Abstract
Characterization of temporal properties is the original purpose of inventing of temporal logics. In this paper, we show that the property like “some event holds periodically” is not omega-regular. Such property is called “periodicity”, which plays an important role in task scheduling and system design. To give a characterization of periodicity, we present the logic QPLTL, which is an extension of LTL via adding quantified step variables. Based on the decomposition theorem, we show that the satisfiability problem of QPLTL is PSPACE-complete.
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Notes
- 1.
Note that we here have an extra \(\mathsf {X}\) in the formula, because k can be assigned to 0.
- 2.
Remind that step variables cannot be instantiated as concrete numbers in such logic, hence we have both \(\mathsf {X}f\) and \(\mathsf {X}^k f\) in the grammar.
- 3.
That is, f involves no free variable.
- 4.
The encoding of \(f_q\) is enlightened by [Sch10].
- 5.
We do not consider the “releases” (\(\mathsf {R}\)) operator here, which is the duality of \(\mathsf {U}\). Indeed, \(f_1\mathsf {R}f_2\) is equivalent to \(f_2\,\mathsf {W}(f_1\wedge f_2)\).
- 6.
For example, when dealing with \((p_1\vee p_2\wedge \mathsf {F}p_3)\mathsf {W}\, p_4\), we need first transform the first operand into disjunctive normal form — i.e., rewrite it as \(((p_1\vee \mathsf {F}p_3)\wedge (p_2\vee \mathsf {F}p_3))\mathsf {W}\, p_4\), and then conduct the transformation with the first rule and the third rule. Similarly, for the second operand, we need first transform it into conjunctive normal form.
- 7.
Note that the operator \(\mathsf {G}\) is derived from \(\mathsf {W}\).
- 8.
Because, the previous transformations could guarantee that: If the outermost operator of \(f_{i,j}\) is not \(\mathsf {F}\), then no \(\mathsf {F}\) occurs in \(f_{i,j}\).
- 9.
Remind that each automaton can be equivalently transformed into another one having a unique initial state, and the transformation is linear (cf. Theorem 2). Indeed, to obtain a finite alphabet, here we may temporarily take \(\mathcal {P}\) as the set constituted with propositions occurring in \(f_1\) or \(f_2\).
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Acknowledgement
The first author would thank Normann Decker, Daniel Thoma and Martin Leucker for the fruitful discussion on this problem. We would also thank the anonymous reviewers for their valuable comments on an earlier version of this paper. Wanwei Liu is supported by Natural Science Foundation of China (No. 61103012, No. 61379054, and No. 61532007). Fu Song is supported by Natural Science Foundation of China (No. 61402179 and No. 61532019).
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Liu, W., Song, F., Zhou, G. (2017). Reasoning About Periodicity on Infinite Words. In: Larsen, K., Sokolsky, O., Wang, J. (eds) Dependable Software Engineering. Theories, Tools, and Applications. SETTA 2017. Lecture Notes in Computer Science(), vol 10606. Springer, Cham. https://doi.org/10.1007/978-3-319-69483-2_12
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