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Domain mu-calculus

Published online by Cambridge University Press:  15 January 2004

Guo-Qiang Zhang
Guo-Qiang Zhang*
Affiliation:
Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106, USA.; gqz@eecs.cwru.edu.
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Abstract

The basic framework of domain μ-calculus was formulated in [39] more than ten years ago. This paper provides an improved formulation of a fragment of the μ-calculus without function space or powerdomain constructions, and studies some open problems related to this μ-calculus such as decidability and expressive power. A class of language equations is introduced for encoding μ-formulas in order to derive results related to decidability and expressive power of non-trivial fragments of the domain μ-calculus. The existence and uniqueness of solutions to this class of language equations constitute an important component of this approach. Our formulation is based on the recent work of Leiss [23], who established a sophisticated framework for solving language equations using Boolean automata (a.k.a. alternating automata [12,35]) and a generalized notion of language derivatives. Additionally, the early notion of even-linear grammars is adopted here to treat another fragment of the domain μ-calculus.

Keywords

Domain theorymu-calculusformal languagesBoolean automata.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 37 , Issue 4: Fixed Points in Computer Science (FICS'02) , October 2003 , pp. 337 - 364
Copyright
© EDP Sciences, 2003

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References

Abramsky, S., Domain theory in logical form. Ann. Pure Appl. Logic 51 (1991) 1-77 . CrossRef
Abramsky, S. and Jung, A., Domain theory. Clarendon Press, Handb. Log. Comput. Sci. 3 (1995) 1-168.
Abramsky, S., A domain equation for bisimulation. Inf. Comput. 92 (1991) 161-218. CrossRef
Arnold, A., The mu-calculus alternation-depth hierarchy is strict on binary trees. RAIRO Theoret. Informatics Appl. 33 (1999) 329-339. CrossRef
Amar, V. and Putzolu, G., On a family of linear grammars. Inf. Control 7 (1964) 283-291. CrossRef
Amar, V. and Putzolu, G., Generalizations of regular events. Inf. Control 8 (1965) 56-63. CrossRef
S. Bloom and Z. Ésik, Equational axioms for regular sets. Technical Report 9101, Stevens Institute of Technology (1991).
Bonsangue, M. and Kok, J.N., Towards an infinitary logic of domains: Abramsky logic for transition systems. Inf. Comput. 155 (1999) 170-201. CrossRef
Bradfield, J.C., Simplifying the modal mu-calculus alternation hierarchy. Lecture Notes in Comput. Sci. 1373 (1998) 39-49. CrossRef
S. Brookes, A semantically based proof system for partial correctness and deadlock in CSP, in Proceedings, Symposium on Logic in Computer Science. Cambridge, Massachusetts (1986) 58-65.
Brzozowski, J. and Leiss, E., On equations for regular languages, finite automata, and sequential networks. Theor. Comput. Sci. 10 (1980) 19-35 . CrossRef
Chandra, A.K., Kozen, D. and Stockmyer, L., Alternation. Journal of the ACM 28 (1981) 114-133 . CrossRef
Ésik, Z., Completeness of Park induction. Theor. Comput. Sci. 177 (1997) 217-283 (MFPS'94). CrossRef
A. Fellah, Alternating finite automata and related problems. Ph.D. thesis, Department of Mathematics and Computer Science, Kent State University (1991).
Fellah, A., Jurgensen, H. and Constructions, S. Yu for alternating finite automata. Int. J. Comput. Math. 35 (1990) 117-132. CrossRef
C. Gunter and D. Scott, Semantic domains. Jan van Leeuwen edn., Elsevier, Handb. Theoretical Comput. Sci. B (1990) 633-674.
D. Janin and I. Walukiewicz, On the expressive completeness of the propositional mu-calculus with respect to monadic second order logic. Lecture Notes in Comput. Sci. (CONCUR'96) 1119 (1996) 263-277.
Jensen, T., Disjunctive program analysis for algebraic data types. ACM Trans. Programming Languages and Systems 19 (1997) 752-804. CrossRef
P.T. Johnstone, Stone Spaces. Cambridge University Press (1982).
Kozen, D., Results on the propositional mu-calculus. Theor. Comput. Sci. 27 (1983) 333-354 . CrossRef
Kozen, D., A completeness theorem for Kleene algebras and the algebra of regular events. Inf. Comput. 110 (1994) 366-390. CrossRef
Leiss, E., Succinct representation of regular languages by Boolean automata. Theor. Comput. Sci. 13 (1981) 323-330. CrossRef
E. Leiss, Language Equations. Monographs in Computer Science, Springer-Verlag, New York (1999).
Lubarsky, R.S., μ-definable sets of integers. J. Symb. Log. 58 (1993) 291-313. CrossRef
D. Niwinski, Fixed points vs. infinite generation. IEEE Computer Press Logic in Computer Science (1988) 402-409.
Niwinski, D., Fixed point characterization of infinite behaviour of finite state systems. Theor. Comput. Sci. 189 (1997) 1-69 . CrossRef
Okhotin, A., Automaton representation of linear conjunctive languages. Proceedings of DLT 2002, Lecture Notes in Comput. Sci. 2450 (2003) 393-404. CrossRef
Okhotin, A., On the closure properties of linear conjunctive languages. Theor. Comput. Sci. 299 (2003) 663-685 . CrossRef
Park, D., Concurrency and automata on infinite sequences. Lecture Notes in Comput. Sci. 154 (1981) 561-572.
G. Plotkin, The Pisa Notes. Department of Computer Science, University of Edinburgh (1981).
Plotkin, G., A powerdomain construction. SIAM J. Computing 5 (1976) 452-487. CrossRef
V.R. Pratt, A decidable mu-calculus: Preliminary report, Proc. of IEEE 22nd Annual Symposium on Foundations of Computer Science (1981) 421-427.
Presburger, M., On the completeness of a certain system of arithmetic of whole numbers in which addition occurs as the only operator. Hist. Philos. Logic 12 (1991) 225-233 (English translation of the original paper in 1930). CrossRef
Rabin, M.O. and Scott, D., Finite automata and their decision problems. IBM J. Res. 3 (1959) 115-125. CrossRef
Vardi, M.Y., Alternating automata and program verification. Computer Science Today - Recent Trends and Developments, Lecture Notes in Comput. Sci. 1000 (1995) 471-485. CrossRef
Walukiewicz, I., Completeness of Kozen's axiomatisation of the propositional μ-calculus. Inf. Comput. 157 (2000) 142-182. CrossRef
G. Winskel, The Formal Semantics of Programming Languages. MIT Press (1993).
S. Yu, Regular Languages. Handbook of Formal Languages, Rozenberg and Salomaa, Springer-Verlag (1997) 41-110.
G.-Q. Zhang, Logic of Domains. Birkhauser, Boston (1991).