2.5 Properties of Relative Fairness
We defined the notion of a relatively fair path of a transition system in Sect.
2.3 and the corresponding notion of relatively fair firing sequence of a Petri net in Sect. 2.4. We decided to use the non-standard notion of relative fairness, since it suffices to derive our results. As we will see in what follows, the more commonly considered notions like weak fairness and strong fairness are more restrictive.
In the sequel F, F1, F2 ⊆ T denote fairness constraints and Fair ⊆ 2T a set of fairness constraints. We also fix a Petri netN.
Relative Fairness, Weak Fairness, Strong Fairness We now compare our notion of relative fairness to the notions of weak and strong fairness. A firing sequence σ is strongly fair w.r.t. a set of transitions F iff whenever infinitely often transitions inF are enabled, then transitions in F occur in σ infinitely often. σisweakly fairw.r.t. a set of transitions F iff wheneverF is eventually permanently enabled (i.e. from some point onward permanently transitions inF are enabled), then transitions inF occur inσinfinitely often.
Definition 2.5.1 (Weak Fairness, Strong Fairness) Let N be a Petri net, andM0 a marking of Σ. Let F ⊆T be a set of transitions andFair⊆2T be a set of fairness constraints. Let σ = t1t2t3... be an infinite firing se-quence with Mi[ti+1iMi+1,∀i≥0.
σ isstrongly fair w.r.t. F iff
whenever ∀i ∈N:∃j ∈N, j ≥i:∃t ∈F :Mj[ti,
then ∀i∈N:∃j ∈N, j ≥i:∃t′ ∈F :Mj[tj+1iMj+1∧tj+1 =t′. σ isweakly fair w.r.t. F iff
whenever ∃i ∈N:∀j ∈N, j ≥i:∃t ∈F :Mj[ti,
then ∀i∈N:∃j ∈N, j ≥i:∃t′ ∈F :Mj[tj+1iMj+1∧tj+1 =t′.
Any finite, maximal firing sequence σ is strongly and weakly fair w.r.t. F.
20 2. Preliminaries Notation Fss(Fair)(M) denotes the set of firing sequences from M that are strongly fair w.r.t. every F ∈ Fair and Fsw(Fair)(M) denotes the set of firing sequences from M, that are weakly fair w.r.t. every F ∈Fair.
Strong fairness is more restrictive then weak fairness, i.e. every strongly fair firing sequence is weakly fair but a weakly fair firing sequence is not necessarily strongly fair. For place/transition nets (P/T nets)—the kind of Petri nets we consider—weak or strong fairness is usually assumed w.r.t.
singletons. The more general form introduced here accords to the definition in [6].
For convenience, we repeat the definition of relative fairness (cf. Def.
2.4.3):
An infinite σ is relatively fairw.r.t. F iff
whenever ∃t ∈F :∃i∈N:∀j ∈N, j ≥i:Mj[ti,
then ∀i∈N:∃j ∈N, j ≥i:∃t′ ∈F :Mj[tj+1iMj+1∧tj+1 =t′ 2. If σ is finite and maximal, it is relatively fair w.r.t. F.
Let us now compare the different fairness notions for the same fairness constraint F ⊆ T: Obviously our notion is less restrictive than strong fair-ness, as relative fairness only rules out infinite firing sequences where a trans-ition in F is eventually permanently enabled and F is fired finitely often only3. In contrast, strong fairness already rules out infinite firing sequences where transitions are infinitely often enabled and F is fired finitely often.
The difference between weak fairness and relative fairness is more subtle.
Both notions refer to permanent enabledness. Loosely speaking, weak fair-ness rules out certain infinite firing sequences where the set of transitions F is eventually permanently enabled: From some point onward every marking enables a transition in F; consecutive markings do not necessarily enable the same transition. Our notion of relative fairness only rules out infinite firing sequences where at leastone transition of F is eventually permanently enabled and F is only fired finitely often.
2As F ⊆ T is finite, this is equivalent to ∃t′ ∈ F : ∀i ∈ N : ∃j ∈ N, j ≥ i : Mj[tj+1iMj+1∧tj+1=t′.
3More precisely: Transitions ofF are fired a finite number of times.
2.5. Properties of Relative Fairness 21 Let us contrast the three notions by means of an example. Consider the net in Fig. 2.3 (a). The firing sequence σ = t1t2t1t2... is not strongly fair w.r.t. {t3}, becauset3 is enabled infinitely often and never fired. However, σ is weakly fair and relatively fair w.r.t. {t3}, sincet3 is not eventually perman-ently enabled. σ is not weakly fair w.r.t. {t3, t4}, because permanently either t3 or t4 are enabled and neither t3 nor t4 are fired infinitely often. σ is rel-atively fair w.r.t. {t3, t4}, since neither t3 nor t4 are eventually permanently enabled.
(a)
t1
t2
t3 t4
(b)
t1
t2 t3
p1
p2 p3
Figure 2.3: Two simple Petri nets.
The above example shows that relative fairness does not imply weak fair-ness. The following proposition summarises the relations of the three fairness notions. As discussed, strong fairness implies weak fairness, which implies relative fairness, but in general not vice versa. Strong, weak and relative fairness coincide for special fairness constraints.
Proposition 2.5.2 Let N be a Petri net, M a marking of N and Fair⊆2T a set of fairness constraints.
(i) FsN,s(Fair)(M)⊂FsN,w(Fair)(M)⊆FsN,Fair(M) for any N, M, Fair, but there are N, M, Fair such thatFsN,w(Fair)(M)6⊆FsN,s(Fair)(M) and there are N, M,Fair such that FsN,Fair(M)6⊆FsN,w(Fair)(M).
(ii) For singleton fairness constraints, weak fairness equals relative fairness.
FsN,{{t}}(M) =FsN,w({{t}})(M) for any t ∈T.
(iii) Relative fairness, weak fairness and strong fairness coincide for the fairness constraint T.
FsN,{T}(M) =FsN,w({T})(M) =FsN,s({T})(M) =FsN,max(M).
22 2. Preliminaries Proof For (i) we only show that FsN,w(Fair)(M) ⊆ FsN,Fair(M). Let σ be a firing sequence that is not relatively fair w.r.t. F ∈ Fair. So there is a transition t ∈ F eventually permanently enabled but only finitely many transitions in F are fired. Since t ∈ F is eventually permanently enabled, alsoF is eventually permanently enabled, and henceσis not weakly fair w.r.t.
F. Similarly it can be shown that strong fairness implies weak fairness. We have seen above examples showing that relative fairness does not imply weak fairness and weak fairness does not imply strong fairness. Straight-forwardly (ii) and (iii) follow from the fairness definitions. 2
Basic Properties To get a better intuition for relative fairness, we briefly summarise its basic properties.
Proposition 2.5.3 Let M be a marking of N. Let F1, F2 ⊆ T be fairness constraints.
(i) FsN,{F1,F2} ⊆ FsN,{F1}(M) holds, but in general FsN,{F1}(M) ⊆ FsN,{F1,F2}(M) does not hold.
(ii) Neither FsN,{F1∪F2}(M) ⊆ FsN,{F1}(M)
nor FsN,{F1}(M) ⊆ FsN,{F1∪F2}(M) hold in general.
(iii) FsN,{F1,F2}(M) ⊆ FsN,{F1∪F2}(M) holds, but in general FsN,{F1∪F2}(M) ⊆ FsN,{F1,F2}(M) does not hold.
Proof (i) It follows directly from Def. 2.4.3 that Fs{F1,F2} ⊆ Fs{F1}. Let us consider the Petri net of Fig. 2.3 (b) and the firing sequence σ =t1t2t2.... σ is relatively fair w.r.t. {{t2}}but not relatively fair w.r.t. {{t2},{t3}}.
(ii) σ is also relatively fair w.r.t. {{t2, t3}}but is not relatively fair w.r.t.
{{t3}}, and σ is relatively fair w.r.t. {{t1}} but not relatively fair w.r.t.
{{t1, t3}}.
(iii) Let σ be fair w.r.t. F1 and F2. If there is at∈F1 (or F2) eventually permanently enabled, then a transition t′ ∈ F1 (F2) is fired infinitely often.
Hence σ is fair w.r.t. F1∪F2. 2