A False History of True Concurrency

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1 A Fals History of Tru Concurrncy Javir Esparza Sofwar Rliability and Scurity Group Institut for Formal Mthods in Computr Scinc Univrsity of Stuttgart.

2 Th arly 6s. 2

3 Abstract Modls of Computation in th arly 6s Lambda calculus (Church 35) Turing machins (Turing 36) Finit automata (Kln 56, Moor 56, Maly 56, Scott and Rabin 59) Pushdown automata (Ottingr 6, Chomsky 62, Evy 63, Schutznbrgr 63). 3

4 Smantics: xcutions Stats: currnt configurations of th machin On or mor initial stats Possibly som distinguishd final stats Transitions: movs btwn configurations Lambda calculus (λx.xx)(λy.y) (λy.y)(λz.z) Turing machin q q 2 Finit automaton q a q 2 Pushdown automaton (q, XYYZ ) a (q 2, XYXYYZ ) Excutions: altrnating squncs of stats and transitions. 4

5 Physics and Computation Abstract machins ar implmntd as physical systms. 5

6 Physics and Computation Abstract machins ar implmntd as can simulat physical systms SIMULA projct (Nygaard and Dahl) startd in

7 Physics and Computation Abstract machins A plan (physical systm) ar implmntd as can simulat physical systms SIMULA projct (Nygaard and Dahl) startd in

8 Physics and Computation Abstract machins A plan (physical systm) ar implmntd as can simulat physical systms SIMULA projct (Nygaard and Dahl) startd in 962 can b simulatd by a plan simulator (abstract machin). 5

9 Physics and Computation Abstract machins A plan (physical systm) ar implmntd as can simulat physical systms SIMULA projct (Nygaard and Dahl) startd in 962 can b simulatd by a plan simulator (abstract machin) which can b implmntd in a vido consol (physical systm). 5

10 Physics and Computation Abstract machins A plan (physical systm) ar implmntd as can simulat physical systms SIMULA projct (Nygaard and Dahl) startd in 962 can b simulatd by a plan simulator (abstract machin) which can b implmntd in a vido consol (physical systm) which can b simulatd by a hardwar simulator (abstract machin). 5

11 Physics and Computation Abstract machins A plan (physical systm) ar implmntd as can simulat physical systms SIMULA projct (Nygaard and Dahl) startd in 962 can b simulatd by a plan simulator (abstract machin) which can b implmntd in a vido consol (physical systm) which can b simulatd by a hardwar simulator (abstract machin) which is implmntd in a PC (physical systm).... 5

12 Ptri s qustion C.A. Ptri points out a discrpancy btwn how Thortical Physics and Thortical Computr Scinc dscribd systms in 962: Thortical Physics dscribs systms as a collction of intracting particls (subsystms), without a notion of global clock or simultanity Thortical Computr Scinc dscribs systms as squntial virtual machins going through a tmporally ordrd squnc of global stats Ptri s qustion: Which kind of abstract machin should b usd to dscrib th physical implmntation of a Turing machin?. 6

13 Ptri Nts A graphical rprsntation of intracting finit automata: s r q a b b c d d s r q. 7

14 Ptri Nts A graphical rprsntation of intracting finit automata: s r q a b b c d d b b s r q. 7

15 Ptri Nts A graphical rprsntation of intracting finit automata: s r q a b b b c d d s r q. 7

16 Ptri Nts A graphical rprsntation of intracting finit automata: s r q a b b b c d d s r q. 7

17 Ptri Nts A graphical rprsntation of intracting finit automata: s r q a b b b c d d d s r q. 7

18 Ptri Nts A graphical rprsntation of intracting finit automata: s r q a b b b c d d d s r q. 7

19 Ptri Nts A graphical rprsntation of intracting finit automata: s r q a a b b b c c d d d s r q. 7

20 Ptri Nts A graphical rprsntation of intracting finit automata: s r q s r q a a b b b c c d d d s r q s r q. 7

21 Th intrlaving smantics of Ptri nts An xcution smantics Stat: marking (distribution of tokns) Transitions: M a M a Excutions: M a M M

22 s r q a b c d s r q. 9

23 s r q a b c d s r q s r q. 9

24 s r q a b c d s r q s r q. 9

25 s r q a b c d s r q c s r q. 9

26 s r q a b c d s r q c s r q. 9

27 s r q a b c d s r q c s r q b. 9

28 s r q a b c d s r q c s r q b. 9

29 s r q a b c d s r q c s r q b. 9

30 s r q s r q a b c d c b d s r q c b. 9

31 s r q s r q a b c d c b d s r q c b d. 9

32 s r q s r q a b c d c b d s r q c b d. 9

33 s r q s r q a b c d c b d s r q c b d. 9

34 s r q s r q a b c d c b d a s r q c b d. 9

35 s r q s r q a b c d c b d a s r q c b d a. 9

36 s r q s r q a b c d c b d a b s r q c b d a. 9

37 s r q s r q a b c d c b d a b s r q c b d a b. 9

38 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q.

39 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s r q.

40 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s r q.

41 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s r c r q.

42 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s r c r q.

43 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s s r c r b r q.

44 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s s r c r b r q.

45 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s s r c r b r q q.

46 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s s r c r b r q q.

47 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s s r c r b r r q q d q.

48 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s s r c r b r r q q d q.

49 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s s r c r b r r q q d q q.

50 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s s r c r b r r q q d q q.

51 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s s a s r c r b r r q q d q q.

52 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s s a s r c r b r r q q d q q.

53 Th tru concurrncy smantics of Ptri nts s r q a b c d s r q s s a s s r c r b r r b r q q d q q.

54 Intrlaving vs. tru concurrncy Th intrlaving thsis: Th total ordr assumption is a rasonabl abstraction, adquat for practical purposs, and lading to nic mathmatics Th tru concurrncy thsis: Th total ordr assumption dos not corrspond to physical rality and lads to awkward rprsntations of simpl phnomna.

55 Th standard xampl In intrlaving smantics, a systm composd of n indpndnt componnts a a n has n! diffrnt xcutions Th automaton accpting thm has 2 n stats In tru concurrncy smantics, it has only on nonsquntial xcution. 2

56 3 yars of concurrncy thory in on slid Intrlaving smantics Ptri nts/vctor addition systms (Hack, Kosaraju, Mayr,... 7s 8s) Procss algbras (Milnr 8) Tmporal logic (Pnuli 77) Modl chcking (Clark, Emrson, Quill, Sifakis 8) Tru concurrncy Axiomatic concurrncy thory (Bst, Frnandz, Ptri... 7s 8s) Trac thory (formal languags, Mazurkiwicz 77) Evnt structurs (domain thory, Winskl 8) Tru concurrncy smantics of procss algbras (Montanari, Winskl... 8s) Partial ordr modl chcking (E., Godfroid, Pld, Wolpr 9s) Tmporal logics for tru concurrncy (9s s). 3

57 Tmporal Logics for Tru Concurrncy. 4

58 LTL: a tmporal logic for squntial runs Syntax: ϕ ::= tru ϕ ϕ ψ a ϕ F ϕ G ϕ ϕ ϕ U ψ whr a blongs to a finit st Act of actions Formulas intrprtd on runs ovr Act : lmnts of Act ω Smantics: ρ = a ϕ if ρ = aρ and ρ = ϕ ρ = F ϕ if ρ = ϕ for som suffix ρ of ρ ρ = G ϕ if ρ = ϕ for all suffixs ρ of ρ ρ = ϕ U ψ if ρ = ψ for som suffix ρ of ρ and ρ = φ for all suffixs ρ btwn ρ and ρ. 5

59 Exampls Invariants: G ϕ G( a tru... a n tru) dadlock frdom Rspons, rcurrnc: G(ϕ F ψ) G( rqust tru F takn tru) G F activ tru vntual accss to a rsourc procss rmains activ Ractivity: G F ϕ G F ψ G F( rqust tru takn 2 tru) G F takn tru strong fairnss. 6

60 Modl chcking Fix a systm S with action alphabt Act W us LTL ovr Act to spcify proprtis of S S satisfis a formula ϕ, dnotd S = LTL ϕ, if all its xcutions satisfy ϕ Th modl chcking problm: givn S and ϕ, dcid if T = LTL ϕ. 7

61 Rsults on LTL Kamp s thorm: LTL has th sam xprssivity as th first-ordr thory of runs FO(Act) ::= R a (x) x y ϕ φ ψ x.ϕ Th satisfiability and modl-chcking problms ar PSPACE-complt Construct a Büchi automaton of siz 2 O( ϕ ) accpting th runs satisfying ϕ (Intrsct it with an automaton accpting all xcutions of S) Chck for mptinss. 8

62 LTrL: Intrprting LTL on nonsquntial runs Fix a distributd alphabt Act = (Act,..., Act n ) of actions a Act i Act j mans that a is a joint action of th i-th and th j-th agnts Dnot by NS(Act) th st of nonsquntial runs ovr Act Th lin of th i-th componnt only contains actions of Act i Joint actions synchroniz th lins of its agnts Exampl: Act = {a, b}, Act 2 = {a, d}, Act 3 = {c, d} 2 3 a c b d. 9

63 LTL now intrprtd on NS(Act) Sam smantics as LTL, but with a nw notion of suffix Prfixs of a nonsquntial run: All minimal placs blong to th prfix If a transition blongs to th prfix, so do its output placs Suffixs: complmnts of prfixs a b a b c d c d. 2

64 Smantics: ν = a ϕ if ν can b xtndd by an a-lablld vnt such that ν {} = ϕ ν = F ϕ if ν = ϕ for som suffix ν of ν ν = G ϕ if ν = ϕ for all suffixs ν of ν ν = ϕ U ψ if ν = ψ for som suffix ν of ρ and ν = φ for all suffixs ν btwn ν and ν. 2

65 Whr is th diffrnc? a tru b tru unsatisfiabl in LTL, satisfiabl in LTrL Exampl satisfis G F( a tru) as a formula of LTrL, but not as a formula of LTL s r q a b c d s r q Bttr spcification of rst stats. 22

66 Modl chcking Fix a systm S = (S,..., S n ) with distributd alphabt Act, W us LTL ovr Act to spcify proprtis of S S satisfis a formula ϕ if all its nonsquntial xcutions satisfy ϕ Th modl chcking problm: givn S and ϕ, dcid if T = LTrL ϕ. 23

67 Rsults on LTrL Exprssivly complt for th first ordr thory of nonsquntial runs FO(Act) ::= R a (x) x y ϕ φ ψ x.ϕ intrprtd on partial ordrs ovr Act that rspct th distribution Thiagarajan and Walukiwicz, LICS 97: LTrL + P a modalitis Dikrt and Gastin, CSL 99: LTrL + XA modalitis Dikrt and Gastin, ICALP Non-lmntary satisfiability and modl chcking problms (Walukiwicz ICALP 98) LTL allows to spcify 2 n -countrs with formulas of lngth O(n) LTrL allows to spcify Towr(2, n)-countrs with formulas of lngth O(n). 24

68 Local LTL: intrprting LTL on local stats Local stat of a componnt: a position in its tim lin Idntify a local stat (plac) with th prfix dtrmind by all its prdcssors A componnt always has complt information about its causal past Componnts xchang full information whn thy synchroniz Exampl: Act = {a, b}, Act 2 = {a, d}, Act 3 = {c, d} 2 3 a c b d. 25

69 Syntax: ϕ ::= tru ϕ ϕ ψ a i ϕ F i ϕ G i ϕ ϕ U i ψ whr a is an action and i n Smantics: a i ϕ mans ϕ holds at i s nxt local stat F i ϕ mans ϕ holds vntually at i s timlin G i ϕ mans ϕ holds always along i s timlin ϕ U i ψ mans ϕ holds until ψ holds along i s timlin Gts intrsting whn formulas us svral indics: F G 2 a tru. 26

70 Rsults on Local LTL PSPACE-complt satisfiability and modl chcking problms (Thiagarajan, LICS 94) Gnralization of th Büchi automaton construction Tchnical problm: to kp track of th latst gossip Exprsivnss still unclar Compltnss rsults for logics with similar flavours (Gastin, Mukund, Kumar MFCS 3) Difficult to spcify with. 27

71 Outlook Intrlaving smantics dfault smantics in practic Tru concurrncy brought in whn intrlaving fails Challng: automatic synthsis of distributd systms Intrlaving logics insnsitiv to distribution rquirmnts Mayb th killr application for tru concurrncy?. 28

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