Csr2011 june17 12_00_morin

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Csr2011 june17 12_00_morin

  1. 1. Shared-Memory Systems and Charts Remi MORIN ´ Universite de la M´diterran´e ´ e e Laboratoire d’Informatique Fondamentale de MarseilleCSR 2011 June 2011
  2. 2. Peterson’s mutual exclusion protocolProcess 1 Process 2 repeat: repeat: f[1] ← true; f[2] ← true; Turn ← 2; Turn ← 1; wait (f[2] = false wait (f[1] = false or Turn = 1); or Turn = 2); Critical Section(1); Critical Section(2); f[1] ← false; f[2] ← false; 1
  3. 3. Executions as partial orders (pomsets) Process 1 Process 2 f[1]←true f[2]←true Turn←1 Turn←2 Turn=2Time C.S.(2) f[2]←false f[2]=false C.S.(1) 2
  4. 4. Foreword Model and semantics Expressive power MSO logic Specifications with automata Checking SMC specifications 2
  5. 5. A simple model for shared-memory systemsLet Σ be a fixed alphabet.A shared-memory system consists of • a set of registers R, a set of data D, • a set of processes P • for each action a ∈ Σ: a non-empty subset Loc(a) ⊆ P • an initial configuration ı ∈ Q • some final configurations F ⊆ Q. • and for each action a ∈ Σ: a set of rules ∆a • A configuration is a mapping q : R → D • A rule is a triple ρ = (ν, a, ν′ ) where ν, ν′ : R ⇀ D Guard Update 3
  6. 6. Sequential operational semanticsFor any two states q, q′ ∈ Q and any rule ρ = (ν, a, ν′ ) ∈ ∆a ,we denote by • aρ = a the action performed by ρ • Rρ = dom(ν) the subset of registers read by ρ • Wρ = dom(ν′ ) the subset of registers modified by ρ ρWe put q→q′ if • q|Rρ = ν (the rule is enabled in q) • q′ |Wρ = ν′ (the rule is applied in q′ ) • q′ (r) = q(r) for all r ∈ R Wρ (nothing else happens inbetween) 4
  7. 7. Special case [Zielonka, RAIRO, 1987]An asynchronous automaton is an SMS such that • P = R and • for all rules ρ ∈ ∆a , Rρ = Loc(a) = Wρ . 5
  8. 8. May-Occur-Concurrently relationLet ρ, ρ′ ∈ ∆ be two rules.We put ρ ρ′ if • Loc(aρ ) ∩ Loc(aρ′ ) = ∅, • Wρ ∩ (Rρ′ ∪ Wρ′ ) = ∅ and • Wρ′ ∩ (Rρ ∪ Wρ ) = ∅.Intuitively, two rules may occur concurrently if they correspond toactions occurring on disjoint sets of processes and if each rule doesnot modify the registers read or written by the other. 6
  9. 9. Partial-order semantics (1/2)Let t = (E, , η) be a labeled partial order, i.e. a partiallyordered multiset (for short: a pomset) over Σ.A run of t is a mapping ρ : E → ∆ such thatR0 : For all e ∈ E, aρ(e) = η(e) (rule action matches event action)R1 : For all e1 , e2 ∈ E with ρ(e1 ) ρ(e2 ), e1 e2 or e2 e1 (dependent rules cannot occur concurrently)R2 : For all e1 , e2 ∈ E with e1 — 2 , ρ(e1 ) ρ(e2 ) ≺e (waiting means rule dependency)where x— means: x ≺ y and x ≺ z ≺y y implies z = y. 7
  10. 10. Partial-order semantics (2/2)Let H be a downward-closed subset of events (a prefix of t).The configuration qρ,H reached after H with run ρ is such that  ν′ (r) if e = max{f ∈ H | r ∈ Wρ(e) } ρ(e) qρ,H (r) = qρ,H (r) = ı(r) if there is no such eventA run ρ of t is applicable if the rule ρ(e) is enabled in qρ,↓e{e}for all events e ∈ E.An applicable run of t = (E, , η) is accepting if qρ,E ∈ F.DefinitionThe language L(S) recognized by S collects all pomsets whichadmit some accepting run. 8
  11. 11. Foreword Model and semantics Expressive power MSO logic Specifications with automata Checking SMC specifications 8
  12. 12. Question!Let Σ = {p, c} and L be the set of all ladders. p c p c p c Does any SMS recognize this language? 9
  13. 13. MSO logicThe language of all ladders is MSO-definable by the conjunc-tion of the following sentences: ∀y : Pc (y) → ∃x.(Pp (x) ∧ x—≺y) ∀x, z : Pp (x) → ∃y.(Pc (y) ∧ x—≺y)) ∀x, y : (Pp (y) ∧ x y) → Pp (x) 10
  14. 14. Cut-bounded languagesThe (universal) cut-width of t = (E, , η) isCW(t) = max #{ (h, e) ∈ H x (E H) | h—≺e } p c H prefix of tDefinitionThe cut-bound B ∈ N ∪ {∞} of L is sup{ CW(t) | t ∈ L }L is cut-bounded if its cut-bound is ∞.ExampleThe language of all ladders is not cut-bounded. 11
  15. 15. First result MSO-definable languages ¢ Cut-bounded languages ¢Pomset languagesTheorem (Expressive power of shared-memory systems)A pomset language is recognized by some finite SMS iffit is MSO-definable and cut-bounded . 12
  16. 16. Deterministic rules and weak unambiguityA shared-memory system has deterministic rules if for allactions a ∈ Σ and all valuations ν ∈ V there exists at mostone rule (ν, a, ν′ ) ∈ ∆.A shared-memory system is weakly-unambiguous if eachpomset from L(S) admits a unique accepting run.Theorem (Expressive power equivalence)The language of any finite shared-memory system is thelanguage of a weakly-unambiguous finite shared-memorysystem with deterministic rules. 13
  17. 17. Unambiguous case MSO-definable languages ¢ Media-bounded languages ¢ Cut-bounded languages ¢Pomset languagesTheorem ≃ [M., IJFCS, 2010]A pomset language is recognized by some unambiguous finiteSMS iff it is MSO-definable and media-bounded . 14
  18. 18. Deterministic case MSO-definable languages ¢ Media-bounded languages ¢ Cut-bounded languages ¢Pomset languages Consistent and coherent languagesTheorem ≃ [M., CONCUR’08]A pomset language is recognized by some deterministic finiteSMS iff it is MSO-definable, media-bounded, coherent andconsistent . 15
  19. 19. Foreword Model and semantics Expressive power MSO logic Specifications with automata Checking SMC specifications 15
  20. 20. How to concatenate two pomsets? a a ?b x = a b a We have to distinguish between a’s 16
  21. 21. How to concatenate two pomsets?a a a ? a aa a b ; a a a b aa a We have to distinguish between a’s 17
  22. 22. Pomsets with gatesLet G be a finite and non-empty set of gates .We consider the extended alphabet Γ = Σ x 2G {∅}.We put (a, H) Γ (a′ , H′ ) if H ∩ H′ ≠ ∅ or a = a′ .Definition (Pomsets with gates)A shared-memory chart (an SMC) is a pomset t = (E, , η) overΓ such that we have either e1 e2 or e2 e1 for any twoevents e1 and e2 with η(e1 ) Γ η(e2 ).We denote by SMC the set of all SMCs. 18
  23. 23. Product of SMCs a, {x} a, {x} b, {x} x = a, {y} b, {x} a, {y}Definition (Product of pomsets with gates)Given two SMCs t1 = (E1 , 1 , η1 ) and t2 = (E2 , 2 , η2 ) theasynchronous product t1 · t2 is the pomset t = (E, , η) whereE = E1 ∪ E2 , η = η1 ∪ η2 , and is the transitive closure of 1 ∪ 2 ∪{(e1 , e2 ) ∈ E1 x E2 | η(e1 ) Γ η(e2 )} 19
  24. 24. Rational SMC languages a a, {x} a a, {x, y} a a a, {x} a, {x} b, {y} ; a a b a a, {x} aDefinition (Automata over pomsets with gates)An SMC specification is an automaton A = (Q, ı, →, F) where Qis a finite set of states, with initial state ı, → ⊆ Q x SMC x Qis a finite set of transitions labeled by SMCs, and F ⊆ Q is asubset of final states. 20
  25. 25. SMCs vs. Mazurkiewicz traces and MSCsAny Mazurkiewicz trace can be regarded as an SMC whereeach action is a gate and each event labeled by a is associatedwith the set of actions dependent with a.Similarly any message sequence chart can be regarded as anSMC where gates are processes and each event is associatedwith the (single) process where it occurs.Moreover these identifications preserve the product oftraces and MSCsIn that way, SMCs appear as a formal generalizationof both Mazurkiewicz traces and message sequencecharts. 21
  26. 26. Foreword Model and semantics Expressive power MSO logic Specifications with automata Checking SMC specifications 21
  27. 27. How to detect unbounded specifications?DefinitionLet t = (E, , η) be an SMC. The communication graph of t is thedirected graph CG(t) = (V, →) over the set V = e∈E π2 (η(e))of active gates in t such that g → g′ if there are e, e′ ∈ E forwhich g ∈ π2 (η(e)), g′ ∈ π2 (η(e′ )) and • either η(e) Γ η(e′ ) • or e— ′ . ≺e 22
  28. 28. Checking unboundednessTheoremThe pomset language LΣ (A) of an SMC specification A is t1 tncut-bounded iff for any loop q0 →...→qn = q0 , all connectedcomponents of the communication graph CG(t1 · ... · tn ) arestrongly connected.Consequently checking for cut-boundedness of a given SMC specification isdecidable. It is actually easy to show that this problem is co-NP-complete. 23
  29. 29. How to detect non-implementable specifications? We cannot decide whether an SMC specification describes an implementable language, since this question is already undecidable for Mazurkiewicz traces. Definition An SMC specification is loop-connected if for all loops t1 tn q0 →...→qn = q0 the communication graph of the SMC t1 · ... · tn is connected. Theorem A cut-bounded language is MSO-definable if and only if it is the language of a loop-connected SMC specification. 24
  30. 30. Conclusion• We have presented a characterization of the expressive power of shared-memory systems 1. in terms of logic definability and cut-boundedness 2. in terms of automata over pomsets with gates.• This model of concurrency and this algebraic framework generalize the theory of Mazurkiewicz traces and message sequence charts.• These results should be extended soon to systems with autoconcurrency .• A simpler notion of communication graph may be designed. 25
  31. 31. Questions?
  32. 32. Unambiguity determinismAn SMS is unambiguous if each pomset admits at most oneapplicable run.An SMS is deterministic if for each a ∈ Σ and each reachableconfiguration q, there exists at most one rule ρ ∈ ∆a such that ρq→q′ .Clearly: Any deterministic SMS is unambiguous. 26

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