The Concurrency Hierarchy and Algorithms for Unbounded Concurrency Eli Gafni Michael Merritt Gadi Taubenfeld

Motivation & Overview  Exploring the consequences of the a priori bound assumption on processes and concurrency  Extends “Computing with infinitely many processes” [MT2000]  Wait-free algorithms using atomic registers  Part 1: Computability: the Concurrency Hierarchy  Part 2: Algorithms:  Understanding one-shot/long-lived relationships  Infinity+Symmetry  Adaptivity  General algorithmic techniques for unbounded concurrency

Concurrency: Simultaneously Active Processes  k -bounded – single bound k over all runs  bounded – in each run an upper bound exists  unbounded – finite at any point, no upper bound  infinite – within a single state (not considered)

2 -concurrency bounded concurrency unbounded concurrency 1 -concurrency Part 1: The Concurrency Hierarchy …

Reminder: Snapshot algorithms e r p v a yd t s e returns {a,e,p,r,v,x} s returns {a,d,e,p,r,s,t,v,x,y} x returns {x} Every process sees itself The snapshots are nested x

Lemma--Any snapshot algorithm has an all-inclusive run ( n is known). p w u q t x z e j a Adaptively rename to 2k-1 names 2k-1 2n … …then there is a 2n-2 renaming algorithm, contradicting [HS99] Suppose not... 2k’-1 Adaptively rename to 2k’-1 names k+k’=n

Corollary--Any snapshot algorithm for unbounded concurrency has the following run: i new processes in the i’ th set

For bounded concurrency, the concurrency bound c limits the maximum difference between successive snapshots: because at most c new processes in each set

The Bounded Snapshot Problem: in complete runs, there is an upper bound on the difference between successive snapshots Impossible in unbounded concurrencySolvable for bounded concurrency?

The Concurrency Hierarchy k -concurrency ( k+1 )-concurrency bounded concurrency unbounded concurrency … bounded snapshot … k-snapshot

Part 2: Algorithms for Unbounded Concurrency unbounded concurrency adaptive one-shot 2k-1 renaming adaptive one-shot snapshot adaptive one-shot renaming long-lived snapshot one-shot snapshot adaptive long-lived renaming adaptive long-lived collect adaptive long-lived snapshot [AST99] [AF99, AF2000] [AST99]

One-shot Snapshot for Unbounded Concurrency... Use diagonalization to make long-lived.... Take a second snapshot (which must include me):...if the road goes on forever, someone paved it after driving past my house....

Adaptive One-shot Renaming [MA95, AF98] Infinite array of splitters: May not terminate...

Adaptive One-shot Renaming for Unbounded Concurrency via “Interleaving” Infinite array of odd-numbered splitters: As soon as p i fails to acquire any name j > 2i  p i takes name 2i p7p7 14

Adaptive One-shot Snapshot for Unbounded Concurrency Adaptive renaming One-shot snapshot

Adaptive, Optimal One-shot Renaming 2k-1 one-shot renaming [BD89] collect for Unbounded Concurrency One-shot snapshot

Summary 1-concurrency 2-concurrency bounded concurrency unbounded concurrency … … 2-snapshot bounded snapshot 1-snapshot a. collect a. snapshot a. 1-shot 2k-1 renaming a. renaming

Conclusions and Open Problems  Concurrency hierarchy  Algorithms for unbounded concurrency (work for unknown number/concurrency)  General techniques  one-shot + infinite-arrival  long-lived (diagonalization)  interleaving, help first, doorways  Are there natural problems outside unbounded concurrency?  Efficiency (space and time)  Separating problems that are “tasks”

One-shot Renaming “black-box” Transformation A1A1 A2A2 A3A3 ApAp … … k processes g(k) names At most k algorithms entered Eran Yahav