By Dan Lawesson.

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2. The diagnosis abstraction DiAbs(SD) of a system description SD is DiAbs(SD) = runabs(fair (runs(SD))). That is, the diagnosis abstraction of all fair runs of SD. 1. 1, a bus, a server and a client. For brevity we call the bus component b, the server is called s and we use c to denote the client component. We also use the convention of describing mappings as sets. The initial system state is thus σ0 = { c → (init, ∅), s → (ready, ∅), b → (wait, ∅) } Mathematical Foundation 39 First assume that the bus does not enter its down state.

A CTL formula is always evaluated in a state, a vertex in the computation tree, and we write q |= F to denote the fact that a formula F holds in state q, or in other words F holds for the computation tree with q as root. The semantics of the CTL operators AG, AF, EG, EF and EX can be described as follows. These operators form a subset of CTL that is sufficient for the purposes of the thesis. The others are not needed, and will therefore not be further described here. In a computation tree rooted at q, we say that q q q q q |= AG(F ) |= AF (F ) |= EG(F ) |= EF (F ) |= EX(F ) iff iff iff iff iff F holds in all states of the tree, in all paths from q, F holds in some state, in some path from q, F is true in all states, in some path from q, F is true in some state, F holds for some immediate successor state of q.

Sn−1 → nent c = (Σ, S , → , s0 , id) is called the reachable sub-component of c = (Σ, S, →, s0 , id) iff S are the potentially reachable states of c and → contains all (s, e, s ) ∈ → such that s ∈ S . In what follows we assume, unless otherwise stated, that all components are reduced to their reachable sub-component. 2. 2. 3, the diagram describes the behavior of an interpolator (modeling) component oipol . To compute the corresponding memory component cipol we find that Mathematical Foundation 29 idle pos?