<<

UvA-DARE (Digital Academic Repository)

Modal fixpoint : some model theoretic questions

Fontaine, G.M.M.

Publication date 2010

Link to publication

Citation for published version (APA): Fontaine, G. M. M. (2010). Modal fixpoint logic: some model theoretic questions. Institute for Logic, Language and Computation.

General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons).

Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible.

UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl)

Download date:26 Sep 2021 Bibliography

[AJ94] and Achim Jung. Domain Theory. In Samson Abramsky, Dov M. Gabbay Dov M., and Tom S. E. Maibaum, editors, Handbook for Logic in Computer Science.1994.

[AN01] Andr´eArnold and Damian Niwi´nski. Rudiments of µ-calculus,volume 146 of Studies in Logic.2001.

[Arn99] Andr´eArnold. The µ-calculus alternation-depth hierarchy is strict on binary trees. Theoretical Informatics and Applications,33(4–5),1999.

[Bak80] Jaco de Bakker. Mathematical Theory of Program Correctness.1980.

[Bar93] Michael Barr. Terminal coalgebras in well-founded set theory. Theo- retical Computer Sciences,114:299–315,1993.

[BC96] Girish Bhat and Rance Cleaveland. Efficient model checking via the equational µ-calculus. In Proceedings of LICS’96,pages304–312,1996.

[Ben76] Johan van Benthem. Modal correspondence theory.PhDthesis,Insti- tute of Logic, Language and Computation, Amsterdam, 1976.

[Ben84] Johan van Benthem. Correspondence theory. In Dov M. Gabbay and Franz Guenthner, editors, Handbook of Philosophical Logic, volume 2, pages 167–248. 1984.

[Ben96] Johan van Benthem. Exploring logical dynamics.1996.

[Ben98] Johan van Benthem. Programming operations that are safe for bisim- ulation. Studia Logica,60(2):311–330,1998.

[Ben02] Johan van Benthem. Extensive games as process models. Journal of logic, language and information,11:289–313,2002.

227 228 Bibliography

[Ben05] Johan van Benthem. Guards, bounds, and generalized semantics. Journal of Logic, Language and Information,14:263–279,2005.

[Ben06] Johan van Benthem. Modal frame correspondences and fixed-points. Studia Logica,83:133–155,2006.

[BK08] Michael Benedikt and Christoph Koch. XPath leashed. ACM Com- puting Surveys,41(1),2008.

[BL69] J.Richard B¨uchi and Lawrence H. Landweber. Solving sequential con- ditions by finite state strategies. Transactions of the American Math- ematical Society,138:295–311,1969.

[BM96] Jon Barwise and Lawrence S. Moss. Vicious Circles: On the Math- ematics of Non-Wellfounded Phenomena, volume 60 of CSLI Lecture Notes.1996.

[Boj07] Miko￿laj Boja´nczyk. Two-way unary temporal logic over trees. In Proceedings of LICS,pages121–130,2007.

[BRV01] Patrick Blackburn, Maarten de Rijke, and Yde Venema. Modal logic. 2001.

[BS84] Robert Bull and Krister Segerberg. Basic modal logic. In Dov M. Gab- bay and Franz Guenthner, editors, Handbook of Philosophical Logic, volume 2, pages 1–88. 1984.

[BS07] Julian C. Bradfield and Colin Stirling. Modal µ-calculi. In Patrick Blackburn, Johan van Benthem, and Frank Wolter, editors, Handbook of Modal ,volume3ofStudies in Logic and Practical Reasoning, pages 721–756. 2007.

[BSd04] Falk Bartels, Ana Sokolova, and Erik de Vink. A hierarchy of proba- bilistic system types. Theoretical Computer Science,327:3–22,2004.

[B¨uc60] J. Richard B¨uchi. Weak second-order arithmetic and finite automata. Zeitschrift f¨urMathematische Logik und Grundlagen der Mathematik, 6:60–92, 1960.

[BW06] Miko￿laj Boja´nczyk and Igor Walukiewicz. Characterizing EF and EX tree logics. Theoretical Computer Science,358(255-272),2006.

[BW07] Miko￿laj Boja´nczyk and Igor Walukiewicz. Forest algebras. In J¨org Flum, Erich Graedel, and Thomas Wilke, editors, Automata and logic: history and perspectives, number 2 in Texts in Logic and Games, pages 107–132. 2007. Bibliography 229

[Cat05] Balder ten Cate. Model theory for extended modal languages.PhD thesis, University of Amsterdam, 2005. ILLC Dissertation Series DS- 2005-01. [CE81] Edmund M. Clarke and E. Allen Emerson. Design and synthesis of synchronization skeletons using branching time temporal logic. LNCS, 131:52–71, 1981. [CK73] Chen Chung Chang and H. Jerome Keisler. Model Theory.1973. [CKP09] Corina Cˆırstea, Clemens Kupke, and Dirk Pattinson. EXPTIME tableaux for the coalgebraic µ-calculus. In Proceedings of CSL’09, pages 179–193, 2009. [Cza10] Marek Czarnecki. How fast can the fixpoints in modal µ-calculus be reached? Manuscript accepted at FICS10, 2010. [Dam94] Mads Dam. CTL* and ECTL* as fragments of the modal mu-calculus. Theoretical Computer Science,126(1):77–96,1994. [DH00] Giovanna D’Agostino and Marco Hollenberg. Logical questions con- cerning the µ-calculus: interpolation, Lyndon and Los-Tarski. Journal of Symbolic Logic,pages310–332,2000. [DO09] Anuj Dawar and Martin Otto. Modal characterisation theorems over special classes of frames. Ann. Pure Appl. Logic,161(1):1–42,2009.

1 1 [Doe89] Kees Doets. Monadic Π1-theories of Π1-properties. Notre Dame Jour- nal of Formal Logic,30(2):224–240,1989. [Don70] John Doner. Tree acceptors and some of their applications. Journal of Computer and System Sciences,4(5):77–96,1970. [DV02] Giovanna D’Agostino and Albert Visser. Finality regained: A coal- gebraic study of Scott-sets and multisets. Archive for Mathematical Logic,41:267–298,2002. [EC80] E. Allen Emerson and Edmund M. Clarke. Characterizing correct- ness properties of parallel programs using fixpoints. In Proceedings of ICALP,volume85,pages169–181,1980. [EI08]´ Zoltan Esik´ and Szabolcs Ivan. Some varieties of finite tree au- tomata related to restricted temporal logics. Fundamenta Informati- cae,82:79–103,2008. [EJ88] E. Allen Emerson and Charanjit S. Jutla. The complexity of tree automata and logics of programs (extended abstract). In Proceedings of FOCS,pages328–337,1988. 230 Bibliography

[EJ91] E. Allen Emerson and Charanjit S. Jutla. Tree automata, µ-calculus and determinacy. In Proceedings of FOCS,pages368–377,1991. [EL86] E. Allen Emerson and Chin-Laung Lei. Efficient model checking in fragments of the propositional mu-calculus (extended abstract). In Proceedings of LICS’86,pages267–278,1986. [Elg61] Calvin C. Elgot. Decision problems of finite automata design and related arithmetics. Transactions of the American Mathematical So- ciety,98:21–52,1961. [FBC85] Maurizio Fattorosi-Barnaba and Claudio Cerrato. Graded modalities I. Studia Logica,44:197–221,1985. [Fin72] Kit Fine. In so many possible worlds. Notre Dame Journal of Formal Logic,13:516–520,1972. [FL79] Michael Fischer and Richard Ladner. Propositional dynamic logic of regular programs. Journal of Computer and System Sciences, 18(2):194–211, 1979. [Fon08] Ga¨elle Fontaine. Continuous fragment of the µ-calculus. In Proceed- ings of CSL,pages139–153,2008. [FV59] Solomon Feferman and Robert Lawson Vaught. The first-order prop- erties of algebraic structures. Fundamenta Informaticae,47:57–103, 1959. [FV10] Ga¨elle Fontaine and Yde Venema. Syntactic characterizations of se- mantic properties of the µ-calculus. submitted soon, 2010. [GHK+80] Gerhard Gierz, Karl Heinrich Hofmann, Klaus Keimel, John Lawson, Michael Mislove, and Dana Stewart Scott. A Compendium of Contin- uous Lattices.1980. [GKP05] Georg Gottlob, Christof Koch, and Reinhard Pichler. Efficient algo- rithms for processing XPath queries. ACM Transactions on Database Systems,61(2):444–491,2005. [Gob70] Lou Goble. Grades of modality. Logique et Analyse,13:323–334,1970. [G¨od31] Kurt G¨odel. Uber¨ formal unentscheidbare s¨atze der principia math- ematica und verwandter system i. Monatschefte f¨urMathematik und Physik,38:173–198,1931. [GT75] Rob Goldblatt and S. Thomason. Axiomatic classes in propositional modal logic. In Algebra and Logic,volume450ofLecture Notes in Mathematics,pages163–173.1975. Bibliography 231

[GTW02] Erich Gr¨adel, Wolfgang Thomas, and Thomas Wilke, editors. Au- tomata, Logic, and Infinite Games,volume2500ofLNCS.2002. [Har84] David Harel. Dynamic logic. In and Franz Guenther, editors, Handbook of Philosophical Logic,volume2,pages497–604. 1984. [Hen50] Leon Henkin. Completeness in the theory of types. Journal of Sym- bolic Logic,15(2):81–91,1950. [HK04] Helle Hansen and Clemens Kupke. A coalgebraic perspective on mono- tone modal logic. Electronic Notes in Theoretical Computer Science, 106:121 – 143, 2004. Proceedings of CMCS. [Hol98a] Marco Hollenberg. Characterizations of negative definability in modal logic. Studia Logica,60:357–386,1998. [Hol98b] Marco Hollenberg. Logic and Bisimulation. PhD thesis, Utrecht Uni- versity, 1998. Zeno Institute of Philosophy. [HP78] David Harel and Vaughan R. Pratt. Non-determinism in logics of programs. In Proceedings of POPL,pages203–213,1978. [Jan96] David Janin. Propri´et´eslogiques du non-d´eterminisme et mu-calcul modal. PhD thesis, Universit´ede Bordeaux, 1996. [Jan97] David Janin. Automata, tableaus and a reduction theorem for fixpoint calculi in arbitrary complete lattices. In Proceedings of LICS,1997. [Jan06] David Janin. Contributions to formal methods: games, logic and automata. Habilitation thesis, 2006. [JL03] David Janin and Giacomo Lenzi. On the relationship between moandic and weak monadic second order logic on arbitrary trees, with applica- tion to the mu-calculus. Fudamenta Informaticae,61:247–265,2003. [Jur98] Marcin Jurdzi´nski. Deciding the winner in parity games is in UP co- UP. Information Processing Letters, 68(3):119–124, November 1998.∩ [Jur00] Marcin Jurdzi´nski. Small progress measures for solving parity games. In Proceedings of STACS, volume LNCS 1770, pages 290–301, 2000. [JW95a] David Janin and Igor Walukiewicz. Automata for the modal µ-calculus and related results. In Proceedings of MFCS,pages552–562,1995. LNCS 969. [JW95b] David Janin and Igor Walukiewicz. Automata for the modal µ-calculus and related results. In Proceedings of MFCS’95,pages552–562,1995. 232 Bibliography

[JW96] David Janin and Igor Walukiewicz. On the expressive completeness of the propositional modal mu-calculus and related results. In Pro- ceedings of CONCUR,pages263–277,1996.

[Kai97] Roope Kaivola. Using Automata to Characterize Fixed Point Tempo- ral Logics. PhD thesis, Department of Computer Science, University of Edinburgh, 1997.

[Koz83] Dexter Kozen. Results on the propositional µ-calculus. Theoretical Computer Science,27(3):333–354,1983.

[Koz95] Dexter Kozen. Results on the propositional µ-calculus. Lecture Notes in Computer Science,962,1995.

[KV05] Clemens Kupke and Yde Venema. Closure properties of coalgebra automata. In Proceedings of LICS 2005,pages199–208,2005.

[KV08] Clemens Kupke and Yde Venema. Coalgebraic automata theory: basic results. Logical Methods in Computer Science,4:1–43,2008.

[KV09] Christian Kissig and Yde Venema. Complementation of coalgebra automata. In Proceedings of CALCO’09,pages81–96,2009.

[KVW00] Orna Kupferman, Moshe Y. Vardi, and Pierre Wolper. An automata- theoretic approach to branching-time model checking. Journal of the ACM,47(2):312–360,2000.

[McN66] Robert McNaughton. Testing and generating infinite sequences by a finite automaton. Information and Control,9(5):521–530,1966.

[MdR05] Maarten Marx and Maarten de Rijke. Semantic characterizations of navigational XPath. ACM SIGMOD Report,34(3):41–46,2005.

[Mos74] Yiannis Nicholas Moschovakis. Elementary induction on abstract structures,volume77ofStudies in Logic and the Foundations of Math- ematics.1974.

[Mos91] Andrzej Mostowski. Games with forbidden positions. Technical Re- port 78, University of Gda´nsk, 1991.

[Mos99] Lawrence S. Moss. Coalgebraic logic. Annals of Pure and Ap- plied Logic,96:277–317,1999.(ErratumpublishedAPAL 99:241–259, 1999).

[MS87] David Muller and Paul Schupp. Alternating automata on infinite trees. Theoretical Computer Science,54:267–276,1987. Bibliography 233

[Niw88] Damian Niwi´nski. Fixed points vs. infinite generation. In Proceedings of LICS,pages402–0409,1988.

[Niw97] Damian Niwi´nski. Fixed point characterization of infinite behavior of finite-state systems. Theoretical Computer Science,189:1–69,1997.

[NW96] Damian Niwi´nski and Igor Walukiewicz. Games for the mu-calculus. Theoretical Computer Science,163(1&2):99–116,1996.

[Ott99] Martin Otto. Eliminating Recursion in the mu-Calculus. In Proceed- ings of STACS,1999.

[Pap94] Christos H. Papadimitriou. Computational complexity.1994.

[Par69] David Park. Fixed point induction and proof of program properties. Machine Intelligence,5:59–78,1969.

[Par80] David Park. On the semantics of fair parallelism,volume86,pages 504–526. LNCS, 1980.

[Par81] David Park. Concurrency and automata on infinite sequences. LNCS, 154:561–572, 1981.

[Pat03] Dirk Pattinson. Coalgebraic modal logic: Soundness, completeness and decidability of local consequence. Theoretical Computer Science, 309:177–193, 2003.

[Pit06] Nir Piterman. From nondeterministic B¨uchi and Streett automata to deterministic parity automata. In Proceedings of LICS,pages255–264, 2006.

[Pnu77] . The temporal logic of programs. In Proceedings of FOCS, pages 46–57, 1977.

[Pra76] Vaughan R. Pratt. Semantical considerations on floyd-hoare logic. In Proceedings of FOCS,pages109–121,1976.

[Pra81] Vaughan R. Pratt. A decidable µ-calculus: Preliminary report. In Proceedings of FOCS,pages421–427,1981.

[PS09] Thomas Place and Luc Segoufin. A decidable characterization of lo- cally testable tree languages. In Proceedings of ICALP,pages285–296, 2009.

[PS10] Thomas Place and Luc Segoufin. Deciding definability in FO2(<) (or XPath) on trees. In Proceedings of LICS’10),pages253–262,2010. 234 Bibliography

[Rab69] Michael O. Rabin. Decidability of second-order theories and automata on infinite trees. Transactions of the American Mathematical Society, 141:1–35, 1969.

[Rab77] Michael O. Rabin. Decidable theories. In Jon Barwise, editor, Hand- book of Mathematical Logic,pages595–629.1977.

[Rij00] Maarten de Rijke. A note on graded modal logic. Studia Logica, 64(2):271–283, 2000.

[Saf92] Shmuel Safra. Exponential determinization for ω-automata with strong-fairness acceptance condition. In Proceedings of STOC,pages 275–282, 1992.

[Sah75] Henrik Sahlqvist. Correspondence and completeness in the first- and second-order semantics for modal logic. In Proceedings of the Third Scandinavian Logic Symposium,pages110–143,1975.

[Sal70] Andrzej Salwicki. Formalized algorithmic languages. Bull. Acad. Polon. Sci., Ser. Sci. Math. Astron. Plys.,18:227–232,1970.

[SdB69] Dana Scott and Jaco de Bakker. A theory of programs. IBM Vienna, 1969.

[SE89] Robert S. Streett and E. Allen Emerson. An automata theoretic de- cision procedure for the propositional µ-calculus. Information and Computation,81(3):249–264,1989.

[Smo85] Craig Smorynski. Self-reference and modal logic.1985.

[Sti95] Colin Stirling. Local model checking games. In Proceedings of CON- CUR,volume962,pages1–11,1995.

[Str81] Robert S. Streett. Propositional dynamic logic of looping and con- verse. In Proceedings of STOC,pages375–383,1981.

[Str82] Robert S. Streett. Propositional dynamic logic of looping and converse is elementarily decidable. Information and Control,54(1/2):121–141, 1982.

[Tar55] Alfred Tarski. A lattice-theoretical fixpoint theorem and its applica- tion. Pacific Journal of Mathematics,5:285–309,1955.

[Tho97] Wolfgang Thomas. Languages, automata and logic. In Grzegorz Rozenberg and Arto Salomaa, editors, Handbook of formal languages, volume 3, pages 389–455. 1997. Bibliography 235

[TW68] James W. Thatcher and Jesse B. Wright. Generalized finite automata theory with an application to a decision problem for second-order logic. Mathematical Systems Theory,2(1):57–81,1968.

[Ven93] Yde Venema. Derivation rules as anti-axioms in modal logic. Journal of Symbolic Logic,58:1003–1054,1993.

[Ven06a] Yde Venema. Algebras and coalgebras. In Patrick Blackburn, Johan van Benthem, and Frank Wolter, editors, Handbook of Modal Logic, volume 3 of Studies in Logic and Practical Reasoning,pages331–426. 2006.

[Ven06b] Yde Venema. Automata and fixed point logic: a coalgebraic perspec- tive. Information and Computation,204:637–678,2006.

[Ven08a] Yde Venema. Lecture notes on µ-calculus. http://staff.science. uva.nl/~yde/teaching/ml/,2008. [Ven08b] Yde Venema. PDL as a fragment of the modal µ-calculus. unpublished manuscript, 2008.

[VW07] Moshe Y. Vardi and Thomas Wilke. Automata: from logics to al- gorithms. In J¨org Flum, Erich Graedel, and Thomas Wilke, editors, Logic and Automata: History and Perspectives,number2inTextsin Logic and Games, pages 629–736. 2007.

[W3C] W3C. XML path language (XPath): Version 1.0. W3C recommenda- tion. http://www.w3.org/TR/xpath.html.

[Wal95] Igor Walukiewicz. Completeness of Kozen’s axiomatization of the propositional µ-calculus. In Proceedings of LICS,pages14–24,1995.

[Wal02] Igor Walukiewicz. Monadic second order logic on tree-like structures. Theoretical Computer Science,275:311–346,2002.

[Wu07] Zhilin Wu. A note on the characterization of TL[EF]. Information Processing Letter,102:28–54,2007.