By Taolue Chen, Wan Fokkink, Sumit Nain (auth.), Luca Aceto, Anna Ingólfsdóttir (eds.)
This publication constitutes the refereed complaints of the ninth foreign convention on Foundations of software program technology and Computation buildings, FOSSACS 2006, held in Vienna, Austria in March 2006 as a part of ETAPS.
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Extra resources for Foundations of Software Science and Computation Structures: 9th International Conference, FOSSACS 2006, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2006, Vienna, Austria, March 25-31, 2006. Proceedings
And a separate set of names, Names, ranged over by n, m, . . , which is divided into locations, A Theory for Observational Fault Tolerance 19 Table 1. Syntax of typed DπF Types T ::= chv P˜ | locsv (stateful types) s ::= a | d (status) U ::= chv P˜ | locv (stateless types) v ::= p | c (visibility) P ::= chp P˜ | locp (public stateless types) Processes P, Q ::= u! P else Q (status testing) Systems M, N, O ::= l[[P]] (located process) | N|M (parallel) | (ν n : T)N (hiding) Locs, ranged over by l, k, .
We discuss two possibilities. If (v @ w) ∈ (x @ y) and w ∈ x the contexts Λ(x @ y)(u @ v) and Λ(x @ y)(u @ w) are equivalent and we can apply the same arguments as for the previous case. If (v @ v) ∈ (x @ y) consider the context Λ(u @ vρ)(x @ y ) where (x @ y ) is obtained by substituting (v @ v) with (v @ u) in (x @ y). Note that Λ(x @ y)(u @ v) u v implies Λ(u @ vρ)(x @ y ) u v so we can apply item 1 followed by the inductive hypothesis and obtain Λ(u @ vρ) P ρ. From this we derive Λ (u @ vρ)P ρ, which is equivalent to Λ ((u @ v)P )ρ.
While this is the case for (a @ a)(u @ u)(v @ u), it is not the case for (a @ a)(u @ u). The condition Λ xρ yρ in the substitution establishes that co-located names remain colocated after having been substituted. Therefore, if we insist in replacing v with a, we must also map u to a. In this case the substitution lemma may be applied and we obtain: (a @ a) a&a 0 Theorem 1 (subject reduction). If (x @ y) P and (x @ y)P → (x @ y)Q then (x @ y) Q. In particular, if P is distributable and P → Q then Q is distributable as well.