Timed Data Streams Sample Clauses

Timed Data Streams. Before introducing the notion of constraint automata, we need to introduce some preliminary notations and the notions of timed data streams, their tuples and languages. 3.1 Let V be any set. We define the sets V ∗ and V ω as the sets of all finite and infinite sequences (words or strings) over V , respectively. Obviously, we can define the set V ω of all streams (infinite sequences) over V as the set of functions w :N V . For a stream w V ω we call w (0) the initial value of w . The (stream) derivative w ′ of a stream w is defined as w ′(k ) = w (k + 1). We write w (i) for the i-th derivative of w which is defined by w (0) = w and w (i+1) = w (i)′. Note that w (i)(k ) = w (i + k ), for all k, i ≥ 0. Definition 3.2 Let be a fixed and nonempty set of data that can be sent or received via channels. A timed data stream is an ordered pair α, a , in which, α is an infinite sequence of data and a is a time stream consisting of increasing positive real numbers that go to infinity. We denote the set of all timed data streams by TDS . Thus, more formally we have: + TDS = {⟨α, a⟩ ∈ Dω × Rω | ∀n ≥ 0 (a(n)⟨a(n + 1) and lim a(n) = ∞)}, where R+ = [0, ∞) is the set of all positive real numbers including zero. +