By FranÃ§ois-Eric Racicot
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The work required to show that nβ is sound with respect to nD is similar to earlier up-to β-moves work discussed in Section 4: we have to show that βmove conﬂuence (similar to Lemma 1) is also preserved for the new action fail; we also have to show that after a β-move, the redex and reduct conﬁgurations are counting-bisimilar (similar to Proposition 1). Finally we prove the following proposition Proposition 4 (Inclusion of fault tolerant simulation up-to β-moves). If Γ1 M1 nβ Γ2 M2 then Γ1 M1 nD Γ2 M2 Proof.
Lemma 1 (Conﬂuence of β-moves). −→β observes the diamond property: Γ N µ Γ τ / Γ M implies Γ N µ N τ β Γ M Γ /Γ M β µ N τ +3 ≡ Γ β f M or µ = τ and Γ M = Γ N A Theory for Observational Fault Tolerance 27 Table 7. β-Equivalence Rules for Typed DπLoc Γ |= N|M Γ |= (N|M)|M Γ |= N|l[] Γ |= (ν n : T)(N|M) Γ |= (ν n : T)(ν m : U)N Γ |= (ν n : T)N Γ |= l[[P]] (bs-comm) (bs-assoc) (bs-unit) (bs-extr) (bs-flip) (bs-inact) (bs-dead) ≡f ≡f ≡f ≡f ≡f ≡f ≡f M|N N|(M|M ) N N|(ν n : T)M (ν m : U)(ν n : T)N N l[[Q]] n fn(N) n fn(N) Γ l : alive Proof.
This is outlined in Section 2, where we also formally deﬁne the language we use, DπLoc, give its reduction semantics, and also outline the behavioural equivalence ∼ =; this last is simply an instance of reduction barbed congruence, , modiﬁed so that observations can only be made at public locations. In Section 3 we give our formal deﬁnition of faulttolerance; actually we give two versions of (1) above, called static and dynamic fault tolerance; we also motivate the diﬀerence with examples. Proof techniques for establishing fault tolerance are given in Section 4; in particular we give a complete co-induction characterisation of ∼ =, using labelled actions, and some useful up-to techniques for presenting witness bisimulations.