Geodesic
Restricted interpolation over modal logic~$\mathrm S4$
Algebra i logika, Tome 52 (2013) no. 4, pp. 461-501.

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The problem of restricted interpolation and definability in normal extensions of modal logic $\mathrm S4$ is investigated.We specify necessary conditions for the restricted interpolation property IPR in the systems under consideration, and prove that there exist only finitely many logics possessing IPR or the projective Beth property PB2. These logics are all residually finite and recognizable over $\mathrm S4$. As a consequence, the restricted interpolation problem and the projective Beth property are decidable over $\mathrm S4$.
Keywords: modal logic $\mathrm S4$, restricted interpolation, projective Beth property. 
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L. L. Maksimova. Restricted interpolation over modal logic~$\mathrm S4$. Algebra i logika, Tome 52 (2013) no. 4, pp. 461-501. http://geodesic.mathdoc.fr/item/AL_2013_52_4_a3/

[1] W. Craig, “Three uses of Herbrand–Gentzen theorem in relating model theory and proof theory”, J. Symb. Log., 22:3 (1957), 269–285 | DOI | MR | Zbl

[2] J. Barwise, S. Feferman (eds.), Model-theoretic logics, Perspect. Math. Log., Springer-Verlag, New York etc., 1985 | MR | Zbl

[3] D. M. Gabbay, L. Maksimova, Interpolation and definability. Modal and intuitionistic logics, Oxford Logic Guides, 46, Clarendon Press, Oxford Sci. Publ., Oxford, 2005 | MR | Zbl

[4] D. Gabbay, Investigations in modal and tense logics with applications to problems in philosophy and linguistics, Synthese Library, 92, D. Reidel Publ. Co., Dordrecht–Boston, 1976 | MR | Zbl

[5] E. W. Beth, “On Padoa's method in the theory of definitions”, Indag. Math., 15:4 (1953), 330–339 | MR

[6] L. L. Maksimova, “Proektivnye svoistva Beta v modalnykh i superintuitsionistskikh logikakh”, Algebra i logika, 38:3 (1999), 316–333 | MR | Zbl

[7] L. Maksimova, “Restricted interpolation in modal logics”, Advances in modal logic, v. 4, eds. Ph. Balbiani et al., King's College Publ., London, 2003, 297–311 | MR | Zbl

[8] L. Maksimova, “Interpolation and joint consistency”, We will show them!, Essays in honour of Dov Gabbay, v. 2, eds. S. Artemov et al., King's College Publ., London, 2005, 293–305

[9] A. V. Karpenko, “Interpolyatsiya v slabo tranzitivnykh modalnykh logikakh”, Algebra i logika, 51:2 (2012), 197–215 | MR | Zbl

[10] A. V. Chagrov, “Nerazreshimye svoistva rasshirenii logiki dokazuemosti”, Algebra i logika, 29:3 (1990), 350–367 | MR | Zbl

[11] L. L. Maksimova, “Interpolyatsionnye teoremy v modalnykh logikakh i amalgamiruemye mnogoobraziya topobulevykh algebr”, Algebra i logika, 18:5 (1979), 556–586 | MR | Zbl

[12] L. L. Maksimova, “Klassifikatsiya rasshirenii modalnoi logiki $\mathrm S4$”, Sib. matem. zh., sdana v zhurnal

[13] L. Maksimova, “Intuitionistic logic and implicit definability”, Ann. Pure Appl. Logic, 105:1–3 (2000), 83–102 | DOI | MR | Zbl

[14] L. Maksimova, “Problem of restricted interpolation in superintuitionistic and some modal logics”, Log. J. IGPL, 18:3 (2010), 367–380 | DOI | MR | Zbl

[15] L. L. Maksimova, “Projective Beth's properties in infinite slice extensions of the modal logic $\mathrm K4$”, Advances in modal logic, v. 3, eds. F. Wolter et al., World Sci. Publ., Singapore, 2002, 349–363 | MR | Zbl

[16] L. L. Maksimova, “Slabaya forma interpolyatsii v ekvatsionalnoi logike”, Algebra i logika, 47:1 (2008), 94–107 | MR | Zbl

[17] A. V. Karpenko, “Slaboe interpolyatsionnoe svoistvo v rasshireniyakh logik $\mathrm S4$ i $\mathrm K4$”, Algebra i logika, 47:6 (2008), 705–722 | MR | Zbl

[18] L. Maksimova, “Projective Beth property in extensions of Grzegorczyk logic”, Stud. Log., 83:1–3 (2006), 365–391 | DOI | MR | Zbl

[19] K. Fine, “An ascending chain of $\mathrm S4$ logics”, Theoria, 40 (1974), 110–116 | DOI | MR

[20] V. A. Yankov, “O svyazi mezhdu vyvodimostyu v intuitsionistskom ischislenii vyskazyvanii i konechnymi implikativnymi strukturami”, Dokl. AN SSSR, 151:6 (1963), 1293–1294 | Zbl

[21] L. L. Maksimova, V. V. Rybakov, “O reshetke normalnykh modalnykh logik”, Algebra i logika, 13:2 (1974), 188–216 | MR | Zbl

[22] H. Rasiowa, R. Sikorski, The mathematics of metamathematics, PWN, Warszawa, 1963 ; E. Raseva, R. Sikorskii, Matematika metamatematiki, Nauka, M., 1972 | MR | MR

[23] W. Blok, Varieties of interior algebras, PhD Thesis, Univ. Amsterdam, 1976

[24] L. L. Maksimova, “Modalnye logiki konechnykh sloev”, Algebra i logika, 14:3 (1975), 304–319 | MR | Zbl

[25] T. Hosoi, “On intermediate logics. I”, J. Fac. Sci. Univ. Tokyo, Sec. Ia, 14 (1967), 293–312 | MR

[26] L. L. Maksimova, “Razreshimost proektivnogo svoistva Beta v mnogoobraziyakh geitingovykh algebr”, Algebra i logika, 40:3 (2001), 290–301 | MR | Zbl

[27] L. L. Maksimova, “Opredelimost v normalnykh rasshireniyakh logiki $\mathrm S4$”, Algebra i logika, 43:4 (2004), 387–410 | MR | Zbl