On the Axiomatisability of Parallel Composition
Luca Aceto, Valentina Castiglioni, Anna Ingolfsdottir, Bas Luttik, Mathias R. Pedersen
aa r X i v : . [ c s . L O ] F e b ON THE AXIOMATISABILITY OF PARALLEL COMPOSITION
LUCA ACETO, VALENTINA CASTIGLIONI, ANNA ING ´OLFSD ´OTTIR, BAS LUTTIK,AND MATHIAS RUGGAARD PEDERSENReykjavik University, Iceland; Gran Sasso Science Institute (GSSI), Italy e-mail address : [email protected] University, Iceland e-mail address : [email protected] University, Iceland e-mail address : [email protected] University of Technology, The Netherlands e-mail address : [email protected] University, Iceland e-mail address : [email protected]
Abstract.
This paper studies the existence of finite equational axiomatisations of the interleaving parallel composition operator modulo the behavioural equivalences in vanGlabbeek’s linear time-branching time spectrum . In the setting of the process algebraBCCSP over a finite set of actions, we provide finite , ground-complete axiomatisations forvarious simulation and (decorated) trace semantics. We also show that no congruence overBCCSP that includes bisimilarity and is included in possible futures equivalence has afinite, ground-complete axiomatisation; this negative result applies to all the nested traceand nested simulation semantics. Introduction
Process algebras [5, 7] are prototype specification languages allowing for the description andanalysis of concurrent and distributed systems, or simply processes . These languages offer avariety of operators to specify composite processes from components one has already built.Notably, in order to model the concurrent interaction between processes, the majority ofprocess algebras include some form of parallel composition operator, also known as merge.Following Milner’s seminal work on CCS [26], the semantics of a process algebra is oftendefined according to a two-step approach. In the first step, the operational semantics [31]of a process is modelled via a labelled transition system (LTS) [24], in which computationalsteps are abstracted into state-to-state transitions having actions as labels.Behavioural equivalences have then been introduced, in the second step, as simple andelegant tools for comparing the behaviour of processes. These are equivalence relations
Key words and phrases:
Axiomatisation, Parallel composition, Linear time-branching time spectrum.A preliminary version of this paper appeared as [1].
Preprint submitted toLogical Methods in Computer Science © L. Aceto, V. Castiglioni, A. Ingólfsdóttir, B. Luttik, and M. R. Pedersen CC (cid:13) Creative Commons
L. ACETO, V. CASTIGLIONI, A. ING ´OLFSD ´OTTIR, B. LUTTIK, AND M. R. PEDERSEN defined on the states of LTSs allowing one to establish whether two processes have thesame observable behaviour . Different notions of observability correspond to different levelsof abstraction from the information carried by the LTS, which can either be consideredirrelevant in a given application context, or be unavailable to an external observer.In [19], van Glabbeek presented the linear time-branching time spectrum , i.e., a taxon-omy of behavioural equivalences based on their distinguishing power. He carried out hisstudy in the setting of the process algebra BCCSP, which consists of the basic operatorsfrom CCS [26] and CSP [23], and he proposed ground-complete axiomatisations for most ofthe congruences in the spectrum over this language. (An axiomatisation is ground-completeif it can prove all the valid equations relating terms that do not contain variables.) The pre-sented ground-complete axiomatisations are finite if so is the set of actions. For the readysimulation, ready trace and failure trace equivalences, the axiomatisation in [19] made useof conditional equations; Blom, Fokkink and Nain gave purely equational, finite axioma-tisations in [8]. Then, the works in [2], on nested semantics, and in [11], on impossiblefutures semantics, completed the studies of the axiomatisability of behavioural congruencesover BCCSP by providing negative results: neither impossible futures nor any of the nestedsemantics have a finite, ground-complete axiomatisation over BCCSP.Obtaining a complete axiomatisation of a behavioural congruence is a classic, key prob-lem in concurrency theory, as an equational axiomatisation characterises the semantics ofa process algebra in a purely syntactic fashion. Hence, this characterisation becomes inde-pendent of the details of the definition of the process semantics of interest, allowing one tocompare semantics that may have been defined in very different styles via a collection ofrevealing axioms.All the results mentioned so far were obtained over the algebra BCCSP, which doesnot include any operator for the parallel composition of processes. Considering the crucialrole of such an operator, it is natural to ask which of those results would still hold over aprocess algebra including it.In the literature, we can find a wealth of studies on the axiomatisability of parallelcomposition modulo bisimulation semantics [30]. Briefly, in the seminal work [22], Hennessyand Milner proposed a ground-complete axiomatisation of the recursion-free fragment ofCCS modulo bisimilarity. That axiomatisation, however, included infinitely many axioms,which corresponded to instances of the expansion law used to express equationally thesemantics of the merge operator. Then, Bergstra and Klop showed in [6] that a finiteground-complete axiomatisation modulo bisimilarity can be obtained by enriching CCSwith two auxiliary operators, i.e., the left merge and the communication merge | . Later,Moller proved that the use of auxiliary operators is indeed necessary to obtain a finiteequational axiomatisation of bisimilarity in [27–29].To the best of our knowledge, no systematic study of the axiomatisability of the parallelcomposition operator modulo the other semantics in the spectrum has been presented sofar. Our contribution.
We consider the process algebra BCCSP k , i.e., BCCSP enriched withthe interleaving parallel composition operator, and we study the existence of finite equa-tional axiomatisations of the behavioural congruences in the linear time-branching timespectrum over it. Our results delineate the boundary between finite and non-finite axioma-tisability of the congruences in the spectrum over the language BCCSP k . (See Figure 1.) N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 3
We start by providing a finite , ground-complete axiomatisation for ready simulation semantics [9]. The axiomatisation is obtained by extending the one for BCCSP with afew axioms expressing equationally the behaviour of interleaving modulo the consideredcongruence. The added axioms allow us to eliminate all occurrences of the interleavingoperator from BCCSP k processes, thus reducing ground-completeness over BCCSP k toground-completeness over BCCSP [8, 19]. Since the axioms for the elimination of paral-lel composition modulo ready simulation equivalence are, of course, sound with respect toequivalences that are coarser than ready simulation equivalence, the “reduction to ground-completeness over BCCSP” works for all behavioural equivalences in the spectrum belowready simulation equivalence. Nevertheless, for those equivalences, we shall offer more el-egant axioms to equationally eliminate parallel composition from closed terms. We shallthen observe a sort of parallelism between the axiomatisations for the notions of simula-tion and the corresponding decorated trace semantics: the axioms used to equationallyexpress the interplay between the interleaving operator and the other operators of BCCSPin a decorated trace semantics can be seen as the linear counterpart of those used in thecorresponding notion of simulation semantics. For instance, while the axioms for ready sim-ulation impose constraints on the form of both arguments of the interleaving operator tofacilitate equational reductions, those for ready trace equivalence impose similar constraintsbut only on one argument.Finally, we complete our journey in the spectrum by showing that nested simulation and nested trace semantics do not have a finite axiomatisation over BCCSP k . To this end,firstly we adapt Moller’s arguments to the effect that bisimilarity is not finitely based overCCS to obtain the negative result for possible futures equivalence , also known as 2- nestedtrace equivalence . Informally, the negative result is obtained by providing an infinite familyof equations that are all sound modulo possible futures equivalence but that cannot allbe derived from any finite, sound axiom system. Then, we exploit the soundness modulobisimilarity of the equations in the family to extend the negative result to all the congruencesthat are finer than possible futures and coarser than bisimilarity, thus including all nestedtrace and nested simulation semantics.All the results mentioned so far are obtained for a parallel composition operator thatimplements interleaving without synchronisation between parallel components. As naturalextension, we then discuss the effect of extending our results to parallel composition withCCS-style synchronisation. Organisation of contents.
After reviewing some basic notions on behavioural equiva-lences and equational logic in Section 2, we start our journey in the spectrum by providinga finite, ground-complete axiomatisation for ready simulation equivalence over BCCSP k inSection 3. In Section 4 we discuss how it is possible to refine the axioms for ready simulationto obtain finite, ground-complete axiomatisations for completed simulation and simulationequivalences. Then, in Section 5 similar refinements are provided for the (decorated) traceequivalences, thus completing the presentation of our positive results. We end our journeyin Section 6 with the presentation of the negative results, namely that the nested simula-tion and nested trace equivalences do not have a finite axiomatisation over BCCSP k . InSection 7 we modify the semantics of parallel composition to allow processes running inparallel to synchronise and we discuss the effect of this extension on the results obtained inSections 3–6. Finally, in Section 8 we draw some conclusions and discuss avenues for futurework. L. ACETO, V. CASTIGLIONI, A. ING ´OLFSD ´OTTIR, B. LUTTIK, AND M. R. PEDERSEN a.x a −→ x x a −→ x ′ x + y a −→ x ′ y a −→ y ′ x + y a −→ y ′ x a −→ x ′ x k y a −→ x ′ k y y a −→ y ′ x k y a −→ x k y ′ Table 1.
Operational semantics of BCCSP k . What’s new.
A preliminary version of this paper appeared as [1]. We have enriched ourprevious contribution as follows:(1) We provide the full proofs of our results.(2) We extend the results presented in [1] from purely interleaving parallel composition toparallel composition with synchronisation `a la CCS. (Section 7).(3) We present model constructions showing that the specific axioms we provide to axioma-tise parallel compositions in ready simulation equivalence cannot be derived from theaxiomatisations of completed simulation equivalence and ready trace equivalence. Simi-larly, we show that the axiomatisations of completed simulation and failure equivalencecannot be derived from that of completed trace equivalence.2.
Background
The language.
The language BCCSP k extends BCCSP with parallel composition. For-mally, BCCSP k consists of basic operators from CCS [26] and CSP [23], with the purely interleaving parallel composition operator k , and is given by the following grammar: t ::= | x | a.t | t + t | t k t where a ranges over a set of actions A and x ranges over a countably infinite set of variables V . In what follows, we assume that the set of actions A is finite and non-empty.We shall use the meta-variables t, u, . . . to range over BCCSP k terms, and write var( t )for the collection of variables occurring in the term t . We also adopt the standard conventionthat prefixing binds strongest and + binds weakest. Moreover, trailing ’s will often beomitted from terms. We use a summation P i ∈{ ,...,k } t i to denote the term t = t + · · · + t k ,where the empty sum represents . We can also assume that the terms t i , for i ∈ { , . . . , k } ,do not have + as head operator, and refer to them as the summands of t . The size of aterm t , denoted by size( t ), is the number of operator symbols in it.A BCCSP k term is closed if it does not contain any variables. We shall, sometimes, referto closed terms simply as processes . We let P denote the set of BCCSP k processes and let p, q, . . . range over it. We use the Structural Operational Semantics (SOS) framework [31]to equip processes with an operational semantics. A literal is an expression of the form t a −→ t ′ for some process terms t, t ′ and action a ∈ A . It is closed if both t, t ′ are closedterms. The inference rules for prefixing a. , nondeterministic choice + and interleavingparallel composition k are reported in Table 1. A substitution σ is a mapping from variablesto terms. It extends to terms, literals and rules in the usual way. A substitution is closed if it maps every variable to a process.The inference rules in Table 1 induce the A - labelled transition system [24] ( P , A , −→ )whose transition relation −→ ⊆ P × A × P contains exactly the closed literals that can bederived using the rules in Table 1. As usual, we write p a −→ p ′ in lieu of ( p, a, p ′ ) ∈ −→ . For N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 5 ( e ) t ≈ t ( e ) t ≈ uu ≈ t ( e ) t ≈ u u ≈ vt ≈ v ( e ) t ≈ uσ ( t ) ≈ σ ( u ) ( e ) t ≈ ua.t ≈ a.u ( e ) t ≈ u t ′ ≈ u ′ t + t ′ ≈ u + u ′ ( e ) t ≈ u t ′ ≈ u ′ t k t ′ ≈ u k u ′ . Table 2.
The rules of equational logiceach p ∈ P and a ∈ A , we write p a −→ if p a −→ p ′ holds for some p ′ , and p a −→6 otherwise. The initials of p are the actions that label the outgoing transitions of p , that is, I ( p ) = { a | p a −→} .For a sequence of actions α = a · · · a k ( k ≥ p, p ′ , we write p α −→ p ′ if andonly if there exists a sequence of transitions p = p a −−→ p a −−→ · · · a k −−→ p k = p ′ . If p α −→ p ′ holds for some process p ′ , then α is a trace of p , and p ′ is a derivative of p . Moreover, we saythat α is a completed trace of p if I ( p ′ ) = ∅ . We let T ( p ) denote the set of traces of p , andwe use CT ( p ) ⊆ T ( p ) for the set of completed traces of p . We write ε for the empty trace ; | α | stands for the length of trace α . It is well known, and easy to show, that T ( p ) is finite and CT ( p ) is non-empty for each BCCSP k process p . It follows that we can define the depth ofa process p , denoted by depth( p ), as the length of a longest completed trace of p . Formally,depth( p ) = max {| α | | α ∈ CT ( p ) } . Similarly, the norm of a process p , denoted by norm( p ),is the length of a shortest completed trace of p , i.e. norm( p ) = min {| α | | α ∈ CT ( p ) } . Behavioural equivalences.
Behavioural equivalences have been introduced to establishwhether the behaviours of two processes are indistinguishable for their observers . Roughly,they allow us to check whether the observable semantics of two processes is the same . In theliterature we can find several notions of behavioural equivalence based on the observationsthat an external observer can make on the process. In his seminal article [19], van Glabbeekgave a taxonomy of the behavioural equivalences discussed in the literature on concurrencytheory, which is now called the linear time-branching time spectrum (see Figure 1).One of the main concerns in the development of a meta-theory of process languages isto guarantee their compositionality , i.e., that the replacement of a component of a systemwith an R -equivalent one, for a chosen behavioural equivalence R , does not affect thebehaviour of that system. In algebraic terms, this is known as the congruence property of R with respect to all language operators, which consists in verifying whether f ( t , . . . , t n ) R f ( t ′ , . . . , t ′ n ) for every n -ary operator f whenever t i R t ′ i for all i = 1 , . . . , n. Since BCCSP k operators are defined by inference rules in the de Simone format [15],by [17, Theorem 4] we have that all the equivalences in the spectrum in Figure 1 arecongruences with respect to them. Our aim in this paper is to investigate the existence ofa finite equational axiomatisation of BCCSP k modulo all those congruences. Equational Logic. An axiom system E is a collection of equations t ≈ u over BCCSP k .An equation t ≈ u is derivable from an axiom system E , notation E ⊢ t ≈ u , if there is an equational proof for it from E , namely if t ≈ u can be inferred from the axioms in E usingthe rules of equational logic , which express reflexivity, symmetry, transitivity, substitutionand closure under BCCSP k contexts and are reported in Table 2. In equational proofs, weshall write p (A) ≈ q to highlight that the axiom denoted by A is used in that step of the proof. L. ACETO, V. CASTIGLIONI, A. ING ´OLFSD ´OTTIR, B. LUTTIK, AND M. R. PEDERSEN bisimulation ( ∼ B )2-nested simulation ( ∼ S )failure simulation ( ∼ FS ) = ready simulation ( ∼ RS )ready trace ( ∼ RT )failure trace ( ∼ FT ) readies ( ∼ R )failures ( ∼ F )completed trace ( ∼ CT )trace ( ∼ T )completed simulation ( ∼ CS )simulation ( ∼ S ) possible futures ( ∼ PF ) Figure 1.
The linear time-branching time spectrum [19]. For the equiva-lence relations in blue we provide a finite, ground-complete axiomatization.For the ones in red, we provide a negative result. The case of bisimulationis known from the literature [27–29]. (A0) x + ≈ x (P0) x k ≈ x (A1) x + y ≈ y + x (P1) x k y ≈ y k x (A2) ( x + y ) + z ≈ x + ( y + z ) (A3) x + x ≈ x Table 3.
Basic axioms for BCCSP k . We define E = { A0 , A1 , A2 , A3 } and E = E ∪ { P0 , P1 } .We are interested in equations that are valid modulo some congruence relation R overclosed terms. The equation t ≈ u is said to be sound modulo R if σ ( t ) R σ ( u ) for allclosed substitutions σ . For simplicity, if t ≈ u is sound modulo R , then we write t R u .An axiom system is sound modulo R if, and only if, all of its equations are sound modulo R . Conversely, we say that E is ground-complete modulo R if p R q implies E ⊢ p ≈ q forall closed terms p, q . We say that R has a finite ground-complete axiomatisation, if thereis a finite axiom system E that is sound and ground-complete modulo R .In Table 3 we present some basic axioms for BCCSP k that are sound with respect toall the behavioural equivalences in Figure 1. Henceforth, we will let E = { A0 , A1 , A2 , A3 } ,and we will denote by E the axiom system consisting of all the axioms in Table 3, namely E = E ∪ { P0 , P1 } .To be able to eliminate the interleaving parallel composition operator from closed termswe will make use of two refinements EL1 and EL2 of EL3, which is the classic expansionlaw [22] (see Table 4). We remark that the actions occurring in the three axioms in Table 4are not action variables. Hence, when we write that an axiom system E includes one ofthese axioms, we mean that it includes all possible instances of that axiom with respectto the actions in A . In particular, EL3 is a schema that generates infinitely many axioms,regardless of the cardinality of the set of actions. This is due to the fact that we can have N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 7 (EL1) ax k by ≈ a ( x k by ) + b ( ax k y ) (EL2) P i ∈ I a i x i k P j ∈ J b j y j ≈ P i ∈ I a i ( x i k P j ∈ J b j y j ) + P j ∈ J b j ( P i ∈ I a i x i k y j )with a i = a k whenever i = k and b j = b h whenever j = h , ∀ i, k ∈ I, ∀ j, h ∈ J (EL3) P i ∈ I a i x i k P j ∈ J b j y j ≈ P i ∈ I a i ( x i k P j ∈ J b j y j ) + P j ∈ J b j ( P i ∈ I a i x i k y j ) Table 4.
The different instantiations of the expansion law.arbitrary summations in the two arguments of the parallel composition in the left hand sideof EL3. On the other hand, when the set of actions is assumed to be finite, we are guaranteedthat there are only finitely many instances of EL1 and EL2. Indeed, EL1 is a particularinstance of EL2, i.e., the one in which both summations are over singletons. The reason forconsidering both is that, as we will see, EL1 is enough to obtain the elimination result whencombined with axioms allowing us to reduce any process of the form ( P i ∈ I a i p i ) k ( P j ∈ J b j q j )to P i ∈ I,j ∈ J ( a i p i k b j q j ). Axiom EL2 is needed when this reduction is not sound modulo theconsidered semantics. 3. Ready simulation
In this section, we begin our journey in the spectrum by studying the equational theory of ready simulation equivalence, whose formal definition is recalled below together with thoseof completed simulation and simulation equivalence.
Definition 3.1 (Simulation equivalences) . • A simulation is a binary relation R ⊆ P × P such that, whenever p R q and p a −→ p ′ , thenthere is some q ′ such that q a −→ q ′ and p ′ R q ′ . We write p ⊑ S q if there is a simulation R such that p R q . We say that p is simulation equivalent to q , notation p ∼ S q , if p ⊑ S q and q ⊑ S p . • A completed simulation is a simulation R such that, whenever p R q and I ( p ) = ∅ , then I ( q ) = ∅ . We write p ⊑ CS q if there is a completed simulation R such that p R q . We saythat p is completed simulation equivalent to q , notation p ∼ CS q , if p ⊑ CS q and q ⊑ CS p . • A ready simulation is a simulation R such that, whenever p R q then I ( p ) = I ( q ). Wewrite p ⊑ RS q if there is a ready simulation R such that p R q . We say that p is readysimulation equivalent to q , notation p ∼ RS q , if p ⊑ RS q and q ⊑ RS p .In [18] the notion of failure simulation was also introduced as a simulation R such that,whenever p R q and I ( p ) ∩ X = ∅ , for some X ⊆ A , then I ( q ) ∩ X = ∅ . Then, in [17] it wasproved that the notion of failure simulation coincides with that of ready simulation.Our aim is to provide a finite , ground-complete axiomatisation of BCCSP k moduloready simulation equivalence. To this end, we recall that in [19] it was proved that theaxiom system consisting of E together with axiom RS in Table 5 is a ground-completeaxiomatisation of BCCSP, i.e., the language that is obtained from BCCSP k if k is omitted,modulo ∼ RS . Hence, to obtain a finite, ground-complete axiomatisation of BCCSP k modulo ∼ RS it suffices to enrich the axiom system E ∪ { RS } with finitely many axioms allowingone to eliminate all occurrences of k from closed BCCSP k terms. In fact, by letting E RS denote the axiom system E ∪ { RS } suitably enriched with such elimination axioms, theelimination result consists in proving that for every closed BCCSP k term p there is a closed L. ACETO, V. CASTIGLIONI, A. ING ´OLFSD ´OTTIR, B. LUTTIK, AND M. R. PEDERSEN (RS) a ( bx + by + z ) ≈ a ( bx + by + z ) + a ( bx + z ) (RSP1) ( ax + ay + u ) k ( bz + bw + v ) ≈ ( ax + u ) k ( bz + bw + v ) + ( ay + u ) k ( bz + bw + v )++( ax + ay + u ) k ( bz + v ) + ( ax + ay + u ) k ( bw + v ) (RSP2) (cid:0)P i ∈ I a i x i (cid:1) k ( by + bz + w ) ≈ (cid:0)P i ∈ I a i x i (cid:1) k ( by + w ) + (cid:0)P i ∈ I a i x i (cid:1) k ( bz + w )++ P i ∈ I a i ( x i k ( by + bz + w ))where a j = a k whenever j = k for j, k ∈ I E RS = E ∪ { RS, RSP1, RSP2, EL2 } Table 5.
Additional axioms for ready simulation equivalence.BCCSP term q (i.e., without any occurrence of k in it) such that E RS ⊢ p ≈ q . Then, thecompleteness of the proposed axiom system over BCCSP k is a direct consequence of thatover BCCSP proved in [19].Clearly, EL3 would allow us to obtain the desired elimination, but, as previously men-tioned, it is a schema that finitely presents an infinite collection of equations, and thus anaxiom system including it is infinite. Instead, we include EL2, which is a variant of EL3that generates only finitely many axioms (see Table 4), and the schemata RSP1 and RSP2that characterise the distributivity of k over + modulo ∼ RS (see Table 5).First of all, we notice that the axiom system E RS = E ∪{ RS , RSP1 , RSP2 , EL2 } is soundmodulo ready simulation equivalence. Theorem 3.2 ( E RS soundness) . The axiom system E RS is sound for BCCSP k modulo readysimulation equivalence, namely whenever E RS ⊢ p ≈ q then p ∼ RS q . Let us focus now on ground-completeness. Intuitively, RSP1 and RSP2 have beenconstructed in such a way that the set of initial actions of the two arguments of k is preserved,while the initial term is reduced to a sum of terms of smaller size. Briefly, according to themain features of ready simulation semantics, axiom RSP1 allows us to distribute k over +when both arguments of k have nondeterministic choices among summands having the sameinitial action. Conversely, axiom RSP2 deals with the case in which only one argument of k has summands with the same initial action. In order to preserve the branching structureof the process, which is fundamental to guarantee the soundness of the axioms modulo ∼ RS ,both RSP1 and RSP2 take into account the behaviour of both arguments of k : the terms inthe right-hand side of both axioms are such that whenever the initial nondeterministic choiceof one argument of k is resolved, the entire behaviour of the other argument is preserved.In fact, we stress that a simplified version of, e.g., RSP1 in which only one argument of k distributes over + would not be sound modulo ∼ RS . Consider, for instance, the process p = ( a + aa + b ) k c . It is immediate to verify that p RS ( a + b ) k c + ( aa + b ) k c (since p RS ( a + b ) k c + ( aa + b ) k c ).The idea is that by (repeatedly) applying axioms RSP1 and RSP2, from left to right, weare able to reduce a process of the form ( P i ∈ I p i ) k ( P j ∈ J p j ) to one of the form P k ∈ K p k suchthat whenever p k has k as head operator then p k = P h ∈ H a h p h k P l ∈ L b l p l , with a h = a h ′ for h = h ′ , and b l = b l ′ for l = l ′ , for some closed BCCSP k terms p h , p l . The eliminationof k from these terms can then proceed by means of the finitary refinement EL2 of theexpansion law presented in Table 4. In particular, we notice that RSP2 is needed because N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 9
RSP1 alone does not allow us to reduce all processes of the form ( P i ∈ I p i ) k ( P j ∈ J p j ) intoa sum of processes to which EL2 can be applied. This is mainly due to the fact that, inorder to be sound modulo ∼ RS , RSP1 imposes constraints on the form of both argumentsof a process ( P i ∈ I p i ) k ( P j ∈ J p j ).We can then proceed to prove the elimination result, starting from a useful remark onthe form of closed BCCSP terms. Remark 3.3 (General form of BCCSP processes) . Given any closed BCCSP term p , wecan assume, without loss of generality, that p = P i ∈ I a i p i for some finite index set I , actions a i ∈ A , and closed BCCSP terms p i , for i ∈ I . In fact, in case p is not already in this shape,then by applying axioms A0 and A1 in Table 3 we can remove superfluous occurrences of summands. In particular, we remark that this transformation does not increase the numberof operator symbols occurring in p . Lemma 3.4.
For all closed
BCCSP terms p and q there exists a closed BCCSP term r such that E RS ⊢ p k q ≈ r .Proof. The proof is by induction on size( p ) + size( q ). Since p, q are closed BCCSP terms,we can assume that p = P i ∈ I a i p i and q = P j ∈ J b j q j (see Remark 3.3). We proceed by acase analysis according to the cardinalities of the sets I and J .(1) Case | I | = 0 or | J | = 0. In that case we can apply axioms P0 and P1 in Table 3 toobtain that either p k q ≈ p or p k q ≈ q .(2) Case | I | = | J | = 1. Let I = { i } and J = { j } . In this case we have that p k q (EL2) ≈ a i ( p i k b j q j ) + b j ( a i p i k q j ) , so by induction hypothesis there exist closed BCCSP terms r i and r j such that p i k b j q j ≈ r i and a i p i k q j ≈ r j . We can conclude that p k q ≈ a i r i + b j r j , where a i r i + b j r j is a closed BCCSPterm.(3) Case | I | = 1 and | J | >
1. We consider two sub-cases: • There exist j , j ∈ J such that j = j and b j = b j . In this case we have that p k q (A2) ≈ p k b j q j + b j q j + X j ∈ J \{ j ,j } b j q j (RSP2) ≈ a i p i k ( b j q j + b j q j + X j ∈ J \{ j ,j } b j q j ) + p k ( b j q j + X j ∈ J \{ j ,j } b j q j ) + p k ( b j q j + X j ∈ J \{ j ,j } b j q j ) . By the induction hypothesis there exist closed BCCSP terms r , r , and r such that a i p i k ( b j q j + b j q j + X j ∈ J \{ j ,j } b j q j ) ≈ r
10 L. ACETO, V. CASTIGLIONI, A. ING ´OLFSD ´OTTIR, B. LUTTIK, AND M. R. PEDERSEN p k ( b j q j + X j ∈ J \{ j ,j } b j q j ) ≈ r p k ( b j q j + X j ∈ J \{ j ,j } b j q j ) ≈ r . We have therefore obtained that p k q ≈ a i r + r + r for the closed BCCSP term a r + r + r . • For all j , j ∈ J such that j = j we have b j = b j . In this case we have that p k q (EL2) ≈ a i ( p i k X j ∈ J b j q j ) + X j ∈ J b j ( p k q j ) , and the induction hypothesis then gives, for each j ∈ J , a closed BCCSP term r j such that p k q j ≈ r j as well as a closed BCCSP term r ′ such that p i k X j ∈ J b j q j ≈ r ′ . Therefore, p k q is equivalent to the closed BCCSP term a i r ′ + P j ∈ J b j r j .(4) Case | I | > | J | = 1. This case can be handled symmetrically to the case where | I | = 1 and | J | > | I | > | J | >
1. We consider four sub-cases: • There exist i and i such that i = i and a i = a i , and there exist j and j such that j = j and b j = b j . In this case we have that p k q ≈ ( a i p i + a i p i + X i ∈ I \{ i ,i } a i p i ) k ( b j q j + b j q j + X j ∈ J \{ j ,j } b j q j ) (RSP1) ≈ ( a i p i + X i ∈ I \{ i ,i } a i p i ) k ( b j q j + b j q j + X j ∈ J \{ j ,j } b j q j )+ ( a i p i + X i ∈ I \{ i ,i } a i p ) k ( b j q j + b j q j + X j ∈ J \{ j ,j } b j q j )+ ( a i p i + a i p i + X i ∈ I \{ i ,i } a i p i ) k ( b j q j + X j ∈ J \{ j ,j } b j q j )+ ( a i p i + a i p i + X i ∈ I \{ i ,i } a i p i ) k ( b j q j + X j ∈ J \{ j ,j } b j q j ) . By the induction hypothesis there are closed BCCSP terms r , r , r , and r suchthat r ≈ ( a i p i + X i ∈ I \{ i ,i } a i p i ) k ( b j q j + b j q j + X j ∈ J \{ j ,j } b j q j ) r ≈ ( a i p i + X i ∈ I \{ i ,i } a i p ) k ( b j q j + b j q j + X j ∈ J \{ j ,j } b j q j ) N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 11 r ≈ ( a i p i + a i p i + X i ∈ I \{ i ,i } a i p i ) k ( b j q j + X j ∈ J \{ j ,j } b j q j ) r ≈ ( a i p i + a i p i + X i ∈ I \{ i ,i } a i p i ) k ( b j q j + X j ∈ J \{ j ,j } b j q j ) . Hence p k q is equivalent to the closed BCCSP term r + r + r + r . • There exist i and i such that i = i and a i = a i , and for all j and j such that j = j we have b j = b j . In this case we have that p k q ≈ ( a i p i + a i p i + X i ∈ I \{ i ,i } a i p i ) k ( X j ∈ J b j q j ) (P1) ≈ ( X j ∈ J b j q j ) k ( a i p i + a i p i + X i ∈ I \{ i ,i } a i p i ) (RSP2) ≈ X j ∈ J b j ( q j k ( a i p i + a i p i + X i ∈ I \{ i ,i } a i p i ))+ ( X j ∈ J b j q j ) k ( a i p i + X i ∈ I \{ i ,i } a i p i ) + ( X j ∈ J b j q j ) k ( a i p i + X i ∈ I \{ i ,i } a i p i ) , so the induction hypothesis gives, for each j ∈ J , a closed BCCSP term r j such that r j ≈ q j k ( a i p i + a i p i + X i ∈ I \{ i ,i } a i p i ) , as well as closed BCCSP terms r ′ and r ′′ such that r ′ ≈ ( X j ∈ J b j q j ) k ( a i p i + X i ∈ I \{ i ,i } a i p i ) r ′′ ≈ ( X j ∈ J b j q j ) k ( a i p i + X i ∈ I \{ i ,i } a i p i ) . Thus p k q is equivalent to the closed BCCSP term P j ∈ J b j r j + r ′ + r ′′ . • For all i and i such that i = i we have a i = a i , and there exist j and j such that j = j and b j = b j . This case follows by applying a symmetrical argument to that used in the previousitem and it is therefore omitted. • For all i and i such that i = i we have a i = a i , and for all j and j such that j = j we have b j = b j . In this case we have that p k q ≈ X i ∈ I a i p i k X j ∈ J b j q j and all the conditions for an application of axiom EL2 in Table4 are satisfied. Hence p k q (EL2) ≈ X i ∈ I a i ( p i k X j ∈ J b j q j ) + X j ∈ J b j ( X i ∈ I a i p i k q j ) , and by the induction hypothesis there exist, for each i ∈ I , a closed BCCSP term r i such that r i ≈ p i k X j ∈ J b j q j , and for each j ∈ J , a closed BCCSP term r ′ j such that r ′ j ≈ X i ∈ I a i p i k q j . Therefore p k q is equivalent to the closed BCCSP term P i ∈ I a i r i + P j ∈ J b j r ′ j . Proposition 3.5 ( E RS elimination) . For every closed
BCCSP k term p there exists a BCCSP term q such that E RS ⊢ p ≈ q .Proof. Straightforward by induction on the structure of p , using Lemma 3.4 in the case that p is of the form p k p for some processes p and p .The ground-completeness of E RS then follows from the ground-completeness of E ∪{ RS } over BCCSP [19]. Theorem 3.6 ( E RS completeness) . The axiom system E RS is a ground-complete axioma-tisation of BCCSP k modulo ready simulation equivalence, i.e., whenever p ∼ RS q then E RS ⊢ p ≈ q . We remark that since axioms RSP1, RSP2, and EL2 are sound modulo ready simula-tion equivalence, they are automatically sound modulo all the equivalences in the spectrumthat are coarser than ∼ RS , namely the completed simulation, simulation, and (decorated)trace equivalences. Hence, we can easily obtain finite, ground-complete axiomatisationsof BCCSP k modulo each of those equivalences by adding RSP1, RSP2 and EL2 to therespective ground-complete axiomatisations of BCCSP that have been proposed in the lit-erature [8, 19]. However, for each of those equivalences we can provide stronger axiomsthat give a more elegant characterisation of the distributivity properties of k over +. Inparticular, the axiom schema RSP2 yields |A| · |A| equational axioms and EL2 yields 2 |A| equational axioms. By exploiting the various forms of distributivity of parallel compositionover choice, we can obtain more concise ground-complete axiomatisations of BCCSP k mod-ulo the coarser equivalences. We devote the next two sections to the presentation of theseresults. 4. Completed simulation and simulation
In this section we refine the axiom system E RS to obtain finite, ground-complete axioma-tisations of BCCSP k modulo completed simulation and simulation equivalences. To thisend, we replace RSP1 and RSP2 with new axioms, tailored for the considered semantics,that allow us to obtain the elimination of k from closed BCCSP k terms, while using lessrestrictive forms of distributivity of k over +.Let us focus first on completed simulation equivalence. We can use axioms CSP1 andCSP2 in Table 6 to characterise restricted forms of distributivity of k over + modulo ∼ CS .Intuitively, CSP1 is the completed simulation counterpart of RSP1, and CSP2 is that ofRSP2. Notice that both CSP1 and CSP2 are such that, when distributing k over +, we never N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 13 (CS) a ( bx + y + z ) ≈ a ( bx + y + z ) + a ( bx + z ) (CSP1) ( ax + by + u ) k ( cz + dw + v ) ≈ ( ax + u ) k ( cz + dw + v ) + ( by + u ) k ( cz + dw + v )++( ax + by + u ) k ( cz + v ) + ( ax + by + u ) k ( dw + v ) (CSP2) ax k ( by + cz + w ) ≈ a ( x k ( by + cz + w )) + ax k ( by + w ) + ax k ( cz + w ) E CS = E ∪ { CS, CSP1, CSP2, EL1 } (S) a ( x + y ) ≈ a ( x + y ) + ax (SP1) ( x + y ) k ( z + w ) ≈ x k ( z + w ) + y k ( z + w ) + ( x + y ) k z + ( x + y ) k w (SP2) ax k ( y + z ) ≈ a ( x k ( y + z )) + ax k y + ax k z E S = E ∪ { S, SP1, SP2, EL1 } Table 6.
Additional axioms for (completed) simulation equivalence.obtain as an argument of k , thus guaranteeing their soundness modulo ∼ CS . Moreover,we stress that CSP1 and CSP2 are not sound modulo ready simulation equivalence. Thisis due to the fact that both axioms allow for distributing k over + regardless of the initialactions of the summands. It is then immediate to check that, for instance, a k ( b + c ) RS a k b + a k c + a k ( b + c ), whereas a k ( b + c ) ∼ CS a k b + a k c + a k ( b + c ). Interestingly, due tothe relaxed constraints on distributivity, by (repeatedly) applying CSP1 and CSP2, fromleft to right, we are able to reduce a BCCSP k process of the form ( P i ∈ I p i ) k ( P j ∈ J p j ) to aBCCSP k process of the form P k ∈ K p k such that whenever p k has k as head operator then p k = a k q k k b k q ′ k for some q k , q ′ k . We can then use the refinement EL1 of the expansion lawto proceed with the elimination of k from these terms.Consider the axiom system E CS = E ∪ { CS,CSP1,CSP2,EL1 } . We can formalise theelimination result for ∼ CS as a direct consequence of the following result. Lemma 4.1.
For all closed
BCCSP terms p and q there exists a closed BCCSP term r such that E CS ⊢ p k q ≈ r .Proof. The proof is by induction on size( p )+size( q ). First note that, since p and q are closedBCCSP terms, we may assume that p = P i ∈ I a i p i and q = P j ∈ J b j q j (see Remark 3.3).We proceed by a case analysis according to the cardinalities of the sets I and J .(1) Case | I | = 0 or | J | = 0. First note that if | J | = 0, i.e., J = ∅ , then q = , so p k q ≈ p by P0, and p is the required closed BCCSP term. Similarly, if | I | = 0, i.e., I = ∅ , then p = , so p k q ≈ q k p ≈ q by P1 and P0 in Table 3, and q is the required closed BCCSPterm.(2) Case | I | = | J | = 1. Let I = { i } and J = { j } , then p k q (EL1) ≈ a i ( p i k q ) + b j ( p k q j ) . Since size( p i ) < size( p ) and size( q j ) < size( q ), by the induction hypothesis there existclosed BCCSP terms r i and r j such that p i k q ≈ r i and p k q j ≈ r j . It follows that p k q ≈ a i r i + b j r j and clearly a i r i + b j r j is a closed BCCSP term. (3) Case | I | = 1 and | J | >
1. We can then assume that I = { i } and there exist j , j ∈ J such that j = j , then p k q (CSP2) ≈ a i ( p i k q ) + p k (cid:16)P j ∈ J \{ j } b j q j (cid:17) + p k (cid:16)P j ∈ J \{ j } b j q j (cid:17) . Since size( p i ) < size( p ), size( P j ∈ J \{ j } b j q j ) < size( q ) and size( P j ∈ J \{ j } b j q j ) < size( q ), by the induction hypothesis there exist closed BCCSP terms r i , r j and r j such that p i k q ≈ r i p k X j ∈ J \{ j } b j q j ≈ r j and p k X j ∈ J \{ j } b j q j ≈ r j . So we have p k q ≈ a i r i + r j + r j and a i r i + r j + r j is a closed BCCSP term.(4) Case | I | > | J | = 1. The proof is similar as in the previous case, with an additionalapplication of axiom P1 in Table 3.(5) Case | I | , | J | >
1. In this case there exist i , i ∈ I with i = i and j , j ∈ J with j = j . Then p k q (CSP1) ≈ X i ∈ I \{ i } a i p i k q + X i ∈ I \{ i } a i p i k q + p k X j ∈ J \{ j } b j q j + p k X j ∈ J \{ j } b j q i . Note that size( P i ∈ I \{ i } a i p i ) < size( p ), size( P i ∈ I \{ i } a i p i ) < size( p ), size( P j ∈ J \{ j } b j q j ) < size( q ) and size( P j ∈ J \{ j } b j q i ) < size( q ), so by the induction hypothesis there exist r i , r i , r j and r j such that X i ∈ I \{ i } a i p i k q ≈ r i , X i ∈ I \{ i } a i p i k q ≈ r i p k X j ∈ J \{ j } b j q j ≈ r j p k X j ∈ J \{ j } b j q i ≈ r j . It follows that p k q ≈ r i + r i + r j + r j and r i + r i + r j + r j is a closed BCCSPterm. Proposition 4.2 ( E CS elimination) . For every closed
BCCSP k term p there exists a BCCSP term q such that E CS ⊢ p ≈ q .Proof. Straightforward by induction on the structure of p , using Lemma 4.1 in the case that p is of the form p k p for some p and p .A similar reasoning could be applied to obtain the elimination result for simulationequivalence. Although this result could be directly derived by the soundness of CSP1and CSP2 modulo simulation equivalence, stronger distributivity properties for parallelcomposition over summation hold modulo ∼ S . Instead of CSP1 and CSP2, we include SP1and SP2 in Table 6, letting E S = E ∪ { S,SP1,SP2,EL1 } . By showing that the axioms CS, N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 15
CSP1 and CSP2 are derivable from E S , we can obtain the elimination result for simulationequivalence as an immediate corollary of that for completed simulation. Lemma 4.3.
The axioms of the system E CS are derivable from the axiom system E S , namely: (1) E S ⊢ CS , (2) E S ⊢ CSP1 , and (3) E S ⊢ CSP2 .Proof.
We start with the derivation of CS. We have the following equational derivation: a ( bx + y + z ) (A1) , (A2) ≈ a ( bx + z + y ) (S) ≈ a ( bx + z + y ) + a ( bx + z ) (A1) , (A2) ≈ a ( bx + y + z ) + a ( bx + z ) . In the case of CSP1, we have the following equational derivation:( ax + by + u ) k ( cz + dw + v ) (A1) , (A3) ≈ ( ax + u + by + u ) k ( cz + v + dw + v ) (SP1) ≈ ( ax + u ) k ( cz + v + dw + v ) + ( by + u ) k ( cz + v + dw + v )+ ( ax + u + by + u ) k ( cz + v ) + ( ax + u + by + u ) k ( dw + v ) (A1) , (A3) ≈ ( ax + u ) k ( cz + dw + v ) + ( by + u ) k ( cz + dw + v )+ ( ax + by + u ) k ( cz + v ) + ( ax + by + u ) k ( dw + v ) . Finally, for CSP2 we have the following equational derivation: ax k ( by + cz + w ) (A1) , (A3) ≈ ax k ( by + w + cz + w ) (SP2) ≈ a ( x k ( by + w + cz + w )) + ax k ( by + w ) + ax k ( cz + w ) (A1) , (A3) ≈ a ( x k ( by + cz + w )) + ax k ( by + w ) + ax k ( cz + w ) . Proposition 4.4 ( E S elimination) . For every closed
BCCSP k term p there exists a closed BCCSP term q such that E S ⊢ p ≈ q . Remark 4.5.
As we showed in Lemma 4.3, the axiom system E S proves all the equationsin E CS . A natural question at this point is whether all the equations in E RS can be derivedfrom E CS . Indeed, that result would allow one to infer Proposition 4.2 (the eliminationresult for completed simulation equivalence) from Proposition 3.5 (the elimination resultfor ready simulation equivalence). The answer is negative, as it is not possible to deriveEL2 from E CS . To prove this claim, we have used Mace4 [25] to generate a model for E CS in which EL2 does not hold (if there are at least two actions). The code can be found inAppendix A. Below, we only present the model and show that it does not satisfy EL2. Theverification that the model indeed satisfies the axioms of E CS is lengthy and tedious. We a : 0 1 2 3 42 2 3 4 4 b : 0 1 2 3 43 2 3 4 4 k : 0 1 2 3 40 0 1 2 3 41 1 0 2 1 22 2 2 3 4 43 3 1 4 4 44 4 2 4 4 4 +: 0 1 2 3 40 0 1 2 3 41 1 1 2 3 42 2 2 2 4 43 3 3 4 3 44 4 4 4 4 4 Table 7.
A model for E CS and E CT .used the mCRL2 toolset [10, 20] to double check the correctness of the model produced byMace4.Let A = { a, b } , and consider the model with carrier set { , , , , } and the operationsdefined in Table 7. This model does not satisfy( ax + by ) k ( az + bw ) ≈ a ( x k ( az + bw )) + b ( y k ( az + bw )) + a (( ax + by ) k z ) + b (( ax + by ) k w )for, e.g., the valuation x = 0, y = 0, z = 1, and w = 1.In light of the results above, and those in [19] showing that E ∪ { CS } and E ∪ { S } aresound and ground-complete axiomatisations of BCCSP modulo ∼ CS and ∼ S , respectively,we can infer that E CS and E S are ground-complete axiomatisations of BCCSP k modulocompleted simulation equivalence and simulation equivalence, respectively. Theorem 4.6 (Soundness and completeness of E CS and E S ) . Let X ∈ { CS , S } . The axiomsystem E X is a sound, ground-complete axiomatisation of BCCSP k modulo ∼ X , i.e., p ∼ X q if and only if E X ⊢ p ≈ q . At the end of Section 3 we noticed that the size of E RS is exponential in A . Now, as afinal remark, we observe that the size of E CS is polynomial in |A| , and the size of E S is linearin |A| . In detail, E CS contains |A| equations that are instances of CS, |A| equations thatare instances of CSP1, and |A| equations arising from (CSP2). On the other hand, in E S ,the axiom schemata S and SP2 yield |A| equations each, whereas SP1 is just one equation.5. Linear semantics: from ready traces to traces
We continue our journey in the spectrum by moving to the linear-time semantics. In thissection we consider trace semantics and all of its decorated versions, and we provide a finite,ground-complete axiomatisation for each of them (see Table 8).From a technical point of view, we can split the results of this section into two parts:(1) those for ready trace, failure trace, ready, and failures equivalence, and(2) those for completed trace, and trace equivalence.In both parts we prove the elimination result only for the finest semantics, namely readytrace (Proposition 5.3) and completed trace (Proposition 5.11) respectively. We then obtainthe remaining elimination results by showing that all the axioms in E X are provable from E Y , where X is finer than Y in the considered part. N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 17 (RT) a (cid:16)P |A| i =1 ( b i x i + b i y i ) + z (cid:17) ≈ a (cid:16)P |A| i =1 b i x i + z (cid:17) + a (cid:16)P |A| i =1 b i y i + z (cid:17) (FP) ( ax + ay + w ) k z ≈ ( ax + w ) k z + ( ay + w ) k z E RT = E ∪ { RT , FP , EL2 } (FT) ax + ay ≈ ax + ay + a ( x + y ) E FT = E ∪ { FT , RS , FP , EL2 } (R) a ( bx + z ) + a ( by + w ) ≈ a ( bx + by + z ) + a ( by + w ) E R = E ∪ { R, FP, EL2 } (F) ax + a ( y + z ) ≈ ax + a ( x + y ) + a ( y + z ) E F = E ∪ { F, R, FP, EL2 } Table 8.
Additional axioms for ready (trace) and failure (trace) equiva-lences.5.1.
From ready traces to failures.
First we deal with the decorated trace semanticsbased on the comparison of the failure and ready sets of processes.
Definition 5.1 (Readiness and failures equivalences) . • A failure pair of a process p is apair ( α, X ), with α ∈ A ∗ and X ⊆ A , such that p α −→ q for some process q with I ( q ) ∩ X = ∅ . We denote by F ( p ) the set of failure pairs of p . Two processes p and q are failuresequivalent , denoted p ∼ F q , if F ( p ) = F ( q ). • A ready pair of a process p is a pair ( α, X ), with α ∈ A ∗ and X ⊆ A , such that p α −→ q for some process q with I ( q ) = X . We let R ( p ) denote the set of ready pairs of p . Twoprocesses p and q are ready equivalent , written p ∼ R q , if R ( p ) = R ( q ). • A failure trace of a process p is a sequence X a X . . . a n X n , with X i ⊆ A and a i ∈ A ,such that there are p , . . . , p n ∈ P with p = p a −−→ p a −−→ . . . a n −−→ p n and I ( p i ) ∩ X i = ∅ for all 0 ≤ i ≤ n . We write FT ( p ) for the set of failure traces of p . Two processes p and q are failure trace equivalent , denoted p ∼ FT q , if FT ( p ) = FT ( q ). • A ready trace of a process p is a sequence X a X . . . a n X n , for X i ⊆ A and a i ∈ A , suchthat there are p , . . . p n ∈ P with p = p a −−→ p a −−→ . . . a n −−→ p n and I ( p i ) = X i for all0 ≤ i ≤ n . We write RT ( p ) for the set of ready traces of p . Two processes p and q are ready trace equivalent , denoted p ∼ RT q , if RT ( p ) = RT ( q ).We consider first the finest equivalence among those in Definition 5.1, namely readytrace equivalence. This can be considered as the linear counterpart of ready simulation: wefocus on the current execution of the process and we require that each step is mimickedby reaching processes having the same sets of initial actions. Interestingly, we can find asimilar correlation between the axioms characterising the distributivity of k over + modulothe two semantics. Consider axiom FP in Table 8. We can see this axiom as the linear counterpart of RSP1: since in the linear semantics we are interested only in the currentexecution of a process, we can characterise the distributivity of k over + by treating thetwo arguments of k independently from one another. To obtain the elimination result for ∼ RT we do not need to introduce the linear counterpart of axiom RSP2. In fact, FP imposesconstraints on the form of only one argument of k . Hence, it is possible to use it to reduce any process of the form ( P i ∈ I p i ) k ( P j ∈ J p j ) into a sum of processes to which EL2 can beapplied. We can in fact prove that the axioms in the system E RT = E ∪ { RT , FP , EL2 } aresufficient to eliminate all occurrences of k from closed BCCSP k terms. Lemma 5.2.
For all closed
BCCSP terms p, q there exists a closed
BCCSP term r suchthat E RT ⊢ p k q ≈ r .Proof. The proof proceeds by induction on the total number of operator symbols occurringin p and q together, not counting parentheses. First of all we notice that, since p and q are closed BCCSP terms, we can assume that p = P i ∈ I a i p i and q = P j ∈ J b j q j (seeRemark 3.3).We proceed by a case analysis according to the cardinalities of the sets I and J .(1) Case | I | = 0 or | J | = 0. In case that | J | = 0, i.e., J = ∅ , then q = , so E RT ⊢ p k q ≈ p by P0, and p is the required closed BCCSP term. Similarly, if | I | = 0, i.e., I = ∅ , then p = , so E RT ⊢ p k q ≈ q k p ≈ q by P0 and P1 in Table 3, and q is the required closedBCCSP term.(2) Case | I | = | J | = 1. Let I = { i } and J = { j } , then p k q (EL2) ≈ a i ( p i k q ) + b j ( p k q j ) . Since both p i and q j have fewer symbols than p and q , respectively, by the inductivehypothesis there exist closed BCCSP terms r i and r j such that E RT ⊢ p i k q ≈ r i and E RT ⊢ p k q j ≈ r j . It follows that E RT ⊢ p k q ≈ a i r i + b j r j , where a i r i + b j r j is aclosed BCCSP term.(3) Case | I | = 1 and | J | >
1. We can then assume that I = { i } and there exist j , j ∈ J such that j = j . We can now distinguish two cases, according to whether b j = b j forsome j , j ∈ J with j = j , or not. • There are some j , j ∈ J such that j = j and b j = b j . In this case we have that p k q (P1) ≈ X j ∈ J b j q j k p (FP) ≈ X j ∈ J \{ j } b j q j k p + X j ∈ J \{ j } b j q j k p . Since both P j ∈ J \{ j } b j q j and P j ∈ J \{ j } b j q j have fewer operator symbols than q , bythe inductive hypothesis there exist closed BCCSP terms r j and r j such that E RT ⊢ X j ∈ J \{ j } b j q j k p ≈ r j and E RT ⊢ X j ∈ J \{ j } b j q j k p ≈ r j . Therefore, we get that E RT ⊢ p k q ≈ r j + r j , where r j + r j is a closed BCCSP term. • For all j , j ∈ J such that j = j it holds that b j = b j . In this case wehave that p k q (EL2) ≈ a i ( p i k q ) + X j ∈ J b j ( p k q j ) , where p i has fewer operator symbols than p and each q j has fewer operator symbolsthan q . Hence, by the inductive hypothesis, we obtain that there is a closed BCCSPterm r i such that E RT ⊢ p i k q ≈ r i , N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 19 and for each j ∈ J there is a closed BCCSP term r j such that E RT ⊢ p k q j ≈ r j . We can thus conclude that E RT ⊢ p k q ≈ a i r i + P j ∈ J b j r j , where the right hand sideof the equation is a closed BCCSP term.(4) Case | I | > | J | = 1. The proof is similar as in the previous case, with an additionalinitial application of axiom P1 in Table 3.(5) Case | I | , | J | >
1. In this case there exist i , i ∈ I with i = i and j , j ∈ J with j = j . The proof follows the same reasoning used in the case | I | = 1 and | J | >
1; wedistinguish three cases: • There are i , i ∈ I such that i = i and a i = a i . In this case we have that p k q (FP) ≈ X i ∈ I \{ i } a i p i k q + X i ∈ I \{ i } a i p i k q . As both P i ∈ I \{ i } a i p i and P i ∈ I \{ i } a i p i have fewer operator symbols than p , by theinductive hypothesis we get that there are closed BCSSP terms r i and r i such that E RT ⊢ X i ∈ I \{ i } a i p i k q ≈ r i and E RT ⊢ X i ∈ I \{ i } a i p i k q ≈ r i . Consequently, we get that E RT ⊢ p k q ≈ r i + r i , where r i + r i is a closed BCCSPterm. • There are some j , j ∈ J such that j = j and b j = b j . This case can be obtained from the previous one, by an additional initial applicationof axiom P1 in Table 3. • For all i , i ∈ I such that i = i we have that a i = a i and for all j , j ∈ J such that j = j we have that b j = b j . In this case we have that p k q (EL2) ≈ X i ∈ I a i ( p i k q ) + X j ∈ J b j ( p k q j ) . Since for each i ∈ I , the term p i has fewer operator symbols than p , by the inductivehypothesis we get that there is a closed BCCSP term r i such that E RT ⊢ p i k q ≈ r i . Similarly, for each j ∈ J , the term q j has fewer operator symbols than q , so that bythe inductive hypothesis we get that there is a closed BCCSP term r j such that E RT ⊢ p k q j ≈ r j . Therefore, we can conclude that E RT ⊢ p k q ≈ P i ∈ I a i r i + P j ∈ J b j r j , where the righthand side of the equation is a closed BCCSP term. Proposition 5.3 ( E RT elimination) . For every closed
BCCSP k term p there is a closed BCCSP term q such that E RT ⊢ p ≈ q .Proof. The proof follows by an easy induction on the structure of p , using Lemma 5.2 inthe case that p is of the form p k p for some p and p . a: 0 1 20 2 2 b: 0 1 20 0 0 k : 0 1 20 0 1 21 1 0 12 2 1 2 +: 0 1 20 0 1 21 1 1 22 2 2 2 Table 9.
A model for E RT and E CT . Remark 5.4.
Similarly to the case of completed simulation (cf. Remark 4.5), the reasonwhy we propose to prove directly the elimination result for ready trace equivalence is thataxiom RSP2 cannot be derived from those in E RT , even though all its closed instantiationscan be derived. Table 9 presents a model, found using Mace4, of E RT from which RSP2 isnot derivable: e.g., it does not satisfy ax k ( ay + az + w ) = a ( x k ( ay + az + w )) + ax k ( ay + w ) + ax k ( az + w ))for x = 0, y = 0, z = 0 and w = 1.We refer the interested reader to Appendix B for the Mace4 code.Interestingly, axiom FP also characterises the distributivity of k over + modulo ∼ FT , ∼ R and ∼ F . Consider the axiom systems E FT = E ∪ { FT,RS,FP,EL2 } , E R = E ∪ { R,FP,EL2 } and E F = E ∪ { F,R,FP,EL2 } . The following derivability relations among them and E RT arethen easy to check. Lemma 5.5. (1)
The axioms in the system E RT are derivable from E FT , namely E FT ⊢ RT . (2) The axioms in the system E RT are derivable from E R , namely E R ⊢ RT . (3) The axioms in the system E FT are derivable from E F , namely, (a) E F ⊢ FT , and (b) E F ⊢ RS .Moreover, also the axioms in the system E R are derivable from E F .Proof. (1) To simplify notation, we introduce some abbreviations: let |A| = n , and for each i ∈ { , . . . , n } let I i = { , . . . , n } \ { i } .First of all, we notice that if we can prove that E FT ⊢ a n X i =1 b i x i + z ! + a n X i =1 b i y i + z ! ≈ a n X i =1 b i x i + z ! + a n X i =1 b i y i + z ! ++ a n X i =1 (cid:0) b i x i + b i y i (cid:1) + z ! (5.1) E FT ⊢ a n X i =1 (cid:0) b i x i + b i y i (cid:1) + z ! ≈ a n X i =1 (cid:0) b i x i + b i y i (cid:1) + z ! + a n X i =1 b i x i + z ! , (5.2)then E FT ⊢ RT, easily follows as a n X i =1 (cid:0) b i x i + b i y i (cid:1) + z ! (5.2) ≈ a n X i =1 (cid:0) b i x i + b i y i (cid:1) + z ! + a n X i =1 b i x i + z ! N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 21(A1),(A3) , (5.2) ≈ a n X i =1 (cid:0) b i x i + b i y i (cid:1) + z ! + a n X i =1 b i x i + z ! + a n X i =1 b i y i + z ! ( A , (5.1) ≈ a n X i =1 b i x i + z ! + a n X i =1 b i y i + z ! . We now proceed to prove the Equations (5.1) and (5.2) separately.(a) Equation (5.1) directly follows by a n X i =1 b i x i + z ! + a n X i =1 b i y i + z ! (FT) ≈ a n X i =1 b i x i + z ! + a n X i =1 b i y i + z ! + a (cid:16) n X i =1 b i x i + z (cid:17) + (cid:16) n X i =1 b i y i + z (cid:17)! (A1),(A2) ≈ a n X i =1 b i x i + z ! + a n X i =1 b i y i + z ! + a n X i =1 (cid:0) b i x i + b i y i (cid:1) + z ! . (b) Before proceeding to the proof of Equation (5.2), we notice that a ( bx + by + z ) (RS) ≈ a ( bx + by + z ) + a ( bx + z ) (A1),(RS) ≈ a ( bx + by + z ) + a ( bx + z ) + a ( by + z ) (FT),(A3) ≈ a ( bx + z ) + a ( by + z ) , so that E FT ⊢ a ( bx + by + z ) ≈ a ( bx + z ) + a ( by + z ) . (5.3)Using Equation (5.3), we can establish Equation (5.2) holds for all n ≥ n .If n = 1, then a n X i =1 ( b i x i + b i y i ) + z ! (5.3) ≈ a ( b x + z ) + a ( b y + z ) (A1),(A2),(A3) ≈ a ( b x + z ) + a ( b y + z ) + a ( b x + z ) (5.3) ≈ a n X i =1 ( b i x i + b i y i ) + z ! + a n X i =1 b i x i + z ! Let n ≥
1, and suppose that Equation (5.2) holds (IH). Then a n +1 X i =1 (cid:0) b i x i + b i y i (cid:1) + z ! (A1) , (A2) , (5.3) ≈ a n X i =1 (cid:0) b i x i + b i y i (cid:1) + b n +1 x n +1 + z ! + a n X i =1 (cid:0) b i x i + b i y i (cid:1) + b n +1 y n +1 + z ! ≈ a n X i =1 (cid:0) b i x i + b i y i (cid:1) + b n +1 x n +1 + z ! + a n X i =1 b i x i + b n +1 x n +1 + z ! + a n X i =1 (cid:0) b i x i + b i y i (cid:1) + b n +1 y n +1 + z ! (A1) , (A2) , (5.3) ≈ a n +1 X i =1 (cid:0) b i x i + b i y i (cid:1) + z ! + a n +1 X i =1 b i x i + z ! We conclude that Equation (5.1) holds for all n ≥
1, and hence, in particular, for n = |A| .(2) To prove that E R ⊢ RT, we first establish that for every n ≥ E R ⊢ a n X i =1 b i x i + z ! + a n X i =1 b i y i + w ! ≈ a n X i =1 ( b i x i + b i y i ) + z ! + a n X i =1 b i y i + w ! . (5.4)We proceed by induction on n ≥ n = 1, then a n X i =1 b i x i + z ! + a n X i =1 b i y i + w ! (R) ≈ a n X i =1 ( b i x i + b i y i ) + z ! + a n X i =1 b i y i + w ! Let n ≥
1, and suppose that Equation (5.4) holds (IH). Then a n +1 X i =1 b i x i + z ! + a n +1 X i =1 b i y i + w ! (A1) , (A2) ≈ a b n +1 x n +1 + n X i =1 b i x i + z ! + a b n +1 y n +1 + n X i =1 b i y i + w ! (R) ≈ a b n +1 x n +1 + b n +1 y n +1 + n X i =1 b i x i + z ! + a b n +1 y n +1 + n X i =1 b i y i + w ! (A1) , (A2) ≈ a n X i =1 b i x i + b n +1 x n +1 + b n +1 y n +1 + z ! + a n X i =1 b i y i + b n +1 y n +1 + w ! (IH) ≈ a n X i =1 ( b i x i + b i y i ) + b n +1 x n +1 + b n +1 y n +1 + z ! + a n X i =1 b i y i + b n +1 y n +1 + w ! (A1) , (A2) ≈ a n +1 X i =1 ( b i x i + b i y i ) + z ! + a n +1 X i =1 b i y i + w ! We conclude that Equation (5.4) holds for all n ≥ N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 23
We can now derive axiom (RT) as follows: a |A| X i =1 ( b i x i + b i y i ) + z (A3) ≈ a |A| X i =1 ( b i x i + b i y i ) + z + |A| X i =1 ( b i x i + b i y i ) + z (A1) , (A2) ≈ a |A| X i =1 ( b i x i + b i y i ) + z + |A| X i =1 b i x i + |A| X i =1 b i y i + z (A1) , (A2) , (5.4) ≈ a |A| X i =1 b i y i + z + |A| X i =1 b i x i + |A| X i =1 b i y i + z (A1) , (A2) ≈ a |A| X i =1 ( b i x i + b i y i ) + z + a |A| X i =1 b i y i + z (5.4) ≈ a |A| X i =1 b i x i + z + a |A| X i =1 b i y i + z . (3) The second claim, namely that the axioms in E R are derivable from E F , follows directlyfrom E R ⊆ E F .To prove the first claim, we start by showing that the axiom FT in Table 8 is derivablefrom the axiom system E F . ax + ay (A0) ≈ ax + a ( y + ) (F) ≈ ax + a ( x + y ) + a ( y + ) (A0),(A1) ≈ ax + ay + a ( x + y ) . We now proceed to prove that the axiom RS in Table 8 is derivable from the axiomsystem E F . a ( bx + by + z ) + a ( bx + z ) (A1),(A2) ≈ a ( bx + z ) + a ( by + ( bx + z )) (R) ≈ a ( bx + by + z ) + a ( by + ( bx + z )) (A2),(A3) ≈ a ( bx + by + z ) . The next proposition is then a corollary of Proposition 5.3 and Lemma 5.5.
Proposition 5.6 ( E FT , E R , E F elimination) . Let X ∈ { FT , R , F } . For every BCCSP k term p there is a closed BCCSP term q such that E X ⊢ p ≈ q . In [8] it was proved that, under the assumption that A is finite, the axiom system E ∪ { RT } is a ground-complete axiomatisation of BCCSP modulo ∼ RT . Moreover, it was (CT) a ( bx + z ) + a ( cy + w ) ≈ a ( bx + cy + z + w ) (CTP) ( ax + by + w ) k z ≈ ( ax + w ) k z + ( by + w ) k z E CT = E ∪ { CT, CTP, EL1 } (T) ax + ay ≈ a ( x + y ) (TP) ( x + y ) k z ≈ x k z + y k z E T = E ∪ { T, TP, EL1 } Table 10.
Additional axioms for completed trace and trace equivalences.also proved that E ∪ { FT,RS } is a ground-complete axiomatisation of BCCSP modulo ∼ FT .The ground-completeness of E ∪ { R } , modulo ∼ R , and that of E ∪ { F , R } , modulo ∼ F , overBCCSP were proved in [19]. Consequently, the soundness and ground-completeness of theproposed axioms systems can then be derived from the elimination results above and thecompleteness results given in [8, 19]. Theorem 5.7 (Soundness and completeness of E RT , E FT , E R and E F ) . Let X ∈ { RT , FT , R , F } .The axiom system E X is a sound, ground-complete axiomatisation of BCCSP k modulo ∼ X ,i.e., p ∼ X q if and only if E X ⊢ p ≈ q . Completed traces and traces.
It remains to consider completed trace equivalenceand trace equivalence.
Definition 5.8 (Trace and completed trace equivalences) . Two processes p and q are traceequivalent , denoted p ∼ T q , if T ( p ) = T ( q ). If, in addition, it holds that CT ( p ) = CT ( q ), then p and q are completed trace equivalent , denoted p ∼ CT q . Remark 5.9.
Since we are considering only BCCSP k processes with finite traces, then toprove completed trace equivalence it is enough to check whether two processes have thesame sets of completed traces. In fact, in our setting, CT ( p ) = CT ( q ) implies T ( p ) = T ( q ) forall processes p, q .Consider the axiom systems E CT = E ∪ { CT,CTP,EL1 } and E T = E ∪ { T,TP,EL1 } ,presented in Table 10. In the same way that axiom FP is the linear counterpart of RSP1and RSP2, we have that CTP is the linear counterpart of CSP1 and CSP2, and TP is thatof SP1 and SP2. It is then easy to check that we can use the axioms in E CT to obtain theelimination result for ∼ CT . Lemma 5.10.
For every closed
BCCSP terms p, q there is a closed
BCCSP term r suchthat E CT ⊢ p k q ≈ r .Proof. The proof proceeds by induction on the total number of operator symbols occurringin p and q together, not counting parentheses. Since p and q are closed BCCSP terms, wecan assume that p = P i ∈ I a i p i and q = P j ∈ J b j q j (see Remark 3.3).We proceed by a case analysis according to the cardinalities of the sets I and J .(1) Case | I | = 0 or | J | = 0. In case that | J | = 0, i.e., J = ∅ , then q = , so E CT ⊢ p k q ≈ p by P0, and p is the required closed BCCSP term. Similarly, if | I | = 0, i.e., I = ∅ , then N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 25 p = , so E CT ⊢ p k q ≈ q k p ≈ q by P0 and P1, and q is the required closed BCCSPterm.(2) Case | I | = | J | = 1. Let I = { i } and J = { j } , then p k q (EL1) ≈ a i ( p i k q ) + b j ( p k q j ) . Since both p i and q j have fewer symbols than p and q , respectively, by the inductivehypothesis there exist closed BCCSP terms r i and r j such that E CT ⊢ p i k q ≈ r i and E CT ⊢ p k q j ≈ r j . It follows that E CT ⊢ p k q ≈ a i r i + b j r j , where a i r i + b j r j is aclosed BCCSP term.(3) Case | I | = 1 and | J | >
1. We can then assume that I = { i } and there exist j , j ∈ J such that j = j , then p k q (A2),(P1) ≈ b j q j + b j q j + X j ∈ J \{ j ,j } b j q j k p (CTP) ≈ b j q j + X j ∈ J \{ j ,j } b j q j k p + b j q j + X j ∈ J \{ j ,j } b j q j k p (A2),(P1) ≈ X j ∈ J \{ j } b j q j k p + X j ∈ J \{ j } b j q j k p . Since P j ∈ J \{ j } b j q j and P j ∈ J \{ j } b j q j have fewer symbols than q , by the inductivehypothesis there exist closed BCCSP terms r j and r j such that E CT ⊢ p k X j ∈ J \{ j } b j q j ≈ r j E CT ⊢ p k X j ∈ J \{ j } b j q j ≈ r j . So we have E CT ⊢ p k q ≈ r j + r j and r j + r j is a closed BCCSP term.(4) Case | I | > | J | = 1. This case can be obtained as the previous one, without theinitial application of axiom P1.(5) Case | I | , | J | >
1. In this case there exist i , i ∈ I with i = i and j , j ∈ J with j = j . Then p k q (A2),(P1),(CTP) ≈ X i ∈ I \{ i } a i p i k q + X i ∈ I \{ i } a i p i k q ++ p k X j ∈ J \{ j } b j q j + p k X j ∈ J \{ j } b j q i . Note that P i ∈ I \{ i } a i p i and P i ∈ I \{ i } a i p i have fewer symbols than p , and P j ∈ J \{ j } b j q j and P j ∈ J \{ j } b j q i have fewer symbols than q , so by the inductive hypothesis there exist r i , r i , r j and r j such that E CT ⊢ X i ∈ I \{ i } a i p i k q ≈ r i E CT ⊢ X i ∈ I \{ i } a i p i k q ≈ r i E CT ⊢ p k X j ∈ J \{ j } b j q j ≈ r j E CT ⊢ p k X j ∈ J \{ j } b j q i ≈ r j . Then E CT ⊢ p k q ≈ r i + r i + r j + r j and r i + r i + r j + r j is a closed BCCSP term. Proposition 5.11 ( E CT elimination) . For every closed
BCCSP k term p there is a closedBCCSP term q such that E CT ⊢ p ≈ q .Proof. The proof follows by an easy induction on p , using Lemma 5.10 in the case that p isof the form p k p for some p and p .Moreover, the elimination for ∼ T follows from the fact that the axioms in E CT arederivable from those in E T . Lemma 5.12.
The axioms in the system E CT are derivable from E T , namely, (1) E T ⊢ CT , and (2) E T ⊢ CTP .Proof.
In the case of CT, we have the following equational derivation: a ( bx + z ) + a ( cy + w ) (T) ≈ a ( bx + z + cy + w ) . For CTP, we have the following equational derivation:( ax + by + w ) k z (A1) , (A3) ≈ ( ax + w + by + w ) k z (TP) ≈ ( ax + w ) k z + ( by + w ) k z . Proposition 5.13 ( E T elimination) . For every closed
BCCSP k term p there exists a closed BCCSP term q such that E T ⊢ p ≈ q . Remark 5.14.
We can use the models in Tables 7 and 9 to prove that, respectively, neitheraxiom EL2 nor axiom CSP2 are derivable from E CT . In detail, in the case of EL2 thecounterexample presented in Remark 4.5 holds also in the case of E CT . In the case of CSP2,we notice that the instance of RSP2 used in Remark 5.4 is in fact also an instance of CSP2,and thus the counterexample for RSP2 holds also in this case. We refer the interested readerto Appendix C for the Mace4 codes generating the desired models.In light of Proposition 5.11, the ground-completeness of E CT over BCCSP k modulo ∼ CT follows from that of E ∪ { CT } over BCCSP provided in [19]. Similarly, the ground-completeness of E ∪ { T } over BCCSP proved in [19] and Proposition 5.13 give us theground-completeness of E T over BCCSP k . Theorem 5.15 (Soundness and completeness of E CT and E T ) . Let X ∈ { CT , T } . The axiomsystem E X is a ground-complete axiomatisation of BCCSP k modulo ∼ X , i.e., p ∼ X q if andonly if E X ⊢ p ≈ q . N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 27 The negative results
We dedicate this section to the negative results: we prove that all the congruences betweenpossible futures equivalence ( ∼ PF ) and bisimilarity ( ∼ B ) do not admit a finite, ground-complete axiomatisation over BCCSP k . This includes all the nested trace and nested sim-ulation equivalences. In [2] it was shown that, even if the set of actions is a singleton, thenested semantics admit no finite axiomatisation over BCCSP. Indeed, the presence of theadditional operator k might, at least in principle, allow us to finitely axiomatise the equa-tions over closed BCCSP terms that are valid modulo the considered equivalences. Hence,we prove these results directly.In detail, firstly we focus on the negative result for possible futures semantics, corre-sponding to the 2-nested trace semantics [22]. To obtain it, we apply the general techniqueused by Moller to prove that interleaving is not finitely axiomatisable modulo bisimilar-ity [27–29]. Briefly, the main idea is to identify a witness property . This is a specificproperty of BCCSP k terms, say W N for N ≥
0, that, when N is large enough , is an invari-ant that is preserved by provability from finite, sound axiom systems. Roughly, this meansthat if E is a finite set of axioms that are sound modulo possible futures equivalence, theequation p ≈ q can be derived from E , and N is larger than the size of all the terms inthe equations in E , then either both p and q satisfy W N , or none of them does. Then, weexhibit an infinite family of valid equations, called the witness family of equations , in which W N is not preserved, namely it is satisfied only by one side of each equation.Afterwards, we exploit the soundness modulo bisimilarity of the equations in the witnessfamily to lift the negative result for ∼ PF to all congruences between ∼ B and ∼ PF .Differently from the aforementioned negative results over BCCSP, ours are obtainedassuming that the set of actions contains at least two distinct elements. In fact, when theaction set is a singleton, and only in that case, the axiom ax k ( ay + az ) ≈ ax k ( ay + a ( y + z )) + ax k ( az + a ( y + z ))is sound modulo ∼ PF . Due to this axiom we were not able to prove the negative result for ∼ PF in the case that |A| = 1, which we leave as an open problem for future work.6.1. Possible futures equivalence.
According to possible futures equivalence [32] twoprocesses are deemed equivalent if, by performing the same traces, they reach processesthat are trace equivalent. For this reason, possible futures equivalence is also known as the2- nested trace equivalence [22].
Definition 6.1 (Possible futures equivalence) . A possible future of a process p is a pair( α, X ) where α ∈ A ∗ and X ⊆ A ∗ such that p α −→ p ′ for some p ′ with X = T ( p ′ ). We write PF ( p ) for the set of possible futures of p . Two processes p and q are said to be possiblefutures equivalent , denoted p ∼ PF q , if PF ( p ) = PF ( q ).Our order of business is to prove the following result. Theorem 6.2.
Assume that |A| ≥ . Possible futures equivalence has no finite, ground-complete, equational axiomatisation over the language BCCSP k . In what follows, for actions a, b ∈ A and i ≥
0, we let b a denote a. b i +1 a standfor b ( b i a ). Consider now the infinite family of equations { e N | N ≥ } given, for a = b , by: p N = N X i =1 b i a ( N ≥ e N : a k p N ≈ ap N + N X i =1 b ( a k b i − a ) ( N ≥ . Notice that the equations e N are sound modulo ∼ PF for all N ≥ e N is,alone, possible futures equivalent to a k p N . However, we now proceed to show that, when N is large enough, having a summand possible futures equivalent to a k p N is an invariantunder provability from finite sound axiom systems, and it will thus play the role of witnessproperty for our negative result.To this end, we introduce first some basic notions and results on ∼ PF .Firstly, as a simplification, we can focus on the absorption properties of BCCSP k .Informally, we can restrict the axiom system to a collection of equations that do not intro-duce unnecessary terms that are possible futures equivalent to in the equational proofs,namely summands and factors . (We refer the interested reader to Appendix D forfurther details.) Definition 6.3.
We say that a BCCSP k term t has a factor if it contains a subterm ofthe form t k t , where either t or t is possible futures equivalent to .Next, we characterise closed BCCSP k terms that are possible futures equivalent to p N . Lemma 6.4.
Let q be a BCCSP k term that does not have summands or factors and suchthat CT ( q ) = CT ( p N ) for some N ≥ . Then q does not contain any occurrence of k .Proof. That q ∼ CT p N implies that q does not contain any occurrence of k directly followsby observing that the completed traces of p N , and thus those of q , contain exactly oneoccurrence of a , and this occurrence must be as the last action in the completed trace. Lemma 6.5.
Let q be a BCCSP k term that does not have summands or factors. Then q ∼ PF p N , for some N ≥ , if and only if q = P j ∈ J q j for some terms q j such that none ofthem has + as head operator and: • for each i ∈ { , . . . , N } there is some j ∈ J such that b i a ∼ PF q j ; • for each j ∈ J there is some i ∈ { , . . . , N } such that q j ∼ PF b i a .Proof. Due to a slight difference in the technical development of the proof, we treat thecase of N = 1 separately. • Case
N > . ( ⇒ ) Let q ∼ PF p N .First of all, we show that q cannot be of the form q = b.q ′ for some term q ′ . In fact, inthat case, since N >
1, we have that p N b −→ a and p N b −→ ba . Therefore, as PF ( q ) = PF ( p N ),it follows that ( b, { ε, a } ) = ( b, T ( q ′ )) = ( b, { ε, b, ba } ), which is a contradiction.We have therefore obtained that q = P i ∈ J q j for some J with | J | ≥
2, and some q j , j ∈ J of the form b.q ′ j . Let i ∈ { , . . . , N } . Since p N b −→ b i − a , we have that ( b, T ( b i − a ))is a possible future of p N . Since q is possible futures equivalent to p N , there is some j N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 29 with T ( q ′ j ) = T ( b i − a ). We claim that q ′ j is possible futures equivalent to b i − a , whichyields that q j = b.q ′ j is possible futures equivalent to b i a . To see this, assume, towards acontradiction, that T ( q ′ j ) = T ( b i − a ) but q ′ j is not possible futures equivalent to b i − a . Thismeans that there are some k ≤ i − q ′ such that q ′ j b k −−→ q ′ and T ( q ′ ) is strictly includedin T ( b i − − k a ). Since T ( q ′ ) is prefix closed, this means that T ( q ′ ) = { b h | ≤ h ≤ ℓ } forsome ℓ ≤ i − − k . This means that q has ( b k +1 , T ( q ′ )) as one of its possible futures.On the other hand, p N has no such possible future, due to the missing trailing a , whichcontradicts our assumption that q ∼ PF p N .Assume now, towards a contradiction, that there is a j ∈ J such that q j PF b i a forall i ∈ { , . . . , N } . As q j is of the form b.q ′ j , for some q ′ j , we get that q b −→ q ′ j and( b, T ( q ′ j )) ∈ PF ( q ). We can distinguish two cases according to the form of CT ( q ′ j ): – CT ( q ′ j ) includes one completed trace of the form b k a for some k ∈ { , . . . , N − } . Since p N is possible futures equivalent to q , there is some i ∈ { , . . . , N } with T ( b i − a ) = T ( q ′ j ).In this case, the same reasoning applied above yields a contradiction with q ∼ PF p N . – There are b k a, b h a ∈ CT ( q ′ j ) for some k = h, k, h ∈ { , . . . , N − } . Since there is no p ′ such that p N b −→ p ′ and CT ( p ′ ) = CT ( q ′ j ), we get an immediate contradiction with q ∼ PF p N .( ⇐ ) Assume now that q = P j ∈ J q j , for some terms q j that do not have + as headoperator and such that(1) for each i ∈ { , . . . , N } there is a j ∈ J such that b i a ∼ PF q j , and(2) for each j ∈ J there is a i ∈ { , . . . , N } such that q j ∼ PF b i a .Since possible futures equivalence is a congruence with respect to summation and, more-over, summation is an idempotent operator (axiom A3 in Table 3), from these assumptionswe can directly conclude that q ∼ PF p N . • Case N = 1 . ( ⇒ ) Let q ∼ PF p . Hence PF ( q ) = n(cid:16) ε, { ε, b, ba } (cid:17) , (cid:16) b, { ε, a } (cid:17) , (cid:16) ba, { ε } (cid:17)o Assume that q = b.q ′ for some term q ′ . Then ( b, T ( q ′ )) ∈ PF ( q ) implies T ( q ′ ) = { ε, a } , andhence q ′ ∼ PF a . This gives q = P j ∈ J q j for some J with | J | = 1 and q j ∼ PF ba . The twoproperties relating the summands of q and p are then immediate.Assume now that q = P j ∈ J q j for some J with | J | ≥ q j of the form b.q ′ j .Assume, towards a contradiction, that there is some q j such that q j PF ba . Since q ∼ PF p implies T ( q ) = T ( ba ), we get that q j , and thus q , has ( b, { ε } ) as a possible future. Thisgives an immediate contradiction with q ∼ PF p .( ⇐ ) The proof of this implication follows as in the case of N > k terms that are possible futures equivalent to a k p N . To this end, we needfirst to lift the notions of transition, norm and depth from closed terms to terms. Theaction-labelled transition relation over BCCSP k terms contains exactly the literals that canderived using the rules in Table 1. clearly, t a −→ t ′ implies σ ( t ) a −→ σ ( t ′ ) for all substitutions σ . The norm and depth of a term are defined exactly as the norm and depth of processes,by replacing the transition relation over processes with that over terms. Equivalently, thenorm and depth of terms can be defined inductively over the structure of terms as follows: • norm( ) = 0; • norm( x ) = 0; • norm( a.t ) = 1 + norm( t ); • norm( t + u ) = min { norm( t ) , norm( u ) } ; • norm( t k u ) = norm( t ) + norm( u ). • depth( ) = 0; • depth( x ) = 0; • depth( a.t ) = 1 + depth( t ); • depth( t + u ) = max { depth( t ) , depth( u ) } ; • depth( t k u ) = depth( t ) + depth( u ).For k ≥
0, we denote by var k ( t ) the set of variables occurring in the k -derivatives of t ,namely var k ( t ) = { x ∈ var( t ′ ) | t α −→ t ′ , | α | = k } . Finally, we write t ∼ PF u if σ ( t ) ∼ PF σ ( u )for all closed substitutions σ . Lemma 6.6.
Let t, u be two
BCCSP k terms. If t ∼ CT u then: (1) For each k ≥ it holds that var k ( t ) = var k ( u ) . (2) t has a summand x , for some variable x , if and only if u does. (3) norm( t ) = norm( u ) and depth( t ) = depth( u ) .Proof. (1) Assume, towards a contradiction, that for some k ≥ x suchthat x ∈ var k ( t ) \ var k ( u ). In particular, this means that there is a term t ′ such that t α −→ t ′ for some trace α with | α | = k and x ∈ var( t ′ ). However, there is no u ′ such that u β −→ u ′ for some trace β with | β | = k and x ∈ var( u ′ ). We can assume, without loss ofgenerality, that k is the largest natural number such that x ∈ var k ( t ).Let n > depth( t ) , depth( u ). Consider the closed substitution σ defined by σ ( y ) = ( a n if y = x otherwise.Then t ∼ CT u implies t ∼ T u and thus t α −→ t ′ implies that u α −→ u ′ for some u ′ . Let usproceed by a case analysis on the structure of t ′ to obtain the desired contradiction. • t ′ = x + w for some term w . Then we get that σ ( t ) has αa n as a completed trace.However, due to the choices of n and σ , we have that σ ( u ) cannot perform thecompleted trace a n after α , thus contradicting t ∼ CT u . By the assumption that k isthe largest natural number such that x ∈ var k ( t ) is suffices to consider two cases: • t ′ = ( x + w ) k w ′ for some terms w, w ′ with w ′ = . From t ∼ T u , we infer that forall β ∈ CT ( σ ( w ′ )) we have that αβ is a trace of σ ( t ) and thus of σ ( u ). However, wecan proceed as in the previous case and argue that αa n β ∈ CT ( t ) \ CT ( u ), which is acontradiction with t ∼ CT u .(2) Assume now that t has a summand x for some variable x . Then, from item (1) of thislemma, x ∈ var ( t ) implies x ∈ var ( u ). By applying similar reasoning as in the proofof the first statement of the lemma we obtain that it cannot be the case that all theoccurrences of x in u are in the scope of prefixing. Hence, u has, at least, one summand u ′ such that either u ′ = x or u ′ = ( x + w ) k w ′ , for some terms w, w ′ with w ′ = . Ourorder of business is to show that it cannot be the case that all summands u ′ of u with x ∈ var ( u ′ ) are of the latter form. To this end, assume, towards a contradiction, thatwhenever x ∈ var ( u ′ ) then u ′ = ( x + w ) k w ′ . Let n > depth ( u ) and consider the closed N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 31 substitution σ ′ defined by σ ′ ( y ) = ( a n if y = xb otherwise.Since t has a summand x , we obtain that a n ∈ CT ( σ ′ ( t )). However, a n CT ( σ ′ ( u )). Infact, w ′ = and σ ′ ( y ) = for all variables y ∈ V give that I ( σ ′ ( w ′ )) = ∅ for all terms w ′ occurring in the summands u ′ of u such that x ∈ var ( u ′ ). Therefore a n CT ( σ ′ ( u ′ ))for all such summands. Due to choices of σ ′ and n , we can conclude that there is nosummand of σ ′ ( u ) having a n has a completed trace, so that a n CT ( σ ′ ( u )). This givesa contradiction with σ ′ ( t ) ∼ CT σ ′ ( u ) and thus with t ∼ CT u .(3) norm( t ) = norm( u ) and depth( t ) = depth( u ) follow immediately from t ∼ CT u anditems (1), (2) of this lemma. Remark 6.7.
Since for all BCCSP k terms t, u we have that t ∼ PF u implies t ∼ CT u ,Lemma 6.6 holds also for all pairs of possible futures equivalent terms. Proposition 6.8.
Assume that p, q are two
BCCSP k processes such that p, q PF , p, q donot have summands or factors, and p k q ∼ PF a k p N , for some N > . Then either p ∼ PF a and q ∼ PF p N , or p ∼ PF p N and q ∼ PF a .Proof. By Lemma 6.6.3, from p k q ∼ PF a k p N (6.1)we can infer that norm( p k q ) = 3. Hence, since p, q PF , we have that either norm( p ) = 1and norm( q ) = 2, or vice versa. We consider only the former case and prove that then p ∼ PF a and q ∼ PF p N . In the other case it can be proved that p ∼ PF p N and q ∼ PF a byanalogous reasoning.So assume that norm( p ) = 1 and norm( q ) = 2. From Equation (6.1) we get that I ( p ) , I ( q ) ⊆ { a, b } . First of all we argue that it cannot be the case that a ∈ I ( p ) ∩ I ( q ). Infact, in that case p k q would be able to perform the trace a , whereas a k p N cannot do so,thus giving a contradiction with Equation (6.1). We now proceed to show that a ∈ I ( p ).Assume now, towards a contradiction, that this is not the case and I ( p ) = { b } . This,together with norm( p ) = 1, gives that p has a summand b . Hence, given any α ∈ CT ( q ) wehave that αb ∈ CT ( p k q ). However, αb CT ( a k p N ), as any completed trace of a k p N mustend with an a , thus giving that p k q CT a k p N , which contradicts Equation (6.1). We havetherefore obtained that a ∈ I ( p ) , a I ( q ) and, moreover, p does not have a summand b .Since p has norm 1, we conclude that it has a summand a .We now proceed to show that also depth( p ) = 1, and thus p ∼ PF a . Since p has asummand a , we have that p k q a −→ k q ∼ PF q . By Equation (6.1), we get a k p N a −→ r forsome r such that q ∼ T r . Since a k p N has a unique initial a -transition a k p N a −→ k p N , weget that r ∼ PF p N and q ∼ T p N . As a consequence, we obtain that depth( q ) = N + 1. Thenwe have depth( p ) = depth( p k q ) − depth( q )= depth( a k p N ) − depth( q ) (by Eq. (6.1) and Lem. 6.6.3)= N + 2 − ( N + 1)= 1 . Therefore, we have obtained that p ∼ PF a and thus, since possible futures equivalence is acongruence with respect to parallel composition, from Equation (6.1) it follows that a k q ∼ PF a k p N . (6.2)Our order of business will now be to show that q ∼ PF p N . We already know that T ( q ) = T ( p N ). Moreover, as an initial a -transition of a k q cannot stem from q , from ( ab i a, ∅ ) ∈ PF ( a k p N ) for all i ∈ { , . . . , N } , we get that CT ( p N ) ⊆ CT ( q ). Assume, towards a contradiction,that CT ( q ) CT ( p N ). As q ∼ T p N , this implies that q has a completed trace of the form b j for some 1 ≤ j ≤ N . Hence, ( ab j , ∅ ) ∈ PF ( a k q ). However, no completed trace of a k p N hasa b as last symbol and thus a k p N has no possible future of the form ( αb, ∅ ) for some trace α . This gives a contradiction with Equation (6.2). Therefore, we have that T ( q ) = T ( p N )and CT ( q ) = CT ( p N ), thus giving q ∼ CT p N . Notice that, by Lemma 6.4, this implies that q does not contain any occurrence of k and q can be written in the general form q = P j ∈ J q j where none of the q j has + as head operator.Assume, towards a contradiction, that q PF p N . By Lemma 6.5 this implies that eitherthere is an i ∈ { , . . . , N } such that b i a PF q j for all j ∈ J , or there is a j ∈ J such that q j PF b i a for all i ∈ { , . . . , N } . In both cases, we can proceed as in the proof of Lemma 6.5and obtain a contradiction with a k p N ∼ PF a k q .We have therefore obtained that q ∼ PF p N , and the proof is thus concluded.The following lemma characterises the open BCCSP k terms whose substitution in-stances can be equivalent in possible futures semantics to terms having at least two sum-mands of p N ( N >
1) as their summands.
Lemma 6.9.
Let t be a BCCSP k term that does not have + as head operator. Let m > and σ be a closed substitution such that σ ( t ) has no summands or factors. If σ ( t ) ∼ PF P mk =1 b i k a , for some ≤ i < . . . < i m , then t = x for some variable x .Proof. For simplicity of notation, we let q m denote P mk =1 b i k a . We proceed by showing thatthe remaining possible forms for t give a contradiction with σ ( t ) ∼ PF q m . We remark that I ( q m ) = { b } , and σ ( t ) ∼ PF q m implies CT ( σ ( t )) = CT ( q m ). • t = b.t ′ for some term t ′ . Then CT ( σ ( t ′ )) = { b i k − a | k ∈ { , . . . , m }} , and since m > | CT ( t ′ ) | ≥
2. It is then immediate to verify that ( b, T ( σ ( t ′ ))) ∈ PF ( σ ( t )), whereas( b, T ( σ ( t ′ ))) PF ( q m ), since whenever q m b −→ r then T ( r ) includes only one trace of theform b j a for some j ∈ { i − , . . . , i m − } . This gives a contradiction with σ ( t ) ∼ PF q m . • t = t ′ k t ′′ for some terms t ′ , t ′′ . Since σ ( t ) has no summands or factors, neitherdoes t thus giving t ′ , t ′′ , σ ( t ′ ) , σ ( t ′′ ) PF . Hence, we directly get a contradiction with CT ( σ ( t )) = CT ( q m ), since all completed traces of q m have exactly one occurrence of a andthis occurrence is as the last action in the completed trace. No process of the form p k q with I ( p ) , I ( q ) = ∅ can satisfy the same property.We now have all the ingredients necessary to prove Theorem 6.2. To streamline ourpresentation, we split the proof of into two parts: Proposition 6.11 deals with the preserva-tion of the witness property under provability from the substitution rule of equational logic.Theorem 6.12 builds on Proposition 6.11 and proves the witness property to be an invariantunder provability from finite sound axiom systems.The following lemma is immediate. N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 33
Lemma 6.10.
Let t be a BCCSP k term, and let σ be a closed substitution. If x ∈ var( t ) then depth( σ ( t )) ≥ depth( σ ( x )) . Proposition 6.11.
Let t ≈ u be an equation over BCCSP k that is sound modulo ∼ PF . Let σ be a closed substitution with p = σ ( t ) and q = σ ( u ) . Suppose that p and q have neither summands nor factors, and that p, q ∼ PF a k p N for some N larger than the sizes of t and u . If p has a summand possible futures equivalent to a k p N , then so does q .Proof. Since σ ( t ) and σ ( u ) have no summands or factors, neither do t and u . We cantherefore assume that, for some finite non-empty index sets I, J , t = X i ∈ I t i and u = X j ∈ J u j , (6.3)where none of the t i ( i ∈ I ) and u j ( j ∈ J ) is or has + as its head operator. Note that,as t and u have no summands or factors, neither do t i ( i ∈ I ) and u j ( j ∈ J ).Since p = σ ( t ) has a summand which is possible futures equivalent to a k p N , there isan index i ∈ I such that σ ( t i ) ∼ PF a k p N . Our aim is now to show that there is an index j ∈ J such that σ ( u j ) ∼ PF a k p N , proving that q = σ ( u ) has the required summand. This we proceed to do by a case analysison the form t i may have.(1) Case t i = x for some variable x . In this case, we have that σ ( x ) ∼ PF a k p N and t has x as a summand. As t ≈ u is sound with respect to possible futures equivalence,from t ∼ PF u we get t ∼ CT u . Hence, by Lemma 6.6.2, we obtain that u has a summand x as well, namely there is an index j ∈ J such that u j = x . It is then immediate toconclude that q = σ ( u ) has a summand which is possible futures equivalent to a k p N .(2) Case t i = ct ′ for some action c ∈ { a, b } and term t ′ . This case is vacuous because,since σ ( t i ) = cσ ( t ′ ) c −→ σ ( t ′ ) is the only transition afforded by σ ( t i ), this term cannotbe possible futures equivalent to a k p N .(3) Case t i = t ′ k t ′′ for some terms t ′ , t ′′ . We have that σ ( t i ) = σ ( t ′ ) k σ ( t ′′ ) ∼ PF a k p N .As σ ( t i ) has no factors, it follows that σ ( t ′ ) PF and σ ( t ′′ ) PF . Thus, byProposition 6.8, we can infer that, without loss of generality, σ ( t ′ ) ∼ PF a and σ ( t ′′ ) ∼ PF p N . Notice that σ ( t ′′ ) ∼ PF p N implies CT ( σ ( t ′′ )) = CT ( p N ). Now, t ′′ can be written in thegeneral form t ′′ = v + · · · + v l for some l >
0, where none of the summands v h is or a sum. By Lemma 6.5, σ ( t ′′ ) ∼ PF p N implies that for each i ∈ { , . . . , N } there is asummand r i of σ ( t ′′ ) such that b i a ∼ PF r i , and for each summand r of σ ( t ′′ ) there is an i r ∈ { , . . . , N } such that r ∼ PF b i a . Observe that, since N is larger than the size of t ,we have that l < N . Hence, there must be some h ∈ { , . . . , l } such that σ ( v h ) ∼ PF m X k =1 b i k a for some m > ≤ i < . . . < i m ≤ N . The term σ ( v h ) has no summands orfactors, or else, so would σ ( t ′′ ) and σ ( t ). By Lemma 6.9, it follows that v h can only be a variable x and σ ( x ) ∼ PF m X k =1 b i k a . (6.4)Observe, for later use, that, since t ′ has no factors, the above equation yields that x var( t ′ ), or else σ ( t ′ ) PF a due to Lemma 6.10. So, modulo possible futuresequivalence, t i has the form t ′ k ( x + t ′′′ ), for some term t ′′′ , with x var( t ′ ), σ ( t ′ ) ∼ PF a and σ ( x + t ′′ ) ∼ PF p N .Our order of business will now be to show that σ ( u ) has a summand u j that ispossible futures equivalent to a k p N . We recall that t ∼ PF u implies t ∼ CT u . Thus, byLemma 6.6.1 we obtain that var k ( t ) = var k ( u ) for all k ≥
0. Hence, from x ∈ var ( t i )we get that there is at least one j ∈ J such that x ∈ var ( u j ).So, firstly, we show that x cannot occur in the scope of prefixing in u j , namely u j cannot be of the form c.u ′ or ( c.u ′ + u ′′ ) k u ′′′ for some c ∈ { a, b } and u ′ with x ∈ var( u ′ ).We proceed by a case analysis:(a) c = b and u j = ( b.u ′ + u ′′ ) k u ′′′ for some u ′ , u ′′ , u ′′′ ∈ BCCSP k with x ∈ var( u ′ ).As σ ( u ) does not have summands or factors we have that σ ( u ′′′ ) PF . Let D = max { d | x ∈ var d ( u ′ ) } . From σ ( x ) ∼ PF P mk =1 b i k a (Equation (6.4)) and CT ( σ ( u )) = CT ( a k p N ) we can infer that the completed traces of σ ( u ′′′ ) are of theform b i a , for some i ∈ { , . . . , N − i m − D − } . In fact, since σ ( x ) can performat least one completed trace of the form b i k a , for some 1 ≤ i k ≤ N , and thecompleted traces of a k p N contain exactly two occurrences of a , of which oneas the final action of the trace, we can infer that the completed traces of σ ( u ′′′ )have to contain exactly one occurrence of a , and this occurrence has to be asthe last symbol of the completed trace. Let α ∈ T ( σ ( u ′ )) be such that | α | = D and u ′ α −→ w with x ∈ var( w ). By the choice of D , we can infer that x doesnot occur in the scope of prefixing in w , and thus T ( σ ( x )) ⊆ T ( σ ( w )). Thenwe get that ( b i abα, T ( σ ( w ))) ∈ PF ( σ ( u )), where b i a ∈ CT ( σ ( u ′′′ )). However, as m ≥
2, there is no p ′ such that a k p N b i abα −−−−→ p ′ and T ( σ ( x )) ⊆ T ( p ′ ), thus giving( b i abα, T ( σ ( w ))) PF ( a k p N ). This gives a contradiction with σ ( u ) ∼ PF a k p N .(b) c = b and u j = b.u ′ for some BCCSP k term u ′ with x ∈ var( u ′ ). The proof of thiscase is similar to, actually simpler than, that of the previous case and it is thereforeomitted.(c) c = a and u j = ( a.u ′ + u ′′ ) k u ′′′ for some u ′ , u ′′ , u ′′′ ∈ BCCSP k with x ∈ var( u ′ ).As σ ( u ) does not have summands or factors we have that σ ( u ′′′ ) PF . From σ ( x ) ∼ PF P mk =1 b i k a we infer that T ( a.σ ( u ′ )) includes traces having two occurrencesof action a . Since σ ( u ) ∼ PF a k p N , this implies that there is no α ∈ T ( σ ( u ′′′ )) suchthat α contains an occurrence of action a , for otherwise σ ( u ) could perform atrace having 3 occurrences of that action. In particular, this implies that the lastsymbol in each trace of σ ( u ′′′ ) must be action b . This gives that there is at least onecompleted trace of σ ( u j ), and thus of σ ( u ), whose last symbol is action b . Hencewe get CT ( σ ( u )) = CT ( a k p N ), thus giving a contradiction with σ ( u ) ∼ PF a k p N .(d) c = a and u j = a.u ′ for some BCCSP k term u ′ with x ∈ var( u ′ ). In this case weare going to prove a slightly weaker property, namely that not all summands u j with x ∈ var( u j ) can be of this form. Despite being weaker, this property is enoughbecause, as shown above, all other possibilities for an occurrence of x in a summand u j have already been excluded. To this end, consider the closed substitution σ ′ N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 35 defined by σ ′ ( y ) = ( ap N if y = xσ ( y ) otherwise.Then we have that σ ′ ( t i ) = σ ′ ( t ′ ) k σ ′ ( x ) + σ ′ ( t ′′′ ) a −→ σ ( t ′ ) k p N ∼ PF a k p N . Since σ ′ ( t ) ∼ PF σ ′ ( u ) then there is a process r such that σ ′ ( u ) a −→ r and T ( r ) = T ( a k p N ).In particular, this means that depth( r ) = N + 2. Hence, from the choices of N, σ and σ ′ , we can infer that such an a -move by σ ′ ( u ) can only stem from a summand u j such that x ∈ var( u j ). Assume, towards a contradiction, that all such summands u j are of the form a.u ′ j for some BCCSP k term u ′ j with x ∈ var( u ′ j ) and r = σ ′ ( u ′ j ).As depth( σ ′ ( u ′ j )) = N + 2 = depth( σ ′ ( x )), by Lemma 6.10 we get that u ′ j can onlybe of the form u ′ j = x + w j for some BCCSP k term w j with depth( σ ′ ( w j )) ≤ N + 2.Notice that T ( σ ′ ( x )) ⊂ T ( a k p N ). Hence σ ′ ( w j ) = . More precisely, σ ′ ( x ) = ap N implies that { bα | bα ∈ T ( a k p N ) } ⊆ T ( σ ′ ( w j )) ⊆ T ( a k p N ). Clearly, no trace startingwith action b can stem from σ ′ ( x ) and we can then infer, in light of Lemma 6.10,that x var( w j ), as depth( σ ′ ( w j )) ≤ N + 2. This implies that σ ′ ( w j ) = σ ( w j ) andthus { bα | bα ∈ T ( a k p N ) } ⊆ T ( σ ( w j )) ⊆ T ( a k p N ). In particular, σ ( w j ) can performat least one (completed) trace of the form bα where α contains two occurrences ofaction a . From σ ( u j ) = a. ( σ ( x ) + σ ( w j )), we then get that ( abα, ∅ ) ∈ PF ( σ ( u )),namely σ ( u ) can perform at least one (completed) trace containing 3 occurrencesof action a . This gives a contradiction with σ ( u ) ∼ PF a k p N .We have therefore obtained that x does not occur in the scope of prefixing in (at leastone) u j . We proceed now by a case analysis on the possible forms of this summand.(a) u j = x . Then σ ( u ) has a summand which is possible futures equivalent to P mk =1 b i k a .We show that this gives a contradiction with σ ( u ) ∼ PF a k p N . This follows directlyby noticing that, due to the summand b i a , we have that ( b i a, ∅ ) ∈ PF ( σ ( u )). How-ever, ( b i a, ∅ ) PF ( a k p N ), since a k p N by performing the trace b i a can reacheither a process that can perform an a (in case the first b -move is performed by thesummand b i a of p N ) or a b (in case the first b -move is performed by a summand b i a of p N such that i > i ).(b) u j = ( x + w ) k w ′ , for some terms w, w ′ with w ′ PF . From σ ( u ) ∼ PF a k p N , weinfer that CT ( σ ( u j )) ⊆ CT ( a k p N ). We recall that no completed trace of a k p N has b as last symbol and, moreover, in all the completed traces of a k p N there areexactly two occurrences of a . Hence, all (nonempty) completed traces of σ ( x ) , σ ( w )and σ ( w ′ ) must have exactly one occurrence of a and this occurrence must be asthe last symbol in the completed trace.We now proceed to show that σ ( w ′ ) has a summand a and a I ( σ ( x ) + σ ( w )). Westart by noticing that it cannot be the case that a ∈ I ( σ ( x ) + σ ( w )) ∩ I ( σ ( w ′ )),for otherwise we would have a ∈ T ( σ ( u j )) ⊆ T ( σ ( u )), thus contradicting σ ( u ) ∼ PF a k p N . Assume now, towards a contradiction, that I ( σ ( w ′ )) = { b } . Then, due tosummand b i m a of σ ( x ), we have that σ ( u j ) b im − −−−−→ ba k σ ( w ′ ) and aα T ( ba k σ ( w ′ ))for any trace α ∈ A ∗ . Clearly, ( b i m − , T ( ba k σ ( w ′ ))) ∈ PF ( σ ( u j )), and thus it isalso a possible future of σ ( u ). However, ( b i m − , T ( ba k σ ( w ′ ))) PF ( a k p N ), asthe interleaving of p N with a guarantees that after an initial trace of an arbitrarynumber of b -transitions it is always possible to perform a trace starting with a .This gives a contradiction with σ ( u ) ∼ PF a k p N . We have therefore obtained that a ∈ I ( σ ( w ′ )). More precisely, from the constraints on the completed traces of σ ( w ′ ),we can infer that σ ( w ′ ) has a summand a .Our order of business will now be to show that σ ( w ′ ) ∼ PF a . Since σ ( w ′ ) a −→ , wehave that σ ( u j ) a −→ ( σ ( x )+ σ ( w )) k ∼ PF σ ( x )+ σ ( w ). Thus, σ ( u ) ∼ PF a k p N impliesthat a k p N a −→ r for some r with T ( r ) = T ( σ ( x ) + σ ( w )). Since a k p N has only onepossible initial a -transition, namely a k p N a −→ k p N , we get that r ∼ PF p N and thus T ( σ ( x )+ σ ( w )) = T ( p N ). In particular, this implies that depth( σ ( x )+ σ ( w )) = N +1.Therefore, we have1 ≤ depth( σ ( w ′ )) = depth( σ ( u j )) − depth( σ ( x ) + σ ( w ))= depth( σ ( u j )) − ( N + 1) ≤ depth( σ ( u )) − ( N + 1)= depth( a k p N ) − ( N + 1) (by Lem. 6.6.3)= N + 2 − ( N + 1)= 1and we can therefore conclude that σ ( w ′ ) ∼ PF a . Furthermore, it is not difficult toprove that CT ( σ ( x ) + σ ( w )) = CT ( p N ), for otherwise we get a contradiction with σ ( u ) ∼ PF a k p N .So far we have obtained that, modulo possible futures equivalence, σ ( u j ) ∼ PF m X k =1 b i k a + σ ( w ) ! k a and CT ( m X k =1 b i k a + σ ( w )) = { b i a | i ∈ { , . . . , N }} . To conclude the proof, we need to show that P mk =1 b i k a + σ ( w ) ∼ PF p N . Let I m = { i , . . . , i m } and I N = { , . . . , N } . Assume, towards a contradiction, that P mk =1 b i k a + σ ( w ) PF p N . Since CT ( σ ( x ) + σ ( w )) = CT ( p N ), from Lemma 6.4 wecan infer that σ ( w ) does not contain any occurrence of k . In particular, σ ( w ) can bewritten in the general form σ ( w ) = P l ∈ L q l for some terms q l that do not have + ashead operator nor contain any occurrence of k . Moreover, as P mk =1 b i k a + σ ( w ) PF p N , by Lemma 6.5, this means that either there is an i ∈ I N \ I m such that b i a PF q l for any l ∈ L , or that there is a summand q l of σ ( w ) such that q l PF b i a for any i ∈ I N . In both cases, we obtain that there is (at least) a summand q l of σ ( w ) suchthat b k a, b h a ∈ CT ( q l ) for some k = h, h, k ∈ I N . We can then proceed as in theproof of Lemma 6.5 to prove that this gives the desired contradiction. We havetherefore obtained that P mk =1 b i k a + σ ( w ) ∼ PF p N . Hence, since possible futuresequivalence is a congruence with respect to parallel composition, we get that σ ( u j ) ∼ PF a k p N and we can therefore conclude that σ ( u ) has the desired summand.This concludes the proof. Theorem 6.12.
Let E be a finite axiom system over BCCSP k that is sound modulo ∼ PF . Let N be larger than the size of each term in the equations in E . Assume that p and q are closedterms that contain no occurrences of as a summand or factor, and that p, q ∼ PF a k p N .If E ⊢ p ≈ q and p has a summand possible futures equivalent to a k p N , then so does q . N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 37
Proof.
Assume that E is a finite axiom system over the language BCCSP k that is soundmodulo possible futures equivalence, and that the statements (1)–(4) below hold, for someclosed terms p and q and positive integer N larger than the size of each term in the equationsin E :(1) E ⊢ p ≈ q ,(2) p ∼ PF q ∼ PF a k p N ,(3) p and q contain no occurrences of as a summand or factor, and(4) p has a summand possible futures equivalent to a k p N .We prove that q also has a summand possible futures equivalent to a k p N by induction onthe depth of the closed proof of the equation p ≈ q from E . Recall that, without loss ofgenerality, we may assume that the closed terms involved in the proof of the equation p ≈ q have no summands or factors (by Proposition D.6, as E may be assumed to be saturated),and that applications of symmetry happen first in equational proofs (that is, E is closedwith respect to symmetry).We proceed by a case analysis on the last rule used in the proof of p ≈ q from E .The case of reflexivity is trivial, and that of transitivity follows immediately by using theinductive hypothesis twice. Below we only consider the other possibilities. • Case E ⊢ p ≈ q , because σ ( t ) = p and σ ( u ) = q for some equation ( t ≈ u ) ∈ E and closed substitution σ . Since σ ( t ) = p and σ ( u ) = q have no summands orfactors, and N is larger than the size of each term mentioned in equations in E , the claimfollows by Proposition 6.11. • Case E ⊢ p ≈ q , because p = cp ′ and q = cq ′ for some p ′ , q ′ such that E ⊢ p ′ ≈ q ′ ,and for some action c . This case is vacuous because p = cp ′ PF a k p N , and thus p does not have a summand possible futures equivalent to a k p N . • Case E ⊢ p ≈ q , because p = p ′ + p ′′ and q = q ′ + q ′′ for some p ′ , q ′ , p ′′ , q ′′ such that E ⊢ p ′ ≈ q ′ and E ⊢ p ′′ ≈ q ′′ . Since p has a summand possible futures equivalent to a k p N , we have that so does either p ′ or p ′′ . Assume, without loss of generality, that p ′ hasa summand possible futures equivalent to a k p N . Since p is possible futures equivalent to a k p N , so is p ′ . Using the soundness of E modulo possible futures equivalence, it followsthat q ′ ∼ PF a k p N . The inductive hypothesis now yields that q ′ has a summand possiblefutures equivalent to a k p N . Hence, q has a summand possible futures equivalent to a k p N ,which was to be shown. • Case E ⊢ p ≈ q , because p = p ′ k p ′′ and q = q ′ k q ′′ for some p ′ , q ′ , p ′′ , q ′′ such that E ⊢ p ′ ≈ q ′ and E ⊢ p ′′ ≈ q ′′ . Since the proof involves no uses of as a summand or afactor, we have that p ′ , p ′′ PF and q ′ , q ′′ PF . It follows that q is a summand of itself.By our assumptions, q ′ k q ′′ ∼ PF a k p N which, by Proposition 6.8 gives that either q ′ ∼ S a and q ′′ ∼ S p N , or q ′ ∼ S p N and q ′′ ∼ S a . In both cases, we can conclude that q has itselfas summand of the required form.This completes the proof of Theorem 6.12 and thus of Theorem 6.2.As the left-hand side of equation e N , i.e., the term a k p N , has a summand possiblefutures equivalent to a k p N , whilst the right-hand side, i.e., the term ap N + P Ni =1 b ( a k b i − a ),does not, we can conclude that the collection of infinitely many equations e N ( N ≥
1) isthe desired witness family. This concludes the proof of Theorem 6.2.
Extending the negative result.
It is easy to check that the equations e N ( N ≥
1) inthe witness family of the negative result for ∼ PF are all sound modulo bisimilarity, i.e., thelargest symmetric simulation. Consequently, they are also sound modulo any congruence R such that ∼ B ⊆ R ⊆ ∼ PF . Hence, the negative result for all these equivalences can bederived from that for ∼ PF , by exploiting this fact and that any finite axiom system that issound modulo R is also sound modulo ∼ PF . Theorem 6.13.
Assume that |A| ≥ . Let R be a congruence such that ∼ B ⊆ R ⊆∼ PF . Then R has no finite, ground-complete, equational axiomatisation over the language BCCSP k .Proof. Let E be a finite equational axiomatisation for BCCSP k that is sound modulo R .Since R is included in ∼ PF , we have that the axiom system E is sound modulo ∼ PF . Let N be larger than the size of each term in the equations in E . Theorem 6.12 implies that theequation a k p N ≈ ap N + N X i =1 b ( a k b i − a )cannot be derived from E . Since this equation is sound modulo R , namely a k p N R ap N + N X i =1 b ( a k b i − a )it follows that E is not complete modulo R .Theorem 6.13 can be applied to establish for n ≥ n -nested trace and simu-lation semantics have no finite, ground-complete equational axiomatisation over BCCSP k .The n -nested trace equivalences were introduced in [22] as an alternative tool to definebisimilarity. The hierarchy of n -nested simulations, namely simulation relations containedin a (nested) simulation equivalence, was introduced in [21]. Definition 6.14 ( n -nested semantics) . For n ≥
0, the relation ∼ n T over P , called the n - nested trace equivalence , is defined inductively as follows: • p ∼ T q for all p, q ∈ P , • p ∼ n +1 T q if and only if for all traces α ∈ A ∗ : – if p α −→ p ′ then there is a q ′ such that q α −→ q ′ and p ′ ∼ n T q ′ , and – if q α −→ q ′ then there is a p ′ such that p α −→ p ′ and p ′ ∼ n T q ′ .For n ≥
0, the relation ⊑ n S over P is defined inductively as follows: • p ⊑ S q for all p, q ∈ P , • p ⊑ n +1 S q if and only if p R q for some simulation R , with R − included in ⊑ n S . n - nested simulation equivalence is the kernel of ⊑ n S , i.e., the equivalence ∼ n S = ⊑ n S ∩ ( ⊑ n S ) − .Note that ∼ T corresponds to trace equivalence, ∼ T is possible futures equivalence, and ∼ S is simulation equivalence. The following theorem is a corollary of Theorems 6.2 and 6.13. Theorem 6.15.
Assume that |A| ≥ . Let n ≥ . Then, n -nested trace equivalence and n -nested simulation equivalence admit no finite, ground-complete, equational axiomatisationover the language BCCSP k . N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 39 x α −→ x ′ x k y α −→ x ′ k y y α −→ y ′ x k y α −→ x k y ′ x a −→ x ′ y ¯ a −→ y ′ x k y τ −→ x ′ k y ′ x ¯ a −→ x ′ y a −→ y ′ x k y τ −→ x ′ k y ′ Table 11.
Operational semantics of parallel composition with CCS com-munication. (ELC1) αx k βy ≈ α ( x k βy ) + β ( αx k y ) if α = β , or α = τ , or β = τ (ELC1 τ ) αx k βy ≈ α ( x k βy ) + β ( αx k y ) + τ ( x k y ) if α = β (ELC2) P i ∈ I α i x i k P j ∈ J β j y j ≈ P i ∈ I α i ( x i k P j ∈ J β j y j ) + P j ∈ J β j ( P i ∈ I α i x i k y j )++ P i ∈ I,j ∈ Jαi = βj τ ( x i k y j )with α i = α k whenever i = k and β j = β h whenever j = h , ∀ i, k ∈ I, ∀ j, h ∈ J (ELC3) P i ∈ I α i x i k P j ∈ J β j y j ≈ P i ∈ I α i ( x i k P j ∈ J β j y j ) + P j ∈ J β j ( P i ∈ I α i x i k y j )++ P i ∈ I,j ∈ Jαi = βj τ ( x i k y j ) Table 12.
The different instantiations of the expansion law when commu-nication is considered.7.
Adding CCS synchronisation
The negative results provided above conclude our analysis of the axiomatisability of thepurely interleaving parallel composition operator modulo the congruences in the linear time-branching time spectrum.The most natural extension of our work consists in allowing the parallel componentsto synchronise. In particular, we are interested in the
CCS-style communication . It presup-poses a bijection · on A such that a = a and a = a for all a ∈ A . Following [26], the specialaction symbol τ
6∈ A , will result from the synchronised occurrence of the complementaryactions a and ¯ a . Let A τ = A ∪ { τ } . Then, we let the metavariables α, β, . . . range over A τ .The rules in Table 11 define the operational semantics of the parallel compositionoperator when also synchronisation is taken into account.Our order of business for this section will then be to show if and how the analysis wecarried out in the previous sections is affected by the addition of synchronisation `a la CCS.7.1. The positive results.
It is not difficult to see that the arguments we used to provethe existence of finite, ground-complete axiomatisations still hold also with synchronisation.The only changes we need to apply are reported in Table 12. For Y ∈ { , , } , the axiomschema ELCY simply adds the terms related to communication to ELY.We remark that, by convention, P i ∈∅ t i = . A finite, ground-complete axiomatisationover BCCSP k modulo ∼ X , for X ∈ { T , CT , F , R , FT , RT , S , CS , RS } is given by the axiom system E c X obtained from E X as follows: • we include { ELC1 , ELC1 τ } instead of EL1; • we include ELC2 instead of EL2; • all other axioms are unchanged (although the action variables occurring in them nowrange over A τ ).For instance, a finite, ground-complete axiomatisation over BCCSP k modulo ready simula-tion equivalence is given by the axiom system E c RS = E ∪ { RS , RSP1 , RSP2 , ELC2 } .7.2. The negative results.
In [27] it was proved that bisimilarity does not admit a finite,ground-complete axiomatisation over BCCSP k with CCS-style synchronisation. We nowproceed to show that, by applying similar arguments to those used in Section 6.1, we canobtain the same negative result for possible futures equivalence. More precisely, we provethe following: Theorem 7.1.
Assume that |A τ | ≥ . Possible futures equivalence has no finite, ground-complete, equational axiomatisation over the language BCCSP k with CCS synchronisation. To this end, consider the infinite family of equations { e cN | N ≥ } given by: q N = N X i =1 τ i a ( N ≥ e cN : a k q N ≈ aq N + N X i =1 τ ( a k τ i − a ) ( N ≥ . Clearly, this family of equations coincides with that used to prove the negative result inSection 6.1 where we have substituted all occurrences of action b with the special action τ .We notice that the equations e cN are sound not only modulo possible futures equivalence, butalso modulo bisimilarity (each e cN is in fact a distinct closed instance of ELC3 in Table 12).This means that if we can obtain the negative result for possible futures, then we can proceedexactly as in Section 6.2 to extend it to all the congruences ∼ such that ∼ B ⊆∼⊆∼ PF .Since τ cannot communicate with any action, hence, in particular, it does not com-municate with a , the analysis we carried out in Section 6.1 (with b replaced by τ ) stillapplies. Remark 7.2.
Most results in Section 6.2 rely on the assumption that terms do not contain summands or factors. Notice that due to undefined communications, the expansion lawschemas in Table 12 may actually introduce terms that are possible futures equivalent to in the equational proofs.However, we remark that this is not an issue. In fact, in general, we could extend theresults in Appendix D to deal with the possible summands introduced by the expansionlaws. Briefly, whenever a summand is introduced, we can assume that axioms A0 and P0(Table 3) are also applied in order to get rid of the unnecessary summands and factors.Hence, the proof of Theorem 7.1 follows from Theorem 6.12 and the fact that none of thesummands in the right-hand side of the equations e cN is possible futures equivalent to a k p N .8. Concluding remarks
We have studied the finite axiomatisability of the language BCCSP k modulo the behaviouralequivalences in the linear time-branching time spectrum. On the one hand we have obtained N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 41 finite, ground-complete axiomatisations modulo the (decorated) trace and simulation seman-tics in the spectrum. On the other hand we have proved that for all equivalences that arefiner than possible futures equivalence and coarser than bisimilarity a finite ground-completeaxiomatisation does not exist.Since our ground-completeness proof for ready simulation equivalence proceeds via elim-ination of k from closed terms (Proposition 3.5), and all behavioural equivalences in thelinear time-branching time spectrum that include ready simulation have a finite ground-complete axiomatisation over BCCSP, it immediately follows from the elimination resultthat all these behavioural equivalences have a finite ground-complete axiomatisation overBCCSP k . Exploiting various forms of distributivity of parallel composition over choice, wewere able to present more concise and elegant axiomatisations for the coarser behaviouralequivalences.In this paper we have considered both, a parallel composition operator that implementsinterleaving without synchronisation between the parallel components, and a parallel com-position operator with CCS-style synchronisation. It is natural to consider extensions ofour result to parallel composition operators with other forms of synchronisation. We expectthe extensions with ACP-style or CSP-style synchronisation to be less straightforward thanthe extension with CCS-style synchronisation presented in this paper, especially in the caseof the negative results, and we leave these as topics for future investigations.As previously outlined, in [2] it was proved that the nested semantics admit no finiteaxiomatisation over BCCSP. However, our negative results cannot be reduced to a merelifting of those in [2], as the presence of the additional operator k might, at least in principle,allow us to finitely axiomatise the equations over BCCSP processes that are valid modulo theconsidered nested semantics. Indeed, auxiliary operators can be added to a language in orderto obtain a finite axiomatisation of some congruence relation (see, e.g. the classic examplegiven in [6]). Understanding whether it is possible to lift non-finite axiomatisability resultsamong different algebras, and under which constraints this can be done, is an interestingresearch avenue and we aim to investigate it in future work. A methodology for transferringnon-finite-axiomatisability results across languages was presented in [4], where a reduction-based approach was proposed. However, that method has some limitations and thus furtherstudies are needed.A behavioural equivalence is finitely based if it has a finite equational axiomatisationfrom which all valid equations between open terms are derivable. In [16] and [3] finite basesfor bisimilarity with respect to PA and BCCSP k extended with the auxiliary operators leftmerge and communication merge were presented. Furthermore, in [12] an overview wasgiven of which behavioural equivalences in the linear time-branching time spectrum arefinitely based with respect to BCCSP. The negative results in Section 6 imply that noneof the behavioural equivalences between possible futures equivalence and bisimilarity isfinitely based with respect to BCCSP k . An interesting question is which of the behaviouralequivalences including ready simulation semantics is finitely based with respect to BCCSP k .In [14] an alternative classification of the equivalences in the spectrum with respectto [19] was proposed. In order to obtain a general, unified, view of process semantics, thespectrum was divided into layers, each corresponding to a different notion of constrainedsimulation [13]. There are pleasing connections between the different layers and the parti-tioning they induce of the congruences in the spectrum, as given in [14], and the relationshipsbetween the axioms for the interleaving operator we have presented in this study. Acknowledgement
This work has been supported by the project ‘
Open Problems in the Equational Logic ofProcesses ’ (OPEL) of the Icelandic Research Fund (grant No. 196050-051).We thank Rob van Glabbeek for a fruitful discussion on the axiomatisability of failuresequivalence.
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Proceedings of Annual Symposium on Foundations of Computer Science , pages 140–149,1981. doi:10.1109/SFCS.1981.36 . Appendix A. The Mace4 code for E CS The following is the Mace4 code we used to generate a model for E CS in which EL2 (as givenin formulas(goals) below) does not hold.s e t ( v e r b o s e ) .a s s i g n ( max megs , 1 0 0 0 ) .a s s i g n ( d o m a i n s i z e , 5 ) .op ( 7 5 0 , p r e f i x , ” a ” ) .op ( 7 5 0 , p r e f i x , ” b ” ) .op ( 8 5 0 , i n f i x , ” p l u s ” ) .op ( 9 5 0 , i n f i x , ” p ar ” ) .f o r m u l a s ( a s s u m p t i o n s ) .( x p l u s 0 ) = x .( x p l u s y ) = ( y p l u s x ) .( ( x p l u s y ) p l u s z ) = ( x p l u s ( y p l u s z ) ) .( x p l u s x ) = x .( x p ar 0 ) = x .( x p ar y ) = ( y p ar x ) .% Axiom EL1( a x p ar a y ) = ( a ( x p ar a y ) p l u s a ( y p ar a x ) ) .( a x p ar b y ) = ( a ( x p ar b y ) p l u s b ( y p ar a x ) ) .( b x p ar b y ) = ( b ( x p ar b y ) p l u s b ( y p ar b x ) ) .% Axiom CS( a ( a x p l u s ( y p l u s z ) ) ) = ( a ( a x p l u s ( y p l u s z ) ) p l u s a ( ax p l u s z ) ) .( a ( b x p l u s ( y p l u s z ) ) ) = ( a ( b x p l u s ( y p l u s z ) ) p l u s a ( bx p l u s z ) ) .( b ( a x p l u s ( y p l u s z ) ) ) = ( b ( a x p l u s ( y p l u s z ) ) p l u s b ( ax p l u s z ) ) .( b ( b x p l u s ( y p l u s z ) ) ) = ( b ( b x p l u s ( y p l u s z ) ) p l u s b ( bx p l u s z ) ) .% Axiom CSP1( ( a x p l u s ( a y p l u s u ) ) p ar ( a z p l u s ( a w p l u s v ) ) ) =( ( ( a x p l u s u ) p ar ( a z p l u s ( a w p l u s v ) ) )p l u s ( ( ( a y p l u s u ) p ar ( a z p l u s ( a w p l u s v ) ) )p l u s ( ( ( a x p l u s ( a y p l u s u ) ) p ar ( a z p l u s v ) )p l u s ( ( a x p l u s ( a y p l u s u ) ) p ar ( a w p l u s v ) )) ) ) .( ( a x p l u s ( a y p l u s u ) ) p ar ( a z p l u s ( b w p l u s v ) ) ) =( ( ( a x p l u s u ) p ar ( a z p l u s ( b w p l u s v ) ) )p l u s ( ( ( a y p l u s u ) p ar ( a z p l u s ( b w p l u s v ) ) ) N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 45 p l u s ( ( ( a x p l u s ( a y p l u s u ) ) p ar ( a z p l u s v ) )p l u s ( ( a x p l u s ( a y p l u s u ) ) p ar ( b w p l u s v ) )) ) ) .( ( a x p l u s ( b y p l u s u ) ) p ar ( a z p l u s ( b w p l u s v ) ) ) =( ( ( a x p l u s u ) p ar ( a z p l u s ( b w p l u s v ) ) )p l u s ( ( ( b y p l u s u ) p ar ( a z p l u s ( b w p l u s v ) ) )p l u s ( ( ( a x p l u s ( b y p l u s u ) ) p ar ( a z p l u s v ) )p l u s ( ( a x p l u s ( b y p l u s u ) ) p ar ( b w p l u s v ) )) ) ) .( ( a x p l u s ( b y p l u s u ) ) p ar ( b z p l u s ( b w p l u s v ) ) ) =( ( ( a x p l u s u ) p ar ( b z p l u s ( b w p l u s v ) ) )p l u s ( ( ( b y p l u s u ) p ar ( b z p l u s ( b w p l u s v ) ) )p l u s ( ( ( a x p l u s ( b y p l u s u ) ) p ar ( b z p l u s v ) )p l u s ( ( a x p l u s ( b y p l u s u ) ) p ar ( b w p l u s v ) )) ) ) .( ( b x p l u s ( b y p l u s u ) ) p ar ( b z p l u s ( b w p l u s v ) ) ) =( ( ( b x p l u s u ) p ar ( b z p l u s ( b w p l u s v ) ) )p l u s ( ( ( b y p l u s u ) p ar ( b z p l u s ( b w p l u s v ) ) )p l u s ( ( ( b x p l u s ( b y p l u s u ) ) p ar ( b z p l u s v ) )p l u s ( ( b x p l u s ( b y p l u s u ) ) p ar ( b w p l u s v ) )) ) ) .% Axiom CSP2( a x p ar ( a y p l u s ( a z p l u s w) ) ) =( a ( x p ar ( a y p l u s ( a z p l u s w) ) )p l u s ( ( a x p ar ( a y p l u s w) )p l u s ( ( a x p ar ( a z p l u s w) )) ) ) .( a x p ar ( a y p l u s ( b z p l u s w) ) ) =( a ( x p ar ( a y p l u s ( b z p l u s w) ) )p l u s ( ( a x p ar ( a y p l u s w) )p l u s ( ( a x p ar ( b z p l u s w) )) ) ) .( a x p ar ( b y p l u s ( b z p l u s w) ) ) =( a ( x p ar ( b y p l u s ( b z p l u s w) ) )p l u s ( ( a x p ar ( b y p l u s w) )p l u s ( ( a x p ar ( b z p l u s w) )) ) ) .( b x p ar ( a y p l u s ( a z p l u s w) ) ) =( b ( x p ar ( a y p l u s ( a z p l u s w) ) )p l u s ( ( b x p ar ( a y p l u s w) )p l u s ( ( b x p ar ( a z p l u s w) )) ) ) . ( b x p ar ( a y p l u s ( b z p l u s w) ) ) =( b ( x p ar ( a y p l u s ( b z p l u s w) ) )p l u s ( ( b x p ar ( a y p l u s w) )p l u s ( ( b x p ar ( b z p l u s w) )) ) ) .( b x p ar ( b y p l u s ( b z p l u s w) ) ) =( b ( x p ar ( b y p l u s ( b z p l u s w) ) )p l u s ( ( b x p ar ( b y p l u s w) )p l u s ( ( b x p ar ( b z p l u s w) )) ) ) .e n d o f l i s t .f o r m u l a s ( g o a l s ) .( ( a x p l u s b y ) p ar ( a z p l u s b w) ) = ( a ( x p ar ( a z p l u s b w) )p l u s ( b ( y p ar ( a z p l u s b w) ) p l u s ( a ( z p ar ( a x p l u s b y ) )p l u s b (w p ar ( a x p l u s b y ) ) ) ) ) .e n d o f l i s t .
Appendix B. The Mace4 code for E RT The following is the Mace4 code we used to generate a model for E RT in which RSP2 (asgiven in formulas(goals) below) does not hold.s e t ( v e r b o s e ) .a s s i g n ( max megs , 1 0 0 0 ) .op ( 7 5 0 , p r e f i x , ” a ” ) .op ( 7 5 0 , p r e f i x , ” b ” ) .op ( 8 5 0 , i n f i x , ” p l u s ” ) .op ( 9 5 0 , i n f i x , ” p ar ” ) .f o r m u l a s ( a s s u m p t i o n s ) .( x p l u s 0 ) = x .( x p l u s y ) = ( y p l u s x ) .( ( x p l u s y ) p l u s z ) = ( x p l u s ( y p l u s z ) ) .( x p l u s x ) = x .( x p ar 0 ) = x .( x p ar y ) = ( y p ar x ) .% Axiom RT( a ( ( ( a x p l u s a y ) p l u s ( a u p l u s a v ) ) p l u s z ) ) =( a ( ( a x p l u s a u ) p l u s z )p l u s a ( ( a y p l u s a v ) p l u s z )) .( a ( ( ( a x p l u s a y ) p l u s ( b u p l u s b v ) ) p l u s z ) ) =( a ( ( a x p l u s b u ) p l u s z ) N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 47 p l u s a ( ( a y p l u s b v ) p l u s z )) .( a ( ( ( b x p l u s b y ) p l u s ( b u p l u s b v ) ) p l u s z ) ) =( a ( ( b x p l u s b u ) p l u s z )p l u s a ( ( b y p l u s b v ) p l u s z )) .( b ( ( ( a x p l u s a y ) p l u s ( a u p l u s a v ) ) p l u s z ) ) =( b ( ( a x p l u s a u ) p l u s z )p l u s b ( ( a y p l u s a v ) p l u s z )) .( b ( ( ( a x p l u s a y ) p l u s ( b u p l u s b v ) ) p l u s z ) ) =( b ( ( a x p l u s b u ) p l u s z )p l u s b ( ( a y p l u s b v ) p l u s z )) .( b ( ( ( b x p l u s b y ) p l u s ( b u p l u s b v ) ) p l u s z ) ) =( b ( ( b x p l u s b u ) p l u s z )p l u s b ( ( b y p l u s b v ) p l u s z )) .% Axiom FP( ( a x p l u s ( a y p l u s w) ) p ar z ) = ( ( ( a x p l u s w) p ar z ) p l u s( ( a y p l u s w) p ar z ) ) .% Axiom EL2( a x p ar a y ) = ( a ( x p ar a y ) p l u s a ( y p ar a x ) ) .( a x p ar b y ) = ( a ( x p ar b y ) p l u s b ( y p ar a x ) ) .( b x p ar b y ) = ( b ( x p ar b y ) p l u s b ( y p ar b x ) ) .( ( a x p l u s b y ) p ar ( a z p l u s b w) ) =( a ( x p ar ( a z p l u s b w) )p l u s ( b ( y p ar ( a z p l u s b w) )p l u s ( a ( ( a x p l u s b y ) p ar z )p l u s ( b ( ( a x p l u s b y ) p ar w)) ) ) ) .e n d o f l i s t .f o r m u l a s ( g o a l s ) .( a x p ar ( b u p l u s ( b v p l u s w) ) ) =( ( a x p ar ( b u p l u s w ) )p l u s ( ( a x p ar ( b v p l u s w ) ) p l u s a ( x p ar ( b u p l u s ( b v p l u s w) ) )) ) .e n d o f l i s t .
Appendix C. The Mace4 codes for E CT The following is the Mace4 code we used to generate a model for E CT in which EL2 (as givenin formulas(goals) below) does not hold.s e t ( v e r b o s e ) .a s s i g n ( max megs , 1 0 0 0 ) .a s s i g n ( d o m a i n s i z e , 5 ) .op ( 7 5 0 , p r e f i x , ” a ” ) .op ( 7 5 0 , p r e f i x , ” b ” ) .op ( 8 5 0 , i n f i x , ” p l u s ” ) .op ( 9 5 0 , i n f i x , ” p ar ” ) .f o r m u l a s ( a s s u m p t i o n s ) .( x p l u s 0 ) = x .( x p l u s y ) = ( y p l u s x ) .( ( x p l u s y ) p l u s z ) = ( x p l u s ( y p l u s z ) ) .( x p l u s x ) = x .( x p ar 0 ) = x .( x p ar y ) = ( y p ar x ) .% Axiom EL1( a x p ar b y ) = ( a ( x p ar b y ) p l u s b ( y p ar a x ) ) .( a x p ar a y ) = ( a ( x p ar a y ) p l u s a ( y p ar a x ) ) .( b x p ar b y ) = ( b ( x p ar b y ) p l u s b ( y p ar b x ) ) .% Axiom CT( a ( a x p l u s z ) p l u s a ( a y p l u s w) ) = ( a ( a x p l u s ( a y p l u s ( zp l u s w) ) ) ) .( a ( a x p l u s z ) p l u s a ( b y p l u s w) ) = ( a ( a x p l u s ( b y p l u s ( zp l u s w) ) ) ) .( a ( b x p l u s z ) p l u s a ( a y p l u s w) ) = ( a ( b x p l u s ( a y p l u s ( zp l u s w) ) ) ) .( a ( b x p l u s z ) p l u s a ( b y p l u s w) ) = ( a ( b x p l u s ( b y p l u s ( zp l u s w) ) ) ) .( b ( a x p l u s z ) p l u s a ( a y p l u s w) ) = ( b ( a x p l u s ( a y p l u s ( zp l u s w) ) ) ) .( b ( a x p l u s z ) p l u s a ( b y p l u s w) ) = ( b ( a x p l u s ( b y p l u s ( zp l u s w) ) ) ) .( b ( b x p l u s z ) p l u s a ( a y p l u s w) ) = ( b ( b x p l u s ( a y p l u s ( zp l u s w) ) ) ) . N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 49 ( b ( b x p l u s z ) p l u s a ( b y p l u s w) ) = ( b ( b x p l u s ( b y p l u s ( zp l u s w) ) ) ) .% Axiom CTP( ( a x p l u s ( a y p l u s w) ) p ar z ) = ( ( ( a x p l u s w) p ar z ) p l u s ( (a y p l u s w) p ar z ) ) .( ( a x p l u s ( b y p l u s w) ) p ar z ) = ( ( ( a x p l u s w) p ar z ) p l u s ( (b y p l u s w) p ar z ) ) .( ( b x p l u s ( b y p l u s w) ) p ar z ) = ( ( ( b x p l u s w) p ar z ) p l u s ( (b y p l u s w) p ar z ) ) .e n d o f l i s t .f o r m u l a s ( g o a l s ) .( ( a x p l u s b y ) p ar ( a z p l u s b w) ) = ( a ( x p ar ( a z p l u s b w) )p l u s ( b ( y p ar ( a z p l u s b w) ) p l u s ( a ( z p ar ( a x p l u s b y ) )p l u s b (w p ar ( a x p l u s b y ) ) ) ) ) .e n d o f l i s t .Finally, by combining the formulas(assumptions) given above with the formulas(goals) presented below, we can generate a model for E CT in which CSP2 does not hold.f o r m u l a s ( g o a l s ) .( a x p ar ( b u p l u s ( b v p l u s w) ) ) =( ( a x p ar ( b u p l u s w ) )p l u s ( ( a x p ar ( b v p l u s w ) )p l u s a ( x p ar ( b u p l u s ( b v p l u s w) ) )) ) .e n d o f l i s t . Appendix D. Simplifying the equational theory: saturated systems
The axioms A0 and P0 in Table 3 (used from left to right) are enough to establish that eachBCCSP k term that is possible futures equivalent to is also provably equal to . Lemma D.1.
Let t be a BCCSP k term. Then t ∼ PF if and only if the equation t ≈ isprovable using axioms A0 and P0 in Table 3 from left to right.Proof. The “if” implication is an immediate consequence of the soundness of the equationsA0 and P0 with respect to ∼ PF . To prove the “only if” implication, define, first of all, thecollection NIL of BCCSP k terms as the set of terms generated by the following grammar: t ::= | t + t | t k t , We claim that each BCCSP k term t is ∼ PF equivalent to if and only if t ∈ NIL. Usingthis claim and structural induction on t ∈ NIL, it is a simple matter to show that if t ∼ PF ,then t ≈ is provable using axioms A0 and P0 from left to right, which was to be shown.To complete the proof, it therefore suffices to show the above claim. To establish the“if” implication in the statement of the claim, one proves, using structural induction on t and the congruence properties of ∼ PF , that if t ∈ NIL, then σ ( t ) ∼ PF for every closed substitution σ . To show the “only if” implication, we establish the contrapositive statement,namely that if t NIL, then σ ( t ) PF for some closed substitution σ . To this end, itsuffices only to show, using structural induction on t , that if t NIL, then σ a ( t ) a −→ forsome action a ∈ A , where σ a is the closed substitution mapping each variable to the closedterm a . The details of this argument are not hard, and are therefore left to the reader.In light of the above result, we shall assume, without loss of generality, that each axiomsystem we consider includes the equations in Table 3. This assumption means, in particular,that our axiom systems will allow us to identify each term that is possible futures equivalentto with .We recall that a BCCSP k term t has a factor if it contains a subterm of the form t k t , where either t or t is possible futures equivalent to .It is easy to see that, modulo the equations in Table 3, every BCCSP k term t has theform P i ∈ I t i , for some finite index set I , and terms t i ( i ∈ I ) that are not and do not havethemselves the form t ′ + t ′′ , for some terms t ′ and t ′′ . The terms t i ( i ∈ I ) will be referredto as the summands of t . Moreover, again modulo the equations in Table 3, each of the t i can be assumed to have no factors.We can now introduce the notion of saturated system , namely an axiom system suchthat if a closed equation that relates two terms containing no occurrences of as a summandor factor, then there is a closed proof for it in which all of the terms have no occurrences of as a summand or factor (cf. [27, Proposition 5.1.5]). Definition D.2.
For each BCCSP k term t , we define t/ thus: / = x/ = x at/ = a ( t/ )( t + u ) / = u/ if t ∼ PF t/ if u ∼ PF ( t/ ) + ( u/ ) otherwise ( t k u ) / = u/ if t ∼ PF t/ if u ∼ PF ( t/ ) k ( u/ ) otherwise.Intuitively, t/ is the term that results by removing all occurrences of as a summandor factor from t .The following lemma collects the basic properties of the above construction. Lemma D.3.
For each
BCCSP k term t , the following statements hold: (1) The equation t ≈ t/ can be proven using the equations in Table 3, and therefore t ∼ PF t/ . (2) The term t/ has no summands or factors. (3) t/ = t , if t has no occurrence of as a summand or factor. (4) σ ( t/ ) / = σ ( t ) / , for each substitution σ .Proof. Immediate by structural induction over t . Definition D.4 (Saturated system) . We say that a substitution σ is a -substitution ifand only if σ ( x ) = x implies that σ ( x ) = , for each variable x . Let E be an axiom system.We define the axiom system cl( E ) thus:cl( E ) = E ∪ { σ ( t ) / ≈ σ ( u ) / | ( t ≈ u ) ∈ E , σ a -substitution } . An axiom system E is saturated if E = cl( E ).The following lemma collects some basic sanity properties of the closure operator cl( · ). N THE AXIOMATISABILITY OF PARALLEL COMPOSITION 51
Lemma D.5.
Let E be an axiom system. Then the following statements hold. (1) cl( E ) = cl(cl( E )) . (2) cl( E ) is finite, if so is E . (3) cl( E ) is sound modulo possible futures equivalence, if so is E . (4) cl( E ) is closed with respect to symmetry, if so is E . (5) cl( E ) and E prove the same equations, if E contains the equations in Table 3.Proof. We limit ourselves to sketching the proofs of statements (1) and (5) in the lemma.In the proof of statement (1), the only non-trivial thing to check is that the equation σ ( σ ′ ( t ) / )) / ≈ σ ( σ ′ ( u ) / )) / is contained in cl( E ), whenever ( t ≈ u ) ∈ E and σ, σ ′ are -substitutions. This follows fromLemma D.3.(4) because the collection of -substitutions is closed under composition.To show statement (5), it suffices only to argue that each equation t ≈ u that is provablefrom cl( E ) is also provable from E , if E contains the equations in Table 3. This can be doneby induction on the depth of the proof of the equation t ≈ u from cl( E ), using Lemma D.3.(1)for the case in which t ≈ u is a substitution instance of an axiom in cl( E ).We are now ready to state our counterpart of [27, Proposition 5.1.5]. Proposition D.6.
Assume that E is a saturated axiom system. Suppose furthermore thatwe have a closed proof from E of the closed equation p ≈ q . Then replacing each term r in that proof with r/ yields a closed proof of the equation p/ ≈ q/ . In particular, theproof from E of an equation p ≈ q , where p and q are terms not containing occurrences of as a summand or factor, need not use terms containing occurrences of as a summandor factor.Proof. The proof follows the lines of that of [27, Proposition 5.1.5], and is therefore omitted.
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