Abstract
In proving the complexity of the satisfiability problem there was for many modal and temporal logics until now still a gap between upper and lower bounds, mainly expressed by the nondeterministic and the deterministic version of the same complexity class. By showing that the (non-) emptiness problem of tree automata and alternating tree automata of Büchi-, Rabin-1- and Muller-1-type can be solved in polynomial time (thereby getting polynomial time completeness for these problems) we are now able to close these gaps therefore showing these satisfiability problems complete for the corresponding complexity classes. For Rabin-tree-automata of arbitrary type testing (non-) emptiness is shown to be in NP∩cc-NP.
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Chandra, A.K.; Kozen, D.C.; Stockmeyer, L.J.: Alternation Journal of the ACM 28, 1981, pp. 114–133
Emerson, E.A.: Automata, Tableaux and Temporal Logics Proc. Workshop on Logics of Programs, LNCS 193, 1985
Emerson, E.A.; Halpern, J.Y.: “Sometimes” and “Not Never” Revisited: On Branching versus Linear Time Proc. 10th ACM Symposium on Principles of Programming Languages, 1983
Emerson, E.A.; Lei, C.: Temporal Model Checking under Generalized Fairness Constraints Proc. 18th Annual Hawai International Conference on System Sciences, 1985, pp. 277–288
Fischer, M.J.; Ladner, R.E.: Propositional Dynamic Logic of Regular Programs Journal of Computer and System Sciences 18, 1979, pp. 194–211
Harel, D.: Dynamic Logic in: Handbook of Philosophical Logic (ed. D. Gabbay, F. Guenthner) vol. II, 1984, pp. 497–604
Hossley, R.; Rackoff, C.W.: The Emptiness Problem for Automata on Infinite Trees Proc. 13th IEEE Symposium on Switching and Automata Theory, 1972, pp. 121–124
Jones, N.D.; Laaser, W.T.: Complete Problems for Deterministic Polynomial Time Theoretical Computer Science 3, 1976, pp. 105–117
Kozen, D.C.: Results on the Propositional μ-Calculus Theoretical Computer Science 27, 1983, pp. 333–354
Muller, D.E.; Schupp, P.E.: Alternating Automata on Infinite Trees Theoretical Computer Science 54, 1987, pp. 267–276
Parikh, R: Propositional Game Logic Proc. 25th IEEE Symp. on Found. of Computer Science, 1983, pp. 195–200
Rabin, M.O: Weakly Definable Relations and Special Automata Proc. Symp. Math. Logic and Foundations of Set Theory (Y. Bar-Hilllel, ed.), North-Holland, 1970, pp. 1–23
Street, R.S.: Propositional Dynamic Logic of Looping and Converse Techn. Report MIT/LCS/TR-263 Massachusetts Institute of Technology, 1981
Vardi, M.Y.; Stockmeyer, L.: Improved Upper and Lower Bounds for Modal Logics of Programs: Preliminary Report Proc. 17th Annual ACM Symp. on Theory of Computing, 1985, pp. 240–251
Vardi, M.Y.; Wolper, P.: Yet another Process Logic Proc. Workshop on Logics of Programs, LNCS 164, 1983, pp. 501–512
Vardi, M.Y.; Wolper, P.: Automata Theoretic Techniques for Modal Logics of Programs Journal of Computer and System Sciences 32, 1986, pp. 183–220
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© 1989 Springer-Verlag Berlin Heidelberg
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Wagner, H. (1989). On the emptiness problem of tree automata and completeness of modal logics of programs. In: Börger, E., Büning, H.K., Richter, M.M. (eds) CSL '88. CSL 1988. Lecture Notes in Computer Science, vol 385. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0026315
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DOI: https://doi.org/10.1007/BFb0026315
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