Refinement-Robust Fairness

Transcription

1 Refinement-Robust Fairness Hagen Völzer Institut für Theoretische Informatik Universität zu Lübeck May 10,

2 Overview 1. Problem 2. Formalization 3. Solution 4. Remarks 1

3 Problem weak fairness wrt t: enabled(t) taken(t) strong fairness wrt t: enabled(t) taken(t) a 2 a 2 c 2 a c c b a 1 b a 1 b c 1 weak and strong fairness are not robust under refinement 2

4 Two refinements T-refinement: a a 1 a 2 P-refinement: p p 1 p 2 3

5 Strong fairness and P-refinement B b D A a c E e C d F B B b D A a c E e C C d F 4

6 Outline recall refinement-robust version of weak fairness (Progress) introduce refinement-robust version of strong fairness (Simple Fairness) present other interesting refinement-robust fairness notions notions form a hierarchy hierarchy is based on conflict structure (free choice, asymmetric choice etc.) 5

7 Overview 1. Problem 2. Formalization 3. Solution 4. Remarks 6

8 Preliminaries C A (safe) system: b c A a d E D B A non-sequential run: A a C b A D c B d E Recall: co-set (of conditions) 7

9 Preliminaries (contd.) p t *p p* *t t* X = x X x X = x X x 8

10 Refinement-Robustness Refinement: ϕ : Nets Nets (lifted: ϕ : Net systems Net systems) Fairness assumption: f : Σ subset of non-sequential runs of Σ Refinement-robustness: f is ϕ-robust if for all Σ, we have f(ϕ(σ)) = ϕ(f(σ)) 9

11 Refinement-Robustness Implication: Linear-time semantics with refinement-robust fairness is preserved under refinement: f, g are ϕ-robust f(σ 1 ) = g(σ 2 ) f(ϕ(σ 1 )) = g(ϕ(σ 2 )) In other words: Partial-order linear time semantics with refinementrobust fairness is a congruence for the algebra (Net systems; T-refinement, P-refinement). 10

12 Weak and strong fairness for non-sequential runs Recall: A non-sequential run represents a set of sequential runs (its interleavings or observations). b a c A d A a b A a B c B Definition: A non-sequential run is weakly fair (strongly fair ) if it has an interleaving that is weakly fair (strongly fair). 11

13 Weak and strong fairness Now we can formally prove: Theorem: Weak and strong fairness are not refinement-robust. 12

14 Overview 1. Problem 2. Formalization 3. Solution Refinement-robust versions of weak and strong fairness Hyperfairness 4. Remarks 13

15 of a co-set D of conditions Persistence and Coexistence // // // persistent coexistent neither nor D = t D D t persistence coexistence observer-independent refinement-robust 14

16 Progress wrt t = t is not persistent (i.e. there is no persistent co-set labelled with t) B b a A A a B b A a C c D C progress is known, e.g., as maximality of partial-order runs weak fairness progress 15

17 Free Fairness wrt t coexistent( t) taken(t) B b a A c C A a B b A a free fairness progress 16

18 Free Fairness wrt t (contd.) coexistent( t) taken(t) B b D d A a C c E e F B b D B A a d A a C c E C 17

19 Simple Fairness wrt t = Q t : persistent(q) coexistent( t \ Q) taken(t) b a c A d A a b A a B c B strong fairness simple fairness free fairness fair arcs is a special case of simple fairness 18

20 Relationships progress weak fairness free fairness simple fairness strong fairness 19

21 Refinement-Robust Fairness Theorem: Progress, free fairness, and simple fairness are refinement-robust. 20

22 Refinement-Robust Versions of Weak and Strong Fairness Theorem: Given Σ and transition t of Σ. 1. Obtain Σ by P-refining all p t. Then progress weak fairness (wrt t in Σ ) simple fairness strong fairness 2. Obtain Σ by T-refining all conflict transitions of t. Then progress weak fairness 21

23 Overview 1. Problem 2. Formalization 3. Solution Refinement-robust versions of weak and strong fairness Hyperfairness 4. Remarks 22

24 Hyperfairness wrt t = t taken(t) b a A B c d e A a b A a d B e d B hyperfairness strong fairness 23

25 Hyperfairness rules out conspiracies that are due to race-conditions other notions with the same goal -fairness [Best 89] = reachable(t) taken(t) hyperfairness [Attie, Francez, Grumberg 93] strongly tied to a particular process language hyperfairness [Lamport 2000] identical with -fairness 24

26 Relationships extended progress weak fairness free fairness simple fairness strong fairness hyperfairness -fairness wrt t [Best 89] = reachable(t) taken(t) 00-fairness 25

27 Overview 1. Problem 2. Formalization 3. Solution 4. Remarks hierarchy and conflict structure Aspects of fairness 26

28 Conflict structure wrt t (1/4) \\ t t is loop-conflict free: p, q t : p p {t} t is loop-conflict free (weak fairness wrt t progress wrt t) 27

29 Conflict structure wrt t (2/4) \\ // t t is conflict free: p, q t : p = {t} A resource of t cannot be taken away t is conflict free (free fairness wrt t progress wrt t) 28

30 Conflict structure wrt t (3/4) t t is free: p, q t : p = q A resource of t cannot be taken away unless t is enabled t is free (simple fairness wrt t free fairness wrt t) 29

31 Conflict structure wrt t (4/4) t t is simple: p, q t : p q q p // induces order on resources such that: If t has a token and all smaller tokens then that token cannot be taken away unless t is enabled t is simple (hyperfairness wrt t simple fairness wrt t) conflict-free free simple 30

32 progress Collapse loop conflict free weak fairness conflict free free fairness free simple fairness simple simple simple conflict free strong fairness hyperfairness (**) (*) 00-fairness (*) = finite net; -fairness assumed for all transitions; (**) = (*) + simple 31

33 Aspects of Fairness fairness wrt concurrency = transition is not delayed by concurrent events = progress fairness wrt choice = each outcome of a recurrent choice is recurrent = free fairness \ progress in simple fairness and hyperfairness, these aspects overlap ( confusion ) 32

34 Synchronization assumptions x: A // t B x: A t B x: y: A t B y: C // D y: C // D C D Progress Simple fairness Hyperfairness 33

35 An intuition 34

36 Conclusion Summary: refinement-robust fairness can be derived from partial-order semantics Additional remarks: the hierarchy separates computational power 35

37 progress, free fairness, and simple fairness can be defined on sequential runs Future work: fairness for unsafe and high-level nets fairness and composition 35