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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
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 \(f_2\).
References
Alpern, B., Schneider, F.B.: Defining liveness. Inf. Process. Lett. 21, 181–185 (1985)
Alpern, B., Schneider, F.B.: Recognizing safety and liveness. Distrib. Comput. 2(3), 117–126 (1987)
Banieqbal, B., Barringer, H.: Temporal logic with fixed points. In: Banieqbal, B., Barringer, H., Pnueli, A. (eds.) Temporal Logic in Specification. LNCS, vol. 398, pp. 62–74. Springer, Heidelberg (1989). doi:10.1007/3-540-51803-7_22
Büchi, J.R.: On a decision method in restricted second order arithmetic. In: Proceedings of the International Congress on Logic, Method and Philosophy of Science 1960, pp. 1–12, Palo Alto. Stanford University Press (1962)
Church, A.: A note on the Entscheidungsproblem. J. Symb. Log. 1, 101–102 (1936)
Clarke, E.M., Schlingloff, B.: Model checking. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning. vol. 2, Chapt. 24, pp. 1369–1522. MIT and Elsevier Science Publishers (2001)
Diekert, V., Gastin, P.: First-order definable languages. In: Flum, J., Grädel, E., Wilke, T. (eds.) Logic and Automata: History and Perspectives [in Honor of Wolfgang Thomas], vol. 2, pp. 261–306. Amsterdam University Press (2008)
Gödel, K.: Über formal unentscheidbare sätze der principia mathematica und verwandter system I. Monatshefte für Methematik und Physik 38, 173–198 (1931)
Leucker, M., Sánchez, C.: Regular linear temporal logic. In: Jones, C.B., Liu, Z., Woodcock, J. (eds.) ICTAC 2007. LNCS, vol. 4711, pp. 291–305. Springer, Heidelberg (2007). doi:10.1007/978-3-540-75292-9_20
McNaughton, R.: Testing and generating infinite sequences by a finite automaton. Inf. Comput. 9, 521–530 (1966)
Pnueli, A.: The temporal logic of programs. In: Proceedings of 18th IEEE Symposium on Foundation of Computer Science (FOCS 1977), pp. 46–57. IEEE Computer Society (1977)
Rabin, M.O.: Decidability of second order theories and automata on infinite trees. Trans. AMS 141, 1–35 (1969)
Schweikardt, N.: On the expressive power of monadic least fixed point logic. Theor. Comput. Sci. 350(2–3), 1123–1135 (2010)
Sebastiani, R., Tonetta, S.: “More deterministic” vs. “smaller” Büchi automata for efficient LTL model checking. In: Geist, D., Tronci, E. (eds.) CHARME 2003. LNCS, vol. 2860, pp. 126–140. Springer, Heidelberg (2003). doi:10.1007/978-3-540-39724-3_12
Taurainen, H., Heljanko, K.: Testing LTL formula translation into Büchi automata. STTT 4, 57–70 (2002)
Thomas, W.: Star-free regular sets of omega-sequences. Inf. Control 42, 148–156 (1979)
Vardi, M.Y., Wolper, P.: Reasoning about infinite computations. Inf. Comput. 115(1), 1–37 (1994)
Wolper, P.: Temporal logic can be more expressive. Inf. Control 56(1–2), 72–99 (1983)
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).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-69483-2_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-69482-5
Online ISBN: 978-3-319-69483-2
eBook Packages: Computer ScienceComputer Science (R0)