An Algorithm for Ensuring Fairness and Liveness in Non-deterministic Systems Based on Multiparty Interactions

An Algo i hm o Ensu ing Fai ness and Li eness in Non-de e minis ic Sys ems Based on Mul ipa y In e ac ions Da id Ruiz, Ra ael Co chuelo, Jos´eA.P´e ez, and Miguel To o Uni e sidad de Se illa, E. T. S. Ingenie os In o m´a icos, A . de la Reina Me cedes s/n, Se illa, E–41012, Spain [email p o ec ed], h p:// dg.lsi.us.es Abs ac . S ong ai ness is a no ion we can use o ensu e ha an ele- men ha is enabled infini ely o en in a non–de e minis ic p og amme, will e en ually be selec ed o execu ion so ha i can p og ess. Un o u- na ely, “e en ually” is oo weak o induce he in ui i e idea o li eliness and leads o anomalies ha a e no desi able, namely ai fini eness and conspi acies. In his pape , we ocus on non–de e minis ic p og ammes based on mul ipa y in e ac ions and we p esen a new c i e ia o selec - ing in e ac ions called s ong k– ai ness ha imp o es on o he p oposals in ha i add esses bo h anomalies simul aneously, and kmay be se a p io i o con ol i s goodness. We also show ou no ion is easible, and p esen an algo i hm o scheduling in e ac ions in a s ongly k– ai man- ne using a heo e ical amewo k o suppo he mul ipa y in e ac ion model. Ou algo i hm does no equi e o ans o m he sou ce code o he p ocesses ha compose he sys em; u he mo e, i can deal wi h bo h e mina ing and non– e mina ing p ocesses. 1 In oduc ion Fai ness is an impo an li eliness concep ha becomes essen ial when he ex- ecu ion o a p og amme is non–de e minis ic [8]. This may be a esul o he inhe en ly non–de e minis ic cons uc s ha he language we used o code i offe s, o a esul o he in e lea ing o a omic ac ions in a concu en and/o dis ibu ed en i onmen . In ui i ely, an execu ion o a p og amme is ai iff e e y elemen unde con- side a ion ha is enabled sufficien ly o en is execu ed sufficien ly o en, which p e en s undesi able execu ions in which an enabled elemen is neglec ed o - e e . The elemen s unde conside a ion may ange om al e na i es in a non– de e minis ic mul i–choice command o high–le el business ules, and combined wi h a p ecise defini ion o “sufficien ly o en” lead o a ich la ice o ai ness no ions ha do no collapse, i.e., a e no equi alen each o he [3]. This a icle was suppo ed by he Spanish In e minis e ial Commission on Science and Technology unde g an TIC2000-1106-C02-01. The e is no a p e ailing defini ion, bu many esea che s ag ee in ha so called s ong ai ness dese es special a en ion [8] because i may induce e - mina ion o e en ual esponse o an e en . Technically, an execu ion is said o be s ongly ai iff e e y elemen ha is enabled indefini ely o en is execu ed infini ely o en, i.e., i p e en s elemen s ha a e enabled infini ely o en, bu no necessa ily pe manen ly, om being neglec ed o e e . In his pape , we ocus on concu en and/o dis ibu ed p og ammes ha use he mul ipa y in e ac ion model as he sole means o p ocess1synch oni- sa ion and communica ion. This in e ac ion model is used in se e al academic p og amming languages like Sc ip s [8], Raddle [7] o IP [9] and in comme cial p og amming en i onmen s like Mic oso .NET O ches a ion [4] oo. In his pape , we ocus on IP because i is in ended o ha e a dual ole: on he one hand, i is in ended o be a dis ibu ed sys em specifica ion language equipped wi h sound seman ics ha u n i in o a language amenable o o mal easoning, a a he impo an p ope y; on he o he hand, i is in ended o be an assemble language suppo ing mo e sophis ica ed high–le el specifica ion languages such as LOTOS, ESTELLE, SDL [10] o CAL [5]. Nex , we epo on hose issues, p esen some app oaches o add ess hem and gi e he eade a bi d’s–eye iew o he es o he pape . 1.1 Known Issues Figu e 1 shows a solu ion o he well–known dining philosophe s p oblem in IP. This classic mul i-p ocess synch onisa ion p oblem consis s o fi e philosophe s si ing a a able who do no hing bu hink and ea . The e is a single o k be ween each philosophe , and hey need o pick bo h o ks up in o de o ea . In addi ion, each philosophe should be able o ea as much as he es , i.e., he whole p ocess should be ai . This p oblem is he co e o a la ge class o p oblems whe e a p ocess needs o acqui e a se o esou ces in mu ual exclusion. Ge iand Relideno e a numbe o h ee–pa y in e ac ions ha allow each philosophe Pi o ge i s co esponding o ks Fiand Fi+1 mod Nin mu ual exclu- sion wi h i s neighbou s (i=1,2,...,N). Fo an in e ac ion o become enabled, he se o p ocesses ha may e en ually eady i , i.e., may e en ually be willing o pa icipa e in he join ac ion i ep esen s, need o be eadying i simul ane- ously. The only way o ensu e ha e e y philosophe ha is hung y will e en ually ea is by in oducing a no ion o ai ness in he implemen a ion o he language. Howe e , s ong ai ness is no p ac ical enough because o he ollowing inhe en p oblems: Fai Fini eness: E e y fini e execu ion is s ongly ai by defini ion. Figu e 2.a shows a simple execu ion ace o an ins an ia ion o he p eceding p o- g amme in which N= 5. The no a ion p.χ means ha p ocess p eadies he 1The e m p ocess e e s o any au onomous, single– h eaded compu ing a e ac . I may be a p ocess in an ope a ing sys em, a h ead, o e en a ha dwa e de ice. ge 2 el 2 ge 3 el 3 ge 4 el 4 ge 5 el 5 ge 1 el 1 P 1 P 2 P 3 P 4 P 5 F 1 F 5 F 4 F 3 F 2 N=5 S::[N i=1FiN i=1Ti] Pi:: *[Ge i[] →ea ; Reli[]; hink] Fi:: *[ Ge i[] →Reli[] [] Ge (i+1 mod N)[] →Rel(i+1 mod N)[] ] (a) (b) Fig. 1. A solu ion o he dining philosophe s p oblem in IP. se o in e ac ions χ. No ice ha o any fini e n, his execu ion is echni- cally s ongly ai , despi e Ge 2being enabled n imes bu ne e selec ed. I n= 10, his execu ion may be conside ed ai om an in ui i e poin o iew, bu i n= 1000 i is no so in ui i e o conside his beha iou ai . Conspi acies: I does no ake in o accoun conspi acies in which an in e ac- ion ne e ge s enabled because o an un o una e in e lea ing o independen a omic ac ions. Fo ins ance, he execu ion shown in Figu e 2.b is s ongly ai o any n≥0, bu no ice ha due o an un o una e in e lea ing, in e - ac ion Ge 2is ne e eadied by all o i s pa icipan s a he same ime and hus ne e ge s enabled. The abo e p oblems show ha s ong ai ness (and o he no ions ha ely on infini eness and e en uali y) ails o cap u e he in ui i e idea o inducing li eliness. Al hough i may be he only way o p oo e mina ion o e en ual esponse o an e en du ing an infini e execu ion, “e en ual” is usually oo weak o p ac ical pu poses because any p ac ical unning p og amme mus necessa ily s op o be s opped a day. 1.2 Rela ed Wo k These issues mo i a ed se e al au ho s o esea ch s onge no ions. He e we ocus on wo app oaches called s ong fini a y ai ness [1] and hype ai ness [2]. An execu ion is s ongly fini a ily ai iff he e exis s a na u al numbe k (no known a p io i) such ha e e y in e ac ion ha is enabled infini ely o en is execu ed a leas e e y ks eps. Al hough his no ion in oduces addi ional li eliness because i bounds he numbe o imes an enabled in e ac ion may be neglec ed, i has se e al d awbacks: (i) kis no known a p io i, and hus i canno be used o fine- une a po en ial schedule depending on he na u e o he sys em i is scheduling; (ii) i does no p e en un ai fini eness; (iii) i does no p e en conspi acies; and, o he bes o ou knowledge, (i ) no gene al algo i hm implemen ing i has been p oduced. (The au ho s do only p esen a P1.{Ge 1},P 2.{Ge 2}, (P5.{Ge 5, Ge 1},P 2.{Ge 2, Ge 3},P 1.{Ge 1, Ge 2}, Ge 1[], P1.{Rel1},P 1.{Rel1,Rel 2},P 5.{Rel5,Rel 1},Rel 1[],P 1.{Ge 1})n (a) P1.{Ge 1},P 2.{Ge 2}P3.{Ge 3}, (P5.{Ge 5, Ge 1},P 3.{Ge 3, Ge 4}, P1.{Ge 1, Ge 2}, Ge 1[],P 2.{Ge 2, Ge 3}, Ge 3[],P 1.{Rel1}, P1.{Rel1,Rel2},P 5.{Rel5,Rel 1},Rel 1[],P 3.{Rel3}, P2.{Rel2,Rel 3},P 3.{Rel3,Rel 4},Rel 3[],P 1.{Ge 1},P 3.{Ge 3})n (b) Fig. 2. S ong ai ness anomalies. ans o ma ional app oach sui able o be used in he con ex o B¨uchi au oma a [12,11].) Hype ai ness also dese es a en ion because i alle ia es he second p ob- lem. Technically, an execu ion is hype ai iff i is fini e o e e y in e ac ion ha may ge enabled infini ely o en, becomes enabled infini ely o en. I is impo - an o no ice ha his defini ion di e ges om classical no ions in ha he la e imply e en ual execu ion o an in e ac ion i i ge s enabled sufficien ly o en, whe eas hype ai ness does only imply e en ual enablemen . Subsequen execu ion is unde he c i e ion o an implied unde lying classical ai ness no- ion. Thus, his no ion p e en s conspi acies due o un o una e in e lea ing o independen a omic ac ions bu combined wi h fini a y o s ong ai ness suffe s om ai fini eness. To he bes o ou knowledge, no gene al algo i hm able o implemen hype ai ness has been p oduced. Howe e , he au ho s p esen ed a ans o ma ional app oach by means o which we can ans o m an IP p o- g amme in o an equi alen s ongly hype ai o m, which implies modifica ion o he sou ce code and c ea ion o explici schedule s o each p og amme. This may be accep able in he con ex o esea ch languages, bu i is no p ac ical enough in eal–wo ld languages in which p ocesses o componen s a e a ailable only in bina y o m and need o be scheduled wi hou any knowledge o hei in e nal de ails. Fu he mo e, i does no add ess he issue o ai fini eness. 1.3 O e iew In his pape , we p esen a new no ion called s ong k– ai ness ha sol es ai fini eness and conspi acies. In ui i ely, an execu ion is s ongly k– ai iff no in- e ac ion is execu ed mo e han k imes unless he se o in e ac ions ha sha e p ocesses wi h i is s able, i.e., he p ocesses ha pa icipa e in hem a e wai ing o in e ac ion o finished, and i is he oldes in he g oup, i.e., he one ha has no been execu ed o a longe pe iod o ime. We p esen a heo e ical in e ac ion amewo k o o malize he mul ipa y in e ac ion model. Fu he mo e, we p esen an algo i hm ha uses his ame- wo k o scheduling in e ac ions in a s ongly k– ai manne , and i is no de- penden on he in e nal de ails o he p ocesses ha compose he sys em, i.e., i is no a ans o ma ional app oach. The succeeding sec ions a e o ganised as ollows: Sec ion 2 p esen s ou he- o e ical in e ac ion amewo k; Sec ion 3 p esen s a o mal defini ion o s ong k– ai ness; Sec ion 4 desc ibes a schedule we can use o implemen his no ion; finally, Sec ion 5 epo s on ou main conclusions. 2 A Theo e ical F amewo k o Suppo he Mul ipa y In e ac ion Model Nex , we p esen a o mal defini ion o ou abs ac in e ac ion amewo k. Defini ion 1 (S a ic Cha ac e isa ion o a Sys em) A sys em Σisa2– uple (PΣ,I Σ)in which PΣ=∅is a fini e se o au onomous p ocesses and IΣ= ∅is a fini e se o in e ac ions. We deno e he se o p ocesses ha may e en ually eady in e ac ion xas P(x)(pa icipan s o in e ac ion x). A configu a ion is a ma hema ical objec ha may be iewed as a snapsho o a sys em a un ime. We deno e hem as C, C,C 1,C 2... An e en is a happening ha induces a sys em o ansi om a configu a ion o ano he . In ou model, we ake in o accoun he ollowing kinds o e en s: p.ι, which indica es ha p ocess pexecu es an a omic ac ion ha does only in ol e i s local da a; p.χ, which indica es ha p ocess pis eadying he in e ac ions in se χ(no ice ha when χ=∅, p ocess pa i es a a fixed poin ha we may in e p e as i s e mina ion); and x, which indica es ha in e ac ion xhas been selec ed and he p ocesses pa icipa ing in i can execu e he co esponding join ac ion a omically. Defini ion 2 (Dynamic Cha ac e isa ion o a Sys em) An execu ion o a sys em Σis a 3– uple (C0,α,β)in which C0deno es i s ini ial configu a ion, α=[C1,C 2,C 3,...]is a maximal (fini e o infini e) sequence o configu a ions h ough which i p oceeds, and β=[e1,e 2,e 3,...]is a maximal (fini e o infini e) sequence o e en s esponsible o he ansi ion be ween e e y wo consecu i e configu a ions. Ob iously |α|=|β|. Finally, le λ=(C0,α,β)be an execu ion o sys em Σ.Wecallαi s configu a ion ace and deno e i as λα, and βi s e en ace and deno e i as λβ. We deno e he ule ha cap u es he unde lying seman ics ha con ol he ansi ion be ween configu a ions as L. Fo ins ance, CeLCin- dica es ha he sys em may ansi om configu a ion C o configu a ion C on occu ence o e en e. Thus, gi en an execu ion λ=(C0,[C1,C 2,C 3,...], [e1,e 2,e 3,...]), we usually w i e i as2:C0e1LC1e2LC2e3L··· 2No ice ha he exac o mula ion o Ldepends comple ely on he language in which he sys em unde conside a ion was w i en. Defini ion 3 (S a ic Cha ac e isa ion o a P ocess) P ocess pis wai ing a in e ac ion se Υa he i– h configu a ion in execu ion λiff i has a i ed a a poin in i s execu ion in which execu ing any x∈Υis one o i s possible con inua ions. P ocess pis finished a he i– h configu a ion in execu ion λiff i can nei he execu e any local compu a ion no any in e ac ion. Wai ing(λ, p, Υ, i)⇐⇒ ∃ k∈[1..i]·β(k)=p.χ ∧Υ⊆χ∧j∈[k+1..i]·β(j)=x∧x∈χ Finished(λ, p, i)⇐⇒ ∃ k∈[1..i]·β(k)=p.∅ (1) Defini ion 4 (S a ic Cha ac e isa ion o an In e ac ion) In e ac ion x is enabled a he i– h configu a ion in execu ion λiff all o he p ocesses in P(x)a e eadying xa ha configu a ion. In e ac ion xis s able a he i– h con- figu a ion in execu ion λiff i is ei he enabled o disabled a ha configu a ion. Enabled(λ, x, i)⇐⇒ ∀ p∈P(x)·Wai ing(λ, p, {x},i) S able(λ, x, i)⇐⇒ ∀ p∈P(x)·∃Υ⊆IΣ·Wai ing(λ, p, Υ, i) (2) Defini ion 5 (Dynamic Cha ac e isa ion o an In e ac ion) The se o in e ac ions linked o in e ac ion xa he i– h configu a ion in execu ion λis he se o in e ac ions such ha he e exis s a p ocess ha is eadying xand any o hose in e ac ions simul aneously. We define he execu ion se o in e ac ion xa he i– h configu a ion in execu ion λas he se o indices up o i ha deno e he configu a ions a which in e ac ion xhas been execu ed. Linked(λ, x, i)= {y∈IΣ·∃p∈PΣ·Wai ing(λ, p, {x, y},i)} ExeSe (λ, x, i)= {k≤i·β(k)=x} (3) 3 S ong k–Fai ness In ui i ely, an execu ion is s ongly k– ai iff no in e ac ion is execu ed mo e han k imes unless all o he in e ac ions ha a e linked o i when i is execu ed a e s able and i is he oldes amongs hem. Defini ion 6 (S ongly k–Fai Execu ion) Le λ=(C0,α,β)be an execu- ion o a sys em, and ka non–null na u al numbe . λis s ongly k– ai iff p ed- ica e SKF(λ, k)holds. SKF(λ, k)⇐⇒ ∀ x∈IΣ,i∈ExeSe (λ, x, ∞)·Enabled(λ, x, i)∧ (LS able(λ, x, i)∧LOldes (λ, x, i)∨ ¬LS able(λ, x, i)∧∆(λ, x, i)≤k) (4) This defini ion elies on a numbe o auxilia y p edica es and unc ions we ha e in oduced o he sake o simplici y. LS able is a p edica e we use o de e mine i an in e ac ion and hose ha a e linked o i a e s able a a gi en configu a ion in an execu ion. I s o mal defini ion ollows: LS able(λ, x, i)⇐⇒ ∀ y∈Linked(λ, x, i)∪{x}·S able(λ, y, i)(5) LOldes is a p edica e we use o de e mine i an in e ac ion is olde han any o he in e ac ions o which i is linked o , in he wo s case, is he same age. I s defini ion ollows: LOldes (λ, x, i)⇐⇒ ∀ y∈Linked(λ, x, i)·Age(λ, x, i)≥Age(λ, y, i) (6) The age o an in e ac ion is he numbe o configu a ions ha ha e elapsed since i was execu ed o he las ime, o ∞i i has ne e been execu ed so a . Age(λ, x, i)=i−max ExeSe (λ, x, i) i ExeSe (λ, x, i)=∅ ∞o he wise (7) ∆is a unc ion ha maps an e en ace, an in e ac ion and an index in o he numbe o imes i has execu ed in he p esence o a non–emp y se o linked in e ac ions ha was no s able. I s defini ion ollows: ∆(λ, x, i)=  φ≤k<i (λβ(k)=x∧Linked(λ, x, k)=∅∧¬LS able(λ, x, k)) (8) whe e deno es he coun e quan ifie ( a∈A P(a)|{a∈A·P(a)}|), and φis defined as ollows (no ice ha we deno e he maximum o an emp y se as ⊥): φji j= max{k∈ExeSe (λ, x, i)·LS able(λ, x, k)}∧j=⊥ 1 o he wise (9) 4 A S ongly k–Fai Schedule Ou algo i hm is based on p e ious p oposals which wan o esol e ano he p oblems in he con ex o he mul ipa y in e ac ions [6,14,13]. The idea behind ou algo i hm o scheduling in e ac ions in a s ongly k– ai manne consis s o a anging he se o in e ac ions in o a queue τso ha he close an in e ac ion is o he ea , he less ime has elapsed since i was execu ed o he las ime. Fu he mo e, each in e ac ion has an associa ed coun e δwe use o coun how many imes i has been semi–enabled in p esence o linked in e ac ions. To know he s a e in which a p ocess o an in e ac ion is, we use a map ϕ om he se o in e ac ions in o he se o p ocesses ha a e eadying i . We upda e i each ime he selec ion module de ec s a ansi ion p.χLoccu s, o an in e ac ion is execu ed. Ou algo i hm selec s o execu ion in e ac ion xas long as i is he fi s enabled one in queue τ, and he se o in e ac ions linked o i is s able. We desc ibe he ope a ional seman ics o ou s ong k– ai ness algo i hm using ansi ion ule SKF on configu a ions D,D,D 1,.... These configu- a ions a e composed o he configu a ion Co he p og amme and he da a s uc u es we need o selec in e ac ions. Nex , we define heses da a s uc u es and he unc ions ha allows us o upda e i . The ex ended configu a ions on which ou algo i hm wo ks a e o he o m (τ, ϕ, δ, ϑ), whe e τis an in e ac ion queue, ϕis a eadiness map, δis a semi– enablemen map, and ϑdeno es he se o p ocesses ha a e finished. Defini ion 7 (Da a S uc u es) ϕdeno es a map om in e ac ions in o se s o p ocesses. ϕ(x)deno es he se o p ocesses ha a e eadying x.ϑ⊆PΣ deno es he se o p ocesses ha a e finished, i.e., can nei he execu e local com- pu a ions no eady any in e ac ion. δis a map om he se o in e ac ions in o he se o na u al numbe s. δ(x)deno es he numbe o imes any in e ac ion linked o xhas been execu ed while xwas semi–enabled. The highe δ(x)is, he highe he p obabili y o conspi acy is. τdeno es a queue in which he se o in- e ac ions has been a anged so ha he close hey a e o he ea , he less ime has elapsed since hey we e execu ed o he las ime. As usual, we conside a queue o in e ac ions is a map om a subse o na u al numbe s in o he se o in e ac ions. The ini ial ex ended configu a ion o ou algo i hm is o he o m (τ0,ϕ 0, δ0,ϑ 0). I does no ma e he o de in which in e ac ions a e ini ially a anged in o τ,bu ϕ0mus sa is y ha ∀x∈dom ϕ0·ϕ0(x)=∅,δ0mus sa is y ha ∀x∈dom δ0·δ0(x)=0,andϑ0=∅. Defini ion 8 (Func ions) When p ocess p eadies a se o in e ac ions χ,we use unc ion AddOffe (ϕ, p, χ) o upda e map ϕ, and when in e ac ion xis exe- cu ed, we use unc ion Remo eOffe (ϕ, x). By defini ion, p ocess pfinishes when i eadies an emp y se o in e ac ions, and unc ion AddFinished(ϑ, p)upda es map ϑ. When in e ac ion xis selec ed o execu ion, we use unc ion O de o mo e i o he ea o τ, no necessa ily o he las posi ion. When in e ac ion x is selec ed o execu ion, we use unc ion Upda e(ϕ, δ, x) o c ea e a new semi– enablemen . P edica e S abilised(Υ, ϕ, ϑ)holds iff all o he p ocesses ha may e en ually eady an in e ac ion in Υhas eadied i o a e finished. EnblDisj(τ,ϕ) deno es he se o in e ac ions ha a e enabled and i does no exis s any p e- ceding in e ac ion in τ ha is enabled and linked o hem. AddOffe (ϕ, p, χ)={x→ ϕ(x)·x∈dom ϕ∧x∈ χ}∪ {x→ ϕ(x)∪{p}·x∈dom ϕ∧x∈χ} Remo eOffe (ϕ, x)={x→ ϕ(x) P(x)·x∈dom ϕ} AddFinished(ϑ, p)= ϑ∪{p}i χ=∅ ϑi χ=∅ O de (τ, δ)=τ⇔dom τ= dom τ∧ an τ= an τ∧ ∀x1,x 2∈ an τ·(τ−1(x1)≤τ−1(x2)⇒δ(x1)≥δ(x2)) Upda e(ϕ, δ, x)=δ⊗{x→ 0}⊗{y→ δ(y)+1·y∈S {x}∧¬S abilised(S,ϕ,ϑ)} S abilised(Υ, ϕ, ϑ)⇔∀x∈Υ·∀p∈P(x)·p∈ an ϕ∨p∈ϑ EnblDisj(τ, ϕ)={x∈dom ϕ·P(x)=ϕ(x)∧y∈S·P(y)=ϕ(y)∧τ−1(y)<τ −1(x)} (10) whe e S{z∈dom ϕ·ϕ(z)∩ϕ(x)=∅}. Ou algo i hm is o mally defined by means o he in e ence ules p esen ed in Figu e 3. Rule 11 is s aigh o wa d because i desc ibes how he da a s uc u es a e upda ed each ime p ocess p eadies a se o in e ac ions χ. Rule 12 desc ibes which in e ac ion mus be selec ed so ha he execu ion is s ongly k– ai . The an eceden is complex, bu he easoning behind i is qui e simple. Assume xis an enabled in e ac ion and he e is no a conflic ing enabled in e ac ion be o e xin queue τ, i.e., x∈EnblDisj(τ,ϕ); in his con ex xis selec ed o execu ion iff any o he ollowing h ee condi ions hold: Cp.χLC∧ ϕ= AddOffe (ϕ, p, χ)∧ϑ= AddFinished(ϑ, p) (C, τ, ϕ, δ, ϑ)p.χSKF (C,τ,ϕ δ, ϑ)(11) x∈EnblDisj(τ, ϕ)∧S={z∈dom ϕ·(ϕ(z)∩ϕ(x)=∅)}∧ τ= O de (τ, δ)∧ϕ= Remo eOffe (ϕ, x)∧δ= Upda e(ϕ, δ, x)∧ (S={x}∨S abilised(S,ϕ,ϑ)∨(S ={x}∧ max y∈S {x}δ(y)<k)) (C, τ, ϕ, δ, ϑ)xSKF (C,τ,ϕ ,δ ,ϑ)∧CxLC(12) Fig. 3. Algo i hm o s ongly k– ai schedule . 1. I is no conflic ing wi h o he in e ac ions (S={x}). 2. The in e ac ions which i is conflic ing wi h a e s abilised, i.e., all o hei pa icipan s a e eadying i o finished (S abilised(S,ϕ,ϑ)). 3. The semi–enablemen coun e associa ed wi h each conflic ing in e ac ion is less han k(S ={x}∧ max y∈S {x}δ(y)<k). 5 Conclusions and Fu u e Wo k In his pape , we ha e p esen ed s ong k– ai ness in he con ex o concu en and/o dis ibu ed p og ammes in which mul ipa y in e ac ion models a e he sole mean o p ocess synch onisa ion and communica ion. An impo an con- ibu ion is he concep o semi–enablemen ha we use o o ecas conspi acies and sol e hem. kcan hus be iewed as a semi–enablemen h eshold ha cha - ac e ises he goodness o ou no ion because i makes ou selec ion c i e ion mo e o less demanding. I kis minimum, he pace a which conflic ing in e ac ions a e execu ed depends hen on he pa icipan ha spends mo e ime a doing local compu a ions because, in his case, no in e ac ion can be selec ed o exe- cu ion unless he se o po en ially conflic ing in e ac ions is consolida ed. I k is e y dis an om i s minimum, he algo i hm in oduces li le delay because in e ac ions may be scheduled as soon as hey a e enabled. The exac choice o kdepends on he ea u es o he p og amme unde conside a ion and can only be uned by means o expe imen a ion.