The Mosaic Method for Temporal Logics

On the Mosaic Method for Many-Dimensional Modal Logics: A Case Study Combining Tense and Modal Operators

“…With regard to temporal logics, a first work considering an adaptation of the technique to the linear temporal logic of general time is [16], including, as an application, the development of a mosaic-based semantic tableaux system, along with a method for automated theorem proving. The authors also discuss the generalization of these results to particular flows of time by suggesting possible modifications of the conditions defining mosaics and saturated sets of mosaics.…”

“…In this paper, we consider the method described in [16] for linear-time logic as our starting point and propose an extension able to deal with logics arising from the combination of linear temporal operators with an "orthogonal" S5 -like modality. The resulting logic is described in [31] (in the case of general linear time).…”

“…The resulting logic is described in [31] (in the case of general linear time). Alongside the linear-time mosaics (defined essentially as in [16]), which we will call vertical mosaics, we will consider also "orthogonal" horizontal mosaics. We will show that, as long as no interaction occurs between the two dimensions, the results of [16] extend to our case.…”

“…Alongside the linear-time mosaics (defined essentially as in [16]), which we will call vertical mosaics, we will consider also "orthogonal" horizontal mosaics. We will show that, as long as no interaction occurs between the two dimensions, the results of [16] extend to our case. Namely, we will prove that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of mosaics, and will apply the technique not only in obtaining a completeness proof for the corresponding Hilbert-style axiomatization, but also in the development of a mosaic-based tableau system.…”

“…Though our mosaic definitions do not lead to a proof of decidability when interactions between the vertical and the horizontal components are considered, they still allow for giving non-analytic but interesting tableau systems for the logics. These tableau systems, inspired by the one described in [16] for linear-time, are strongly based on mosaics and are thus quite different from the standard semantic tableau systems for modal logics. The definition of the tableau rules follows very naturally from the mosaics definitions (each rule corresponding to a property that the mosaics are required to satisfy), and allows for an extremely clear and appealing representation of the (counter) model under construction.…”