Fixed point characterization of infinite behavior of finite-state systems

Theoretical Computer Science

Volumn 189, Issue 1-2, 1997, Pages 1-69

  Niwiński, Damian ^a  

^a UNIVERSITY OF WARSAW   (Poland)

Author keywords

Fixed points; Powerset algebras; Rabin automata

Indexed keywords

CODES (SYMBOLS); DIFFERENTIATION (CALCULUS); MATHEMATICAL OPERATORS; POLYNOMIALS; SET THEORY; TREES (MATHEMATICS);

FIXED POINTS; POWERSET ALGEBRA; RABIN AUTOMATA;

FINITE AUTOMATA;

EID: 0346678020     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0304-3975(97)00039-X     Document Type: Review

References (54)

1. A. Arnold, M. Nivat, Nondeterministic recursive programs, in: Proc. Fundamentals of Computation Theory, Lecture Notes in Computer Science, vol. 56, Springer, Berlin, 1977, pp. 12-21.
2. A. Arnold, D. Niwiński, Fixed point characterization of Büchi automata on infinite trees, J. Inform. Process. Cybernet EIK 26 (1990) 453-461.
3. A. Arnold, D. Niwiński, Fixed point characterization of weak monadic logic definable sets of trees, in: M. Nivat, A. Podelski (Eds.), Tree Automata and Languages, Elsevier, Amsterdam, 1992, pp. 159-188.
4. J.C. Bradfield, The modal mu-calculus alternation hierarchy is strict, in: Proc. CONCUR, 1996.
5. J.R. Büchi, On a decision method in restricted second-order arithmetic, in: E. Nagl (Ed.), Logic, Methodology, and Philosophy of Science, Stanford Univ. Press, Stanford, 1960, pp. 1-11.
6. B. Courcelle, Equivalences and transformations of regular systems. Applications to recursive program schemes and grammars, Theoret. Comput. Sci. 42 (1986) 1-122.
7. E.A. Emerson, The role of Büchi's automata in computing science, in: S. MacLane, D. Siefkes (Eds.), The Collected Works of J.R. Büchi, Springer, Berlin, 1990.
8. E.A. Emerson, Temporal and modal logic, in: J. van Leeuven (Ed.), Handbook of Theoretical Computer Science, vol. B, Elsevier, Amsterdam, 1990, pp. 995-1072.
9. E.A. Emerson, E.M. Clark, Characterizing correctness properties of parallel programs using fixpoints, in: Proc. 7th Internat. Coll. on Automata, Languages and Programming, 1980, pp. 169-181.
10. E.A. Emerson, C.S. Jutla, The complexity of tree automata and logics of programs, in: Proc. 29th IEEE Symp. on Foundations of Computer Science, 1988, pp. 328-337.
11. E.A. Emerson, C.S. Jutla, On simultaneously determinizing and complementing ω-automata, in: Proc. 4th IEEE Symp. on Logic in Comput. Sci., 1989, pp. 333-342.
12. E.A. Emerson, C.S. Jutla, Tree automata, mu-calculus and determinacy, in: Proc. 32nd IEEE Symp. on Foundations of Computer Science, 1991, pp. 368-377.
13. E.A. Emerson, C. Lei, Efficient model checking in fragments of the propositional mu-calculus, in: Proc. 1st IEEE Symp. on Logic in Comput. Sci., 1986, pp. 267-278.
14. J. Ferrante, C.W. Rackoff, The computational complexity of logical theories, Lecture Notes in Math., vol. 718, Springer, Berlin, 1979.
15. L. Flon, N. Suzuki, Consistent and complete proof rules for the total correctness of parallel programs, in: Proc. 19th IEEE Symp. on Foundations of Computer Science, 1978.
16. G. Grätzer, Universal Algebra, D. Van Nostrand Co. Inc., Princeton, 1968.
17. I. Guessarian, Algebraic Semantics, Lecture Notes in Computer Science, vol. 99, Springer, Berlin, 1981.
18. Y. Gurevich, L. Harrington, Trees, Automata and games, in: Proc. 14th ACM Symp. on Theory of Computation, 1982, pp. 60-65.
19. T. Hafer, Automaten und Logiken über unendlichen Bäumen, Ph.D. Thesis, RWTH Aachen, 1989.
20. R. Hossley, C. Rackoff, The emptiness problem for automata on infinite trees, in: Proc. 13th IEEE Symp. Switching and Automata Theory, 1972, pp. 121-124.
21. D. Janin, I. Walukiewicz, Automata for the μ-calculus and related results, in: Proc. MFCS'95, Lecture Notes in Computer Science, vol. 969, Springer, Berlin, 1995, pp. 552-562.
22. B. Knaster, Un théorème sur les fonctions des ensembles. Ann. Soc. Polon. Math 6 (1928) 133-134.
23. D. Kozen, Results on the propositional mu-calculus, Theoret. Comput. Sci. 27 (1983) 333-354.
24. D. Kozen, R.J. Parikh, A decision procedure for the propositional mu-calculus, in: 2nd Workshop on Logics of Programs, Lecture Notes in Computer Science, vol. 164, Springer, Berlin, 1983, pp. 313-32:5.
25. D. Kozen, J. Tiuryn, Logics of programs, in: J. van Leeuven (Ed.), Handbook of Theoretical Computer Science, vol. B, Elsevier, Amsterdam, 1990, pp. 789-840.
26. K. Kuratowski, A. Mostowski, Set Theory, North-Holland, Amsterdam, 1976.
27. G. Lenzi, A hierarchy theorem for the μ-calculus, in: Proc. ICALP'96, Lecture Notes in Computer Science, vol. 1099, Springer, Berlin, 1996, pp. 87-109.
28. R. McNaughton, Infinite games played on finite graphs, Ann. Pure Appl. Logic 65 (1993) 149-184.
29. A.W. Mostowski, Regular expressions for infinite trees and a standard form of automata, in: Computation Theory. Lecture Notes in Computer Science, vol. 208, Springer, Berlin, 1985, pp. 157-168.
30. A.W. Mostowski, Games with forbidden positions, Tech. Report 78, University of Gdańsk, 1991.
31. D.E. Muller, Infinite sequences and finite machines, in: Proc. 4th Ann. IEEE Symp. on Switching Circuit Theory and Logical Design, 1963, pp. 3-16.
32. D.E. Muller, P. Schupp, Alternating automata on infinite trees, Theoret. Comput. Sci. 54 (1987) 267-276.
33. D.E. Muller, P. Schupp, Simulating alternating tree automata by nondeterministic automata: new results and new proofs of the theorems of Rabin, McNaughton and Safra, Theoret. Comput. Sci. 141 (1995) 69-107.
34. D. Niwiński, Fixed-point characterization of context-free ∞-languages, Inform. Control 61 (1984) 247-276.
35. D. Niwiński, Equational μ-calculus, in: Computation Theory, Lecture Notes in Computer Science, vol. 208, Springer, Berlin, 1985, pp. 169-176.
36. D. Niwiński, On fixed point clones, in: L. Kott (Ed.), Proc. 13th ICALP'86, Lecture Notes in Computer Science, vol. 226, Springer, Berlin, 1986, pp. 464-473.
37. D. Niwiński, Hierarchy of objects definable in the fixed point calculus, Ph.D. Thesis, University of Warsaw, 1987 (in Polish).
38. D. Niwiński, Fixed points vs infinite generation, in: Proc. 3rd IEEE Symp. on Logic in Comput. Sci., 1988, pp. 402-409.
39. D.M.R. Park, Fixpoint induction and proof of program semantics, in: Machine Intelligence, vol. 5, Edinburgh University Press, Edinburgh, 1969, pp. 59-78.
40. D.M.R. Park, On the semantics of fair parallelism, in: Abstract Software Specification, Lecture Notes in Computer Science, vol. 86. Springer, Berlin, 1980, pp. 504-526.
41. M.O. Rabin, Decidability of second-order theories and automata on infinite trees, Trans. Amer. Soc. 141 (1969) 1-35.
42. M.O. Rabin, Weakly definable relations and special automata, in: Y. Bar-Hillel (Ed.), Mathematical Logic in Foundations of Set Theory, 1970, pp. 1-23.
43. M.O. Rabin, Automata on Infinite Objects and Church's Problem, Amer. Math. Soc., Providence, RI, 1972, pp. 1-22.
44. S. Safra, On the complexity of ω-automata, in: Proc. 29th IEEE Symp. on Foundations of Computer Science, 1988, pp. 319-327.
45. C.P. Stirling, D.J. Walker, Local model checking in the modal mu-calculus, in: Proc. Internat. Joint Conf. in Theory and Practice of Software Development, Lecture Notes in Computer Science, vol. 351, Springer, Berlin, 1989, pp. 369-382.
46. R.S. Streett, E.A. Emerson, An automata theoretic procedure for the propositional mu-calculus, Inform. Comput. 81 (1989) 249-264.
47. M. Takahashi, The greatest fixed points and rational omega-tree languages, Theoret. Comput. Sci. 44 (1986) 259-274.
48. A. Tarski, A lattice theoretical fixpoint theorem and its applications, Pacific J. Math. 5 (1955) 285-309.
49. W. Thomas, Automata on infinite objects, in: J. van Leeuven (Ed.), Handbook of Theoretical Computer Science, vol. B, Elsevier, Amsterdam, 1990, pp. 133-191.
50. W. Thomas, On the synthesis of strategies in infinite games, in: Proc. STACS'95, Lecture Notes in Computer Science, vol. 900, Springer, Berlin, 1995, pp. 1-13.
51. M.Y. Vardi, P. Wolper, Automata-theoretic techniques for modal logic of programs, J. Comput. System Sci. 18 (1986) 183-221.
52. K. Wagner, Eine topologische Charakterisierung einiger Klassen regulärer Folgenmengen, J. Inform. Process. Cybernet EIK 13 (1977) 473-487.
53. K. Wagner, On ω-regular sets, Inform. and Control 43 (1979) 123-177.
54. I. Walukiewicz, Completeness of Kozen's axiomatisation of the propositional μ-calculus, in: Proc. 10th IEEE Symp. on Logic in Comput. Sci., 1995, pp. 14-24.