From lynch@theory.lcs.mit.edu Thu Aug 15 09:45:45 2002 Date: Thu, 15 Aug 2002 09:45:43 -0400 From: Nancy Lynch To: joanne@theory.lcs.mit.edu Subject: [mischu@research.att.com: Re: computing with infinitely many processes] X-Spam-Status: No, hits=1.2 required=5.0 tests=LARGE_HEX,PORN_10 version=2.20 X-Spam-Level: * I need to collect all the papers he alludes to in this message. First, the message is supposed to have several attachments. Please print them all. They are: 1. An updated version of the infinite processes paper. This should be the version put on-line for the class, if Michael says OK. 2. Copy of the PODC 2001 paper, possibly revised over the proceedings version. 3. Two .ppt presentations. Also, Lamport's paper on fast mutual exclusion, ACM TOCS, 1987 (add to end of bib, under misc.) FP87 is a paper by Friedberg and Peterson, An efficient solution to the mutual exclusino problem using weak semaphores, IPL, 1987. (add to end of bib, under misc.) Thanks, N. ------- Start of forwarded message ------- Date: Wed, 14 Aug 2002 15:17:31 -0400 From: Michael Merritt X-Accept-Language: en To: Nancy Lynch Cc: gadi@research.att.com Subject: Re: computing with infinitely many processes Content-Type: multipart/mixed; boundary="------------35169952CA328004EE654592" X-Spam-Status: No, hits=0.0 required=5.0 tests= version=2.20 X-Spam-Level: This is a multi-part message in MIME format. - --------------35169952CA328004EE654592 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Nancy, Gadi and I talked about it, and our thinking is to cover results also from our paper from PODC 2001 with Eli (all.ps, attached), in particular the one-shot snapshot algorithm. We are thinking that in a week, you could cover snapshot, mutual exclusion, and election. Specifically, Theorem 3.1 from the paper with Eli (snapshot), and from the Computing with infinitely many processes paper, Theorem 2.4 (election with 2 registers) and Theorem 3.2 (adaptive starvation-free mutual exclusion.) (The revision we're sending presents this last algorithm much more modularly, paralleling the way we presented it at DISC.) You could use Theorem 2.5 (election with 1 register) as an extra-credit assignment,. Theorem 3.3 (non-adaptive starvation-free mutual exclusion for infinite concurrency using only bits) is pretty simple, and can be used to emphasize that the algorithm of Theorem 3.2 is adaptive. It is also possible to include more algorithms from the paper with Eli, the powerpoint we're sending you demonstrates some of the modularity that can be emphasized in building long-lived and adaptive solutions. Anyway, we're attaching an updated version of the Infinite processes papere, a copy of the paper with Eli, and the two powerpoint presentations we used to present them. Feel free to adapt the powerpoint as you like for your class, and if you have questions, give me a call or send one of us an email. (I'll be away next week, Gadi will be around.) We don't think there are prerequisites that need to be covered in detail for the above algorithms, but certainly Lamport's fast mutual exclusion is a good paper for anybody to know. (We don't know of any later versions of the papers you discuss.) If you're willing to take more than a week, we would encourage you to look at more results from the paper with Eli, some of the lower bounds from the other paper, and theorem 5.2, which is a nice algorithm from FP87. Michael and Gadi Nancy Lynch wrote: > Hi, Michael, > > I'm teaching a course in "Advanced Distributed Algorithms" starting > next month. I'm going to cover lots of important results that > everyone should know but that aren't in my book. More interestingly, > I'm going to cover some new topics, trying to focus on computing in > networks with unknown sets of participating processes. > > I'd like to put together a short unit covering your work on computing > with infinitely many processes. > > 1. Can you tell me exactly which papers these are, and what are the > latest versions? E.g., I have a copy of your paper with Gadi > called "Computing with Infinitely Many Processes", dated May 25, > 2001. It looks like a conference or TR version. Is there a later > version now? > > 2. Can you suggest which of your papers on this topic I should cover > in the course, and in what order? Maybe something supersedes > something else...??? Also, what papers are needed as > prerequisites? I can assume that the students already know > everything in my book (!). > > E.g., on a quick look, it seems that it would be useful, before > reading Merritt-Taubenfeld, for us to read Styer-Peterson89 (I > wonder if they ever published this in a journal), Peterson94 (but > that is only a TR), and Lamport87 (fast mutual exclusion). > Do you think these are all necessary? Which is/are important to > read? > > Think about a week-long unit in a PhD-level course. I expect very > good students in the class. > > 3. I asked some students to read these papers last year, and they > found them rough going. Can you give me some perspective that > might help? E.g., do you have any hints for what are the most > interesting results for me to focus on? 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b(an)f(easy)g(case)g(analysis,)h(in)g(one)g(suc)o(h)f(run)h(no) f(leader)h(is)44 720 y(elected,)c(or)f(there)g(are)g(man)o(y)g(runs)g (in)h(whic)o(h)g(t)o(w)o(o)e(or)h(more)g(leaders)h(are)e(elected,)j(a)d (con)o(tradiction.)104 789 y(The)i(\(compatible\))f(run)h(of)f(the)g (pro)q(cesses)h(in)g Fg(C)i Fn(is)e(constructed)f(as)g(follo)o(ws:)20 b(Let)c Fg(\015)h Fn(=)f Fg(y)1690 796 y Fi(1)1710 789 y Fg(;)8 b(:::;)g(y)1813 796 y Fb(k)1832 789 y Fn(.)44 845 y(First)18 b(run)h(these)g Fg(c)g Fn(pro)q(cess)f(un)o(til)i(eac)o (h)f(one)g(of)f(them)h(is)g(ab)q(out)f(to)g(execute)i(its)f(\014rst)f (write.)30 b(Run)44 902 y(pro)q(cess)17 b Fg(p)227 909 y Fi(1)262 902 y Fn(un)o(til)g(it)g(tak)o(es)e(the)h(last)g(step)g(in)h Fg(\014)r Fn(\()p Fg(p)920 909 y Fi(1)940 902 y Fn(\))e(that)h(lea)o(v) o(es)g(the)g(memory)g(in)h(state)e Fg(y)1648 909 y Fi(1)1668 902 y Fn(.)22 b(Rep)q(eat)44 958 y(for)15 b Fg(p)137 965 y Fi(2)173 958 y Fn(through)g Fg(p)368 965 y Fb(c)385 958 y Fn(,)h(then)g(rep)q(eat)f(this)i(lo)q(op)f(for)f(eac)o(h)h(state) f(in)h Fg(\015)s Fn(.)21 b(The)16 b(resulting)g(\014nite)h(run)f(lea)o (v)o(es)44 1015 y(the)h(shared)f(memory)g(in)h(state)e Fg(s)p Fn(,)i(and)f(eac)o(h)h(of)f(the)g Fg(c)g Fn(pro)q(cesses)h Fg(p)1245 1022 y Fb(i)1275 1015 y Fn(has)f(tak)o(en)g(the)h(steps)f(in) h Fg(\014)r Fn(\()p Fg(p)1800 1022 y Fb(i)1814 1015 y Fn(\).)44 1071 y(Since)g(the)f(state)e Fg(s)i Fn(rep)q(eats)f (in\014nitely)j(often)d(in)i(the)e(remainder)h(of)f(eac)o(h)h(of)f(the) g(solo)h(runs)f Fg(sol)q(o)p Fn(\()p Fg(p)1800 1078 y Fb(i)1814 1071 y Fn(\),)44 1127 y(the)e(run)h(can)f(b)q(e)h(extended,)g (using)g(a)f(round-robin)h(sc)o(hedule,)h(app)q(ending)g(a)e(\014nite)h (sequence)g(of)f(steps)44 1184 y(of)i(eac)o(h)g(pro)q(cess)h(in)g (turn,)e(alw)o(a)o(ys)h(returning)h(the)f(memory)g(to)f(state)g Fg(s)p Fn(.)104 1253 y(Notice)i(that)e(the)h(signatures)h(are)f(used)g (only)h(in)g(the)f(\014rst)g(stage)f(of)h(the)g(construction,)g(when)h (the)44 1309 y(pro)q(cesses)h(run)f(un)o(til)h(the)f(memory)f(state)g (is)i Fg(s)p Fn(,)f(in)h(suc)o(h)f(a)g(w)o(a)o(y)f(that)g(eac)o(h)h (pro)q(cess)g(\\thinks")g(it)g(ran)44 1366 y(solo)h(to)e(get)h(there.) 24 b(After)16 b(reac)o(hing)h(\(together\))e(the)i(recurring)g(state)e Fg(s)p Fn(,)i(eac)o(h)f(pro)q(cess)h(can)g(b)q(e)g(run)44 1422 y(solo)k(un)o(til)g Fg(s)g Fn(recurs.)37 b(Since)22 b(the)e(same)h(state)e Fg(s)i Fn(recurs)g(in\014nitely)i(man)o(y)d (times)h(for)f(eac)o(h)h(of)f(the)44 1479 y(pro)q(cesses)c(in)h Fg(C)s Fn(,)e(w)o(e)g(can)h(sc)o(hedule)h(eac)o(h)f(in)g(turn)g(that)f (has)g(not)h(terminated)g(to)f(tak)o(e)g(\(some)g(\014nite,)44 1535 y(nonzero)g(n)o(um)o(b)q(er\))g(of)g(steps)f(un)o(til)i Fg(s)g Fn(o)q(ccurs)f(next,)f(then)i(sc)o(hedule)g(the)f(next)g(one)g (\(i.e.,)f(round)h(robin)44 1592 y(sc)o(heduling\).)22 b(No)o(w)14 b(ev)o(ery)o(one)h(either)h(runs)g(forev)o(er)e(or)h (terminates)g(as)g(in)h(a)f(solo)g(run.)p 1828 1592 17 38 v 44 1775 a Fl(2.2)56 b(Algorithms)17 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Fk(Theorem)g(2.3)23 b Fj(F)m(or)d(\014nite)f(c)n (oncurr)n(ency)h Fg(c)g Fn(=)h(2)f Fj(with)h Fn([2)p Fg(;)8 b Fh(1)p Fn(])p Fj(-p)n(articip)n(ation,)20 b(ther)n(e)h(ar)n(e) f(adaptive)44 1917 y(symmetric)c(solutions)g(to)h(ele)n(ction)e(and)h (c)n(onsensus)e(using)i(one)g(atomic)h(r)n(e)n(gister.)44 2034 y Fc(Pro)q(of:)27 b Fn(The)20 b(algorithm)f(in)h(Figure)f(1)g(is)h (v)o(ery)e(simple.)33 b(The)20 b(register)f(v)m(alue)h(con)o(tains)f(t) o(w)o(o)f(\014elds,)44 2091 y Fg(Leader)k Fn(and)g Fg(M)5 b(ar)q(k)q(ed)p Fn(,)23 b(resp)q(ectiv)o(ely)l(,)h(where)e Fg(Leader)g Fn(is)g(a)g(pro)q(cess)g(id)g(\(or)f(0\))g(and)g Fg(M)5 b(ar)q(k)q(ed)22 b Fn(is)44 2147 y(b)q(o)q(olean.)27 b(The)17 b(statemen)o(t)f Fk(a)o(w)o(ait)h Fj(c)n(ondition)g Fn(is)g(used)h(as)e(an)h(abbreviation)h(for)f Fk(while)g Fh(:)p Fj(c)n(ondition)44 2204 y Fk(do)e Fj(skip)p Fn(,)g(and)g(hence,) h(ma)o(y)f(in)o(v)o(olv)o(e)g(man)o(y)g(accesses)h(to)e(the)i(shared)f (memory)l(.)104 2272 y(In)e(this)g(v)o(ery)f(simple)h(algorithm)g(the)f (second)h(pro)q(cess)f(to)g(write)g(to)g(the)g(shared)g(register)g(is)h (elected.)44 2329 y(The)21 b(\014rst)g(pro)q(cess)g(to)f(write)h(spins) h(un)o(til)g(the)f(second)h(pro)q(cess)f(writes,)h(then)f(marks)g(the)g (second)44 2385 y(pro)q(cess)16 b(as)e(the)i(winner.)21 b(The)15 b(concurrency)h(b)q(ound)g(guaran)o(tees)f(no)g(other)f(pro)q (cess)i(will)h(in)o(terfere.)104 2454 y(This)d(election)g(algorithm)f (can)g(b)q(e)g(easily)h(con)o(v)o(erted)f(in)o(to)g(a)f(consensus)i (algorithm)f(b)o(y)f(app)q(ending)44 2511 y(the)k(input)h(v)m(alue)h (\(0)d(or)g(1\))h(to)f(eac)o(h)h(pro)q(cess)h(id.)23 b(The)16 b(input)h(v)m(alue)g(of)f(the)g(leader)h(is)f(the)g(consensus) 44 2567 y(v)m(alue.)p 1828 2567 V 104 2685 a(The)f(next)g(algorithm,)f (solving)h(asymmetric)g(election)h(for)e(concurrency)h Fg(c)d Fh(\025)h Fn(2,)h(is)h(similar)h(to)e(the)44 2742 y(previous,)f(in)h(that)d(the)i(last)f(participan)o(t)h(to)f(write)g (the)h Fg(Leader)g Fn(\014eld)h(is)e(then)h Fg(M)5 b(ar)q(k)q(ed)12 b Fn(as)g(the)h(leader.)933 2866 y(7)p eop %%Page: 8 9 8 8 bop 44 86 a Fn(A)18 b(second)h(register,)g Fg(U)5 b(nion)p Fn(,)19 b(is)g(used)g(b)o(y)f(the)g(\014rst)g Fg(c)g Fn(participan)o(ts)h(to)e(ensure)i(that)f(they)g(ha)o(v)o(e)g (all)44 143 y(\014nished)f(writing)e(their)h(id)g(in)o(to)e Fg(Leader)q Fn(.)20 b(\(Note,)14 b(that)g(in)i(doing)f(so,)g(they)g (learn)g(eac)o(h)g(other's)f(id's.\))44 199 y(Because)k Fg(c)g Fn(is)g(also)f(the)h(n)o(um)o(b)q(er)g(of)f(required)h (participan)o(ts,)g(eac)o(h)g(of)f(the)h(\014rst)f Fg(c)g Fn(participan)o(ts)h(can)44 256 y(spin)e(on)g(the)f Fg(U)5 b(nion)15 b Fn(register)g(un)o(til)i(the)e(n)o(um)o(b)q(er)g(of)g(id's) h(in)g(it)f(is)h Fg(c)p Fn(.)44 373 y Fk(Theorem)h(2.4)23 b Fj(F)m(or)e(any)g(\014nite)g(c)n(oncurr)n(ency)g Fg(c)g Fj(with)h Fn([)p Fg(c;)8 b Fh(1)p Fn(])p Fj(-p)n(articip)n(ation,)22 b(ther)n(e)g(ar)n(e)f(adaptive)44 429 y(symmetric)16 b(solutions)g(to)h(ele)n(ction)e(and)h(c)n(onsensus)e(using)i(two)h (atomic)f(r)n(e)n(gisters.)44 547 y Fc(Pro)q(of:)j Fn(The)14 b(algorithm)g(in)h(Figure)g(2)f(uses)g(a)g(k)o(ey)g(idea)h(used)f (previously)i(in)f(a)f(symmetric)g(algorithm)44 603 y(due)j(to)f(A)o (ttiy)o(a,)g Fj(et)i(al)e Fn([A)o(GM98)n(].)24 b(As)17 b(in)g(the)g(A)o(ttiy)o(a,)e Fj(et)j(al)e Fn(algorithm,)h(the)g (\014rst)f(register,)g Fg(U)5 b(nion)p Fn(,)44 659 y(holds)17 b(a)f(set)g(of)f(at)h(most)f Fg(c)h Fn(pro)q(cesses)g(id's.)23 b(Pro)q(cesses)17 b(engage)f(in)g(a)g(phase)h(of)e(reading)i(the)f (curren)o(t)44 716 y(set)d(of)g(v)m(alues,)i(comparing)e(it)h(to)f(the) g(set)h(they)f(ha)o(v)o(e)g(already)h(seen,)g(and)f(writing)h(when)g (they)g(kno)o(w)e(a)44 772 y(v)m(alue)k(not)d(re\015ected)i(in)g(the)f (set)g(read.)20 b(This)14 b(phase)h(ends)f(when)h(a)f(pro)q(cess)g (reads)g Fg(c)g Fn(di\013eren)o(t)g(v)m(alues.)44 829 y(\(In)g(the)g(A)o(ttiy)o(a,)f Fj(et)i(al)f Fn(algorithm,)g(this)g (register)g(holds)g(a)g(set)f(of)h(inputs)g(to)g(a)f(collectiv)o (ely-computed)44 885 y(function.)20 b(This)13 b(phase)f(ends)h(when)g (a)f(pro)q(cess)h(observ)o(es)f(enough)g(inputs)i(to)d(uniquely)k (determine)e(the)44 942 y(function)i(v)m(alue,)h(whic)o(h)f(is)g (written)g(in)g(the)g(second)g(register,)f(pro)o(viding)h(the)g (termination)g(condition)44 998 y(and)h(output)f(v)m(alue)h(for)f(all)h (participating)g(pro)q(cesses.\))104 1067 y(In)h(the)f(algorithm)h(b)q (elo)o(w,)f(the)h(second)g(register)f(is)h(used)f(b)q(oth)h(to)e (\\gate")g(access)i(to)f(the)g Fg(U)5 b(nion)44 1124 y Fn(register,)14 b(and)h(to)f(break)h(symmetry)f(on)g(the)h(set)f(of)g (pro)q(cess)h(id's.)20 b(It)15 b(con)o(tains)g(t)o(w)o(o)e(\014elds:)21 b(a)14 b(pro)q(cess)44 1180 y(id)21 b(\(in)f(the)g Fg(Leader)g Fn(\014eld\))g(and)g(a)g(bit)g(\(the)f Fg(M)5 b(ar)q(k)q(ed)20 b Fn(\014eld\).)34 b(Pro)q(cesses)19 b(access)h(the)g Fg(U)5 b(nion)20 b Fn(\014eld)44 1236 y(only)f(if)h(they)f(read)f(the)h Fg(M)5 b(ar)q(k)q(ed)19 b Fn(\014eld)h(as)e(0.)30 b(In)20 b(addition,)g(no)f(pro)q(cess)g(terminates)g(un)o(til)g(it)g(sets)44 1293 y(the)c Fg(M)5 b(ar)q(k)q(ed)14 b Fn(\014eld)i(to)d(1.)20 b(The)14 b(concurrency)i(b)q(ound)f(guaran)o(tees)f(that)f(the)i Fg(c)8 b Fn(+)h(1'st)14 b(pro)q(cess)g(cannot)44 1349 y(tak)o(e)19 b(a)g(step)h(un)o(til)g(the)g(\014rst)f Fg(c)g Fn(pro)q(cesses)h(terminate,)g(b)o(y)f(whic)o(h)h(time)g(the)g Fg(M)5 b(ar)q(k)q(ed)19 b Fn(\014eld)i(is)f(set.)44 1406 y(Hence,)h(only)e(the)g(\014rst)g Fg(c)f Fn(pro)q(cesses)i(access)f (the)g Fg(U)5 b(nion)19 b Fn(register.)31 b(W)l(e)19 b(pro)o(v)o(e)g(b)q(elo)o(w)g(that)f(one)i(of)44 1462 y(these)13 b(pro)q(cesses)h(m)o(ust)e(ev)o(en)o(tually)i(see)f Fg(c)g Fn(di\013eren)o(t)g(pro)q(cess)g(id's)h(\(and)f(go)f(on)h(to)f (set)h Fg(M)5 b(ar)q(k)q Fn(\).)19 b(Hence,)44 1519 y(the)c(leader)h (will)h(b)q(e)f(the)f Fg(c)p Fn('th)g(pro)q(cess)g(to)g(write)g(to)g Fg(Leader)q Fn(.)104 1588 y(The)23 b(pro)q(of)f(b)q(egins)h(b)o(y)f (arguing)h(that)e(all)i(the)g(pro)q(cesses)f(that)g(terminate)g(elect)h (exactly)g(the)44 1644 y(same)d(participan)o(t)h(as)e(a)h(leader.)36 b(Note)20 b(\014rst)f(that)h(only)g(participating)i(pro)q(cess)e(id's)h (can)f(app)q(ear)44 1701 y(in)h(the)g(shared)f Fg(U)5 b(nion)21 b Fn(register.)35 b(A)21 b(pro)q(cess)f(sets)g Fg(M)5 b(ar)q(k)q(ed)20 b Fn(from)g(0)g(to)g(1)g(and)g(terminates)h (only)44 1757 y(after)14 b(it)h(determines)g(that)f Fg(c)g Fn(pro)q(cesses)h(are)f(already)h(participating.)21 b(Th)o(us,)14 b(the)h(\014rst)f Fg(c)g Fn(pro)q(cesses)h(to)44 1813 y(participate)i(will)g(\014rst)e(see)h(that)f Fg(M)5 b(ar)q(k)q(ed)15 b Fn(is)h(not)g(set,)f(and)h(no)f(later)h(pro)q(cess)g (can)g(participate)g(un)o(til)44 1870 y(one)h(of)e(these)i(terminates,) f(after)g(setting)g Fg(M)5 b(ar)q(k)q(ed)p Fn(.)23 b(Hence,)17 b(only)g(the)f(\014rst)g Fg(c)g Fn(pro)q(cesses)h(will)h(write)44 1926 y(their)d(id's)f(in)o(to)g Fg(Leader)q Fn(.)19 b(Moreo)o(v)o(er,) 12 b(no)i(pro)q(cess)g(sets)f Fg(M)5 b(ar)q(k)q(ed)14 b Fn(un)o(til)h(the)f(last)g(of)f(these)h Fg(c)g Fn(pro)q(cesses)44 1983 y(ha)o(v)o(e)i(written)g(to)f Fg(Leader)q Fn(.)22 b(Hence,)17 b(an)o(y)f(pro)q(cesses)g(that)g(terminate)g(will)h(elect)g (the)f(last)g(pro)q(cess)h(to)44 2039 y(write)e(its)h(id)g(to)e Fg(Leader)q Fn(.)104 2108 y(T)l(o)j(pro)o(v)o(e)f(termination,)i (consider)g(the)f(\014nite)h(partial)f(order)g(on)f(the)i(\014rst)e Fg(c)h Fn(pro)q(cesses)g(imp)q(osed)44 2165 y(b)o(y)h(set)g(inclusion)j (on)d(their)h Fj(lo)n(c)n(al)p 645 2165 14 2 v 16 w(union2)f Fn(v)m(ariables.)30 b(This)19 b(order)f(ev)o(olv)o(es)g(as)g(these)h(v) m(ariables)g(are)44 2221 y(up)q(dated{b)q(ecause)d(these)f(sets)f(are)g (only)g(up)q(dated)i(b)o(y)e(union,)h(pro)q(cesses)g(can)f(only)h(mo)o (v)o(e)e(up)o(w)o(ard)h(in)44 2278 y(this)f(order.)19 b(\(And)13 b(call)h(the)f(set)g(of)f(id's)h(in)h Fj(lo)n(c)n(al)p 878 2278 V 16 w(union2)e Fn(the)h(set)g(that)f(the)h(asso)q(ciated)g (pro)q(cess)g Fj(knows)44 2334 y(ab)n(out)p Fn(.\))28 b(As)18 b(long)g(as)g Fg(M)5 b(ar)q(k)q(ed)17 b Fn(is)i(not)e(set,)h(w) o(e)g(argue)f(that)g(the)h(only)h(stable)f(state)f(of)g(this)i(partial) 44 2390 y(order)d(is)h(one)g(in)g(whic)o(h)g(ev)o(ery)g(pro)q(cess)f (kno)o(ws)g(ab)q(out)g(ev)o(ery)h(other.)23 b(\(That)15 b(is,)i Fh(j)p Fj(lo)n(c)n(al)p 1592 2390 V 20 w(union2)6 b Fh(j)14 b Fn(=)h Fg(c)44 2447 y Fn(for)g(eac)o(h)g(of)g(the)g (\014rst)g Fg(c)g Fn(pro)q(cesses.\))104 2516 y(Consider)f(an)o(y)g (other)f(state.)18 b(It)c(con)o(tains)g(minimal)h(elemen)o(ts:)20 b(pro)q(cesses)14 b(with)f Fh(j)p Fj(lo)n(c)n(al)p 1628 2516 V 20 w(union2)7 b Fh(j)12 b Fg(<)44 2572 y(c)19 b Fn(and)g(suc)o(h)h(that)e(no)i(other)e(pro)q(cess)i(kno)o(ws)f(ab)q (out)g(strictly)g(few)o(er)g(pro)q(cesses.)32 b(Once)21 b(a)e(minimal)44 2629 y(pro)q(cess)g(writes)g(its)g(v)m(alue)h(of)e Fj(lo)n(c)n(al)p 681 2629 V 20 w(union2)25 b Fn(to)18 b Fg(U)5 b(nion)p Fn(,)20 b(no)e(strictly)h(smaller)h(set)e(can)h(b)q (e)g(written.)44 2685 y(Hence,)14 b(eac)o(h)e(of)g(these)g(pro)q (cesses)h(can)g(write)f(at)g(most)f(once)i(without)f(the)h(partial)f (order)h(c)o(hanging,)g(as)44 2742 y(an)o(y)h(read)h(m)o(ust)f(then)h (b)q(e)g(of)f(the)h(same)f(v)m(alue)i(they)f(wrote.)j(But)d(if)g(the)g (partial)g(order)f(isn't)h(c)o(hanged)933 2866 y(8)p eop %%Page: 9 10 9 9 bop 44 130 a Fa(Process)23 b Fg(i)p Fa('s)g(program.)44 186 y Fk(Shared)p Fn(:)271 243 y(\()p Fg(Leader)o(;)8 b(M)d(ar)q(k)q(ed)p Fn(\):)18 b(\(Pro)q(cess)d(id,)g(b)q(o)q(olean\),)h (initially)i(\(0)p Fg(;)8 b Fn(0\))271 299 y Fj(Union)p Fn(:)19 b(set)c(of)g(at)g(most)f Fg(c)h Fn(pro)q(cess)g(id's,)g (initially)j Fh(;)44 356 y Fk(Lo)q(cal)p Fn(:)271 412 y Fg(l)q(ocal)p 370 412 14 2 v 16 w(l)q(eader)q Fn(:)i(Pro)q(cess)15 b(id)271 469 y Fj(lo)n(c)n(al)p 361 469 V 16 w(union1)p Fn(:)20 b(set)15 b(of)g(at)f(most)h Fg(c)f Fn(pro)q(cess)i(id's)271 525 y Fj(lo)n(c)n(al)p 361 525 V 16 w(union2)p Fn(:)k(set)15 b(of)g(at)f(most)h Fg(c)f Fn(pro)q(cess)i(id's,)f(initially)j Fh(f)p Fg(i)p Fh(g)44 606 y Fn(1)68 b Fk(if)16 b Fg(M)5 b(ar)q(k)q(ed)12 b Fn(=)h(0)i Fk(then)g Fn(\()p Fg(Leader)o(;)8 b(M)d(ar)q(k)q(ed)p Fn(\))11 b(:=)h(\()p Fg(i;)c Fn(0\))13 b Fk(\014)44 662 y Fn(2)68 b Fg(l)q(ocal)p 234 662 V 16 w(union)p Fn(1)15 b(:=)g Fg(U)5 b(nion)44 719 y Fn(3)68 b Fk(while)16 b Fn(\()p Fh(j)p Fg(l)q(ocal)p 400 719 V 15 w(union)p Fn(1)p Fh(j)d Fg(<)g(c)p Fn(\))c Fh(^)h Fn(\()p Fg(M)5 b(ar)q(k)q(ed)12 b Fn(=)h(0\))i Fk(do)44 775 y Fn(4)159 b Fk(if)16 b Fh(:)p Fn(\()p Fg(l)q(ocal)p 419 775 V 16 w(union)p Fn(2)c Fh(\022)h Fg(l)q(ocal)p 731 775 V 16 w(union)p Fn(1\))i Fk(then)44 832 y Fn(5)250 b Fg(U)5 b(nion)13 b Fn(:=)f Fg(l)q(ocal)p 617 832 V 16 w(union)p Fn(1)e Fh([)h Fg(l)q(ocal)p 920 832 V 16 w(union)p Fn(2)44 888 y(6)159 b Fk(\014)44 945 y Fn(7)g Fg(l)q(ocal)p 325 945 V 16 w(union)p Fn(2)15 b(:=)g Fg(l)q(ocal)p 655 945 V 16 w(union)p Fn(1)10 b Fh([)h Fg(l)q(ocal)p 958 945 V 16 w(union)p Fn(2)44 1001 y(8)159 b Fg(l)q(ocal)p 325 1001 V 16 w(union)p Fn(1)13 b(:=)f Fg(U)5 b(nion)44 1058 y Fn(9)68 b Fk(o)q(d)44 1114 y Fn(10)45 b Fg(l)q(ocal)p 234 1114 V 16 w(l)q(eader)13 b Fn(:=)g Fg(Leader)44 1171 y Fn(11)45 b(\()p Fg(Leader)o(;)8 b(M)d(ar)q(k)q(ed)p Fn(\))13 b(:=)i(\()p Fg(l)q(ocal)p 691 1171 V 16 w(l)q(eader)o(;)8 b Fn(1\))44 1227 y(12)45 b Fk(return)p Fn(\()p Fj(L)n(e)n(ader)p Fn(\))475 1375 y(Figure)16 b(2:)j(Symmetric)d(election)h(for)d (concurrency)i Fg(c)44 1512 y Fn(otherwise,)j(ev)o(en)o(tually)g(a)e (pro)q(cess)i(that)e(kno)o(ws)g(a)h(di\013eren)o(t)g(id)h(will)h(write) e(to)f(the)h Fg(U)5 b(nion)18 b Fn(register)44 1569 y(and)e(then)f(b)q (e)h(read)f(b)o(y)g(a)g(minimal)i(pro)q(cess.)104 1638 y(It)e(follo)o(ws)g(that)f(at)g(some)h(p)q(oin)o(t)g(a)g(pro)q(cess)g (will)h(learn)g(that)e Fg(c)g Fn(pro)q(cesses)h(are)g(participating)h (\(i.e.,)44 1694 y Fh(j)p Fg(U)5 b(nion)p Fh(j)16 b Fn(=)h Fg(c)p Fn(\))f(and)i(set)f Fj(Marke)n(d)h Fn(to)e(1,)i(allo)o(wing)g (the)f(rest)g(of)g(the)h(pro)q(cessing)g(to)f(terminate.)27 b(This)44 1751 y(completes)16 b(the)f(correctness)g(pro)q(of)g(of)g (the)g(election)i(algorithm.)104 1820 y(As)d(ab)q(o)o(v)o(e,)g(this)g (election)i(algorithm)e(can)h(b)q(e)f(easily)i(con)o(v)o(erted)d(in)o (to)i(a)e(consensus)i(algorithm,)f(b)o(y)44 1876 y(adding)i(the)f (input)i(v)m(alue)f(\(0)f(or)f(1\))h(to)f(eac)o(h)i(pro)q(cess)f(id.)p 1828 1876 17 38 v 44 1995 a(Note)g(that)g(the)h(termination)g(argumen)o (t)e(ab)q(o)o(v)o(e)h(holds)i(ev)o(en)e(if)h(some)g(pro)q(cesses)g (stop)f(up)q(dating)h(the)44 2051 y Fg(U)5 b(nion)11 b Fn(register,)g(as)g(long)g(as)f(the)h(set)f(of)h(activ)o(e)f(pro)q (cesses)i(kno)o(ws)e(ab)q(out)g(ev)o(ery)o(one.)18 b(This)12 b(observ)m(ation)44 2108 y(is)g(imp)q(ortan)o(t)e(to)h(the)g(next)g (algorithm,)g(an)g(asymmetric)g(solution)h(to)f(election)h(\(and)f (consensus\))g(using)44 2164 y(only)19 b(a)e(single)i(atomic)f (register.)28 b(Asymmetric)18 b(algorithms)g(allo)o(w)g(comparisons)g (b)q(et)o(w)o(een)g(pro)q(cess)44 2221 y(id's)c(\(w)o(e)e(assume)h(a)g 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y(0)c(to)g(ac)o(kno)o(wledge)g(the)h(signal.)24 b(Ha)o(ving)16 b(ac)o(kno)o(wledged)h(the)f(signal,)h Fg(p)g Fn(stops)e(participating)j(in)f(the)44 889 y(up)q(dates)g(to)e Fg(U)5 b(nion)p Fn(.)24 b(\(A)o(t)15 b(this)i(p)q(oin)o(t,)f(pro)q (cess)h Fg(p)f Fn(kno)o(ws)f(that)h(the)g(leader)h(kno)o(ws)f(all)h Fg(c)f Fn(initial)i(id's,)44 946 y(and)f(will)h(ev)o(en)o(tually)g (refresh)f Fg(U)5 b(nion)17 b Fn(if)g(necessary)l(.\))24 b(The)17 b(leader)h(itself)f(initiates)h(eac)o(h)f(handshak)o(e)44 1002 y(once)h(it)f(kno)o(ws)g(the)g(id's)g(of)g(all)h Fg(c)f Fn(initial)i(pro)q(cesses.)26 b(Slo)o(w)18 b(pro)q(cesses)f(ma)o (y)f(o)o(v)o(erwrite)h(handshak)o(es)44 1059 y(in)f(progress,)e(whic)o (h)i(the)g(leader)g(\(or)e(higher-ranking)j(pro)q(cesses\))e(will)i (detect)e(and)g(refresh.)104 1127 y(As)h(in)h(the)f(previous)h (algorithm,)f(no)g(pro)q(cess)g(exits)h(the)f(while)i(lo)q(op)e(in)h (line)h(2)e(un)o(til)h Fg(c)f Fn(pro)q(cesses)44 1184 y(ha)o(v)o(e)c(en)o(tered)h(the)f(lo)q(op.)19 b(\(In)13 b(a)f(giv)o(en)g(run,)h(let)g Fg(I)t(nitial)f Fn(denote)h(the)f(set)g (of)g(these)g(initial)j Fg(c)c Fn(pro)q(cesses.\))44 1240 y(W)l(e)16 b(argue)f(b)q(elo)o(w)h(that)f(none)g(of)g(the)h (initial)i(pro)q(cesses)d(terminates)h(un)o(til)g(the)g(shared)f (register)h(con-)44 1297 y(tains)i(the)g(\014nal)g(and)g(stable)g(v)m (alue)h(\()p Fg(I)t(nitial)q(;)8 b(c;)g Fn(1)o(\).)24 b(It)18 b(follo)o(ws)g(that)f(all)h(pro)q(cesses)g(that)f(terminate)44 1353 y(elect)f(the)g(same)e(leader.)104 1422 y(T)l(o)i(argue)g(that)g (ev)o(ery)g(pro)q(cess)g(terminates,)g(the)h(main)g(di\016cult)o(y)g (is)g(arguing)f(that)g(the)g(initial)i Fg(c)44 1479 y Fn(pro)q(cesses)c(terminate.)19 b(Note)13 b(that)g(no)g(pro)q(cess)g (stops)g(up)q(dating)h Fg(U)5 b(nion)14 b Fn(un)o(til)g(it)g(kno)o(ws)f (the)g(id's)h(of)e Fg(c)44 1535 y Fn(participating)j(pro)q(cesses,)e (\(the)g(initial)j(while)e(lo)q(op)g(in)g(line)h(2\))e(and)g(indeed,)i (un)o(til)f(it)g(also)f(kno)o(ws)g(that)44 1592 y(the)i(leader)g(kno)o (ws)f(the)h(id's)g(of)f Fg(c)h Fn(participating)h(pro)q(cesses)f(\(the) f(up)q(date)h(in)h(line)g(16)e(that)g(is)h(skipp)q(ed)44 1648 y(once)i(the)f(leader's)g(signal)h(is)g(ac)o(kno)o(wledged\).)23 b(As)16 b(in)h(the)f(argumen)o(t)g(in)h(the)f(preceding)h(algorithm,)44 1704 y(this)f(implies)h(that)e(all)h(pro)q(cesses)f(in)h Fg(I)t(nitial)g Fn(will)g(ev)o(en)o(tually)h(learn)e(the)h(id's)f(in)h Fg(I)t(nitial)q Fn(.)104 1773 y(By)i(induction)i(on)e(the)g(rank)g Fg(r)g Fh(\025)g Fn(1)g(of)f(pro)q(cess)i Fg(p)f Fn(in)g Fg(I)t(nitial)q Fn(,)g(pro)q(cess)g Fg(p)g Fn(ev)o(en)o(tually)h(reads) f(its)44 1830 y(signal)j(from)f(the)h(leader,)h(the)e(leader)h(ev)o(en) o(tually)h(reads)e(its)g(ac)o(kno)o(wledgemen)o(t,)i(and)e(\(using)h (the)44 1886 y(explicit)e Fg(ack)q(ed)d Fn(\015ag)g(to)g(a)o(v)o(oid)h (unnecessary)g(refreshes)g(of)f(the)g(register\))h(pro)q(cess)f Fg(p)h Fn(nev)o(er)g(executes)44 1943 y(another)f(write)g(op)q (eration.)22 b(Hence,)17 b(the)f(leader)g(will)i(ev)o(en)o(tually)f (set)f(the)g(register)g(to)f(\()p Fg(I)t(nitial)q(;)8 b(c;)g Fn(1)o(\))44 1999 y(and)14 b(terminate.)20 b(The)14 b(other)f(of)h(the)g(initial)i(participan)o(ts)e(\(who)g(ha)o(v)o(e)f (stopp)q(ed)i(writing)f(but)g(are)g(still)44 2056 y(reading\))20 b(will)h(ev)o(en)o(tually)g(read)e Fg(S)s(ig)h Fn(=)g Fg(c)f Fn(and)h(terminate,)g(and)g(as)f(w)o(e)g(argued)h(ab)q(o)o(v)o (e,)g(the)f(later)44 2112 y(pro)q(cesses)d(will)h(all)f(see)f Fg(c)g Fn(di\013eren)o(t)g(v)m(alues)i(in)f Fg(U)5 b(nion)15 b Fn(and)h(terminate)f(immediately)l(.)p 1828 2112 17 38 v 44 2318 a Fm(3)67 b(A)n(tomic)23 b(registers:)29 b(participation)c(is)e(not)f(required)44 2431 y Fn(As)12 b(men)o(tioned)h(ab)q(o)o(v)o(e,)e(an)o(y)h(problem)g(solution)h(in)g (a)e(mo)q(del)i(in)g(whic)o(h)g(participation)g(is)f(not)f(required,)44 2488 y(is)19 b(also)f(correct)f(when)i(participation)g(is)f(required,)i (and)e(hence)h(suc)o(h)g(solutions)f(are)g(more)g(di\016cult)44 2544 y(to)d(construct.)21 b(The)15 b(main)h(new)g(result)g(of)f(this)h (section)g(is)g(a)f(new)h(adaptiv)o(e)g(starv)m(ation-free)f(m)o(utual) 44 2601 y(exclusion)i(algorithm)e(for)g(un)o(b)q(ounded)i(concurrency)l (.)921 2866 y(11)p eop %%Page: 12 13 12 12 bop 44 86 a Fl(3.1)56 b(Consensus,)19 b(election)e(and)i(m)n (utual)f(exclusion)f(for)h(concurrency)g Fd(c)44 185 y Fn(W)l(e)c(next)h(consider)g(the)f(n)o(um)o(b)q(er)h(and)f(size)h(of) f(registers)g(required)i(to)d(solv)o(e)i(consensus,)f(election)i(and)44 241 y(m)o(utual)11 b(exclusion)h(in)g(a)e(mo)q(del)i(in)f(whic)o(h)h (participation)g(is)f(not)f(required)i(\(i.e.,)f([1)p Fg(;)d Fh(1)p Fn(]-participation\).)44 298 y(Earlier)16 b(w)o(ork)e(on)h(these)h(problems)g(for)e(a)h(\014nite)h(n)o(um)o(b)q (er)g(of)f(pro)q(cesses)g(establishes)i(some)e(b)q(ounds:)44 416 y Fk(Theorem)i(3.1)23 b Fj(F)m(or)15 b(c)n(oncurr)n(ency)h(level)f Fg(c)p Fj(,)h(the)g(numb)n(er)h(of)f(atomic)h(r)n(e)n(gisters)e(that)i (ar)n(e:)44 473 y(\(1\))f(ne)n(c)n(essary)e Fn([BL93])i Fj(and)g(su\016cient)g Fn([SP89)o(])g Fj(for)h(solving)e(de)n(ad)r(lo)n (ck-fr)n(e)n(e)g(mutual)j(exclusion,)d(is)h Fg(c)p Fj(;)44 529 y(\(2\))g(ne)n(c)n(essary)e(and)j(su\016cient)e(for)i(solving)e (ele)n(ction,)g(is)h Fn(log)8 b Fg(c)i Fn(+)g(1)p Fj(;)16 b Fn([SP89])44 586 y Fj(\(3\))g(su\016cient)g(for)g(solving)f(c)n (onsensus,)f(is)i Fn(log)8 b Fg(c)i Fn(+)h(1)16 b([SP89)o(])p Fj(.)44 704 y Fc(Pro)q(of:)k Fn(\(1\))15 b(An)o(y)h(deadlo)q(c)o (k-free)h(m)o(utual)e(exclusion)j(algorithm)e(for)f Fg(n)h Fn(pro)q(cesses)g(m)o(ust)f(use)h(at)f(least)44 761 y Fg(n)h Fn(shared)f(registers)g([BL93]:)k(this)d(b)q(ound)g(applies)h (immediately)f(to)f(concurrency)l(.)104 830 y(There)c(is)h(a)f (symmetric)g(deadlo)q(c)o(k-free)h(m)o(utual)f(exclusion)i(algorithm)e (for)f Fg(n)i Fn(pro)q(cesses)f(whic)o(h)h(uses)44 886 y(only)18 b Fg(n)g Fn(shared)f(registers)h([SP89)o(].)26 b(This)18 b(algorithm)g(w)o(orks)e(without)h(mo)q(di\014cation)i(for)e 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b(the)f(same)f(n)o(um)o(b)q(er)i(of)e(registers:)18 b(Eac)o(h)11 b(pro)q(cess)g(app)q(ends)h(its)f(v)o(ote)g(\(0)f(or)h(1\))f(to)g(its)i (id.)19 b(Pro)q(cesses)44 1432 y(c)o(ho)q(ose)c(the)h(leader's)f(v)o (ote)g(as)f(the)i(consensus)g(v)m(alue.)p 1828 1432 17 38 v 104 1550 a(Theorem)f(3.1)e(implies)k(that)d(when)h(the)g (concurrency)h(lev)o(el)g(is)f(not)f(\014nite,)h(an)g(in\014nite)i(n)o (um)o(b)q(er)e(of)44 1607 y(registers)h(are)f(necessary)h(for)f (solving)i(the)f(election)h(and)f(m)o(utual)g(exclusion)i(problems.)k (But)16 b(are)f(an)44 1663 y(in\014nite)21 b(n)o(um)o(b)q(er)e(of)f (registers)g(su\016cien)o(t?)32 b(\(W)l(e)18 b(observ)o(e,)h(based)g (on)g(a)f(result)h(from)f([Y)l(A94)o(],)h(that)44 1720 y(if)e(w)o(e)g(restrict)g(the)f(n)o(um)o(b)q(er)i(of)e(pro)q(cesses)h (that)f(can)h(try)f(to)g(write)h(the)g(same)g(register)f(at)h(the)f (same)44 1776 y(time,)f(the)h(answ)o(er)e(is)i(no.\))44 1910 y Fl(3.2)56 b(Mutual)19 b(exclusion)e(algorithms)f(for)j(un)n(b)r 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(oncurr)n(ency)g(\(and)g(henc)n(e)f(also)h(to)h(ele)n(ction)e(and)i(c)n (onsensus\))d(using)h(an)i(in\014nite)44 2522 y(numb)n(er)16 b(of)h(r)n(e)n(gisters.)44 2641 y Fn(These)e(problems)h(can)f(all)g(b)q (e)h(solv)o(ed)f(b)o(y)f(simple)j(adaptations)d(to)g(the)h(deadlo)q(c)o (k-free)h(m)o(utual)f(exclu-)44 2698 y(sion)h(algorithm)f(presen)o(ted) h(b)q(elo)o(w.)k(This)c(algorithm)f(has)g(the)g(follo)o(wing)h(in)o (teresting)g(prop)q(erties:)921 2866 y(12)p eop %%Page: 13 14 13 13 bop 100 86 a Fn(1.)22 b(It)15 b(w)o(orks)f(for)h(un)o(b)q(ounded) i(concurrency)l(.)100 180 y(2.)22 b(In)e(the)g(absence)h(of)e(con)o (ten)o(tion,)i(only)f(nine)h(accesses)f(to)f(the)h(shared)g(memory)f (are)h(needed)158 237 y(\(eigh)o(t)15 b(in)h(the)f(en)o(try)g(co)q(de)h (and)f(one)g(in)i(the)e(exit)g(co)q(de\).)100 330 y(3.)22 b(It)13 b(is)h(adaptiv)o(e)g({)f(its)h(time)f(complexit)o(y)i(\(ev)o (en)e(in)i(the)e(w)o(orst)f(case,)i(measured)f(from)g(exit)h(of)f(the) 158 387 y(critical)k(section)f(b)o(y)f(a)g(pro)q(cess)g(to)g(the)h (next)f(en)o(trance)g(b)o(y)h(an)o(y)f(pro)q(cess\))g(is)h(a)f (function)h(of)f(the)158 443 y(actual)g(n)o(um)o(b)q(er)h(of)e(con)o (tending)j(pro)q(cesses.)100 537 y(4.)22 b(It)14 b(is)g(symmetric)h({)e (iden)o(ti\014ers)j(are)d(only)i(written,)f(read)g(and)g(compared)g (for)f(equalit)o(y)l(,)i(but)f(are)158 594 y(not)h(ordered)g(or)g(used) g(to)g(index)h(shared)g(registers.)44 700 y(Except)i(for)f(the)h (non-adaptiv)o(e)h(algorithm)e(in)i(the)f(next)g(subsection,)h(w)o(e)e (kno)o(w)h(of)f(no)h(m)o(utual)g(ex-)44 756 y(clusion)j(algorithm)e (\(using)h(atomic)f(registers\))g(satisfying)g(the)g(\014rst)g(prop)q (ert)o(y)l(.)32 b(The)19 b(algorithm)h(is)44 813 y(de\014ned)15 b(formally)e(in)h(Figure)g(6,)f(and)g(is)h(built)g(out)f(of)g(simple)h (building)i(blo)q(c)o(ks)e(called)h Fj(splitters)d Fn(whic)o(h)44 869 y(are)j(in)o(tro)q(duced)h(next.)44 1002 y Fk(3.2.1)52 b(Splitters)44 1100 y Fn(In)13 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b(in)o(tegers,)e(initially)j(\(0)p Fg(;)8 b Fn(0\))44 299 y Fk(Lo)q(cal:)135 356 y Fn(\()p Fg(tick)q(et)267 363 y Fb(i)281 356 y Fg(;)g(v)r(al)q(id)405 363 y Fb(i)417 356 y Fn(\):)20 b(in)o(tegers)44 437 y(1)68 b Fh(h)p Fn(\()p Fg(tick)q(et)285 444 y Fb(i)299 437 y Fg(;)8 b(v)r(al)q(id)423 444 y Fb(i)435 437 y Fn(\))k(:=)h(\()p Fg(tick)q(et;)8 b(v)r(al)q(id)p Fn(\))44 493 y(2)114 b Fg(tick)q(et)13 b Fn(:=)f Fg(tick)q(et)f Fn(+)f(1)p Fh(i)44 550 y Fn(3)68 b Fk(while)16 b Fg(tick)q(et)384 557 y Fb(i)411 550 y Fh(6)p Fn(=)d Fg(v)r(al)q(id)562 557 y Fb(i)590 550 y Fk(do)44 606 y Fn(4)204 b Fh(h)p Fg(v)r(al)q(id)392 613 y Fb(i)417 606 y Fn(:=)13 b Fg(v)r(al)q(id)p Fh(i)g Fk(o)q(d)44 662 y Fn(5)68 b Fj(critic)n(al)16 b(se)n(ction)44 719 y Fn(6)68 b Fh(h)p Fg(v)r(al)q(id)11 b Fn(:=)h Fg(v)r(al)q(id)d Fn(+)h(1)p Fh(i)148 867 y Fn(Figure)15 b(8:)20 b(FIF)o(O)15 b(m)o(utual)g(exclusion)i(using)f (read-mo)q(dify-write:)21 b(The)15 b(tic)o(k)o(et)g(algorithm)104 1004 y(T)l(o)i(pro)o(v)o(e)h(the)f(su\016cien)o(t)i(condition,)g(w)o(e) e(sligh)o(tly)i(mo)q(dify)f(the)g(tic)o(k)o(et)f(algorithm)h(from)f([F) 1745 988 y Fi(+)1773 1004 y Fn(89].)44 1061 y(The)g(algorithm)f(in)h (Figure)g(8)f(w)o(orks)f(as)h(follo)o(ws:)23 b(A)16 b(pro)q(cess)h (wishing)g(to)f(en)o(ter)g(its)h(critical)h(section)44 1117 y(tak)o(es)13 b(the)g(next)g(a)o(v)m(ailable)i(tic)o(k)o(et)e(and) g(w)o(aits)g(un)o(til)h(its)g(tic)o(k)o(et)f(b)q(ecomes)g(v)m(alid,)i (at)e(whic)o(h)h(p)q(oin)o(t)g(it)f(can)44 1174 y(safely)k(en)o(ter)g (its)f(critical)i(section.)25 b(When)17 b(it)g(exits,)g(it)g(discards)g (its)g(tic)o(k)o(et)g(and)f(v)m(alidates)i(the)f(next)44 1230 y(in)o(v)m(alid)h(tic)o(k)o(et)d(in)h(order)f(\(ev)o(en)g(if)g (this)h(tic)o(k)o(et)f(has)g(not)g(b)q(een)h(tak)o(en)f(y)o(et\).)104 1299 y(The)20 b(shared)h(register)e(con)o(tains)i(t)o(w)o(o)d (\014elds,)k Fj(ticket)e Fn(and)g Fj(valid)p Fn(,)h(eac)o(h)f(a)g (non-negativ)o(e)g(in)o(teger.)44 1355 y(The)e(brac)o(k)o(ets)e Fh(h)h(i)g Fn(are)g(used)h(to)f(explicitly)j(mark)d(the)g(b)q(eginning) j(and)d(end)h(of)f(exclusiv)o(e)i(access)f(to)44 1412 y(the)h(shared)g(read-mo)q(dify-write)h(register.)31 b(An)20 b(execution)g(of)e(a)h(brac)o(k)o(eted)g(section)g(is)h (considered)44 1468 y(as)f(an)g(atomic)g(action.)33 b(Eac)o(h)19 b(pro)q(cess)h(\014rst)f(reads)g(the)g(v)m(alue)i(of)e(the)g(shared)g (register,)h(stores)f(its)44 1525 y(comp)q(onen)o(ts)e(in)g(lo)q(cal)h (memory)l(,)e(and)h(incremen)o(ts)h Fg(tick)q(et)f Fn(b)o(y)f(one.)24 b(A)o(t)17 b(an)o(y)f(later)g(p)q(oin)o(t,)i(a)e(pro)q(cess)44 1581 y(b)q(ecomes)d(the)e(\014rst)h(in)g(the)g(\\w)o(aiting)g(line")h (if)f(it)g(learns)h(\(b)o(y)e(insp)q(ecting)j Fj(valid)p Fn(\))d(that)g(its)h(tic)o(k)o(et)g(n)o(um)o(b)q(er)44 1638 y Fg(tick)q(et)158 1645 y Fb(i)188 1638 y Fn(equals)k Fj(valid)p Fn(,)e(in)i(whic)o(h)g(case)g(it)f(can)g(safely)h(en)o(ter)f (its)g(critical)i(section.)p 1828 1638 17 38 v 44 1822 a Fl(5.2)56 b(Semaphores)44 1920 y Fn(Giv)o(en)14 b(the)g(results)g(so) g(far)f(ab)q(out)g(starv)m(ation-free)h(m)o(utual)g(exclusion,)h(it)f (is)h(natural)e(to)h(ask)f(whether)44 1976 y(it)f(can)h(b)q(e)f(solv)o (ed)h(with)f(a)g(b)q(ounded)h(n)o(um)o(b)q(er)f(of)g(semaphores.)18 b(The)13 b(answ)o(er,)e(as)h(presen)o(ted)g(in)h([FP87)o(],)44 2033 y(is)18 b(that)e(using)h(w)o(eak)g(semaphores,)f(it)h(can)g(b)q(e) h(solv)o(ed)f(with)g(small)h(constan)o(t)e(space)h(for)f(un)o(b)q (ounded)44 2089 y(concurrency)l(,)g(but)f(not)g(with)h(in\014nite)h (concurrency)f(lev)o(el.)104 2158 y(A)f(binary)h(semaphore)f(is)h(a)f (shared)g(ob)s(jects)g(that)f(ma)o(y)g(tak)o(e)h(the)g(v)m(alues)h(0)f (or)g(1,)f(and)i(is)f(initially)44 2215 y(set)g(to)g(1.)k(It)c(supp)q (ort)h(t)o(w)o(o)e(op)q(erations,)h(called)h Fj(P)k Fn(and)15 b Fj(V)9 b Fn(:)112 2321 y Fh(\017)23 b Fn(When)17 b(a)f(pro)q(cess)g (p)q(erforms)g(a)h Fj(P)j Fn(op)q(eration,)d(if)g(the)f(v)m(alue)i(is)f (1,)f(then)h(the)f(v)m(alue)i(is)f(set)f(to)g(0;)158 2377 y(otherwise,)h(the)g(pro)q(cess)h(is)f(blo)q(c)o(k)o(ed)h(un)o (til)h(the)e(v)m(alue)h(is)g(1.)25 b(T)l(esting)18 b(and)f(decremen)o (ting)h(the)158 2434 y(semaphore)d(are)g(executed)h(atomically)l(.)112 2528 y Fh(\017)23 b Fn(When)15 b(a)g(pro)q(cess)h(p)q(erforms)f(a)f Fj(V)24 b Fn(op)q(eration,)15 b(the)g(v)m(alue)i(is)e(set)g(to)g(1.)44 2634 y(The)e(result)h(b)q(elo)o(w)f(refers)g(to)f Fj(we)n(ak)h Fn(semaphores,)g(in)h(whic)o(h)g(a)e(pro)q(cess)h(that)g(executes)g(a)g Fj(V)21 b Fn(op)q(eration)44 2690 y(will)16 b(not)d(b)q(e)h(the)g(one)g (to)f(complete)h(the)g(next)g Fj(P)k Fn(op)q(eration)c(on)g(that)f (semaphore,)g(if)i(another)e(pro)q(cess)921 2866 y(21)p eop %%Page: 22 23 22 22 bop 44 86 a Fn(has)19 b(b)q(een)g(blo)q(c)o(k)o(ed)h(at)e(that)g (semaphore.)30 b(Instead,)19 b(one)g(of)f(the)h(blo)q(c)o(k)o(ed)h(pro) q(cesses)f(is)g(allo)o(w)o(ed)g(to)44 143 y(\\pass")c(the)g(semaphore)g (to)g(a)f(blo)q(c)o(k)o(ed)j(pro)q(cess.)44 241 y Fk(Theorem)g(5.2)g (\(F)l(riedb)q(erg)h(and)g(P)o(eterson)e([FP87)o(]\))112 330 y Fh(\017)23 b Fj(Ther)n(e)13 b(is)h(an)g(adaptive)h(symmetric)f (solution)g(to)g(starvation-fr)n(e)n(e)g(mutual)h(exclusion)e(using)h (two)158 386 y(atomic)j(bits)e(and)i(two)g(we)n(ak)f(semaphor)n(es,)g (when)g(the)h(c)n(oncurr)n(ency)e(is)h(unb)n(ounde)n(d.)112 473 y Fh(\017)23 b Fj(Ther)n(e)15 b(is)g(no)h(solution)f(to)h(the)g (starvation-fr)n(e)n(e)g(mutual)g(exclusion)f(pr)n(oblems)g(using)h (any)f(\014nite)158 530 y(numb)n(er)h(of)g(atomic)h(r)n(e)n(gisters)e (and)h(we)n(ak)g(semaphor)n(es,)h(when)f(the)g(c)n(oncurr)n(ency)g(is)f (in\014nite.)44 682 y Fm(6)67 b(Discussion)44 796 y Fn(W)l(e)21 b(ha)o(v)o(e)f(explored)h(ho)o(w)f(v)m(arious)h(assumptions)g(ab)q(out) g(the)f(n)o(um)o(b)q(er)h(of)f(pro)q(cesses,)i(the)f(concur-)44 853 y(rency)16 b(lev)o(el,)h(and)e(the)h(n)o(um)o(b)q(er)g(of)f (participating)h(pro)q(cesses,)g(e\013ect)f(the)h(design)g(of)f(shared) h(memory)44 909 y(algorithms.)31 b(In)19 b(particular,)h(w)o(e)e(ha)o (v)o(e)h(lo)q(ok)o(ed)g(at)f(the)h(case)g(of)f(computing)h(with)g (in\014nitely)j(man)o(y)44 966 y(pro)q(cesses.)f(There)15 b(are)g(man)o(y)g(in)o(teresting)h(questions)g(that)e(are)h(left)h(op)q (en:)k(In)c(the)g(fault-free)f(mo)q(del)44 1022 y(using)h(atomic)e (registers)h(only)l(,)h(are)e(there)h(in)o(teresting)h(problems)f(that) f(can)h(b)q(e)h(solv)o(ed)f(for)g(an)o(y)f(\014nite)44 1078 y(and)20 b Fj(a)g(priori)g Fn(kno)o(wn)g(n)o(um)o(b)q(er)g(of)f (pro)q(cesses)h(but)f(cannot)g(b)q(e)i(solv)o(ed)f(for)e(a)i(\014nite)g (and)g(unkno)o(wn)44 1135 y(n)o(um)o(b)q(er)13 b(of)g(pro)q(cesses)g (\(ev)o(en)f(with)h(un)o(b)q(ounded)i(space\)?)20 b(Are)12 b(there)h(in)o(teresting)h(problems)f(that)f(can)44 1191 y(b)q(e)k(solv)o(ed)g(using)f(a)g Fj(\014nite)g Fn(n)o(um)o(b)q(er)g (of)g(atomic)g(registers)g(b)o(y)g(an)g(in\014nite)i(n)o(um)o(b)q(er)e (of)g(pro)q(cesses)h(with)44 1248 y(b)q(ounded)g(concurrency?)22 b(Designing)16 b(w)o(ait-free)e(algorithms)h(for)f(problems)i(suc)o(h)f (as)f(collect,)i(renam-)44 1304 y(ing)f(and)f(snapshot)f(assuming)i(un) o(b)q(ounded)g(concurrency)g(are)f(c)o(hallenging)h(problems.)20 b(Recen)o(t)15 b(w)o(ork)44 1361 y(in)i(this)g(area)e(demonstrates)g (the)h(existence)i(of)d(a)h(hierarc)o(h)o(y)g(of)g(computable)h(tasks,) e(determined)i(b)o(y)44 1417 y(concurrency)f(b)q(ounds)g([GMT2001)n(].) 104 1486 y(Are)j(there)f(automatic)g(transformations)f(b)q(et)o(w)o (een)h(some)h(algorithms)f(\(suc)o(h)g(as)g(symmetric)h(al-)44 1543 y(gorithms\))c(whic)o(h)i(w)o(ork)d(correctly)i(for)f(a)g (\014nite)i(n)o(um)o(b)q(er)f(of)f(pro)q(cesses,)h(to)f(algorithms)h (whic)o(h)g(w)o(ork)44 1599 y(correctly)g(\(p)q(erhaps)f(assuming)h(b)q (ounded)g(concurrency\))g(for)e(in\014nitely)k(man)o(y)d(pro)q(cesses?) 104 1668 y(Finally)l(,)g(are)f(there)f(algorithms)h(that)f(are)g (correct)g(assuming)h(\014nite)h(concurrency)l(,)f(but)g(whic)o(h)g (fail)44 1724 y(if)j(this)h(constrain)o(t)e(is)i(not)e(met?)26 b(Can)16 b(w)o(e,)h(in)h(general,)f(mo)q(dify)g(suc)o(h)h(algorithms)e (to)h(guaran)o(tee)f(at)44 1781 y(least)g(safet)o(y)e(\(if)h(not)g(liv) o(eness\))i(ev)o(en)f(when)f(the)h(constrain)o(ts)f(on)g(the)g(n)o(um)o (b)q(er)h(of)f(pro)q(cesses)h(and)f(the)44 1837 y(concurrency)h(lev)o (el)h(are)e(not)f(met?)44 1906 y Fk(Ac)o(kno)o(wledgemen)o(t)c Fn(The)i(authors)f(wish)h(to)g(thank)f(an)h(anon)o(ymous)f(referee)h (for)f(man)o(y)g(suggestions)44 1963 y(and)17 b(corrections,)g(and)g (notably)l(,)g(catc)o(hing)h(an)e(error)g(in)i(the)f(conference)h(v)o (ersion)f(of)f(the)h(algorithm)44 2019 y(in)f(Figure)g(3.)44 2172 y Fm(References)44 2285 y Fn([A)l(T96])22 b(R.)d(Alur)h(and)g(G.)f (T)l(aub)q(enfeld.)34 b(Con)o(ten)o(tion-free)19 b(complexit)o(y)i(of)e (shared)g(memory)g(algo-)199 2342 y(rithms.)h Fj(Information)c(and)h (Computation)e Fn(126:1)f(\(1996\))g(62{73.)f(\(Also)i(in)h(PODC)f (1994.\))44 2429 y([A)o(GM98])21 b(H.)d(A)o(ttiy)o(a,)f(A.)h(Gorbac)o (h,)g(and)g(S.)g(Moran.)28 b(Computing)18 b(in)h(totally)g(anon)o (ymous)e(asyn-)199 2485 y(c)o(hronous)f(shared)g(memory)f(systems.)21 b(In)16 b Fj(Pr)n(o)n(c.)h(12th)g(International)f(Symp)n(osium)g(on)h (Dis-)199 2542 y(tribute)n(d)g(Computing)p Fn(,)e(LNCS)h(1499:49-61,)c (Septem)o(b)q(er)k(1998.)44 2629 y([BMT95])21 b(H.)f(Brit,)i(S.)e (Moran,)g(and)g(G.)g(T)l(aub)q(enfeld.)36 b(Public)22 b(data)e(structures:)30 b(coun)o(ters)20 b(as)f(a)199 2685 y(sp)q(ecial)c(case.)i Fj(Pr)n(o)n(c.)d(3r)n(d)g(Isr)n(ael)g(Symp) n(osium)g(on)g(The)n(ory)g(of)h(Computing)f(and)h(Systems,)d Fn(T)l(el)199 2742 y(Aviv,)k(98{110,)d(Jan)o(uary)i(1995.)k(\(Also:)h (accepted)c(to)e(Theoretical)i(Computer)f(Science.\))921 2866 y(22)p eop %%Page: 23 24 23 23 bop 44 86 a Fn([B)89 70 y Fi(+)119 86 y Fn(82])21 b(J.)16 b(E.)f(Burns,)h(P)l(.)f(Jac)o(kson,)h(N.)f(A.)g(Lync)o(h,)h(M.) f(J.)h(Fisc)o(her,)g(and)f(G.)g(L.)h(P)o(eterson.)21 b(Data)14 b(re-)199 143 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/breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /endash 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n -1000 5587 m -1000 -1000 l 6937 -1000 l 6937 5587 l cp clip 0.06000 0.06000 sc /Times-Roman-iso ff 180.00 scf sf 2430 3435 m gs 1 -1 sc ( ) col0 sh gr /Times-Roman-iso ff 165.00 scf sf 3938 2543 m gs 1 -1 sc (1-concurrency) col0 sh gr /Times-Roman-iso ff 165.00 scf sf 3938 2834 m gs 1 -1 sc (2-concurrency) col0 sh gr /Times-Roman-iso ff 165.00 scf sf 3716 3428 m gs 1 -1 sc (bounded concurrency) col0 sh gr /Times-Roman-iso ff 165.00 scf sf 4041 3762 m gs 1 -1 sc (unbounded) col0 sh gr /Times-Roman-iso ff 165.00 scf sf 3999 3964 m gs 1 -1 sc (concurrency) col0 sh gr % Polyline 7.500 slw n 3330 2625 m 3225 2625 3225 4320 105 arcto 4 {pop} repeat 3225 4425 5520 4425 105 arcto 4 {pop} repeat 5625 4425 5625 2730 105 arcto 4 {pop} repeat 5625 2625 3330 2625 105 arcto 4 {pop} repeat cp gs col0 s gr % Polyline n 3514 3229 m 3409 3229 3409 4174 105 arcto 4 {pop} repeat 3409 4279 5329 4279 105 arcto 4 {pop} repeat 5434 4279 5434 3334 105 arcto 4 {pop} repeat 5434 3229 3514 3229 105 arcto 4 {pop} repeat cp gs col0 s gr % Polyline n 3776 3528 m 3671 3528 3671 4023 105 arcto 4 {pop} repeat 3671 4128 5066 4128 105 arcto 4 {pop} repeat 5171 4128 5171 3633 105 arcto 4 {pop} repeat 5171 3528 3776 3528 105 arcto 4 {pop} repeat cp gs col0 s gr % Polyline n 3030 2325 m 2925 2325 2925 4470 105 arcto 4 {pop} repeat 2925 4575 5820 4575 105 arcto 4 {pop} repeat 5925 4575 5925 2430 105 arcto 4 {pop} repeat 5925 2325 3030 2325 105 arcto 4 {pop} repeat cp gs col0 s gr /Times-Roman-iso ff 180.00 scf sf 4424 3000 m gs 1 -1 sc (:) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 4424 3134 m gs 1 -1 sc (:) col0 sh gr $F2psEnd rs %%EndDocument @endspecial 32 1133 a(Figure)g(1:)j(Hierarc)o(h)o(y)e(of)e (concurrency)j(mo)q(dels.)-75 1311 y(The)g(results)f(of)g(this)f (section)i(are)f(essen)o(tially)g(a)f(corollary)g(of)-75 1361 y(the)21 b(follo)o(wing)d(imp)q(ossibilit)o(y)g(result)j(\(the)g (pro)q(of)f(of)g(whic)o(h)-75 1411 y(uses)c(sophisticated)g(top)q (ological)d(c)o(haracterization)j(of)e(asyn-)-75 1461 y(c)o(hronous)h(computabilit)o(y\):)-75 1541 y Fl(Lemma)h(2.1)f (\(Herlih)o(y)f(and)h(Sha)o(vit)f([HS99]\))20 b 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Fm(i)458 -4 y Fv(is)g Ft(i)p Fv(,)h(and)f(construct)i(the) f(fol-)-75 45 y(lo)o(wing)c(in\014nite)h(run)g(of)f Ft(A)p Fv(:)17 b(In)10 b(round)g(1,)g(the)h(single)e(pro)q(cess)j(in)-75 95 y Ft(G)-42 101 y Fn(1)-12 95 y Fv(is)g(activ)n(ated)g(and)f(it)h (runs)g(alone)f(un)o(til)g(it)h(returns)h(a)f(snap-)-75 145 y(shot)g(whic)o(h)g(includes)g(itself.)17 b(In)11 b(round)h(2,)g(the)g(t)o(w)o(o)f(pro)q(cesses)-75 195 y(in)17 b Ft(G)10 201 y Fn(2)45 195 y Fv(are)h(activ)n(ated)f(and)g(sc) o(heduled)i(in)d(suc)o(h)i(a)f(w)o(a)o(y)g(that)-75 245 y(the)i(t)o(w)o(o)f(of)g(them)f(return)j(a)e(snapshot)h(whic)o(h)f (includes)h(all)-75 294 y(the)c(three)h(pro)q(cesses)i(in)c Ft(G)367 300 y Fn(1)395 294 y Fs([)9 b Ft(G)465 300 y Fn(2)483 294 y Fv(.)21 b(This)14 b(construction)i(con-)-75 344 y(tin)o(ues)i(similarly)l(,)d(where)j(in)e(round)h Ft(i)p Fv(,)h(all)e(the)i(pro)q(cesses)h(in)-75 394 y Ft(G)-42 400 y Fm(i)-16 394 y Fv(are)12 b(activ)n(ated)h(and)f(sc)o 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y(rency)j(is)e(strictly)g(more)f(p)q(o)o(w)o(erful)h(than)g(\()p Ft(k)14 b Fv(+)f(1\)-b)q(ounded)-75 1093 y(concurrency)m(.)18 b(This)10 b(is)f(done)g(b)o(y)g(sho)o(wing)g(that)g(for)g(an)o(y)g Ft(k)k Fs(\025)e Fv(1,)-75 1143 y(there)k(exists)e(a)g(stronger)h(v)o (ersion)g(of)e(the)i(b)q(ounded-snapshot)-75 1192 y(problem)k(that)h (is)g(solv)n(able)f(assuming)f Ft(k)q Fv(-b)q(ounded)j(concur-)-75 1242 y(rency)m(,)k(but)d(is)g(not)h(solv)n(able)e(assuming)g(\()p Ft(k)15 b Fv(+)f(1\)-b)q(ounded)-75 1292 y(concurrency)m(.)-75 1402 y(The)e Ft(k)q Fu(-snapshot)g Fv(problem)e(is)h(de\014ned)i(as)e (follo)o(ws:)k(Eac)o(h)d(pro-)-75 1452 y(cess)f Ft(i)f Fv(outputs)g(a)f(set)h(of)f(pro)q(cess)i(id's)e(\(a)g(snapshot\),)i Ft(S)790 1458 y Fm(i)804 1452 y Fv(.)16 b(They)-75 1501 y(m)o(ust)d(ha)o(v)o(e)h(the)g(usual)g(snapshot)g(prop)q(erties:)20 b Ft(i)12 b Fs(2)f Ft(S)779 1507 y Fm(i)793 1501 y Fv(,)i(for)h(all)-75 1551 y Ft(S)-50 1557 y Fm(i)-20 1551 y Fv(and)h Ft(S)87 1557 y Fm(j)105 1551 y Fv(,)h(one)f(is)h(a)f(subset)i(of)e(the)h (other,)g(and)g(if)e Ft(i)i Fv(termi-)-75 1601 y(nates)f(b)q(efore)g Ft(j)h Fv(starts)f(then)g Ft(j)f Fs(62)e Ft(S)499 1607 y Fm(i)513 1601 y Fv(.)19 b(In)14 b(addition,)e(tak)o(e)i(an)o(y)-75 1651 y(\014nite)f(or)g(in\014nite)f(run)h(in)f(whic)o(h)g(all)g (participating)f(pro)q(cesses)-75 1701 y(terminate,)j(and)g(reorder)i (the)f(output)g(sets)g Ft(S)667 1707 y Fn(1)687 1701 y Ft(;)7 b(S)731 1707 y Fn(2)749 1701 y Ft(;)g(:::)13 b Fv(b)o(y)h(set)-75 1751 y(inclusion)g(to)g(get)h Ft(R)252 1757 y Fn(1)270 1751 y Ft(;)7 b(R)321 1757 y Fn(2)339 1751 y Ft(;)g(:::)p Fv(.)18 b(It)c(is)h(required)g(that)g(for)f(ev)o (ery)-75 1800 y(suc)o(h)d(sequence)h Ft(R)p Fv(,)d Fs(j)p Ft(R)280 1806 y Fn(1)298 1800 y Fs(j)i(\024)h Ft(k)f Fv(and)e(for)g(all)g Ft(i)p Fv(,)h Fs(j)p Ft(R)667 1806 y Fm(i)p Fn(+1)722 1800 y Fs(j)q(\000)q(j)p Ft(R)812 1806 y Fm(i)824 1800 y Fs(j)i(\024)f Ft(k)q Fv(.)-75 1850 y(That)17 b(is,)f(no)g(snapshot)h(con)o(tains)f(more)g(than)g Ft(k)h Fv(\\new")g(ele-)-75 1900 y(men)o(ts.)-75 1991 y Fl(Theorem)e(2.2)21 b Fu(The)d Ft(k)q Fu(-snapshot)h(pr)n(oblem)f(is) g(solvable)g(as-)-75 2041 y(suming)h Ft(k)q Fu(-b)n(ounde)n(d)g(c)n (oncurr)n(ency,)g(but)f(is)f(not)i(solvable)f(as-)-75 2090 y(suming)d Fv(\()p Ft(k)c Fv(+)e(1\))p Fu(-b)n(ounde)n(d)16 b(c)n(oncurr)n(ency.)-75 2181 y Fk(Pro)q(of:)i Fv(F)m(or)12 b(the)h Ft(k)q Fv(-b)q(ounded)g(concurrency)i(mo)q(del,)c(the)i(algo-) - -75 2231 y(rithm)f(is)h(simply)e(for)i(eac)o(h)h(pro)q(cess)h(to)e(tak) o(e)g(a)g(snapshot)h(and)-75 2281 y(return)i(it.)21 b(It)16 b(easy)f(to)g(see)h(that)f(in)g(an)o(y)f(run,)h(if)f(w)o(e)i(sort)f (the)-75 2331 y(snapshots)e(b)o(y)e(size)h(then)g(the)g(maxim)n(um)7 b(di\013erence)13 b(b)q(et)o(w)o(een)-75 2380 y(an)o(y)h(t)o(w)o(o)f (successiv)o(e)j(snapshots)f(is)f Ft(k)q Fv(.)-33 2430 y(Next,)30 b(w)o(e)c(sho)o(w)h(that)g(it)f(is)g(imp)q(ossible)f(to)i (solv)o(e)f Ft(k)q Fv(-)-75 2480 y(snapshot)19 b(assuming)d(\()p Ft(k)d Fv(+)g(1\)-b)q(ounded)18 b(concurrency)m(.)32 b(As-)-75 2530 y(sume)20 b(to)g(the)h(con)o(trary)f(that)h(there)g (exists)g(an)f(algorithm,)-75 2580 y(called)15 b Ft(A)p Fv(,)g(that)g(solv)o(es)g(the)h Ft(k)q Fv(-snapshot)f(problem.)20 b(W)m(e)15 b(con-)-75 2629 y(struct)f(a)e(\014nite)h(run)g(of)f Ft(A)g Fv(with)g Ft(k)c Fv(+)e(1)13 b(activ)o(e)f(pro)q(cesses:)20 b(The)-75 2679 y Ft(k)13 b Fv(+)f(1)17 b(pro)q(cesses)j(are)e(activ)n (ated)f(and)g(sc)o(heduled)i(in)e(suc)o(h)h(a)-75 2729 y(w)o(a)o(y)d(that)h(all)e(of)h(them)g(return)i(a)f(snapshot)g(whic)o (h)f(includes)1025 -4 y(all)c Ft(k)d Fv(+)g(1)k(activ)o(e)h(pro)q (cesses.)21 b(The)13 b(fact)g(that)f(suc)o(h)i(a)f(sc)o(hedul-)1025 45 y(ing)g(is)h(p)q(ossible)g(follo)o(ws)e(from)g(Lemma)f(2.2.)17 b(Clearly)m(,)c(in)g(this)1025 95 y(\014nite)i(run,)g Ft(S)1246 101 y Fn(1)1278 95 y Fv(=)e Ft(R)1355 101 y Fn(1)1388 95 y Fv(and)i Fs(j)p Ft(R)1514 101 y Fn(1)1531 95 y Fs(j)e Fv(=)g Ft(k)e Fv(+)f(1.)21 b(This)14 b(con)o(tradicts)1025 145 y(the)i(assumption)f(that)h Ft(A)h Fv(solv)o(es)f(the)h Ft(k)q Fv(-snapshot)f(problem.)p 1025 195 V 1066 294 a(A)j(similar)d(pro)q(of)i(sho)o(ws)h(that)g(a)f(\014nite)h(hierarc)o (h)o(y)g(exists)1025 344 y(within)13 b(eac)o(h)h Ft(n)p Fv(-\014nite)g(arriv)n(als)f(mo)q(del:)1025 427 y Fl(Theorem)h(2.3)21 b Fu(F)m(or)16 b Ft(k)e(<)g(n)p Fu(,)i(in)g(the)g Ft(n)p Fu(-arrivals)e(mo)n(del,)i(the)1025 477 y Ft(k)q Fu(-snapshot)d(pr)n (oblem)f(is)g(solvable)g(assuming)h Ft(k)q Fu(-b)n(ounde)n(d)h(c)n(on-) 1025 527 y(curr)n(ency,)i(but)g(is)g(not)h(solvable)f(assuming)h Fv(\()p Ft(k)11 b Fv(+)f(1\))p Fu(-b)n(ounde)n(d)1025 577 y(c)n(oncurr)n(ency.)1025 714 y Fw(3)67 b(Algorithms)75 b(for)f(Un)n(b)r(ounded)1126 789 y(Concurrency)1025 880 y Fv(The)19 b(results)h(of)e(the)i(previous)f(section)g(establish)g (that)g(un-)1025 929 y(b)q(ounded)10 b(concurrency)i(is)d(the)h(w)o (eak)o(est)g(in)f(an)h(in\014nite)f(hierar-)1025 979 y(c)o(h)o(y)h(of)f(computational)e(mo)q(dels.)15 b(In)10 b(this)g(section,)h(w)o(e)f(demon-)1025 1029 y(strate)i(tec)o(hniques)g (for)f(designing)f(algorithms)f(for)h(this)i(w)o(eak-)1025 1079 y(est)22 b(concurrency)h(mo)q(del.)39 b(The)21 b(\014rst)h (subsection)h(demon-)1025 1129 y(strates)17 b(some)e(simple)f(tec)o (hniques)j(and)f(sho)o(ws)g(that)f(renam-)1025 1179 y(ing,)k(snapshot,) i(\(and)e(hence)i(collect\))f(are)f(solv)n(able)g(in)g(the)1025 1228 y(un)o(b)q(ounded)13 b(concurrency)h(case.)k(Using)12 b(more)f(complex)g(con-)1025 1278 y(structions,)21 b(subsequen)o(t)g (sections)f(sho)o(w)f(that)h(these)g(prob-)1025 1328 y(lems)13 b(not)i(only)f(admit)f(solution)h(in)g(the)i(case)f(of)g(un)o (b)q(ounded)1025 1378 y(concurrency)m(,)f(but)f(indeed)h(ha)o(v)o(e)f (solutions)f(whic)o(h)h(are)h(adap-)1025 1428 y(tiv)o(e)f(to)h(the)h (concurrency)m(.)1025 1544 y Fo(3.1)55 b(Simple)32 b(algorithms)g(and)i (tec)n(hniques)1152 1602 y(for)19 b(un)n(b)r(ounded)g(concurrency)1025 1679 y Fv(The)12 b(results)g(of)f(this)h(subsection)g(establish)g(the)g (solv)n(abilit)o(y)d(of)1025 1728 y(basic)17 b(tasks)h(in)e(the)i(un)o (b)q(ounded)f(concurrency)j(mo)q(del,)15 b(and)1025 1778 y(in)o(tro)q(duce)c(simple)e(tec)o(hniques)j(that)e(will)f(b)q(e)i (applied)f(to)g(more)1025 1828 y(complex)g(constructions)j(in)f(the)g (adaptiv)o(e)f(algorithms)f(of)h(the)1025 1878 y(next)j(subsection.) 1025 1986 y Fl(3.1.1)48 b(Snapshot)13 b(for)i(un)o(b)q(ounded)d (concurrency)1025 2062 y Fv(There)k(are)f(sev)o(eral)g(snapshot)g (algorithms)d(for)j(the)g Ft(n)p Fv(-arriv)n(al)1025 2112 y(mo)q(del)j([AA)1228 2097 y Fn(+)1255 2112 y Fv(93)o(,)h(AR93)o (,)g(AST99].)34 b(These)20 b(algorithms)d(do)1025 2162 y(not)10 b(w)o(ork)f(for)h(un)o(b)q(ounded)h(concurrency)m(.)18 b(Belo)o(w,)10 b(w)o(e)h(presen)o(t)1025 2212 y(a)21 b(new)h(snapshot)g(algorithm)d(that)j(w)o(orks)f(in)g(the)h(case)h(of) 1025 2262 y(un)o(b)q(ounded)16 b(concurrency)h(\(and)f(hence,)g(of)f (course,)i(also)e(for)1025 2311 y(b)q(ounded)f(concurrency\).)1025 2394 y Fl(Theorem)g(3.1)21 b Fu(Ther)n(e)f(is)g(a)g(snapshot)i (algorithm)d(for)h(un-)1025 2444 y(b)n(ounde)n(d)c(c)n(oncurr)n(ency.) 1025 2527 y Fk(Pro)q(of:)h Fv(W)m(e)12 b(\014rst)h(consider)g(the)g Fu(one-shot)g Fv(problem,)e(in)g(whic)o(h)1025 2577 y(eac)o(h)f(pro)q (cess)i(is)d(assumed)h(to)g(in)o(v)o(ok)o(e)e(at)i(most)f(one)h(op)q (eration.)1025 2627 y(Later)i(w)o(e)g(explain)f(ho)o(w)g(our)h (solution)f(can)g(b)q(e)i(trivially)c(made)1025 2677 y Fu(long-live)n(d)p Fv(.)24 b(T)m(o)15 b(simplify)e(the)k(presen)o (tation,)f(w)o(e)h(consider)g(a)1025 2726 y(v)o(ersion)11 b(of)f(the)i(snapshot)f(problem)f(where)i(a)f(pro)q(cess)i(returns)p eop %%Page: 5 5 5 4 bop -75 -4 a Fv(a)14 b(list)f(of)g(activ)o(e)h(pro)q(cess)h(ids,)e (instead)h(of)f(returning)i(a)e(list)g(of)-75 45 y(v)n(alues.)20 b(\(Clearly)m(,)14 b(for)g(the)i(one-shot)f(v)o(ersion)g(a)f(pro)q (cess)j(can)-75 95 y(then)e(read)f(the)g(v)n(alue)f(asso)q(ciated)i 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