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Erschienen in:
Modal Logics with Hard Diamond-Free Fragments Kapitel lesen Erstes Kapitel lesen

2016 | OriginalPaper | Buchkapitel

Interpolation Method for Multicomponent Sequent Calculi

verfasst von : Roman Kuznets

Erschienen in: Logical Foundations of Computer Science

Verlag: Springer International Publishing

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Abstract

The proof-theoretic method of proving the Craig interpolation property was recently extended from sequents to nested sequents and hypersequents. There the notations were formalism-specific, obscuring the underlying common idea, which is presented here in a general form applicable also to other similar formalisms, e.g., prefixed tableaus. It describes requirements sufficient for using the method on a proof system for a logic, as well as additional requirements for certain types of rules. The applicability of the method, however, does not imply its success. We also provide examples from common proof systems to highlight various types of interpolant manipulations that can be employed by the method. The new results are the application of the method to a recent formalism of grafted hypersequents (in their tableau version), the general treatment of external structural rules, including the analytic cut, and the method’s extension to the Lyndon interpolation property.

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Vorheriges Kapitel Sequent Calculus for Intuitionistic Epistemic Logic IEL
Nächstes Kapitel Adjoint Logic with a 2-Category of Modes
Fußnoten
1
In negation normal form, formulas are built from \(\mathbin {\wedge }\), \(\mathbin {\vee }\), \(\top \), \(\bot \), propositional variables p, and their negations \(\overline{p}\); \(\mathbin {\lnot }\) is defined via De Morgan laws; \(\mathbin {\rightarrow }\) is defined via \(\mathbin {\lnot }\).
 
2
Here \(D \mathbin {\rightarrow }E\) means \(\overline{D} \mathbin {\vee }E\), where \(\overline{D}\) is the defined negation of D.
 
3
Despite the homogeneity of the components of hypersequents, maps can only be used unrestrictedly if the worlds of the model are completely homogeneous too, as in the case of the class of all models with \(R = W \times W\), another class of models used for S5.
 
4
We assume the standard semantics, i.e., that exactly one of A or \(\overline{A}\) holds at a world.
 
5
This is true also for \(l_i=0\): the empty conjunction is \(\top \) and \(\mathcal {M}, f(\alpha ) \Vdash \Diamond \top \). However, \(\mathcal {G}^\circ \left( \Diamond \bigwedge _{k=1}^{l_i} C_{ik}\right) \) cannot be dropped: the diamond formula in the disjunct that is forced ensures the existence of an accessible world and the possibility to use (2).
 
6
While the tableaus presented in [10] are not symmetric, the necessary modifications are standard, and thus the completeness results from [10] can be applied here.
 
Literatur
1.
Avron, A.: The method of hypersequents in the proof theory of propositional non-classical logics. In: Hodges, W., Hyland, M., Steinhorn, C., Truss, J. (eds.) Logic: From Foundations to Applications: European Logic Colloquium, pp. 1–32. Clarendon Press (1996)
2.
Brünnler, K.: Nested Sequents. Habilitation thesis, Institut für Informatik und angewandte Mathematik, Universität Bern (2010)
3.
Craig, W.: Three uses of the herbrand-gentzen theorem in relating model theory and proof theory. J. Symbolic Log. 22(3), 269–285 (1957)MathSciNetCrossRefMATH
4.
Fitting, M.: Proof Methods for Modal and Intuitionistic Logics. D. Reidel Publishing Company, Dordrecht (1983)CrossRefMATH
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7.
Indrzejczak, A.: Cut-free hypersequent calculus for S4.3. Bull. Sect. Logic Univ. Lódz 41(1–2), 89–104 (2012)MathSciNetMATH
8.
Kurokawa, H.: Hypersequent calculi for modal logics extending S4. In: Nakano, Y., Satoh, K., Bekki, D. (eds.) JSAI-isAI 2013. LNCS, vol. 8417, pp. 51–68. Springer, Heidelberg (2014)
9.
Kuznets, R.: Craig interpolation via hypersequents. In: Concepts of Proof in Mathematics, Philosophy, and Computer Science. Ontos Verlag, Accepted (2016)
10.
11.
Lahav, O.: From frame properties to hypersequent rules in modal logics. In: 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), pp. 408–417. IEEE Press (2013)
12.
Maehara, S.: On the Interpolation Theorem of Craig (in Japanese). Sugaku 12(4), 235–237 (1960–1961)
13.
Maksimova, L.L.: Absence of the interpolation property in the consistent normal modal extensions of the dummett logic. Algebra Log. 21(6), 460–463 (1982)CrossRefMATH
14.
Restall, G.: Proofnets for S5: sequents and circuits for modal logic. In: Dimitracopoulos, C., Newelski, L., Normann, D., Steel, J.R. (eds.) Logic Colloquium 2005, pp. 151–172. Cambridge University Press (2007)
Metadaten
Titel
Interpolation Method for Multicomponent Sequent Calculi
verfasst von
Roman Kuznets
Copyright-Jahr
2016
Buch
Logical Foundations of Computer Science

Print ISBN: 978-3-319-27682-3

Electronic ISBN: 978-3-319-27683-0

Copyright-Jahr: 2016

DOI
https://doi.org/10.1007/978-3-319-27683-0_15

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