Abstract
This paper deals with restricted star height of regular sets, as defined by L. C. Eggan. Theorems relating the star height of a regular event with the structure of the reduced deterministic automaton recognizing the event are presented. These theorems provide new techniques for establishing star height; in many cases the exact star height of the given regular set can be determined and in others lower bounds to the star height are found.
Some results concerning the star height of sets recognized by reset-free state graphs (i.e., partial state graphs in which two transitions labelled by the same input letter never enter the same state) are also obtained, e.g., the star height of any event recognized by a group-free reset-free state graph equals the cycle rank of the state graph.
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References
L. C. Eggan, Transition graphs and the star height of regular events,Michigan Math. J. 10 (1963), 385–397.
R. McNaughton, Techniques for manipulating regular expressions,Systems and Computer Sciences (J. F. Hart and S. Takasu, eds.), pp. 24–41, University of Toronto Press, Toronto, 1965.
R. McNaughton, The loop complexity of regular events,Information Sci. 1 (1969), 305–328.
R. McNaughton, The loop complexity of pure-group events,Information and Control 11 (1967), 167–176.
R. McNaughton andH. Yamada, Regular expressions and state graphs for automata,IRE Trans. Electronics Computers EC-9 (1960), 39–57; also inSequential Machines: Selected Papers, (E. F. Moore, ed.), pp. 157–174, Addison-Wesley, Reading, Mass., 1963.
J. A. Brzozowski, Derivatives of regular expressions,J. Assoc. Comput. Math. 11 (1964), 481–494.
J. A. Brzozowski andRina Cohen, on decompositions of regular events,J. Assoc. Comput. Mach. 16 (1969), 132–144.
R. S. Cohen andJ. A. Brzozowski, General properties of star height of regular events,J. Comput. System Sci. 4 (1970), 260–280.
R. S. Cohen, Star height of certain families of regular events,J. Comput. System Sci. 4 (1970), 281–297.
R. S. Cohen, Rank-non-increasing transformations on transition graphs, to appear.
R. S. Cohen, Cycle Rank of Transition Graphs and the Star Height of Regular Events, Ph.D. Dissertation, University of Ottawa, Ottawa, 1968.
F. Dejean andM. P. Schützenberger, On a question of Eggan,Information and Control 9 (1966), 23–25.
R. S. Cohen andJ. A. Brzozowski, On the star height of regular events,Proceedings of the Eighth Annual Symposium on Switching and Automata Theory, pp. 265–279, Inst. Elec. Electronics Eng., New York, 1967.
R. S. Cohen, Transition graphs and the star height problem,Proceedings of the Ninth Annual Symposium on Switching and Automata Theory, pp. 383–394, Inst. Elec. Electronics Eng., New York, 1968.
M. O. Rabin andD. Scott, Finite Automata and their Decision Problems.IBM J. Res. Develop. 3 (1959), 114–125.
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Cohen, R.S. Techniques for establishing star height of regular sets. Math. Systems Theory 5, 97–114 (1971). https://doi.org/10.1007/BF01702866
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DOI: https://doi.org/10.1007/BF01702866