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Interpolation in Modal Logic [chapter]

Maarten Marx
1998 Lecture Notes in Computer Science  
For modal logics, this leads to simple, easy{ to{check conditions on the logic which imply interpolation. We apply this result to bering of modal logics and to modal logics of knowledge and belief.  ...  The interpolation property and Robinson's consistency property are important tools for applying logic to software engineering.  ...  Modal logics for knowledge and belief Interpolation in the standard system In this section, we apply theorem 1 to modal logics of knowledge and belief.  ... 
doi:10.1007/3-540-49253-4_13 fatcat:6satcwsg7vc63gzisiq5gwgb6u

Uniform Lyndon interpolation property in propositional modal logics [article]

Taishi Kurahashi
2020 arXiv   pre-print
We prove several propositional modal logics including K, KB, GL and Grz enjoy ULIP. Our proofs are modifications of Visser's proofs of uniform interpolation property using layered bisimulations.  ...  We introduce and investigate the notion of uniform Lyndon interpolation property (ULIP) which is a strengthening of both uniform interpolation property and Lyndon interpolation property.  ...  Is there a logic having both UIP and LIP but does not have ULIP?  ... 
arXiv:1809.00943v3 fatcat:vdpsotfqvvet7j3n3irs2citke

Uniform Interpolation in Coalgebraic Modal Logic [article]

Fatemeh Seifan, Lutz Schröder, Dirk Pattinson
2022 arXiv   pre-print
We study interpolation in the context of coalgebraic modal logics, i.e. modal logics axiomatized in rank 1, restricting for clarity to the case with finitely many modalities.  ...  We introduce a notion of one-step (uniform) interpolation, which refers only to a restricted logic without nesting of modalities, and show that a coalgebraic modal logic has uniform interpolation if it  ...  XX:4 Uniform Interpolation in Monotone Coalgebraic Modal Logic Coalgebraic Modal Logic We briefly recall the syntax and semantics of coalgebraic modal logic.  ... 
arXiv:2205.00448v1 fatcat:6f35cobmszgl5d36l6z5fr2f2q

Modality, bisimulation and interpolation in infinitary logic

Johan van Benthem
1999 Annals of Pure and Applied Logic  
In this contribution, I consider the bridge concept of blsimulation, which links modal logic with computational process theories.  ...  Rohit Parikh has been one of the pioneers at the interface of modal and dynamic logic.  ...  As it happens, this notion first emerged in modal logic.  ... 
doi:10.1016/s0168-0072(98)00029-3 fatcat:kmxgk5pudndcvliy2kcfmdz66m

Failure of Interpolation in Combined Modal Logics

Maarten Marx, Carlos Areces
1998 Notre Dame Journal of Formal Logic  
We investigate transfer of interpolation in such combinations of modal logic which lead to interaction of the modalities. Combining logics by taking products often blocks transfer of interpolation.  ...  Viewing rst order logic as a product of modal logics, we derive a strong counterexample for failure of interpolation in the nite variable fragments of rst order logic.  ...  This work was started when the rst author was invited to teach a course on modal logic at the ECI'96 winter school in Buenos Aires, Argentina.  ... 
doi:10.1305/ndjfl/1039293067 dblp:journals/ndjfl/MarxA98 fatcat:6ve3x4ezp5fclmsqvz4oijfqk4

Uniform interpolation and sequent calculi in modal logic

Rosalie Iemhoff
2018 Archive for Mathematical Logic  
This method is applied to several modal logics and is shown to cover known results from the literature, such as the existence of uniform interpolants for the modal logic K.  ...  The results imply that for modal logics K4 and S4, which are known not to have uniform interpolation, certain sequent calculi cannot exist.  ...  Standard assignment for modal logics In this section we extend the method developed above to modal logic by extending the class of rules to which Corollary 4.6 applies.  ... 
doi:10.1007/s00153-018-0629-0 fatcat:qxqjg4zuvzambfurqocl7xhyuy

Repairing the interpolation theorem in quantified modal logic

Carlos Areces, Patrick Blackburn, Maarten Marx
2003 Annals of Pure and Applied Logic  
While interpolation and Beth's deÿnability theorem fail in a number of well-known quantiÿed modal logics (for example in quantiÿed modal K, T, D, S4, S4.3 and S5 with constant domains), their counterparts  ...  These are special cases of the main result of the paper: the quantiÿed hybrid logic of any class of frames deÿnable in the bounded fragment of ÿrst-order logic has the interpolation property, irrespective  ...  Instead we will closely inspect the proof of the interpolation theorem in ÿrst-order logic and see that we added just enough expressive power to quantiÿed modal logic to ÿnd the interpolant in the right  ... 
doi:10.1016/s0168-0072(03)00059-9 fatcat:g4hs3f233jegxpb4ryntsoueta

Uniform Interpolation and Propositional Quantifiers in Modal Logics

Marta Bílková
2007 Studia Logica: An International Journal for Symbolic Logic  
We investigate uniform interpolants in propositional modal logics from the proof-theoretical point of view.  ...  Our approach is adopted from Pitts' proof of uniform interpolation in intuitionistic propositional logic [15] .  ...  Ghilardi and Zawadowski in [9] proved that uniform interpolation fails for modal logic S4. It follows immediately that it fails for modal logic K4 as well.  ... 
doi:10.1007/s11225-007-9021-5 fatcat:zafkuihx7zaxhhnv3phwvgpmii

Weakly Aggregative Modal Logic: Characterization and Interpolation [chapter]

Jixin Liu, Yanjing Wang, Yifeng Ding
2019 Lecture Notes in Computer Science  
Weakly Aggregative Modal Logic (WAML) is a collection of disguised polyadic modal logics with n-ary modalities whose arguments are all the same.  ...  WAML has some interesting applications on epistemic logic and logic of games, so we study some basic model theoretical aspects of WAML in this paper.  ...  Interpolation By a standard strategy in [19] , we know that the basic polyadic modal logics (PML) have the Craig Interpolation theorem.  ... 
doi:10.1007/978-3-662-60292-8_12 fatcat:do5xgd74nvdibmfdhis22pnheq

Uniform Interpolation by Resolution in Modal Logic [chapter]

Andreas Herzig, Jérôme Mengin
Lecture Notes in Computer Science  
It has been shown that modal logic K has the uniform interpolation property, and a method has recently been proposed for forgetting variables in a modal formula (of mu-calculus) given in disjunctive normal  ...  In this paper, we propose a method, based on an extension of resolution to modal logics, to perform variable forgetting for formulae in conjunctive normal form, in the modal logic K.  ...  Conclusion The results above show that, although it is more complicated than in propositional logic, resolution can be used to compute a uniform interpolant in modal logic K.  ... 
doi:10.1007/978-3-540-87803-2_19 fatcat:dbnaejddivf6nhudxxns5y75ry

Constructive interpolation in hybrid logic

Patrick Blackburn, Maarten Marx
2003 Journal of Symbolic Logic (JSL)  
order modal logics.  ...  In addition, on the class of all frames, the basic algorithm is conservative: on purely modal input it computes interpolants in which the hybrid syntactic machinery does not occur.  ...  Propositional modal logic K has the interpolation property. In this case, the algorithm can be adapted to produce modal formulas as interpolants.  ... 
doi:10.2178/jsl/1052669059 fatcat:3vmjz4lelfgkllut45hxopxtiu

Uniform Interpolation for Propositional and Modal Team Logics [article]

Giovanna D'Agostino
2018 arXiv   pre-print
In this paper we consider Modal Team Logic, a generalization of Classical Modal Logic in which it is possible to describe dependence phenomena between data.  ...  As a consequence, we prove that these fragments enjoy the Uniform Interpolation Property.  ...  Similarly, uniform interpolation for Modal Team Logic can be easily proved from uniform interpolation of standard modal logic, but this easy proof cannot be used for other team modal logics, where we have  ... 
arXiv:1810.05395v1 fatcat:lidh5cvptnek7jue2oidwfukxa

The many faces of interpolation

Johan van Benthem
2008 Synthese  
We present a number of, somewhat unusual, ways of describing what Craig's interpolation theorem achieves, and use them to identify some open problems and further directions.  ...  But matters are tricky in general, especially for logical interpolation in weak logical languages (Hendriks 2000) .  ...  Here are two surprising results from the 1990s: Theorem Basic modal logic has uniform interpolation (Pitt 1992 , Visser 1996 .  ... 
doi:10.1007/s11229-008-9351-5 fatcat:wvg4jrrgsrbohdr6lxzorlseim

8 Modal consequence relations [chapter]

Marcus Kracht
2007 Studies in Logic and Practical Reasoning  
In detail, it shall give an algebraic characterisation of interpolation and ways of establishing interpolation for logics.  ...  Now, even if one is comfortable with classical logic, it is not immediately clear what the correct inferences are in modal logic.  ... 
doi:10.1016/s1570-2464(07)80011-5 fatcat:kigpak25arbyhbfuroe7zupneu

Coherence in Modal Logic [article]

Tomasz Kowalski, George Metcalfe
2019 arXiv   pre-print
In this paper, a more general criterion is obtained and used to prove the failure of coherence and uniform deductive interpolation for a broad family of modal logics, including K, KT, K4, and S4.  ...  In a recent paper by the authors it was shown that coherence forms a key ingredient of the uniform deductive interpolation property for equational consequence in a variety, and a general criterion was  ...  Relationships between uniform interpolation and bisimulation quantifiers for the modal µ-calculus and other fixpoint modal logics have been studied in some depth in [4, 5, 22] .  ... 
arXiv:1902.02540v1 fatcat:twg72iaqlbbsvlgdeij6eohhby
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