Abstract
This last chapter is about the model construction problem in classes of models having relations that are transitive closures of other relations. The main such logics are linear-time temporal logic LTL and propositional dynamic logic PDL. These logics require both blocking and model checking.
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Notes
- 1.
There exist alternative notations for the first three operators:
instead of X
A, □A instead of F
A, and ◊A instead of G
A. - 2.
A more standard formulation uses paths. Let 〈w 0,w 1,…〉 be the sequence of moments that are in the future of w 0=w.
$$\begin{aligned} M ,w_0 \Vdash\mathsf{X} A\quad & \mbox{iff}\ M ,w_1 \Vdash A; \\ M ,w_0 \Vdash\mathsf{F} A\quad & \mbox{iff there is a natural number}\ i\ \mbox{such that}\ M ,w_i \Vdash A; \\ M ,w_0 \Vdash\mathsf{G}A\quad &\mbox{iff for all natural numbers}\ i\ \mbox{we have}\ M ,w_i \Vdash A; \\ M ,w_0 \Vdash A \mathsf{U} B \quad & \mbox{iff there is a natural number}\ i\ \mbox{such that}\ M ,w_i \Vdash B\ \mbox{and} \\ &\mbox{for all}\ 0\leq j<i\ \mbox{we have}\ M,w_j\Vdash A . \end{aligned}$$
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Gasquet, O., Herzig, A., Said, B., Schwarzentruber, F. (2014). Modal Logics with Transitive Closure. In: Kripke’s Worlds. Studies in Universal Logic. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-8504-0_7
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DOI: https://doi.org/10.1007/978-3-7643-8504-0_7
Publisher Name: Birkhäuser, Basel
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