food is 65% more expensive than the European average here.

I just heard that the director of Samskip, one of the local new rich man, has just turned 50. For his birthday, he had Elton John sing for his guests. Cost: over 100 million ISK (about 1,110,789 euros)! This is one sixth of the annual budget of the Icelandic research council. I estimate that, with that money, I could hire at least 11 visiting professors for a year, changing the face of my department.

Life is not fair.

Lack of Funding for Italian Researchers Logic

Not so long ago, I reported with pleasure on a success story for Italian research in logic. Unfortunately, it looks like it is now time for bad news regarding research in logic in Italy.

Over the last few days, I have been reading a sequence of messages posted on the mailing list of the Italian Association for Logic and its Applications (AILA) related to the results of the applications for PRIN projects. (I think that PRIN is an Italian acronym standing for "Projects of National Interest".) The list of PRIN projects funded in the subject area of mathematics and computer science is here. A quick browse through the list gives me the impression that not enough of the funded projects are from computer science. Moreover, the only funded project related to logic in computer science is

OMODEO Eugenio

Università degli Studi di TRIESTE

22.680

Sviluppo su grande scala di dimostrazioni certificate

(Large scale development of certified proofs. Amount: 22.680 euros.)

One can also make the case that the project

DEGANO Pierpaolo

Università degli Studi di PISA

94.500

Sistemi e calcoli di ispirazione biologica e loro applicazioni -- BISCA

impacts on logic in computer science, broadly construed. (By the way, well done Pierpaolo!)

Funding for projects in mathematical logic and its relationships to computer science is conspicuously absent, despite the strength of Italian researchers in this area. Albeit the lack of funding will have very negative consequences on logic research in Italy, this is not the only thing that is bothering Italian logicians. From what I gather, all of the projects proposals from logicians received the same one-line evaluation, namely interesting, but marginal, as well as the same numerical score.

I do not know who the reviewers of the projects were, and I know nothing about the quality of the submitted projects. However, I feel that I have to take a firm and public stance against this uniform, general and vague "scientific" evaluation for 50-page long proposals. Would any of us be happy to write a long paper and have it rejected with a one-liner? And can logic research be at all labelled as being marginal?

Logic is the calculus of computer science! Apart from its intrinsic scientific and cultural interest, this fact alone shows that logic research is by no means marginal. In fact, thanks to its close connections with computer science, logic is the most applied branch of mathematics! (This is not just the opinion of your truly. I invite Italian readers --- assuming they exist, of course :-) --- to check out pages 108-109 of the interview with Enrico Giusti in the excellent book Professione Matematico. Note that Giusti is not a logician.)

I hope that the Italian mathematical community does not consider logic a marginal subject, and that, in the future, the evaluation of project proposals will involve expert reviewers from abroad, as it is done essentially everywhere else. This will be a great step forward towards the use of proper peer review in the distribution of the funding available for scientific research, and will increase the transparency of the whole process.

A Flurry of Prizes in Maths

It is prize time in the mathematical world. The Wolf Foundation has announced that Harry Furstenberg (Hebrew University of Jerusalem) and Stephen Smale (University of California, Berkeley) will share the US$100,000 2006-7 Wolf Prize in Mathematics.

The AMS has instead awarded its prizes for 2007. Once more the AMS-MAA-SIAM Morgan Prize for Outstanding Research by an Undergraduate Student went to an MIT student, Daniel Kane. It is humbling experience to see that this BSc student has already about 16 papers under his belt :-(

The February 2007 issue of the Notices is out. It features an interview with Lennart Carleson coauthored by my former Aalborg colleague Martin Raussen, and a beautiful book review Symmetry and the Monster: One of the Greatest Quests of Mathematics--A Book Review by Robert Griess. I liked it so much that I could not resist buying the book today. I might report on the book when I read it.

Three Billion Down the Drain?

Last Thursday the Icelandic Minister of Education and the rector of the University of Iceland unveiled a , previously very much secretive, plan aimed at bringing the University of Iceland amongst the top 100 best universities in the world. (Yes, you read that right!)

The heart of the plan is an increase in governmental funding to the University of Iceland amounting to 640 million ISK a year over the next five years (roughly 7 million euro a year). It is not completely clear to me yet what milestones the University of Iceland will have to meet to justify this investment of tax payers money. What is, however, abundantly clear is that the main competitor of the university I am working for right now (Reykjavík University) is much more powerful now than it was only four days ago, provided they will make good use of the extra funding. If previous events are anything to go by, this is very unlikely to happen in computer science unless the CS department at the University of Iceland is forced to hire according to the qualifications of the applicants.

Iceland being what it is, even your truly decided to enter the political arena for a few minutes by writing the following email message:

Subject: Financing Universities and Research


Dear Minister of Education,

It is an excellent idea to increase the level of funding for universities and research in Iceland. It is also a positive step to have some form of financing that is not linked to specific, and often narrow, project work. This is done in many other countries with great benefit to the level and quality of research. It is, however, very unclear at best that donating a large sum of money to HI the way the Ministry did yesterday is the best way to promote international quality research in Iceland.

Financial support should go to those individual researchers, research centres and laboratories where high quality academic research has been performed consistently, irrespective of the university where they are located. Allocation of resources should be made based on merit, as established by fair and objective peer review. Centuries of scientific work have shown that this is the best, if not the only, way to ensure that whatever support is available be used in the best possible way.

By way of analogy, I invite you to consider the choice of an investment fund where to invest some of your savings. Before choosing one, you would most likely examine the interests that specific funds have guaranteed their customers over the last few years. Why are political decisions on how to invest in research not based on the same principles?

Sincerely Yours,

Luca Aceto

Do I expect to receive any answer? Certainly not, but, as present head of the CS department at Reykjavík University, I felt like I had to write to the Minister on behalf of my colleagues.

I'll write a post with my first views on the HoD job over the next few days. Not surprisingly, the executive summary will be that the job is a huge time sink :-(

Concurrency Column for February 2007

Earlier this week, I posted the concurrency column for the February issue of the Bulletin of the EATCS. This column will be devoted to a paper entitled Characteristic Formulae: From Automata to Logic co-authored by Anna Ingolfsdottir and me.

The piece is loosely based on three posts I wrote for this blog. (Parts 1, 2 and 3.) This is the first time that I use this blog as a a sort of scratchpad while planning the writing of a paper. I think that I'll experiment with this use of the blog again in the future---assuming that I have anything at all to write about, that is.

Quality of Life of Researchers in Italy

Not so long ago, I wrote a post on research in Italy. In that post, I criticized the lack of sensibility of Italian politicians towards science and basic research, and praised Italian researchers for being very active in my field and in TCS as a whole despite the lack of support they are faced with.

I just read an article in La Repubblica that adds fuel to the fire. (The article is in Italian.) According to a report commissioned by the trade union Nidil Cgil, Italian researchers earn on average between 800 and 1.200 euro per month before taxes. Moreover 31% of the people interviewed by the report writers earns less than 800 euro per month after taxes. It is not surprising to read that those who do not have permanent positions yet are not optimistic about their futures. About 25% of the people who do not have a permanent position are over 35 years old, and half of those are over 40. (At this age, in most other countries other you have a tenured position of some sort or you have found a different type of job altogether.) This uncertainty about the future has obvious repercussions on the quality of life, and on family planning. It turns out that 82% of the interviewees do not have children.

How can this be at all good for my own country? And why doesn't Italian society see that it is undermining its own future by not supporting the researchers who work in Italy? Can Italy ever attract foreign researchers by offering those wages? Of course, money is not everything, but it does help, and is necessary to lead a decent life. Surprise, surprise: researchers are fully fledged human beings, with human needs and desires!

Post Scriptum: The article I am refererring to also has statistics regarding the long working hours of researchers. I chose not to focus on those statistics at all here because, in my modest experience, this is the norm, rather than the exception, throughout. My present dean often tells me that "academic freedom is not the freedom of doing nothing", to which I always reply that academics have often much longer working hours than many other people.

Back to the Future

The new year is three days old, I have been head of department for two, and I have been thinking about doing a 2006 rewind, following the tradition of The Wire, one of my favourite music magazines despite the fact that I stopped subscribing to it last summer after about 15 years. However, the more I thought about it, the less I felt I had anything meaningful to say about concurrency theory in 2006. My colleagues produced several excellent papers, and a look at the proceedings of conferences like CONCUR 2006, FSTTCS 2006, ICALP 2006, and LICS 2006 will suffice to get an idea of some of the best work done last year.

Others have done a great "2006 rewind" for complexity theory and mathematics. (From the latter blog, I encourage you to read the opinions on category theory. Savour John Baez's one liner `I hope most mathematicians continue to fear and despise category theory, so I can continue to maintain a certain advantage over them.')

I encourage all of us to learn from Lance Fortnow's advice on how to deliver a job talk. That piece of advice, and the comments to the post, apply to any talk, not just our job talks.

Finally, I hope that during this year I'll follow the following piece of advice from Allan Bay, an Italian cook and food writer, whose sixth piece of advice in Cuochi Si Diventa (one of my favourite cookbooks) reads:

Cook with calm; haste and anxiety are your worst advisors. If you know you have little time, opt for a quick recipe, one that you can cook as well as you can in the little time you have at your disposal. This virtue is useful well beyond your kitchen....

Indeed, haste and anxiety are usually not conducive to good research work either. (The advice we give others is the advice we ourselves need.)

Hopefully 2007 will be a great year for concurrency theory, ICE-TCS, all of my collaborators and friends, and, why not, a decently productive one for me too. I wish that I could begin by attending this talk :-)

Christmas Reading

Science magazine has chosen Perelman's proof of the Poincare Conjecture as breakthrough of the year for 2006. See this article by Dana Mackenzie. You can also read about the nine runners-up as well as the scientific fraud of the year.

The January 2007 issue of Scientific American features an article by Bill Gates entitled A Robot in Every Home.

Happy reading and merry Christmas.

Sussex in 1987

Julian Rathke is moving from Brighton to Southampton, where he joins Vladimiro Sassone (another former member of the School of Informatics at the University of Sussex). It does not take magical powers to foresee that Southampton will soon have a very strong theory group.

Julian tells me that it is a strange time at Sussex. Half of the department is moving buildings, so the department office and all of the secretaries are gone, the good old debugging room (the unofficial coffee room of the department at a time when the university regulations essentially did not allow for such rooms, hence the name) is also no more. Just an empty shell of a room now. Very sad.

Reading Julian's description prompted me to look back to a windy September-October 1987, when I arrived in Brighton to work for a year as a research assistant under Matthew Hennessy. The department of computer science was very small then. Yet, there was already a tightly knit group of theory people that created a very good atmosphere for intellectual growth. At that time, Marek Bedcnarzyk was putting the final touches to his PhD thesis, and Allen Stoughton was finalizing his monograph on fully abstract models of programming languages. Mark Millington, one of Matthew's former Edinburgh students, was there teaching Software Engineering. (It is a pity that Mark's PhD thesis is not very well known. It was a good piece of work.) I shared an office with Rance Cleaveland, a freshly minted PhD from Cornell, where he had formalized theories of processes using NuPRL. (Rance was working on the development of the Concurrency Workbench, but did a lot more than that---not least, he taught me a lot over beers at our local pub :-)) Across the fire doors leading to the Mathematics department, Andy Pitts was spending some time at Sussex, supported by a grant from the Royal Society.
Already the year after, the theory group grew substantially in size, with the arrival of PhD students from Britain and from abroad, postdocs, and later new members of staff. The rest, as they say, is history. It is a pity that not so much is left of that theory group at Sussex. (The linchpin, Matthew Hennessy, is still there, but many people have come and gone in the meantime.)


This is one of the pros and cons of academic life. One travels from place to place, possibly grows attached to each of the departments/centres one works for, and the colleagues one meets there, but eventually we all leave for somewhere else.

Good luck to Julian for his new job at Southampton, and congratulations to Vladi for having enticed an excellent collaborator like Julian to join his new department.

Being Head of Department: A New Experience

As of 1 January 2007, and for at least about eight months, I'll be the acting head of the CS department at Reykjavík University.

Not surprisingly, I am trying to gather some useful advice on how to survive this new experience, and hopefully live to tell the tale as far as research, teaching and service to the academic community are concerned. So far, I have found some useful information on the CRA web site, but I'd love to hear any advice you might have on how to manage one's time while being head of department, and what aspects of the job one should be particularly aware of. Drop me a line, or use the comments section. I'll be grateful for any piece of information you might provide.

I apologize to those of you who kindly invited me to serve on PCs for conferences during the first half of 2007. I was honoured by your consideration, but decided to politely decline all of those invitations rather than do a bad job. (I must freely admit that the prospect of chairing the department is worrying me somewhat :-() I hope that I made the right choice.

Research in Italy

Yesterday I travelled back to Reykjavík after a very short visit to Italy. During the trip back to the North Atlantic, I was reading La Repubblica, an Italian newspaper. One of the news items that caught my eye dealt with the protests by the rectors of Italian universities, who are very unhappy about the cuts to their institutions' budgets that are part of the latest financial law passed by the Italian government.

I am afraid that I do not know enough about the developments to express an informed opinion about the whole thing. However, I have always sternly refused to buy the argument that, unlike other countries, Italy does not have the cash to support universities and research. How can a member state of the G8 elite be short of money to support its future? Let's not mask lack of interest in science and technology as supposed poverty. By way of example, BRICS and the other research centres of the Danish National Research Foundation were richly funded for five-year periods (14 years for BRICS) with the interests resulting from the privatization of an insurance company! The outcome for Danish science is for everybody to see.

Why shouldn't the same type of investment in basic research be possible in a country like Italy? The reasons must be the same that prevent Italy from investing in its schools and universities. Three figures in the article I was reading yesterday paint an amazingly bleak picture. Italy boasts only about 3 researchers every 1000 workers. This is less than each of the other countries mentioned in the list. We (Italians) spend only about 1.2% of the GNP in research. Only Greece, Spain and Portugal invest less. Last but not least, Italy spends only 8000$ a year per university student. This is the same as Hungary, and half of what Sweden spends. Why are we Italians so masochistic?

Still, if I look at Italian research in TCS, I can only classify it as being very strong, despite the lack of money for research and the sub-optimal support. Yes, I know that I am biased. Even though I have never worked in an Italian university myself, I try to maintain good ties with my colleagues based in Italy. It is, however, an incontrovertible fact that there are many very active and very strong Italian TCS researchers. The strongest Italian CS departments are high class, and Italy exports talent. Off the top of my head, I could come up with the following list of Italian TCS researchers working abroad (with apologies to those whose names have not crossed my mind right now):

1. Luca de Alfaro
   
2. Roberto Amadio
   
3. Antonio Bucciarelli
   
4. Cristiano Calcagno
   
5. Luca Cardelli
6. Ilaria Castellani
   
7. Giuseppe Castagna
   
8. Roberto Di Cosmo
   
9. Luigi Liquori (who is from Pescara like me)
10. Giuseppe Longo
11. Sergio Maffeis
    
12. Pasquale Malacaria
    
13. Silvio Micali
    
14. Catuscia Palamidessi
    
15. Luigi Santocanale
16. Vladimiro Sassone
    
17. Luca Trevisan
    
18. Daniele Varacca
19. Luca Viganò
    
20. Francesco Zappa Nardelli
    

(Notice how many of these researchers work in France, and how many of them are located in Paris!) I believe that the above people would be considered as forming a very strong theory group anywhere in the world, and there are many more strong TCS researchers in Italy itself. I can only encourage the Italian political establishment and Italian universities to give Italian researchers a suitable environment for producing the best research they can. To my mind, this would be a win-win situation for Italy as a whole. Unfortunately for Italian science, it looks like Italy's politicians disagree with me.

Addendum 19/12/2006: Yesterday I read an article by Princeton philosopher Peter Singer, on the subject of philantropy, poverty and ethics. (Thanks to Luca Trevisan for his post on this very interesting article, which is a must read during the crazy Christmas period.)

The following excerpt from that article struck me as being very relevant to this disjointed post of mine:

The Nobel Prize-winning economist and social scientist Herbert Simon estimated that “social capital” is responsible for at least 90 percent of what people earn in wealthy societies like those of the United States or northwestern Europe. By social capital Simon meant not only natural resources but, more important, the technology and organizational skills in the community, and the presence of good government. These are the foundation on which the rich can begin their work.

If instead of wealth, we consider "intellectual output", how much would Italian researchers produce in the presence of better technology and organizational skills in the community, and in the presence of good government? What if they could devote more of their time to not fighting against the system?

Zeilberger's 76th Opinion

Dr. Z strikes again. Read his 76th opinion, entitled Why P Does Not Equal NP and Why Humans Will Never Prove It by Themselves, and form your own counter-opinion. As usual, reading Zeilberger's opinion won't leave you cold.

Zeilberger writes:

All the problems in, say, Garey and Johnson, are essentially one problem, since they are all trivially equivalent.

Well, I am not sure that I agree that those problems are "trivially" equivalent. "Triviality" is in the eye of the beholder, and maybe one has to be a top notch mathematician like Zeilberger in order to find all of the reductions between NP-complete problems trivial. I admit that I do not find most of them to be trivial at all.

I also find the characterization of the work of complexity theorists as "just proving yet-another reduction theorem" somewhat restrictive.

I am expecting to see some comments on this opinion on the complexity theoretic blogs.

Addendum posted on 15 December 2006: Read Luca Trevisan's thoughtful comments.

Let me also mention another statement of Zeilberger's from opinion 76:

There is no hope, as smart and "creative" as you and your cronies are, that you will be able to prove it by yourselves. Your only hope is to enlist that "simple" mechanical device, that ironically, you computer-scientists, do not use very much, not even for spell-checking!

Now that I have algebraic combinatorialists working near me, I can vouch that they use computers a fair amount in their work---possibly more than many theoretical computer scientists. However, in concurrency theory, people construct and use software tools to make models of computing devices and to analyze their behaviour algorithmically. Readers who have never used such tools might wish to play with Uppaal or HyTech to mention but two such tools.

Accepted Papers for FOSSACS 2007

The list of accepted papers for FOSSACS 2007 is now available. (Thanks to Jun Pang for the pointer.) The conference looks interesting, and I hope to report on some of the papers when, if ever, I have time to read them.

Accepted Papers at ETAPS 2007

The list of accepted papers for TACAS 2007 is out, and so is the one for ESOP 2007. TACAS accepted as many as 54 papers, and the list of contributors looks very impressive. Several papers have an all-star list of authors, e.g.,

Multi-Objective Model Checking of Markov Decision Processes by Kousha Etessami, Marta Kwiatkowska, Moshe Y. Vardi, and Mihalis Yannakakis.

ESOP features a list of accepted papers that is definitely of interest to concurrency theorists.

Most accepted papers at TACAS and ESOP have at least three authors, further confirming the trend to higher and higher degrees of collaboration in most fields of TCS.

Happy browsing! I am still awaiting the list of accepted papers for FOSSACS 2007. In any event, the tenth anniversary edition of ETAPS looks already like a conference one should attend. I myself, however, will be attending a marriage of mathematicians that will be held in Chicago at around that time.

RIP (Research In Peace)

The Institut Mittag-Leffler has just celebrated its ninetieth birthday. It is a Nordic research institute for mathematics, under the auspices of the Royal Swedish Academy of Sciences, created by Gösta and Signe Mittag-Leffler who donated their house, library and fortune to the Academy. The Institute runs year-long research programs in specialized areas of mathematics, and I am told that it is a great place to visit.

One of their programmes is called RIP (Research in Peace). This is a programme allowing mathematicians from Scandinavian universities, who are not directly connected to the current scientific programme, to visit the Institute for shorter periods. My colleague, and frequent indirect contributor to this blog, Anders Claesson is presently visiting Mittag-Leffler using the RIP programme.

Maybe some of you will be interested in organizing a programme on some topic in TCS at that institution.

Characteristic Formulae for Timed Automata

The last installment of this series of three posts on characteristic formulae deals with characteristic formulae for behavioural equivalences over timed automata.

The first construction of a characteristic formula for timed bisimilarity over timed automata I am aware of was presented in the paper

François Laroussinie, Kim G. Larsen, and Carsten Weise. From Timed Automata to Logic - and Back. Jirí Wiedermann, Petr Hájek (Eds.): Mathematical Foundations of Computer Science 1995, 20th International Symposium, MFCS'95, Prague, Czech Republic, August 28 - September 1, 1995, Proceedings. Lecture Notes in Computer Science 969 Springer 1995, ISBN 3-540-60246-1.

In that paper, the authors propose a timed version of Hennessy-Milner logic Lnu, and show thateach timed automaton can be uniquely characterized by a single formula in that logic up to timed bisimilarity.

The construction in that paper relied on the region graph of the timed automaton, and thus produces "large" formulae. (See yesterday's post on the number of regions in a timed automaton.) In the setting of timed automata without invariants, the construction by Laroussinie, Larsen and Weise was improved upon in the paper

Luca Aceto, Anna Ingólfsdóttir, Mikkel Lykke Pedersen, and Jan Poulsen. Characteristic Formulae for Timed Automata. ITA 34(6): 565-584 (2000).

That paper offers characteristic formula constructions in the real-time logic Lnu for several behavioural relations between (states of) timed automata. The behavioural relations studied in op. cit. are timed (bi)similarity, timed ready simulation, faster-than bisimilarity and timed trace inclusion. The characteristic formulae delivered by the constructions offered in the aforementioned paper have size which is linear in that of the timed automaton they logically describe.

Finally, the paper

Luca Aceto, Patricia Bouyer, Augusto Burgueño, Kim G. Larsen. The power of reachability testing for timed automata. Theor. Comput. Sci. 300(1-3): 411-475 (2003)

uses a property language characterizing the power of reachability testing over timed automata in the context of "test automata" to provide a definition of characteristic properties with respect to a timed version of the ready simulation preorder, for nodes of τ-free, deterministic timed automata.

I am not aware of a timed counterpart of the results by Browne, Clarke and Grümberg on the characterization of bisimulation-like equivalences over Kripke structures using characteristic formulae in fragments of CTL. Is Timed CTL expressive enough to describe characteristic formulae for timed bisimilarity? This has been an item on my "to do" list for quite some time, but I have never had the guts to start working on the problem yet.

Number of Regions in a Timed Automaton

While working on the book Reactive Systems: Modelling, Specification and Verification, my co-authors and I were looking for a formula counting the number of regions in a timed automaton exactly in order to motivate the introduction of the notion of zone. Thanks to Rajeev Alur and Tom Henzinger, I now know the answer, which I report here in case anybody else is interested.

There is a characterization of the number of regions where all clocks lie between 0 and 1 in terms of Stirling numbers of the second kind in this paper (page 13 bottom).

Peter Kopke's PhD thesis available from Tom's web page at http://mtc.epfl.ch/~tah/Students/kopke.pdf also contains that characterization. The section that starts on page 164 (of the file) offers the following formula:

Regions(2n) = sum_k=1²ⁿ S_k²ⁿ k!

where Regions(2n) is the number of equivalence classes over 2n clocks each constrained to lie in the interval (0,1), and S_k²ⁿ is the number of ways of partitioning a set with 2n elements into k subsets. So Regions(2n) is exactly the number of ways of partitioning a set with 2n elements into k ordered subsets.

As the algebraic combinatorialists here at Reykjavík University know well, I am always very impressed by these counting formulae :-)