{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T00:35:35Z","timestamp":1776472535585,"version":"3.51.2"},"reference-count":139,"publisher":"Elsevier","isbn-type":[{"value":"9780444516909","type":"print"}],"license":[{"start":{"date-parts":[[2007,1,1]],"date-time":"2007-01-01T00:00:00Z","timestamp":1167609600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2007,1,1]],"date-time":"2007-01-01T00:00:00Z","timestamp":1167609600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2007]]},"DOI":"10.1016\/s1570-2464(07)80008-5","type":"book-chapter","created":{"date-parts":[[2007,10,4]],"date-time":"2007-10-04T09:38:50Z","timestamp":1191490730000},"page":"249-329","source":"Crossref","is-referenced-by-count":76,"title":["5 Model theory of modal logic"],"prefix":"10.1016","member":"78","reference":[{"key":"10.1016\/S1570-2464(07)80008-5_bib1","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1023\/A:1004275029985","article-title":"Modal languages and bounded fragments of predicate logic","volume":"27","author":"Andreka","year":"1998","journal-title":"Journal of Philosophical Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib2","series-title":"Rudiments of \u03bc-Calculus","author":"Arnold","year":"2001"},{"key":"10.1016\/S1570-2464(07)80008-5_bib3","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1145\/2402.322389","article-title":"On the desirability of acyclic database schemes","volume":"30","author":"Beeri","year":"1983","journal-title":"Journal of the ACM"},{"key":"10.1016\/S1570-2464(07)80008-5_bib4","series-title":"Graphs and Hypergraphs","author":"Berge","year":"1973"},{"key":"10.1016\/S1570-2464(07)80008-5_bib5","series-title":"Modal Logic","author":"Blackburn","year":"2001"},{"key":"10.1016\/S1570-2464(07)80008-5_bib6","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1007\/BF00649480","article-title":"An incomplete system of modal logic","volume":"14","author":"Boolos","year":"1985","journal-title":"Journal of Philosophical Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib7","doi-asserted-by":"crossref","unstructured":"R. Bull and K. Segerberg. Basic modal logic. In Gabbay and Guenthner [33], pages 1-88.","DOI":"10.1007\/978-94-009-6259-0_1"},{"key":"10.1016\/S1570-2464(07)80008-5_bib8","series-title":"Logical Foundations of Probability","author":"Carnap","year":"1950"},{"key":"10.1016\/S1570-2464(07)80008-5_bib9","first-page":"83","article-title":"Advanced modal logic","volume":"vol 3","author":"Chagrov","year":"2001"},{"key":"10.1016\/S1570-2464(07)80008-5_bib10","series-title":"Modal Logic","author":"Chagrov","year":"1997"},{"key":"10.1016\/S1570-2464(07)80008-5_bib11","doi-asserted-by":"crossref","first-page":"1261","DOI":"10.2307\/2275473","article-title":"An undecidable problem in correspondence theory","volume":"56","author":"Chagrova","year":"1991","journal-title":"Journal of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib12","series-title":"Model Theory","author":"Chang","year":"1973"},{"key":"10.1016\/S1570-2464(07)80008-5_bib13","article-title":"Definability and changing perspectives. The Beth property for three extensions of modal logic","author":"Conradie","year":"2002"},{"key":"10.1016\/S1570-2464(07)80008-5_bib14","first-page":"17","article-title":"Elementary canonical formulae: a survey on syntactic, algorithmic, and model-theoretic aspects","volume":"volume 5","author":"Conradie","year":"2005"},{"key":"10.1016\/S1570-2464(07)80008-5_bib15","article-title":"Modal Logic and Non-Well-Founded Set Theory: Translation, Bisimulation, Inter- polation","author":"D'Agostino","year":"1998"},{"key":"10.1016\/S1570-2464(07)80008-5_bib16","doi-asserted-by":"crossref","first-page":"310","DOI":"10.2307\/2586539","article-title":"Logical questions concerning the \u03bc-calculus: interpolation, Lyndon and o\u015b-Tarski","volume":"65","author":"D'Agostino","year":"2000","journal-title":"Journal of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib17","series-title":"Proc. of LICS'2005","first-page":"21","article-title":"Modal characterisation theorems over special classes of frames","author":"Dawar","year":"2005"},{"key":"10.1016\/S1570-2464(07)80008-5_bib18","article-title":"Extending Modal Logic","author":"de Rijke","year":"1993"},{"key":"10.1016\/S1570-2464(07)80008-5_bib19","unstructured":"M. de Rijke. A Lindstr\u00f6m theorem for modal logic. In Ponse et al. [109], pages 217\u2013230."},{"key":"10.1016\/S1570-2464(07)80008-5_bib20","article-title":"Modal model theory","author":"de Rijke","year":"1995"},{"issue":"2","key":"10.1016\/S1570-2464(07)80008-5_bib21","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1023\/A:1005245900406","article-title":"A note on graded modal logic","volume":"64","author":"de Rijke","year":"2000","journal-title":"Studia Logica"},{"key":"10.1016\/S1570-2464(07)80008-5_bib22","series-title":"Essays on Non-Classical Logic","first-page":"111","article-title":"Global definability in basic modal logic","author":"de Rijke","year":"2001"},{"key":"10.1016\/S1570-2464(07)80008-5_bib23","series-title":"Basic Model Theory","author":"Doets","year":"1996"},{"key":"10.1016\/S1570-2464(07)80008-5_bib24","first-page":"275","article-title":"Higher-order logic","volume":"vol. 1","author":"Doets","year":"1983"},{"key":"10.1016\/S1570-2464(07)80008-5_bib25","series-title":"Perspectives in Mathematical Logic","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03182-7","article-title":"Finite Model Theory","author":"Ebbinghaus","year":"1995"},{"key":"10.1016\/S1570-2464(07)80008-5_bib26","series-title":"Mathematical Logic","author":"Ebbinghaus","year":"1984"},{"key":"10.1016\/S1570-2464(07)80008-5_bib27","doi-asserted-by":"crossref","first-page":"50","DOI":"10.2307\/2272945","article-title":"Probabilities on finite models","volume":"41","author":"Fagin","year":"1976","journal-title":"Journal of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib28","unstructured":"K. Fine. Some connections between elementary and modal logic. In Kanger [81], pages 15-31."},{"key":"10.1016\/S1570-2464(07)80008-5_bib29","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1016\/0022-0000(79)90046-1","article-title":"Propsitional dynamic logic of regular programs","volume":"18","author":"Fischer","year":"1979","journal-title":"Journal of Computer and System Sciences"},{"key":"10.1016\/S1570-2464(07)80008-5_bib30","series-title":"Preprint, Inst, f\u00fcr Math. Logik und Grundl. der Math","article-title":"On the (infinite) model theory of fixed-point logics","author":"Flum","year":"1995"},{"key":"10.1016\/S1570-2464(07)80008-5_bib31","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1111\/j.1755-2567.1970.tb00430.x","article-title":"Selective filtration in modal logic","volume":"30","author":"Gabbay","year":"1970","journal-title":"Theoria"},{"key":"10.1016\/S1570-2464(07)80008-5_bib32","series-title":"Investigations in Modal and Tense Logics with Applications to Problems in Philosophy and Linguistics","author":"Gabbay","year":"1976"},{"key":"10.1016\/S1570-2464(07)80008-5_bib33","volume":"vol. 2","year":"1984"},{"key":"10.1016\/S1570-2464(07)80008-5_bib34","series-title":"Temporal Logic: Mathematical Foundations and Com- putational Aspects","author":"Gabbay","year":"1994"},{"key":"10.1016\/S1570-2464(07)80008-5_bib35","article-title":"Many-dimensional modal logics: theory and applications","volume":"148","author":"Gabbay","year":"2003"},{"key":"10.1016\/S1570-2464(07)80008-5_bib36","article-title":"Interpolation and Definability. Modal and Intuitionistic Logic","volume":"46","author":"Gabbay","year":"2005"},{"key":"10.1016\/S1570-2464(07)80008-5_bib37","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02759729","article-title":"Concerning measures in first-order calculi","volume":"2","author":"Gaifman","year":"1964","journal-title":"Israel J. of Math."},{"key":"10.1016\/S1570-2464(07)80008-5_bib38","series-title":"Logic Colloquium '81","first-page":"105","article-title":"On local and nonlocal properties","author":"Gaifman","year":"1982"},{"key":"10.1016\/S1570-2464(07)80008-5_bib39","series-title":"Trends in Logic","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-015-9936-8_3","article-title":"Sheaves, Games, and Model Completions","author":"Ghilardi","year":"2002"},{"key":"10.1016\/S1570-2464(07)80008-5_bib40","doi-asserted-by":"crossref","first-page":"142","DOI":"10.1007\/BF01071084","article-title":"Range and degree of realizability of formulas in the restricted predicate calculus","volume":"5","author":"Glebskii","year":"1969","journal-title":"Cybernetics"},{"key":"10.1016\/S1570-2464(07)80008-5_bib41","doi-asserted-by":"crossref","first-page":"419","DOI":"10.2178\/jsl\/1052669058","article-title":"Simulating polyadic modal logics by monadic ones","volume":"68","author":"Goguadze","year":"2003","journal-title":"Journal of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib42","doi-asserted-by":"crossref","first-page":"35","DOI":"10.2307\/2272267","article-title":"First-order definability in modal logic","volume":"40","author":"Goldblatt","year":"1975","journal-title":"Journal of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib43","first-page":"41","article-title":"Metamathematics of modal logic I","volume":"6","author":"Goldblatt","year":"1976","journal-title":"Reports on Mathematical Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib44","first-page":"21","article-title":"Metamathematics of modal logic II","volume":"7","author":"Goldblatt","year":"1976","journal-title":"Reports on Mathematical Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib45","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/0168-0072(89)90032-8","article-title":"Varieties of complex algebras","volume":"38","author":"Goldblatt","year":"1989","journal-title":"Annals of Pure and Applied Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib46","article-title":"Logics of Time and Computation","author":"Goldblatt","year":"1992"},{"key":"10.1016\/S1570-2464(07)80008-5_bib47","series-title":"Center for the Study of Language and Information","article-title":"Mathematics of Modality, volume 43 of CSLI Lecture Notes","author":"Goldblatt","year":"1993"},{"key":"10.1016\/S1570-2464(07)80008-5_bib48","unstructured":"R. Goldblatt. Saturation and the Hennessy-Milner property. In Ponse et al. [109], pages 107-129."},{"key":"10.1016\/S1570-2464(07)80008-5_bib49","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1016\/S1570-8683(03)00008-9","article-title":"Mathematical modal logic: A view of its evolution","volume":"1","author":"Goldblatt","year":"2003","journal-title":"Journal of Applied Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib50","doi-asserted-by":"crossref","first-page":"186","DOI":"10.2178\/bsl\/1082986262","article-title":"Erd\u00f6s graphs resolve Fine's canonicity problem","volume":"10","author":"Goldblatt","year":"2004","journal-title":"Bulletin of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib51","series-title":"Algebra and Logic","first-page":"163","article-title":"Axiomatic classes in propositional modal logic","author":"Goldblatt","year":"1974"},{"key":"10.1016\/S1570-2464(07)80008-5_bib52","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1023\/A:1005021313747","article-title":"Axiomatizations with context rules of inference in modal logic","volume":"61","author":"Goranko","year":"1998","journal-title":"Studia Logica"},{"key":"10.1016\/S1570-2464(07)80008-5_bib53","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/s001530100135","article-title":"The modal logic of the countable random frame","volume":"42","author":"Goranko","year":"2003","journal-title":"Archive for Mathematical Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib54","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1093\/logcom\/2.1.5","article-title":"Using the universal modality: Gains and questions","volume":"2","author":"Goranko","year":"1992","journal-title":"Journal of Logic and Computation"},{"key":"10.1016\/S1570-2464(07)80008-5_bib55","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.apal.2005.10.005","article-title":"Elementary canonical formulae.Extending Sahlqvist's theorem","volume":"141","author":"Goranko","year":"2005","journal-title":"Annals of Pure and Applied Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib56","doi-asserted-by":"crossref","first-page":"1719","DOI":"10.2307\/2586808","article-title":"On the restraining power of guards","volume":"64","author":"Gr\u00e4del","year":"1999","journal-title":"Journal of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib57","first-page":"90","article-title":"Why are modal logics so robustly decidable?","volume":"68","author":"Gr\u00e4del","year":"1999","journal-title":"Bulletin of the EATCS"},{"issue":"3","key":"10.1016\/S1570-2464(07)80008-5_bib58","doi-asserted-by":"crossref","first-page":"418","DOI":"10.1145\/507382.507388","article-title":"Back and forth between guarded and modal logics","volume":"3","author":"Gr\u00e4del","year":"2002","journal-title":"ACM Transactions on Computational Logics"},{"key":"10.1016\/S1570-2464(07)80008-5_bib59","doi-asserted-by":"crossref","first-page":"53","DOI":"10.2307\/421196","article-title":"On the Decision Problem for Two-Variable First-Order Logic","volume":"3","author":"Gr\u00e4del","year":"1997","journal-title":"Bulletin of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib60","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/S0304-3975(98)00308-9","article-title":"On Logics with Two Variables","volume":"224","author":"Gr\u00e4del","year":"1999","journal-title":"Theoretical Computer Science"},{"key":"10.1016\/S1570-2464(07)80008-5_bib61","first-page":"213","article-title":"Undecidability Results on Two-Variable Logics","volume":"38","author":"Gr\u00e4del","year":"1999","journal-title":"Archive for Mathematical Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib62","series-title":"Proceedings of 14th IEEE Symposium on Logic in Computer Science LICS '99, Trento","first-page":"45","article-title":"Guarded Fixed Point Logic","author":"Gr\u00e4del","year":"1999"},{"key":"10.1016\/S1570-2464(07)80008-5_bib63","first-page":"180","article-title":"Complexity of the first-order theory of almost all structures","volume":"52","author":"Grandjean","year":"1983","journal-title":"Information and Com-putation"},{"key":"10.1016\/S1570-2464(07)80008-5_bib64","first-page":"90","article-title":"Zero-one laws: The logic in computer science column","volume":"46","author":"Gurevich","year":"1992","journal-title":"Bulletin of the European Asso-ciation for Theoretical Computer Science"},{"key":"10.1016\/S1570-2464(07)80008-5_bib65","series-title":"Proceedings of ICALP 1987","first-page":"269","article-title":"Computation tree logic CTL* and path quantifiers in the monadic theory of the binary tree","author":"Hafer","year":"1987"},{"key":"10.1016\/S1570-2464(07)80008-5_bib66","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/0168-0072(94)90084-1","article-title":"Zero-one laws for modal logic","volume":"69","author":"Halpern","year":"1994","journal-title":"Annals of Pure and Applied Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib67","doi-asserted-by":"crossref","first-page":"1985","DOI":"10.1090\/S0002-9947-99-02374-0","article-title":"Extending partial isomorphisms and the profinite topology on free groups","volume":"352","author":"Herwig","year":"2000","journal-title":"Transactions of the AMS"},{"key":"10.1016\/S1570-2464(07)80008-5_bib68","series-title":"Model Theory","author":"Hodges","year":"1993"},{"key":"10.1016\/S1570-2464(07)80008-5_bib69","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1023\/A:1015178417181","article-title":"Loosely guarded fragment of first-order logic has the finite model property","volume":"70","author":"Hodkinson","year":"2002","journal-title":"Studia Logica"},{"key":"10.1016\/S1570-2464(07)80008-5_bib70","doi-asserted-by":"crossref","first-page":"387","DOI":"10.2178\/bsl\/1058448678","article-title":"Finite conformai hypergraph covers and Gaifman cliques in finite struc-tures","volume":"9","author":"Hodkinson","year":"2003","journal-title":"The Bulletin of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib71","unstructured":"M. Hollenberg. Hennessy-Milner classes and process algebra. In Ponse et al. [109], pages 187-216."},{"key":"10.1016\/S1570-2464(07)80008-5_bib72","series-title":"Proceedings 10th Amsterdam Colloquium","first-page":"359","article-title":"Safety for bisimulation in general modal logic","author":"Hollenberg","year":"1996"},{"key":"10.1016\/S1570-2464(07)80008-5_bib73","article-title":"Definability and Interpolation: Model-theoretic investigations","author":"Hoogland","year":"2001"},{"key":"10.1016\/S1570-2464(07)80008-5_bib74","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/BF00935597","article-title":"Every world can see a reflexive world","volume":"49","author":"Hughes","year":"1990","journal-title":"Studia Logica"},{"key":"10.1016\/S1570-2464(07)80008-5_bib75","series-title":"A New Introduction to Modal Logic","author":"Hughes","year":"1996"},{"key":"10.1016\/S1570-2464(07)80008-5_bib76","doi-asserted-by":"crossref","first-page":"760","DOI":"10.1137\/0216051","article-title":"Languages that capture complexity classes","volume":"16","author":"Immerman","year":"1987","journal-title":"SIAM Journal of Computing"},{"key":"10.1016\/S1570-2464(07)80008-5_bib77","series-title":"Proceedings CONCUR '96","first-page":"263","article-title":"On the expressive completeness of the propositional \u03bc-calculus w.r.t. monadic second-order logic","author":"Janin","year":"1996"},{"key":"10.1016\/S1570-2464(07)80008-5_bib78","doi-asserted-by":"crossref","first-page":"891","DOI":"10.2307\/2372123","article-title":"Boolean algebras with operators, Part I","volume":"73","author":"J\u00f3nsson","year":"1952","journal-title":"American Journal of Mathematics"},{"key":"10.1016\/S1570-2464(07)80008-5_bib79","doi-asserted-by":"crossref","first-page":"127","DOI":"10.2307\/2372074","article-title":"Boolean algebras with operators, Part II","volume":"74","author":"J\u00f3nsson","year":"1952","journal-title":"American Journal of Mathemat-ics"},{"key":"10.1016\/S1570-2464(07)80008-5_bib80","article-title":"Tense Logic and the Theory of Linear Order","author":"Kamp","year":"1968"},{"key":"10.1016\/S1570-2464(07)80008-5_bib81","series-title":"Proceedings of the Third Scandinavian Logic Symposium. Uppsala 1973","year":"1975"},{"key":"10.1016\/S1570-2464(07)80008-5_bib82","series-title":"Languages with Expressions of Infinite Length","author":"Karp","year":"1964"},{"key":"10.1016\/S1570-2464(07)80008-5_bib83","series-title":"Model Theory for Infinitary Logic","author":"Keisler","year":"1971"},{"key":"10.1016\/S1570-2464(07)80008-5_bib84","series-title":"Tools and Techniques in Modal Logic, volume 142 of Studies in Logic","author":"Kracht","year":"1999"},{"key":"10.1016\/S1570-2464(07)80008-5_bib85","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1023\/A:1004900300438","article-title":"Simulation and transfer results in modal logic: a survey","volume":"59","author":"Kracht","year":"1997","journal-title":"Studia Logica"},{"key":"10.1016\/S1570-2464(07)80008-5_bib86","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/2964568","article-title":"A completeness theorem in modal logic","volume":"24","author":"Kripke","year":"1959","journal-title":"Journal of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib87","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1002\/malq.19630090502","article-title":"Semantical analysis of modal logic I.Normal modal propositional calculi","volume":"9","author":"Kripke","year":"1963","journal-title":"Zeitschrift f\u00fcr Mathematische Logik und Grundlagen der Mathematik"},{"issue":"2","key":"10.1016\/S1570-2464(07)80008-5_bib88","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1016\/S0004-3702(98)00109-X","article-title":"Expressiveness of concept expressions in first-order description logics","volume":"107","author":"Kurtonina","year":"1999","journal-title":"Artificial Intelligence"},{"key":"10.1016\/S1570-2464(07)80008-5_bib89","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1111\/j.1755-2567.1974.tb00082.x","article-title":"A note on Thomason's refined structures for tense logics","volume":"40","author":"Lachlan","year":"1970","journal-title":"Theoria"},{"key":"10.1016\/S1570-2464(07)80008-5_bib90","series-title":"Proceedings of LICS'2002","first-page":"225","article-title":"Zero-one law fails for frame satisfiability in propositional modal logic","author":"Le Bars","year":"2002"},{"key":"10.1016\/S1570-2464(07)80008-5_bib91","series-title":"An Introduction to Modal Logic","author":"Lemmon","year":"1977"},{"key":"10.1016\/S1570-2464(07)80008-5_bib92","first-page":"457","article-title":"Intensional logics without iterative axioms","volume":"3","author":"Lewis","year":"1974","journal-title":"Journal of Philisophical Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib93","series-title":"Elements of Finite Model Theory","author":"Libkin","year":"2004"},{"key":"10.1016\/S1570-2464(07)80008-5_bib94","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/S0304-3975(98)00312-0","article-title":"Linear ordering on graphs, anti-founded sets and polynomial time com-putability","volume":"224","author":"Lisitsa","year":"1999","journal-title":"Theoretical Computer Science"},{"key":"10.1016\/S1570-2464(07)80008-5_bib95","doi-asserted-by":"crossref","first-page":"299","DOI":"10.4064\/fm-59-3-299-300","article-title":"On defining well-orderings","volume":"59","author":"Lopez-Escobar","year":"1966","journal-title":"Fundamenta Mathematicae"},{"key":"10.1016\/S1570-2464(07)80008-5_bib96","series-title":"Proceedings of 15th International Workshop on Computer Science Logic, volume 2142 of LNCS","first-page":"247","article-title":"Modal logic and the two-variable fragment","author":"Lutz","year":"2001"},{"key":"10.1016\/S1570-2464(07)80008-5_bib97","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1111\/j.1755-2567.1970.tb00431.x","article-title":"A generalization of the concept of relational model","author":"Makinson","year":"1970","journal-title":"Theoria"},{"issue":"3","key":"10.1016\/S1570-2464(07)80008-5_bib98","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1090\/conm\/131.3\/1175909","article-title":"Definability and interpolation in classical modal logics","volume":"131","author":"Maksimova","year":"1992","journal-title":"Contemporary Mathematics"},{"key":"10.1016\/S1570-2464(07)80008-5_bib99","article-title":"Complexity of modal logics of relations","author":"Marx","year":"1997"},{"key":"10.1016\/S1570-2464(07)80008-5_bib100","series-title":"Automated Reasoning with Analytic Tableaux and Related Methods, LNCS 2796","first-page":"150","article-title":"Xpath and modal logics of finite DAG's","author":"Marx","year":"2003"},{"key":"10.1016\/S1570-2464(07)80008-5_bib101","series-title":"Proceedings of the 14th IEEE Symposium on Logic in Computer Science","first-page":"360","article-title":"On the expressive power of CTL*","author":"M\u00f6ller","year":"1999"},{"key":"10.1016\/S1570-2464(07)80008-5_bib102","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1002\/malq.19750210118","article-title":"On languages with two variables","volume":"21","author":"Mortimer","year":"1975","journal-title":"Zeitschrift f\u00fcr Mathematische Logik und Grundlagen der Mathematik"},{"key":"10.1016\/S1570-2464(07)80008-5_bib103","series-title":"Logic at Work, Essays dedicated to the memory of Helena Rasiowa","first-page":"89","article-title":"A fixpoint approach to second-order quantifier elimination with ap- plications to correspondence theory","author":"Nonnengart","year":"1998"},{"key":"10.1016\/S1570-2464(07)80008-5_bib104","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1016\/S0304-3975(98)00314-4","article-title":"Bisimulation-invariant Ptime and higher-dimensional mu-calculus","volume":"224","author":"Otto","year":"1999","journal-title":"Theoretical Computer Science"},{"key":"10.1016\/S1570-2464(07)80008-5_bib105","article-title":"Elementary proof of the van Benthem-Rosen characterisation theorem","author":"Otto","year":"2004"},{"key":"10.1016\/S1570-2464(07)80008-5_bib106","doi-asserted-by":"crossref","first-page":"173","DOI":"10.1016\/j.apal.2004.04.003","article-title":"Modal and guarded characterisation theorems over finite transition systems","volume":"130","author":"Otto","year":"2004","journal-title":"Annals of Pure and Applied Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib107","article-title":"Bisimulation invariance and finite models","author":"Otto","year":"2005"},{"key":"10.1016\/S1570-2464(07)80008-5_bib108","series-title":"A Course in Model Theory","author":"Poizat","year":"2000"},{"key":"10.1016\/S1570-2464(07)80008-5_bib109","series-title":"Modal Logic and Process Algebra. A Bisimulation Perspective., volume 53 of CSLI Lecture Notes","year":"1995"},{"key":"10.1016\/S1570-2464(07)80008-5_bib110","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1007\/BF00370641","article-title":"Infinitary propositional normal modal logic","volume":"46","author":"Radev","year":"1987","journal-title":"Studia Logica"},{"key":"10.1016\/S1570-2464(07)80008-5_bib111","doi-asserted-by":"crossref","first-page":"1011","DOI":"10.2307\/2695091","article-title":"An axiomatization of full computation tree logic","volume":"66","author":"Reynolds","year":"2001","journal-title":"Journal of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib112","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1023\/A:1008275906015","article-title":"Modal logic over finite structures","volume":"6","author":"Rosen","year":"1997","journal-title":"Journal of Logic, Language and Information"},{"key":"10.1016\/S1570-2464(07)80008-5_bib113","unstructured":"H. Sahlqvist. Completeness and correspondence in the first and second order semantics for modal logic. In Kanger [81], pages 110-143."},{"key":"10.1016\/S1570-2464(07)80008-5_bib114","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/0168-0072(88)90021-8","article-title":"Topology and duality in modal logic","volume":"37","author":"Sambin","year":"1988","journal-title":"Annals of Pure and Applied Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib115","doi-asserted-by":"crossref","first-page":"992","DOI":"10.2307\/2274758","article-title":"A topological proof of Sahlqvist's theorem","volume":"54","author":"Sambin","year":"1989","journal-title":"Journal of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib116","first-page":"477","article-title":"A decision method for validity of sentences in two variables","volume":"27","author":"Scott","year":"1962","journal-title":"Journal of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib117","series-title":"An Essay in Classical Modal Logic","author":"Segerberg","year":"1971"},{"key":"10.1016\/S1570-2464(07)80008-5_bib118","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/0890-5401(89)90031-X","article-title":"An automata theoretic decision procedure for the propositional mu-calculus","volume":"81","author":"Streett","year":"1989","journal-title":"Information and Computation"},{"key":"10.1016\/S1570-2464(07)80008-5_bib119","article-title":"Modale Fragmente von Cwwund CW1W","author":"Sturm","year":"1997"},{"key":"10.1016\/S1570-2464(07)80008-5_bib120","first-page":"190","article-title":"Interpolation and preservation in MLW1","volume":"39","author":"Sturm","year":"1998","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib121","article-title":"Expressivity of second-order propositional modal logic","author":"ten Cate","year":"2005","journal-title":"Journal of Philosophical Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib122","article-title":"Model theory for extended modal languages","author":"ten Cate","year":"2005"},{"key":"10.1016\/S1570-2464(07)80008-5_bib123","doi-asserted-by":"crossref","first-page":"150","DOI":"10.1111\/j.1755-2567.1974.tb00077.x","article-title":"An incompleteness theorem in modal logic","volume":"40","author":"Thomason","year":"1974","journal-title":"Theoria"},{"key":"10.1016\/S1570-2464(07)80008-5_bib124","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1002\/malq.19750210114","article-title":"Reduction of second-order logic to modal logic","volume":"21","author":"Thomason","year":"1975","journal-title":"Zeitschrift f\u00fcr Mathematische Logik und Grundlagen der Mathematik"},{"key":"10.1016\/S1570-2464(07)80008-5_bib125","article-title":"Modal Correspondence Theory","author":"van Benthem","year":"1976"},{"key":"10.1016\/S1570-2464(07)80008-5_bib126","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/2273696","article-title":"Canonical modal logics and ultrafilter extensions","volume":"44","author":"van Benthem","year":"1980","journal-title":"Journal of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib127","series-title":"Modal Logic and Classical Logic","author":"van Benthem","year":"1983"},{"key":"10.1016\/S1570-2464(07)80008-5_bib128","doi-asserted-by":"crossref","unstructured":"J. van Benthem. Correspondence theory. In Gabbay and Guenthner [33], pages 167\u2013247.","DOI":"10.1007\/978-94-009-6259-0_4"},{"key":"10.1016\/S1570-2464(07)80008-5_bib129","first-page":"20","article-title":"Notes on modal definability","volume":"39","author":"van Benthem","year":"1989","journal-title":"Notre Dame Journal of Formal Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib130","article-title":"Exploring Logical Dynamics","author":"van Benthem","year":"1996"},{"key":"10.1016\/S1570-2464(07)80008-5_bib131","article-title":"Modality, bisimulation and interpolation in infinitary logic","author":"van Benthem","year":"1997"},{"key":"10.1016\/S1570-2464(07)80008-5_bib132","doi-asserted-by":"crossref","first-page":"696","DOI":"10.2178\/jsl\/1122038910","article-title":"Minimal predicates, fixed points, and definability","volume":"70","author":"van Benthem","year":"2005","journal-title":"Journal of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib133","article-title":"A new Lindstr\u00f6m theorem for modal logic","author":"van Benthem","year":"2006","journal-title":"ILLC Publication Series Report PP- 2006-06"},{"key":"10.1016\/S1570-2464(07)80008-5_bib134","series-title":"Proceedings of 14th ACM Symposium on Theory of Computing","first-page":"137","article-title":"The complexity of relational query languages","author":"Vardi","year":"1982"},{"key":"10.1016\/S1570-2464(07)80008-5_bib135","series-title":"Descriptive Complexity and Finite Models, volume 31 of DIMACS Series in Discrete Mathematics","first-page":"149","article-title":"Why is modal logic so robustly decidable?","author":"Vardi","year":"1997"},{"key":"10.1016\/S1570-2464(07)80008-5_bib136","series-title":"Proceedings of 25th Symposium on Automata, Language and Programming, volume 1443 of LNCS","first-page":"628","article-title":"Reasoning about the past with two-way automata","author":"Vardi","year":"1998"},{"key":"10.1016\/S1570-2464(07)80008-5_bib137","doi-asserted-by":"crossref","first-page":"1003","DOI":"10.2307\/2275109","article-title":"Derivation rules as anti-axioms in modal logic","volume":"58","author":"Venema","year":"1993","journal-title":"Journal of Symbolic Logic"},{"key":"10.1016\/S1570-2464(07)80008-5_bib138","series-title":"Essays Dedicated to Johan van Benthem on the Occasion of his 50th Birthday","article-title":"Modal definability, purely modal","author":"Venema","year":"1999"},{"key":"10.1016\/S1570-2464(07)80008-5_bib139","article-title":"Bisimulations, model descriptions and propositional quantifiers","author":"Visser","year":"1996"}],"container-title":["Studies in Logic and Practical Reasoning","Handbook of Modal Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1570246407800085?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S1570246407800085?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T23:17:20Z","timestamp":1761607040000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S1570246407800085"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007]]},"ISBN":["9780444516909"],"references-count":139,"URL":"https:\/\/doi.org\/10.1016\/s1570-2464(07)80008-5","relation":{},"ISSN":["1570-2464"],"issn-type":[{"value":"1570-2464","type":"print"}],"subject":[],"published":{"date-parts":[[2007]]}}}