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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.

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