By Yuxin Deng
This booklet discusses the semantic foundations of concurrent structures with nondeterministic and probabilistic behaviour. specific recognition is given to clarifying the connection among checking out and simulation semantics and characterising bisimulations from metric, logical, and algorithmic views. along with proposing contemporary study results in probabilistic concurrency idea, the e-book exemplifies using many mathematical suggestions to resolve difficulties in computing device technological know-how, that's meant to be available to postgraduate scholars in desktop technological know-how and arithmetic. it may possibly even be utilized by researchers and practitioners both for complex examine or for technical reference.
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Example text
Otherwise we would have u R s, which means, together with t R u and the transitivity of R, that t R s, a contradiction to the hypothesis t R s. It then follows that u ∈ As and then we conclude that R(As ) ⊆ As . We have verified that R(s) and As are R-closed sets. 5(2) and obtain that Δ(R(s)) ≤ Θ(R(s)) and Θ(R(s)) ≤ Δ(R(s)), that is Δ(R(s)) = Θ(R(s)). 15) that Δ([s]≡ ) = Θ([s]≡ ). ✷ as we have desired. 2 Note that in the above proof the equivalence classes [s]≡ are not necessarily R-closed. For example, let S = {s, t}, IdS = {(s, s), (t, t)} and the relation R = IdS ∪ {(s, t)}.
9). Therefore, we always have m(s, t)w(s, t) = 0 for any s, t ∈ S. e. m(Δ, ˆ Θ) = 0, and the optimal solution is determined by w. 8) determines a weight function, thus m(Δ, ˆ Θ) = 0 implies ΔR† Θ. ✷ 36 3 Probabilistic Bisimulation The above property will be used in Sect. 7 to give a metric characterisation of probabilistic bisimulation (cf. 11). 8) do entail the same metric on D (S). 8). 5 Let m be a metric over S. Then m ˆ is a metric over D (S). Proof We verify that (D (S), m) ˆ satisfies the definition of pseudometric space.
As in the nonprobabilistic setting, probabilistic bisimilarity can be approximated by a family of inductively defined relations. 6 Let S be the state set of a pLTS. We define: • ∼0 := S × S • s ∼n+1 t, for n ≥ 0, if a a 1. whenever s −→ Δ, there exists some Θ such that t −→ Θ and Δ ( ∼n )† Θ; a a 2. whenever t −→ Θ, there exists some Δ such that s −→ Δ and Δ ( ∼n )† Θ. • ∼ω := n≥0 ∼n In general, ∼ is a strictly finer relation than ∼ω . However, the two relations coincide when limited to image-finite pLTSs where for any state s and action a, the set a {Δ ∈ D (S) | s −→ Δ} is finite.