By Abraham Ginzburg
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Xn ˆ y 1 . . ym = x 1 . . xn y 1 . . ym where x1 . . xn denotes a ﬁnite sequence of length n with x1 ,. . , xn as its elements. For inﬁnite streams r, s: IN → M we deﬁne sˆx, xˆs, sˆr to be inﬁnite streams as follows: sˆx = s, sˆr = s, [ x1 . . xn ˆs](t) = xt s(t + n) if t ≤ n otherwise We may see ﬁnite streams as partial functions IN → M and inﬁnite streams as total functions. Based on concatenation we introduce the preﬁx order preﬁx as a relation on streams s, r ∈ Mω s r ⇔def ∃ z ∈ Mω : sˆz = r as its least element, complete in the (Mω , ) is a partially ordered set with sense that every chain xt ∈ Mω : t ∈ IN has a least upper bound.
In general, in a system several communication streams occur. Therefore we work with channels to refer to individual communication streams. Accordingly, in Focus, a channel is simply an identiﬁer in a system that evaluates to a stream in every execution of the system. 46 M. Broy A channel is an identiﬁer for streams. A channel is a name of a stream. Formally it is an identiﬁer of type Stream T with some type T. The concept of a stream is used to deﬁne the concept of a channel history. A channel history is given by the messages communicated over a channel.
We assume that causality is reﬂected by the timing model. Based on the ideas of an interactive component we deﬁne forms of composition. We basically introduce only one powerful composition operator, namely parallel composition with interaction. For establishing a relation between interactions and services interaction reﬁnement. These notions of reﬁnement typically occur in a systematic top down system development. 2 Central Model of Interaction: Streams A stream is a ﬁnite or inﬁnite sequence of elements of a given set.