A propositional linear time logic with time flow isomorphic to ω2

Primarily guided with the idea to express zero-time transitions by means of temporal propositional language, we have developed a temporal logic where the time flow is isomorphic to ordinal ω 2 (concatenation of ω copies of ω). If we think of ω 2 as lexicographically ordered ω×ω, then any particular zero-time transition can be represented by states whose indices are all elements of some {n} × ω. In order to express non-infinitesimal transitions, we have introduced a new unary temporal operator [ω] (ω-jump), whose effect on the time flow is the same as the effect of α → α + ω in ω 2. In terms of lexicographically ordered ω × ω, [ω]φ is satisfied in i, j-th time instant iff φ is satisfied in i + 1, 0-th time instant. Moreover, in order to formally capture the natural semantics of the until operator U, we have introduced a local variant u of the until operator. More precisely, φ uψ is satisfied in i, j-th time instant iff ψ is satisfied in i, j + k-th time instant for some nonnegative integer k, and φ is satisfied in i, j + l-th time instant for all 0 l < k. As in many of our previous publications, the leitmotif is the usage of infinitary inference rules in order to achieve the strong completeness.