Splittings and the finite model property

An old conjecture of modal logics states that every splitting of the major systems K4, S4 and Grz has the finite model property. In this paper we will prove that all iterated splittings of G have Imp, whereas in the other cases we will give explicit counterexamples. We also introduce a proof technique which will give a positive answer for large classes of splitting frames. The proof works by establishing a rather strong property of these splitting frames namely that they preserve the finite model property in the following sense. Whenever an extension A has fmp so does the splitting A/ f of A by f. Although we will also see that this method has its limitations because there are frames lacking this property, it has several desirable side effects. For example, properties such as compactness, decidability and others can be shown to be preserved in a similar way and effective bounds for the size of models can be given. Moreover, all methods and proofs are constructive.

[1]  Frans Voorbraak,et al.  A Simplification of the Completeness Proofs for Guaspari and Solovay's R , 1986, Notre Dame J. Formal Log..

[2]  W. Rautenberg Der Verband der normalen verzweigten Modallogiken , 1977 .

[3]  Albert Visser,et al.  Peano's Smart Children: A Provability Logical Study of Systems with Built-in Consistency , 1986, Notre Dame J. Formal Log..

[4]  Albert Visser Kunnen wij elke machine verslaan?: Beschouwingen rondom Lucas' Argument , 1986 .

[5]  Kit Fine Logics Containing K4. Part I , 1974, J. Symb. Log..

[6]  Edith Hemaspaandra,et al.  Query Optimization Using Rewrite Rules , 1990, RTA.

[7]  Fer-Jan de Vries A functional program for the fast Fourier transform , 1988, ACM SIGPLAN Notices.

[8]  Frans Voorbraak,et al.  A Preferential Model Semantics For Default Logic , 1991, ECSQARU.

[9]  Wolfgang Rautenberg,et al.  Modal tableau calculi and interpolation , 1983, J. Philos. Log..

[10]  J. Vrancken,et al.  Parallel object-oriented term rewriting : the booleans , 1988 .

[11]  L. Maksimova Pretabular extensions of LewisS4 , 1975 .

[12]  M. B. Kalsbeek Towards the Interpretability Logic of IΔo+EXP , 1991 .

[13]  A. Visser,et al.  Evaluation, provably deductive equivalence in Heyting's arithmetic of substitution instances of propositional formulas , 1985 .

[14]  G. R. Renardel de Lavalette,et al.  Interpolation in natural fragments of intuitionistic propositional logic , 1986 .

[15]  J. C. Mulder,et al.  A modular approach to protocol verification using process algebra , 1986 .

[16]  G. R. Renardel de Lavalette,et al.  Choice in applicative theories , 1989 .

[17]  F. Voorbraak,et al.  Tensed intuitionistic logic , 1987 .

[18]  Gerard R. Renardel de Lavalette,et al.  Interpolation in fragments of intuitionistic propositional logic , 1987, Journal of Symbolic Logic.

[19]  Albert Visser,et al.  Explicit Fixed Points in Interpretability Logic , 1989, Stud Logica.

[20]  H. van Ditmarsch Abstractie in wiskunde, expertsystemen en argumentatie , 1986 .

[21]  Kit Fine,et al.  An incomplete logic containing S4 , 1974 .

[22]  R. Lathe Phd by thesis , 1988, Nature.

[23]  Erik C. W. Krabbe Naess's Dichotomy of Tenability and Relevance , 1986 .

[24]  Marcus Kracht,et al.  Prefinitely Axiomatizable Modal and Intermediate Logics , 1993, Math. Log. Q..

[25]  G. R. Renardel de Lavalette,et al.  Strictness analysis for POLYREC, a language with polymorphic and recursive types , 1988 .

[26]  COUNTEREXAMPLES IN APPLICATIVE THEORIES WITH CHOICE , 2022 .

[27]  F. D. Vries,et al.  Applications of constructive logic to sheaf constructions in toposes , 1987 .

[28]  Fer-Jan de Vries,et al.  Intuitionistic Free Abelian Groups , 1988, Math. Log. Q..

[29]  Marcus Kracht,et al.  An almost general splitting theorem for modal logic , 1990, Stud Logica.

[30]  Kit Fine,et al.  Logics containing K4. Part I , 1974, Journal of Symbolic Logic.

[31]  J.L.M. Vrancken Parallel Object Oriented Terms Rewriting : a first implementation in Pool2 , 1989 .

[32]  Frans Voorbraak,et al.  On the Justification of Dempster's Rule of Combination , 1988, Artif. Intell..

[33]  R. McKenzie Equational bases and nonmodular lattice varieties , 1972 .

[34]  Craig Smorynski,et al.  Arithmetic analogues of McAloon's unique Rosser sentences , 1989, Archive for Mathematical Logic.

[35]  Jan A. Bergstra,et al.  Process algebra semantics for queues , 1983 .

[36]  Albert Visser,et al.  A Course on Bimodal Provability Logic , 1987, Ann. Pure Appl. Log..

[37]  F.-J. de Vries,et al.  A functional program for Gaussian Elimination , 1987 .

[38]  Gerard R. Renardel de Lavalette Strictness Analysis via Abstract Interpretation for Recursively Defined Types , 1992, Inf. Comput..

[39]  Marcus Kracht,et al.  Internal Definability And Completeness In Modal Logic , 1990 .

[40]  W. Blok On the degree of incompleteness of modal logics and the covering relation in the lattice of modal logics , 1978 .

[41]  Marcus Kracht,et al.  Properties of independently axiomatizable bimodal logics , 1991, Journal of Symbolic Logic.

[42]  Michiel Doorman The existence property in the presence of function symbols , 1988 .

[43]  P. Rodenburg,et al.  Specification of the Fast Fourier Transform algorithm as a term rewriting system , 1987 .

[44]  D. de Jonh,et al.  COMPUTATIONS IN FRAGMENTS OF INTUITIONISTIC PROPOSITIONAL LOGIC , 1988 .

[45]  Frans Voorbraak,et al.  The Logic of Objective Knowledge and Rational Belief , 1990, JELIA.

[46]  J. A. Bergstra,et al.  Module algebra for relational specifications , 1986 .

[47]  Wolfgang Rautenberg,et al.  Splitting lattices of logics , 1980, Arch. Math. Log..

[48]  Frans Voorbraak,et al.  A Computationally Efficient Approximation of Dempster-Shafer Theory , 1988, Int. J. Man Mach. Stud..

[49]  Wolfgang Rautenberg,et al.  Klassische und nichtklassische Aussagenlogik , 1979 .

[50]  Albert Visser,et al.  Preliminary notes on interpretability logic , 1988 .

[51]  Albert Visser,et al.  A descending hierarchy of reflection principles , 1988 .

[52]  D. van Dalen,et al.  The war of frogs and mice, or the crisis of the Mathematische Annalen , 1987 .

[53]  Marcus Kracht,et al.  On the Logic of Category Definitions , 1989, CL.

[54]  Frans Voorbraak,et al.  The logic of actual obligation. An alternative approach to deontic logic , 1989 .