Operational Semantics of UML 2.0 Interactions

Operational Semantics of UML2.0InteractionsMar´ıa Victoria Cengarle1and Alexander Knapp21Technische Universit¨a t M¨u nchencengarle@in.tum.de2Ludwig-Maximilians-Universit¨a t M¨u nchenknapp@pst.ifi.lmu.deAbstract.An operational semantics for UML2.0Interactions is defined thatforfinite traces is compliant with the trace-based denotational semantics of aprevious work.To this end,the notion of interactions in the classical sense is used.That is,the operational semantics of UML2.0Interactions is given by composingtheir translation into interactions in the classical sense and the reduction relationsbelow.1PreliminariesWe briefly review the basic definitions on partially ordered,labelled multisets as intro-duced by Pratt[3]for modelling concurrency.In particular,we define sequential and parallel composition operators and the notion of traces and processes.A partially ordered,labelled multiset,or pomset,is the isomorphism class[(X,≤X,λX)]of a labelled partial order(X,≤X,λX)w.r.t.monotone,label-preserving maps.A trace is a pomset whose ordering is total.We write p↓for all possible linearisa-tions of a pomset p,i.e.,all traces that extend the ordering of p:[(X ,≤X ,λX )]∈[(X,≤X,λX)]↓if,and only if X =X,λX =λX,and≤X⊆≤X where x1≤X x2 or x2≤X x1for all x1,x2∈X .The empty pomset,represented by(∅,∅,∅),is denoted byε.Let p=[(X,≤X,λX)] and q=[(Y,≤Y,λY)]be pomsets such that X∩Y=∅.The concurrence of p and q, written as p q,is given by[(X∪Y,≤X∪≤Y,λX∪λY)].The concatenation of p and q, written as p;q,is given by[(X∪Y,(≤X∪≤Y∪(X×Y))∗,λX∪λY)].Given a binary, symmetric relation<>on labels,the<>-concatenation of p and q,written as p;<>q,is given by[(X∪Y,(≤X∪≤Y∪{(x,y)∈X×Y|λX(x)<>λY(y)})∗,λX∪λY)].We write p n for the n-fold iteration of pomset concatenation with n a natural number,i.e.,p0=εand p n+1=p;p n.Furthermore we let p∗denote n≥0p n.Analogously,we write p n<> for the n-fold iteration of<>-concatenation with p0<>=εand p(n+1)<>=p;<>p n<>.Note that concatenation and<>-concatenation are associative,and concurrence is associative and commutative.A process is a set of pomsets.An n-ary function f on pomsets is lifted to processes P1,...,P n by defining f(P1,...,P n)={f(p1,...,p n)|p1∈P1,...,p n∈P n}.2Abstract SyntaxWe define the abstract syntax of a fragment of the language of UML2.0interactions of[2].We assume two primitive domains for instances I and messages M.An event e isInteraction::=None|Empty|Basic|CombinedFragmentCombinedFragment::=strict(Interaction,Interaction)|seq(Interaction,Interaction)|par(Interaction,Interaction)|loop(Nat,(Nat|∞),Interaction)|ignore(Messages,Interaction)|restr(Instances,Interaction)|alt(Interaction,Interaction)|not(Interaction)Table1.Abstract syntax of interactions(fragment).either of the form snd(s,r,m)or of the form rcv(s,r,m),representing the dispatch and the arrival of message m from sender instance s to receiver instance r,respectively. The set E comprises all events over I and M.The message of an event e is denoted byµ(e).We say that the instance s is active for snd(s,r,m)and,similarly,that the instance r is active for rcv(s,r,m);the(singleton)set of instances active for an event e is denoted byα(e).We define a binary,symmetric conflict relation<>on events: e<>e ⇔α(e)∩α(e )=∅.A basic interaction is given by an event-labelled pomset[(E,≤E,λE)]such that conflicting events do not occur concurrently,i.e.,if e1,e2∈E withλE(e1)<>λE(e2), then e1≤E e2or e2≤E e1.The abstract syntax of interactions is given by the grammar in Tab.1.Therein,Basic ranges over the basic interactions,Nat ranges over the natural numbers,Messages over the subsets of M,and Instances over the subsets of I.Note that the interaction constants None and Empty as well as the interaction oper-ators restr and not are not part of the specification of UML2.0interactions as defined in[2].The operator not results from the translation of UML2.0interactions(including the negative operators neg and assert)into“interactions in the classical sense”;see[1]. The operator restr and the constants None and Empty simplify the definition of the op-erational semantics.Interactions of the form seq(alt(B1,B2),B2)motivate the operator restr,where B i are the basic interactions{snd(s i,r i,m i)≤rcv(s i,r i,m i)}(i=1,2) with m1=m2.In case the second operand of the weak sequencing operator seq makes progress by sending the message m2from instance s2to instance r2,then thefirst op-erator of the seq,which is a disjunction,may only choose itsfirst operator;we call that progress a non-local choice.3Denotational Semantics3.1Semantic DomainsThe domain P comprises all basic interactions.The subdomain T of P comprises all pomsets in P that are traces.In particular,the empty pomsetεis in T;also all eventscan be identified with traces of length one.The notion of the set of active instances of an event e ∈E is extended to an event-labelled pomset p =[(X,≤X ,λX )]by setting α(p )= x ∈X α(λX (x )).For a pomset p =[(X,≤X ,λX )]∈P and an event e ∈E we write e ∈min p ,if there is an x ∈X with x ∈min ≤X X and λX (x )=e ;note that x is unique defined,if it exists.If e ∈min p ,we write p \{e }for [(X \{x }),≤X ∩(X \{x })2,λX (X \{x })]with x ∈min ≤X X and λX (x )=e .On pomsets in P and for a set of messages M ,the filtering relation filter (M ):P →℘P removes some elements of a pomset whose labels show a message in M .More pre-cisely,we first define filter (M )on event-labelled sets:Let X be a set and λ:X →E a labelling function;then X ∈filter (M )(X,λ)if X ⊆X and,if x ∈X \X ,then µ(λ(x ))∈M .For an event-labelled partial order (X,≤X ,λX )we set (X ,≤X ∩(X ×X ),λX X )∈filter (M )(X,≤X ,λX )if X ∈filter (M )(X,λX ).Finally,we extend these definitions to event-labelled pomsets by setting [(X ,≤X ,λX )]∈filter (M )([(X,≤X ,λX )])if (X ,≤X ,λX )∈filter (M )(X,≤X ,λX ),which is ob-viously well-defined.For a pomset p ∈P ,we write p M for filter (M )−1(p );and,consequently,for a process P ⊆P ,we write P M for filter (M )−1(P ).Finally,on processes in ℘P and for a set of instances L ,the restriction function restr (L ):℘P →℘P removes all those pomsets from a process which show an event that is active for an instance in L ,i.e.,restr (L )(P )={p ∈P |α(p )∩L =∅}.We also write P [L ]for restr (L )(P ).The process building operators are transferred to traces using the following identi-ties:Lemma 1.Let P,P 1,P 2⊆P be processes,M ⊆M a set of messages,and L ⊆I a set of instances.1.(P 1;P 2)↓=(P 1↓);(P 2↓)2.(P 1;<>P 2)↓=((P 1↓);<>(P 2↓))↓3.(P 1 P 2)↓=((P 1↓) (P 2↓))↓4.(P M )↓=((P ↓) M )↓5.(P [L ])↓=(P ↓)[L ]6.(P 1∪P 2)↓=(P 1↓)∪(P 2↓)7.(P \P )↓=(P \(P ↓))↓3.2Trace-Based SemanticsThe trace-based,denotational semantics of the authors [1]can be rendered as a function P :Interaction →℘T defined as follows:P None =∅P Empty ={ε}P B =B ↓P strict(S ,S )=P S ;P SP seq(S ,S )=(P S ;<>P S)↓P par(S,S )=(P S P S )↓P loop(m,n,S)= m≤i<n+1((P S)i<>)↓P ignore(M,S)=((P S) M )↓P restr(L,S)=(P S)[L]P alt(S,S )=P S∪P SP not(S)=T\P Swhere S and S are interactions,M⊆M,L⊆I,and n a natural number or infinity, where∞+1=∞.In particular,a trace t is positive for an interaction S,written t|=p S if,and only if, t∈P S.We make use of the following syntactical identifications:strict(Empty,S)≡Sseq(Empty,S)≡Spar(Empty,S)≡par(S,Empty)≡Sloop(0,0,S)≡Emptyrestr(L,Empty)≡Emptyignore(∅,Empty)≡Emptyalt(Empty,Empty)≡Empty4ProcessesFor a process P⊆P and an event e∈E,we define the left quotient P/e of P by e to be the process{p∈P|e;p∈P}.This operation is right-adjoined to prefixing pomsets by e with respect to set inclusion,as e;P⊆P if,and only if P⊆P /e. Lemma2.Let P,P1,P2⊆P be processes,e∈E an event,M⊆M a set of messages, and L⊆I a set of instances.1.(P1;P2)/e=((P1/e);P2)∪((P1∩{ε});(P2/e))2.(P1;<>P2)/e=((P1/e);<>P2)∪(P1[α(e)];<>(P2/e))3.(P1 P2)/e=((P1/e) P2)∪(P1 (P2/e))4.(P M )/e=(P/e) M ∪{ε|µ(e)∈M};P M5.(P[L])/e={ε|α(e)∩L=∅};(P/e)[L]6.(P1∪P2)/e=(P1/e)∪(P2/e)7.(P\P)/e=P\(P/e)Proof.By calculation,we have:(1)(P1;P2)/e={p|∃p1,p2.p1∈P1∧p2∈P2∧e;p=p1;p2}={p|∃p 1,p2.e;p 1∈P1∧p2∈P2∧p=p 1;p2∨∃p1,p 2.ε=p1∈P1∧e;p 2∈P2∧p=p1;p 2}=(P1/e);P2∪(P1∩{ε});(P2/e)(2)(P1;<>P2)/e={p|∃p1,p2.p1∈P1∧p2∈P2∧e;p=p1;<>p2}={p|∃p 1,p2.e;p 1∈P1∧p2∈P2∧p=p 1;<>p2∨∃p1,p 2.p1∈P1∧α(p1)∩α(e)=∅∧e;p 2∈P2∧p=p1;<>p 2} =(P1/e);<>P2∪(P1[α(e)];<>(P2/e))(3)(P1 P2)/e={p|∃p1,p2.p1∈P1∧p2∈P2∧e;p=p1 p2}={p|∃p 1,p2.e;p 1∈P1∧p2∈P2∧p=p 1 p2∨∃p1,p 2.p1∈P1∧e;p 2∈P2∧p=p1 p 2}=((P1/e) P2)∪(P1 (P2/e))(4)(P M )/e={p|∃p .e;p ∈P∧p∈p M∨p∈P M ∧µ(e)∈M}=(P/e) M ∪{ε|µ(e)∈M};P M(5)(P[L])/e={p|∃p .e;p ∈P∧α(p )∩α(L)=∅∧α(e)∩α(L)=∅∧p=p }={ε|α(e)∩L=∅};(P/e)[L](6)(P1∪P2)/e={p|e;p∈P1}∪{p|e;p∈P2}=(P1/e)∪(P2/e)(7)(P\P)/e={p|¬(e;p∈P)}=P\(P/e)The following lemma summarises an obvious characterisation of when the empty pomset can result from a process expression:Lemma3.Let P,P1,P2⊆P be processes,M⊆M a set of messages,and L⊆I a set of instances.1.ε∈(P1;P2)⇐⇒(ε∈P1)∧(ε∈P2)2.ε∈(P1;<>P2)⇐⇒(ε∈P1)∧(ε∈P2)3.ε∈(P1 P2)⇐⇒(ε∈P1)∧(ε∈P2)4.ε∈(P M )⇐⇒ε∈P5.ε∈(P[L])⇐⇒ε∈P6.ε∈(P1∪P2)⇐⇒(ε∈P1)∨(ε∈P2)7.ε∈(P\P)⇐⇒ε/∈PProcess expressions satisfy some obvious monotonicity conditions:Lemma4.Let P,P ,P1,P 1,P2,P 2⊆P be processes,M,M ⊆M a set of messages, and L,L ⊆I sets of instances.1.P1⊆P 1∧P2⊆P 2⇒(P1;P2)⊆(P 1;P 2)2.P1⊆P 1∧P2⊆P 2⇒(P1;<>P2)⊆(P 1;<>P 2)3.P1⊆P 1∧P2⊆P 2⇒(P1 P2)⊆(P 1 P 2)4.P⊆P ∧M⊆M ⇒(P M )⊆(P M )5.P⊆P ∧L⊇L ⇒(P[L])⊆(P [L ])6.P1⊆P 1∧P2⊆P 2⇒(P1∪P2)⊆(P 1∪P 2)7.P⊇P ⇒(P\P)⊆(P\P )All these observations can be extended straightforwardly to P n<>/e:P0<>/e=∅P(n+1)<>/e=((P/e);<>P n<>)∪(P[α(e)];<>(P n<>/e))ε∈P n<>⇐⇒(ε∈P)∨(n=0)P⊆P ⇒P n<>⊆P n<>5Operational SemanticsWe define the domain Eτof events and the silent eventτas E∪{τ}.Analogously, Pτis the domain of all pomsets labelled with events from Eτ,and Tτthe subdomain comprising all pomsets in Pτthat are traces.We define the set of active instances ofτasα(τ)=∅.Based on the observations in Lemma3,we define a predicateε(−)on interactions which determines whether an interaction contains the empty trace:ε(None)⇐⇒ffε(Empty)⇐⇒ttε(B)⇐⇒B=εε(strict(S,S ))⇐⇒ε(S)∧ε(S )ε(seq(S,S ))⇐⇒ε(S)∧ε(S )ε(par(S,S ))⇐⇒ε(S)∧ε(S )ε(loop(m,n,S))⇐⇒ε(S)∨(m=0)ε(ignore(M,S))⇐⇒ε(S)ε(restr(L,S))⇐⇒ε(S)ε(alt(S,S ))⇐⇒ε(S)∨ε(S )ε(not(S))⇐⇒¬ε(S)The operational semantics of interactions is given by two ternary relations between interactions S and S and an event e∈Eτ:The positive reduction relation,denoted byS e−→p S ,is defined by the rules in Tab.2.The negative reduction relation,denoted by S e−→n S ,is defined by the rules in Tab.3.In these rules,the variously decoratedmeta-variables range as follows:S over interactions,B over basic interactions,e over Eτ,e over E,m over the natural numbers,n over the natural numbers or infinity.(basic p)B e−→p B\{e}if e∈min B(strict p)S1e−→p S 1strict(S1,S2)e−→p strict(S 1,S2)(seq1p)S1e−→p S 1seq(S1,S2)e−→p seq(S 1,S2)(seq2p)S2e−→p S 2seq(S1,S2)e−→p seq(restr(α(e),S1),S 2)(par1p)S1e−→p S 1par(S1,S2)e−→p par(S 1,S2)(par2p)S2e−→p S 2par(S1,S2)e−→p par(S1,S 2)(loop1p)loop(0,n,S)τ−→p Empty(loop2p)S e−→p Sloop(m,n+1,S)e−→p seq(S ,loop(m˙−1,n,S))(ignore1p)ignore(M,Empty)τ−→p Empty(ignore2p)S e−→p Signore(M,S)e−→p ignore(M,S )(ignore3p)ignore(M,S)e−→p ignore(M,S)ifµ(e)∈M (restr p)S e−→p Srestr(L,S)e−→p restr(L,S )ifα(e)∩L=∅(alt1p)S1e−→p S 1alt(S1,S2)e−→p S 1(alt2p)S2e−→p S 2alt(S1,S2)e−→p S 2(not1p)S e−→n Snot(S)e−→p not(S )(not2p)not(S)τ−→p Empty if¬ε(S) Table2.Positive reduction relation of the operational semantics.6CorrectnessIf S e1−→p S1,S1e2−→p S2,...,S n−1e n−→p S ,we write S t−→p S ,where t= e1;e2;···;e n∈Tτis afinite trace possibly containing one or more occurrences of(empty n)Empty e−→n None(none n)None e−→n None(basic1n)B e−→n B\{e}if e∈min B(basic2n)B e−→n None if e/∈min B (strict n)S1e−→n S 1S2e−→n S 2strict(S1,S2)e−→n alt(strict(S 1,S2),strict(restr(I,S1),S 2))(seq n)S1e−→n S 1S2e−→n S 2seq(S1,S2)e−→n alt(seq(S 1,S2),seq(restr(α(e),S1),S 2))(par n)S1e−→n S 1S2e−→n S 2par(S1,S2)e−→n alt(par(S 1,S2),par(S1,S 2))(loop1n)loop(0,∞,S)e−→n not(None)(loop2n)S e−→n S loop(m˙−1,n,S)e−→n Sloop(m,n+1,S)e−→n alt(seq(S ,loop(m˙−1,n,S)),seq(restr(α(e),S),S ))(ignore1n)S e−→n Signore(M,S)e−→n alt(ignore(M,S ),ignore(M,S))ifµ(e)∈M(ignore2n)S e−→n Signore(M,S)e−→n ignore(M,S )if e=τ∨µ(e)/∈M(restr1n)S e−→n Srestr(L,S)e−→n restr(L,S )ifα(e)∩L=∅(restr2n)restr(L,S)e−→n None ifα(e)∩L=∅(alt n)S1e−→n S 1S2e−→n S 2alt(S1,S2)e−→n alt(S 1,S 2)(not n)S e−→p Snot(S)e−→n not(S )Table3.Negative reduction relation of the operational semantics.the silent eventτ.Given a trace t∈Tτ,we let t denote the trace obtained from t by removing every occurrence of the silent eventτ.The above introduced operational semantics of interactions is correct w.r.t.the de-notational one,that is,given an interaction,traces that lead the interaction to Empty are positive for the interaction:Lemma5.Let S,S be interactions and e∈E.1.If S e−→p S ,then P S ⊆P S/e.2.If Sτ−→p S ,then P S ⊆P S.3.If S e−→n S ,then P S ⊇P S/e.4.If Sτ−→n S ,then P S ⊇P S.Proof.Claims(1)and(3)follow immediately from Lemma2,claims(2)and(4)from Lemma4.Proposition6.Let S be an interaction and t be a trace in Tτ.If S t−→p Empty,then t |=p S.Proof.From Lemma5,and by induction on the length of t,follows that if S t−→p Empty, then t ;P Empty⊆P S,i.e., t ∈P(S).7CompletenessLemma7.Let S be an interaction and e∈E.Then1.(P S)/e= {P(S )|Sτ∗;e−−→p S }2.(T\P S)/e= {T\P(S )|Sτ∗;e−−→n S }−→p Empty3.ε∈P S⇐⇒Sτ∗Proof.Claims(1)and(2)follow by mutual induction on the term structure of the inter-action S from Lemma2.Claim(3)follows by induction on the term structure of the interaction S from the syntactical identifications,the definition ofε(−)and theτ-rules.Proposition8.Let S be an interaction and t∈T be afinite trace.If t∈P(S),then there is a t∈Tτwith t= t and S t−→p Empty.Proof.From Lemma7(1)and by induction follows that for everyfinite trace t∈T and for every interaction S with t∈P(S),there are a t∈Tτwith t= t and an interaction S t such that S t−→p S t andε∈P S t.Thus,by Lemma7(3),for an interaction S and afinite trace t∈T,there is a t∈Tτwith t= t and S t−→p Empty.References1.Mar´ıa Victoria Cengarle and Alexander Knapp.UML2.0Interactions:Semantics and Refine-ment.In Jan J¨u rjens,Eduardo B.Fernandez,Robert France,and Bernhard Rumpe,editors, Proc.3rd Int.Wsh.Critical Systems Development with UML(CSDUML’04),pages85–99.Technical Report TUM-I0415,Institut f¨u r Informatik,Technische Universit¨a t M¨u nchen,2004.2.Object Management Group.UML2.0Superstructure Specification.Final adopted specifica-tion,OMG,2003./cgi-bin/doc?ptc/03-08-02.3.Vaughan Pratt.Modeling Concurrency with Partial 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