For every equivalence in van Glabbeek’s linear time – branching time spectrum I, it has now been determined whether it is finitely based or not. Tab. 4.1 presents an overview, with a + indicating that a finite basis exists and a – indicating that a finite basis does not exist. (The grey shadow indicates that it was open prior to the results described in this chapter.) We distinguish three categories, according to the cardinality of the alphabet A: singleton, finite with at least two actions, and infinite.
|A| = 1 1 < |A| < ∞ |A| = ∞
bisimulation + + +
2-nested simulation – – –
possible futures – – –
ready simulation + – +
completed simulation + – –
simulation + – +
possible worlds + – +
ready traces + – –
failure traces + – +
readies + – +
failures + + +
completed traces + + +
traces + + +
Table 4.1: The existence of finite bases for BCCSP in the linear time – branching time spectrum I
Impossible Futures
5.1 Introduction
In this chapter, we study impossible futures semantics [Vog92, VM01]. This semantics is missing in van Glabbeek’s original spectrum I, because it was only studied seriously from 2001 on, the year that [vG01] appeared. Impossible fu-tures semantics is a natural variant of possible fufu-tures semantics [RB81] (see also Def. 2.1.4). It is also closely related to fair testing semantics [RV07]. In [vGV06] it was shown that weak impossible futures equivalence is the coarsest congruence with respect to choice and parallel composition operators contain-ing weak bisimilarity with explicit divergence that respects deadlock/livelock traces and assigns unique solutions to recursive equations. For the definitions of impossible futures semantics, see Def. 2.1.4 and Def. 2.1.6.
In Chapter 4, a complete categorization of the (in)equational theories for the process algebra BCCSP modulo the semantics in the linear time – branching time spectrum I [vG01] has been given. For each preorder and equivalence it is studied whether a finite, sound, ground-complete axiomatization exists. And if so, whether there exists a finite basis for the equational theory.
So all questions on these matters regarding concrete semantics have been resolved? No, as for impossible futures semantics, the (in)equational theory remained unexplored, with the only exception that the inequational theory of BCCS modulo weak impossible futures preorder was studied in [VM01]. In that paper, Voorhoeve and Mauw offer a finite, sound, ground-complete axiom-atization; their ground-completeness proof relies heavily on the presence of τ . They also prove that their axiomatization is ω-complete (they do not refer to ω-completeness explicitly, but they work on open terms, see [VM01, Thm. 5]).
They implicitly assume an infinite alphabet (at [VM01, page 7] they require a different action for each variable).
As for weak semantics in general, much less is known, compared to concrete semantics. For several of the semantics in the linear time – branching time spectrum II [vG93b], a sound and ground -complete axiomatization has been given, in the setting of BCCS, see, e.g., [vG97]. Moreover a finite basis has been given for weak, delay, η- and branching bisimulation semantics [Mil89b, vG93a].
89
In this chapter, we focus on the axiomatizability of concrete and weak im-possible futures preorders and equivalences in the setting of BCCSP and BCCS.
In summary, we obtain the following results:
1. We prove that there exists a finite, sound, ground-complete axiomatiza-tion for BCCSP modulo concrete impossible futures preorder -IF. By contrast, in [AFvGI04] it was shown that such an axiomatization does not exist modulo possible futures preorder. Thanks to the result established in Section 3.3, a finite, sound, ground-complete axiomatization for weak impossible futures preorder WIF can be obtained for free.
2. We show that BCCS modulo weak impossible futures equivalence ≃WIF
does not have a finite, sound, ground-complete axiomatization. This nega-tive result is based on the following infinite family of equations, for m ≥ 0:
τ a2m0 + τ (am0 + a2m0) ≈ τ (am0 + a2m0) .
Again thanks to the result established in Section 3.3, this negative result carries over to concrete impossible futures equivalence ≃IF. Moreover, in light of this, one can easily establish the nonderivability of the equa-tions a2m+10 + a(am0 + a2m0) ≈ a(am0 + a2m0) from any given finite equational axiomatization sound for ≃WIF. As these equations are valid modulo (concrete) 2-nested simulation equivalence, this negative result applies to all BCCS-congruences that are at least as fine as weak im-possible futures equivalence and at least as coarse as concrete 2-nested simulation equivalence. Note that the corresponding result of [AFvGI04]
can be inferred.
3. We investigate ω-completeness for impossible futures semantics.
First, we prove that if the alphabet of actions is infinite, then the afore-mentioned ground-complete axiomatization for BCCSP modulo -IF is ω-complete. To prove this result, we apply the technique of inverted substi-tutions (see Section 3.4). The result established in Section 3.3 allows this result to carry over to WIF.
Second, we prove that in case of a finite alphabet of actions, the inequa-tional theory of BCCS modulo WIF does not have a finite basis. In case of a singleton alphabet, this negative result is based on the following infinite family of equations, for m ≥ 0:
amx 4 amx + x .
And for finite alphabets with at least two actions, we use the family τ (amx)+τ (amx+x)+X
b∈A
τ (amx+amb0) 4 τ (amx+x)+X
b∈A
τ (amx+amb0) .
The result established in Section 3.3 allows these negative results to carry over to -IF.
4. n-Nested concrete impossible futures semantics, for n ≥ 0, form a natural hierarchy (cf. [AFvGI04]), which coincides with the universal relation for n = 0, trace semantics for n = 1, and impossible futures semantics for n = 2. Using a proof strategy from [AFvGI04], we show that the negative result regarding concrete impossible futures equivalence extends to all n-nested impossible futures equivalences for n ≥ 2, and to all n-nested impossible futures preorders for n ≥ 3. Apparently, (2-nested) impossible futures preorder is the only positive exception.
To achieve these negative results, we mainly exploit the proof-theoretic tech-nique (cf. Section 2.1.5). On top of this, a saturation principle is introduced, to transform a single summand into a large collection of (semi-)saturated sum-mands, which plays a pivotal role in obtaining positive results.
To the best of our knowledge, impossible futures semantics is the first exam-ple that affords a finite, ground-comexam-plete axiomatization for BCCSP (or BCCS) modulo the preorder, while missing a finite, ground-complete axiomatization for BCCSP (or BCCS) modulo the equivalence. This surprising fact suggests that, for instance, if one wants to show p ≃IFq in general, one has to resort to deriving p -IFq and q -IFp separately, instead of proving it directly.
In [AFI07, dFG07] an algorithm is presented which produces, from an axiom-atization for BCCSP modulo a preorder, an axiomaxiom-atization for BCCSP modulo the corresponding equivalence. If the original axiomatization for the preorder is ground-complete or ω-complete, then so is the resulting axiomatization for the equivalence. In Section 3.2, we show that the same algorithm applies equally well to weak semantics. However, that algorithm only applies 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, dFG07] and Section 3.2. Interestingly, our results yield that no such algorithm exists for certain semantics incomparable with (or finer than) ready simulation.
Structure of the chapter. Section 5.2 offers sound, finite, ground-complete axiomatizations for -IF and WIF; it also contains the proof of the negative result for ≃IF and ≃WIF. Section 5.3 is devoted to the proofs of the negative and positive results regarding ω-completeness. Section 5.4 contains the negative results regarding n-nested (concrete) impossible futures semantics. Section 5.5 concludes the chapter with an overview of the positive and negative results achieved in this chapter.