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## ON THE AXIOMATIZABILITY OF IMPOSSIBLE FUTURES (2015)

### Citations

4170 | Communicating Sequential Processes
- Hoare
- 1985
(Show Context)
Citation Context ...bility of concrete and weak semantics. For any semantics at least as coarse as impossible futures semantics, an algorithm is provided to turn a ground-complete axiomatization of the concrete version into a ground-complete axiomatization of the corresponding weak version. Moreover, if the former axiomatization is ω-complete, then so is the latter. (2) As an application of this algorithm, we derive finite, sound, ground- and ω-complete axiomatizations for the weak trace, completed trace and failures preorders and equivalences. Failures semantics plays a prominent role in the process algebra CSP [BHR84]. For convergent processes, it coincides with testing semantics [DH84, RV07], and thus is the coarsest congruence for the CCS parallel composition that respects deadlock behavior. A ground-complete axiomatization for weak failures equivalence was already given in [G97]. (3) We provide a finite, sound, ground-complete axiomatization for BCCSP modulo the concrete impossible futures preorder -IF.3 (By contrast, no such axiomatization exists for the possible futures preorder [AFGI04].) Using (1), a finite, sound, ground-complete axiomatization for the weak impossible futures preorder -WIF is obtai... |

523 | Testing equivalences for processes
- Nicola, Hennessy
- 1984
(Show Context)
Citation Context ...arse as impossible futures semantics, an algorithm is provided to turn a ground-complete axiomatization of the concrete version into a ground-complete axiomatization of the corresponding weak version. Moreover, if the former axiomatization is ω-complete, then so is the latter. (2) As an application of this algorithm, we derive finite, sound, ground- and ω-complete axiomatizations for the weak trace, completed trace and failures preorders and equivalences. Failures semantics plays a prominent role in the process algebra CSP [BHR84]. For convergent processes, it coincides with testing semantics [DH84, RV07], and thus is the coarsest congruence for the CCS parallel composition that respects deadlock behavior. A ground-complete axiomatization for weak failures equivalence was already given in [G97]. (3) We provide a finite, sound, ground-complete axiomatization for BCCSP modulo the concrete impossible futures preorder -IF.3 (By contrast, no such axiomatization exists for the possible futures preorder [AFGI04].) Using (1), a finite, sound, ground-complete axiomatization for the weak impossible futures preorder -WIF is obtained. (4) We prove that BCCS modulo weak impossible futures equivalence 'WIF ... |

367 | The linear time-branching time spectrum
- Glabbeek
- 1990
(Show Context)
Citation Context ... R. VAN GLABBEEK ready traces failures completed traces traces possible worlds bisimulation 2-nested simulation ready simulation readies completed simulation possible futures failure traces simulation impossible futures Figure 1: Linear time-branching time spectrum spectrum I” of behavioral semantics for finitely branching, concrete,1 sequential processes. These semantics are based on simulation notions or on decorated traces. Fig. 1 depicts this spectrum,2 where an arrow from one semantics to another means that the target of the arrow is coarser, i.e. less discriminating, than the source. In [G93a], 155 weak semantics, which take into account the hidden action τ , are surveyed. They constitute the “linear time – branching time spectrum II” for finitely branching, abstract, sequential processes. In this paper, we mainly study impossible futures semantics [V92, VM01], which is a natural variant of possible futures semantics [RB81]. It is also related to fair testing semantics [RV07]. Weak impossible futures equivalence is the coarsest congruence with respect to choice and parallel composition that contains weak bisimilarity with explicit divergence, respects deadlock/livelock traces, and ... |

123 | The linear time-branching time spectrum I. The semantics of concrete, sequential processes
- Glabbeek
- 2001
(Show Context)
Citation Context ... to be ω-complete. If the alphabet is finite, we prove that the inequational theories of BCCS modulo the concrete and weak impossible futures preorder lack such a finite basis. 1. Introduction Labeled transition systems constitute a fundamental model of concurrent computation. Processes are captured by explicitly describing their states and the transitions from state to state together with the actions that produce these transitions. A wide range of notions of behavioral semantics have been proposed, with the aim to identify those states that afford the same observations. Notably, van Glabbeek [G01] presented the “linear time – branching time 2012 ACM CCS: [Theory of computation]: Models of computation—Concurrency—Process calculi. Key words and phrases: Concurrency Theory, Equational Logic, Impossible Futures, Process Algebra. c NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program. LOGICAL METHODSl IN COMPUTER SCIENCE DOI:10.2168/LMCS-11(3:17)2015 c© T. Chen, W. Fokkink, and R. van GlabbeekCC© Creative Commons 2 T. CHEN, W. FOKKINK, AND R. VAN GLABBEEK ready traces failures c... |

96 | Fair testing
- Rensink, Vogler
(Show Context)
Citation Context ...mulation notions or on decorated traces. Fig. 1 depicts this spectrum,2 where an arrow from one semantics to another means that the target of the arrow is coarser, i.e. less discriminating, than the source. In [G93a], 155 weak semantics, which take into account the hidden action τ , are surveyed. They constitute the “linear time – branching time spectrum II” for finitely branching, abstract, sequential processes. In this paper, we mainly study impossible futures semantics [V92, VM01], which is a natural variant of possible futures semantics [RB81]. It is also related to fair testing semantics [RV07]. Weak impossible futures equivalence is the coarsest congruence with respect to choice and parallel composition that contains weak bisimilarity with explicit divergence, respects deadlock/livelock traces, and assigns unique solutions to recursive equations [GV06]. The process algebra BCCSP plays a fundamental role in the study of concrete semantics. It contains only basic process algebraic operators from CCS and CSP, but is sufficiently powerful to express all finite synchronization trees (without τ -transitions). Van Glabbeek [G01] associated with most behavioral equivalences in his spectrum... |

76 |
Modular Construction and Partial Order Semantics of Petri Nets.
- Vogler
- 1992
(Show Context)
Citation Context ...pectrum I” of behavioral semantics for finitely branching, concrete,1 sequential processes. These semantics are based on simulation notions or on decorated traces. Fig. 1 depicts this spectrum,2 where an arrow from one semantics to another means that the target of the arrow is coarser, i.e. less discriminating, than the source. In [G93a], 155 weak semantics, which take into account the hidden action τ , are surveyed. They constitute the “linear time – branching time spectrum II” for finitely branching, abstract, sequential processes. In this paper, we mainly study impossible futures semantics [V92, VM01], which is a natural variant of possible futures semantics [RB81]. It is also related to fair testing semantics [RV07]. Weak impossible futures equivalence is the coarsest congruence with respect to choice and parallel composition that contains weak bisimilarity with explicit divergence, respects deadlock/livelock traces, and assigns unique solutions to recursive equations [GV06]. The process algebra BCCSP plays a fundamental role in the study of concrete semantics. It contains only basic process algebraic operators from CCS and CSP, but is sufficiently powerful to express all finite synchroni... |

71 |
A complete axiomatisation for observational congruence of finite-state behaviours
- Milner
- 1989
(Show Context)
Citation Context ... ground-complete axiomatization exists. And if so, whether a finite basis exists for the (in)equational theory. However, for concrete impossible futures semantics the (in)equational theory remained unexplored. With regard to the axiomatizability of weak semantics, relatively little is known compared to concrete semantics. For some semantics in the “linear time – branching time spectrum II” [G93a], a sound and ground-complete axiomatization has been given, in the setting of BCCS (see, e.g., [G97]). Moreover, a finite basis has been given for weak, delay, η- and branching bisimulation semantics [M89, G93b]. The inequational theory of BCCS modulo the weak impossible futures preorder was studied in [VM01], which offers a finite, sound, ground-complete axiomatization. Voorhoeve and Mauw also proved that their axiomatization is ω-complete. It is worth noting that an infinite alphabet of actions is assumed implicitly [VM01, p. 7], because a different action is required for each variable. The current paper studies the axiomatizability of BCCSP and BCCS for semantics at least as coarse as impossible futures semantics. In summary, we obtain the following results. (1) A link is established between the a... |

32 | A complete axiomatization for branching bisimulation congruence of finitestate behaviours
- Glabbeek
- 1993
(Show Context)
Citation Context ... ground-complete axiomatization exists. And if so, whether a finite basis exists for the (in)equational theory. However, for concrete impossible futures semantics the (in)equational theory remained unexplored. With regard to the axiomatizability of weak semantics, relatively little is known compared to concrete semantics. For some semantics in the “linear time – branching time spectrum II” [G93a], a sound and ground-complete axiomatization has been given, in the setting of BCCS (see, e.g., [G97]). Moreover, a finite basis has been given for weak, delay, η- and branching bisimulation semantics [M89, G93b]. The inequational theory of BCCS modulo the weak impossible futures preorder was studied in [VM01], which offers a finite, sound, ground-complete axiomatization. Voorhoeve and Mauw also proved that their axiomatization is ω-complete. It is worth noting that an infinite alphabet of actions is assumed implicitly [VM01, p. 7], because a different action is required for each variable. The current paper studies the axiomatizability of BCCSP and BCCS for semantics at least as coarse as impossible futures semantics. In summary, we obtain the following results. (1) A link is established between the a... |

30 | Finite equational bases in process algebra: Results and open questions
- Aceto, Fokkink, et al.
- 2005
(Show Context)
Citation Context ...red. ON THE AXIOMATIZABILITY OF IMPOSSIBLE FUTURES 3 An ω-complete axiomatization enjoys the property that if all closed instances of an equation can be derived from it, then the equation itself can be derived from it. In universal algebra, such an axiomatization is referred to as a basis for the equational theory of the algebra it axiomatizes. Groote [G90] developed a technique of “inverted substitutions” to prove that an axiomatization is ω-complete, and proved for some equivalences in the “linear time – branching time spectrum I” that their equational theory in BCCSP has a finite basis. In [AFIL05, CFLN08], for each preorder and equivalence in the “linear time – branching time spectrum I” it was determined whether a finite, sound, ground-complete axiomatization exists. And if so, whether a finite basis exists for the (in)equational theory. However, for concrete impossible futures semantics the (in)equational theory remained unexplored. With regard to the axiomatizability of weak semantics, relatively little is known compared to concrete semantics. For some semantics in the “linear time – branching time spectrum II” [G93a], a sound and ground-complete axiomatization has been given, in the settin... |

23 |
A new strategy for proving ω-completeness with applications in process algebra
- Groote
- 1990
(Show Context)
Citation Context ...ing τ just as any other action, or weak ones, allowing a degree of abstraction from τ steps. 2Impossible futures semantics was missing in the original spectrum I [G01], because it was studied seriously only from 2001 on, the year that [G01] appeared. ON THE AXIOMATIZABILITY OF IMPOSSIBLE FUTURES 3 An ω-complete axiomatization enjoys the property that if all closed instances of an equation can be derived from it, then the equation itself can be derived from it. In universal algebra, such an axiomatization is referred to as a basis for the equational theory of the algebra it axiomatizes. Groote [G90] developed a technique of “inverted substitutions” to prove that an axiomatization is ω-complete, and proved for some equivalences in the “linear time – branching time spectrum I” that their equational theory in BCCSP has a finite basis. In [AFIL05, CFLN08], for each preorder and equivalence in the “linear time – branching time spectrum I” it was determined whether a finite, sound, ground-complete axiomatization exists. And if so, whether a finite basis exists for the (in)equational theory. However, for concrete impossible futures semantics the (in)equational theory remained unexplored. With r... |

22 |
Possible futures, acceptances, refusals and communicating processes
- ROUNDS, BROOKES
- 1981
(Show Context)
Citation Context ...sequential processes. These semantics are based on simulation notions or on decorated traces. Fig. 1 depicts this spectrum,2 where an arrow from one semantics to another means that the target of the arrow is coarser, i.e. less discriminating, than the source. In [G93a], 155 weak semantics, which take into account the hidden action τ , are surveyed. They constitute the “linear time – branching time spectrum II” for finitely branching, abstract, sequential processes. In this paper, we mainly study impossible futures semantics [V92, VM01], which is a natural variant of possible futures semantics [RB81]. It is also related to fair testing semantics [RV07]. Weak impossible futures equivalence is the coarsest congruence with respect to choice and parallel composition that contains weak bisimilarity with explicit divergence, respects deadlock/livelock traces, and assigns unique solutions to recursive equations [GV06]. The process algebra BCCSP plays a fundamental role in the study of concrete semantics. It contains only basic process algebraic operators from CCS and CSP, but is sufficiently powerful to express all finite synchronization trees (without τ -transitions). Van Glabbeek [G01] associa... |

18 | Ready to preorder: get your BCCSP axiomatization for free
- Aceto, Fokkink, et al.
- 2007
(Show Context)
Citation Context ...is necessarily incomplete, so ≡ (resp. v) lacks a finite axiomatization. On top of this, a saturation principle is introduced, to transform a single summand into a large collection of semi-saturated summands. Impossible futures semantics is the first example that, in the context of BCCSP/BCCS, affords a ground-complete axiomatization modulo the preorder, while missing a groundcomplete axiomatization modulo the equivalence. This surprising fact suggests that if one wants to show p 'IF q, in general one has to resort to deriving p -IF q and q -IF p separately, instead of proving it directly. In [AFI07, FGP07] an algorithm is presented which produces from an axiomatization for BCCSP modulo a preorder, an axiomatization for BCCSP modulo the associated equivalence. If the original axiomatization for the preorder is ground-complete or ω-complete, then so is the resulting axiomatization for the equivalence. In [CFG08], we have shown that the same algorithm applies equally well to weak semantics. However, these algorithms apply only to semantics that are at least as coarse as ready simulation semantics. Since impossible futures semantics is incomparable to ready simulation semantics, it falls outside th... |

15 | Nested semantics over finite trees are equationally hard
- Aceto, Fokkink, et al.
- 2004
(Show Context)
Citation Context ... trace and failures preorders and equivalences. Failures semantics plays a prominent role in the process algebra CSP [BHR84]. For convergent processes, it coincides with testing semantics [DH84, RV07], and thus is the coarsest congruence for the CCS parallel composition that respects deadlock behavior. A ground-complete axiomatization for weak failures equivalence was already given in [G97]. (3) We provide a finite, sound, ground-complete axiomatization for BCCSP modulo the concrete impossible futures preorder -IF.3 (By contrast, no such axiomatization exists for the possible futures preorder [AFGI04].) Using (1), a finite, sound, ground-complete axiomatization for the weak impossible futures preorder -WIF is obtained. (4) We prove that BCCS modulo weak impossible futures equivalence 'WIF does not have a finite, sound, ground-complete axiomatization. Likewise, we prove that BCCSP modulo 'IF does not have a finite, sound, ground-complete axiomatization. The infinite families of equations that we use to prove these negative results are also sound modulo (weak resp. concrete) 2-nested simulation equivalence. Therefore these negative results apply to all BCCS- and BCCSP-congruences that are at... |

9 | A finite basis for failure semantics.
- Fokkink, Nain
- 2005
(Show Context)
Citation Context ...(F2) have to be added to make the axiomatization ω-complete. But the latter inequation, ∑ a∈A τxa 4 ∑ a∈A τxa + y, can be derived using W1′ and WIF1. Theorem 4.1. A1-4+WIF1+WIF2′+W1′ is sound and ground-complete for BCCS(A) modulo -WF. If |A |=∞, then it is also ω-complete. If |A |<∞, then the axiomatization becomes ω-complete by adding the (sound) axiom F2. According to [G01], A1-4 together with the axioms FE1 ax+ a(y + z) ≈ ax+ a(x+ y) + a(y + z) FE2∗ a(bx+ u) + a(by + v) ≈ a(bx+ by + u) + a(by + v) constitute a sound and ground-complete axiomatization for BCCSP(A) modulo 'F. As remarked in [FN05], in the presence of FE1, axiom FE2∗ can be simplified to FE2 a(x+ by) + a(x+ by + bz) ≈ a(x+ bx+ bz). Moreover, in [FN05] it was proved that if |A |=∞ then this axiomatization is ω-complete, while if |A |<∞ then a finite basis for the inequational theory of BCCSP(A) modulo -F is obtained by adding the following axiom: FE3 a(x+ ∑ b∈A bzb) + a(x+ y + ∑ b∈A bzb) ≈ a(x+ y + ∑ b∈A bzb). Our algorithm from the previous section produces a ground-complete axiomatization for BCCS(A) modulo 'WF, which consists of A1-4 and WIF1-2 together with FE1′ αx+ α(y + z) ≈ αx+ α(x+ y) + α(y + z) FE2′ α(x+ by) + α... |

8 | On finite alphabets and infinite bases.
- Chen, Fokkink, et al.
- 2008
(Show Context)
Citation Context ...red. ON THE AXIOMATIZABILITY OF IMPOSSIBLE FUTURES 3 An ω-complete axiomatization enjoys the property that if all closed instances of an equation can be derived from it, then the equation itself can be derived from it. In universal algebra, such an axiomatization is referred to as a basis for the equational theory of the algebra it axiomatizes. Groote [G90] developed a technique of “inverted substitutions” to prove that an axiomatization is ω-complete, and proved for some equivalences in the “linear time – branching time spectrum I” that their equational theory in BCCSP has a finite basis. In [AFIL05, CFLN08], for each preorder and equivalence in the “linear time – branching time spectrum I” it was determined whether a finite, sound, ground-complete axiomatization exists. And if so, whether a finite basis exists for the (in)equational theory. However, for concrete impossible futures semantics the (in)equational theory remained unexplored. With regard to the axiomatizability of weak semantics, relatively little is known compared to concrete semantics. For some semantics in the “linear time – branching time spectrum II” [G93a], a sound and ground-complete axiomatization has been given, in the settin... |

7 | On the axiomatizability of impossible futures: Preorder versus equivalence.
- Chen, Fokkink
- 2008
(Show Context)
Citation Context ...nd (in)equational logic. Sect. 3 describes a transformation of an axiomatization for a concrete to an axiomatization for a corresponding weak semantics. In Sect. 4 this transformation is applied to failures, completed trace and trace semantics. Sect. 5 provides finite, sound, ground-complete axiomatizations for -IF and -WIF; it also presents the aforementioned negative result for 'IF and 'WIF. Sect. 6 is devoted to the proofs of the positive and negative results regarding ω-completeness. Sect. 7 concludes the paper. This paper combines and extends two previous papers in conference proceedings [CF08] and [CFG09]. In particular, [CF08] dealt with the concrete impossible futures semantics and [CFG09] extended it to weak impossible futures and weak failures semantics. Here, new results are presented in Sect. 3, which yield a much simplified proof for the results regarding ON THE AXIOMATIZABILITY OF IMPOSSIBLE FUTURES 5 weak failures semantics given in [CFG09], and a unified treatment of concrete and weak impossible futures semantics. 2. Preliminaries 2.1. Labeled transition systems. Let A be a nonempty, countable set of concrete (a.k.a. observable, external, visible) actions, which is ranged... |

7 |
Ready to preorder: an algebraic and general proof.
- Frutos-Escrig, Gregorio-Rodriguez, et al.
- 2008
(Show Context)
Citation Context ...is necessarily incomplete, so ≡ (resp. v) lacks a finite axiomatization. On top of this, a saturation principle is introduced, to transform a single summand into a large collection of semi-saturated summands. Impossible futures semantics is the first example that, in the context of BCCSP/BCCS, affords a ground-complete axiomatization modulo the preorder, while missing a groundcomplete axiomatization modulo the equivalence. This surprising fact suggests that if one wants to show p 'IF q, in general one has to resort to deriving p -IF q and q -IF p separately, instead of proving it directly. In [AFI07, FGP07] an algorithm is presented which produces from an axiomatization for BCCSP modulo a preorder, an axiomatization for BCCSP modulo the associated equivalence. If the original axiomatization for the preorder is ground-complete or ω-complete, then so is the resulting axiomatization for the equivalence. In [CFG08], we have shown that the same algorithm applies equally well to weak semantics. However, these algorithms apply only to semantics that are at least as coarse as ready simulation semantics. Since impossible futures semantics is incomparable to ready simulation semantics, it falls outside th... |

6 | Ready to preorder: The case of weak process semantics.
- Chen, Fokkink, et al.
- 2008
(Show Context)
Citation Context ...ete axiomatization modulo the preorder, while missing a groundcomplete axiomatization modulo the equivalence. This surprising fact suggests that if one wants to show p 'IF q, in general one has to resort to deriving p -IF q and q -IF p separately, instead of proving it directly. In [AFI07, FGP07] an algorithm is presented which produces from an axiomatization for BCCSP modulo a preorder, an axiomatization for BCCSP modulo the associated equivalence. If the original axiomatization for the preorder is ground-complete or ω-complete, then so is the resulting axiomatization for the equivalence. In [CFG08], we have shown that the same algorithm applies equally well to weak semantics. However, these algorithms apply only to semantics that are at least as coarse as ready simulation semantics. Since impossible futures semantics is incomparable to ready simulation semantics, it falls outside the scope of [AFI07, FGP07, CFG08]. Interestingly, our results yield that no such algorithm can exist for impossible futures semantics. The paper is structured as follows. Sect. 2 presents basic definitions regarding the studied semantics, the process algebra’s BCCSP and BCCS, and (in)equational logic. Sect. 3 ... |

4 | Liveness, fairness and impossible futures.
- Glabbeek, Voorhoeve
- 2006
(Show Context)
Citation Context ...e hidden action τ , are surveyed. They constitute the “linear time – branching time spectrum II” for finitely branching, abstract, sequential processes. In this paper, we mainly study impossible futures semantics [V92, VM01], which is a natural variant of possible futures semantics [RB81]. It is also related to fair testing semantics [RV07]. Weak impossible futures equivalence is the coarsest congruence with respect to choice and parallel composition that contains weak bisimilarity with explicit divergence, respects deadlock/livelock traces, and assigns unique solutions to recursive equations [GV06]. The process algebra BCCSP plays a fundamental role in the study of concrete semantics. It contains only basic process algebraic operators from CCS and CSP, but is sufficiently powerful to express all finite synchronization trees (without τ -transitions). Van Glabbeek [G01] associated with most behavioral equivalences in his spectrum a sound axiomatization, to equate closed BCCSP terms that are behaviorally equivalent. These axiomatizations were shown to be ground-complete, meaning that all behaviorally equivalent closed BCCSP terms can be equated. The process algebra BCCS (see, e.g., [G97]) ... |

1 | On finite bases for weak semantics: Failures versus Impossible futures. Full version of current paper. Available at http:// arxiv.org/abs/0810.4904.
- Chen, Fokkink, et al.
- 2008
(Show Context)
Citation Context ...tional logic. Sect. 3 describes a transformation of an axiomatization for a concrete to an axiomatization for a corresponding weak semantics. In Sect. 4 this transformation is applied to failures, completed trace and trace semantics. Sect. 5 provides finite, sound, ground-complete axiomatizations for -IF and -WIF; it also presents the aforementioned negative result for 'IF and 'WIF. Sect. 6 is devoted to the proofs of the positive and negative results regarding ω-completeness. Sect. 7 concludes the paper. This paper combines and extends two previous papers in conference proceedings [CF08] and [CFG09]. In particular, [CF08] dealt with the concrete impossible futures semantics and [CFG09] extended it to weak impossible futures and weak failures semantics. Here, new results are presented in Sect. 3, which yield a much simplified proof for the results regarding ON THE AXIOMATIZABILITY OF IMPOSSIBLE FUTURES 5 weak failures semantics given in [CFG09], and a unified treatment of concrete and weak impossible futures semantics. 2. Preliminaries 2.1. Labeled transition systems. Let A be a nonempty, countable set of concrete (a.k.a. observable, external, visible) actions, which is ranged over by a, ... |

1 |
Notes on the methodology of CCS
- Glabbeek
- 1997
(Show Context)
Citation Context ... [GV06]. The process algebra BCCSP plays a fundamental role in the study of concrete semantics. It contains only basic process algebraic operators from CCS and CSP, but is sufficiently powerful to express all finite synchronization trees (without τ -transitions). Van Glabbeek [G01] associated with most behavioral equivalences in his spectrum a sound axiomatization, to equate closed BCCSP terms that are behaviorally equivalent. These axiomatizations were shown to be ground-complete, meaning that all behaviorally equivalent closed BCCSP terms can be equated. The process algebra BCCS (see, e.g., [G97]) extends BCCSP with τ , playing the same role as BCCSP in the research of weak semantics. 1Concrete processes do not feature the hidden action τ . Abstract processes, with τ , come with strong semantics, treating τ just as any other action, or weak ones, allowing a degree of abstraction from τ steps. 2Impossible futures semantics was missing in the original spectrum I [G01], because it was studied seriously only from 2001 on, the year that [G01] appeared. ON THE AXIOMATIZABILITY OF IMPOSSIBLE FUTURES 3 An ω-complete axiomatization enjoys the property that if all closed instances of an equatio... |

1 | A characterisation of weak bisimulation congruence.
- Glabbeek
- 2005
(Show Context)
Citation Context ...le we chose to ignore preservation of soundness, as the init-τ operator would give rise to some ON THE AXIOMATIZABILITY OF IMPOSSIBLE FUTURES 13 technical complications, in the alternative approach, soundness of E modulo the strong semantics yields soundness of A(E) modulo the weak semantics in a straightforward fashion, provided we assume the strong preorder (or equivalence) is included in the corresponding weak one. However, the price to pay is that this alternative method only works for groundcompleteness (so not for ω-completeness), and assumes the so-called Fresh Atom Principle (see e.g. [G05]). We briefly sketch the idea behind this alternative approach. The axiomatization E is required to be sound and ground-complete for BCCSP(A). The crucial case (2) in the proof of Thm. 3.4 can now be tackled without init-τ . As before we arrive at ∑ i∈I τti vw ∑ j∈J τuj , but as we are dealing with ground-completeness only, the ti and uj are closed terms. Let a be a fresh action which is not in the alphabet A. The last mentioned relation together with the soundness of WIF1 yields ∑ i∈I ati vw ∑ j∈J auj . This implies∑ i∈I ati vc ∑ j∈J auj . Since a is fresh, renaming it into τ yields ∑ i∈I τti... |