Grafting hypersequents onto nested sequents

We introduce a new Gentzen-style framework of grafted hypersequents that combines the formalism of nested sequents with that of hypersequents. To illustrate the potential of the framework, we present novel calculi for the modal logics $\mathsf{K5}$ and $\mathsf{KD5}$, as well as for extensions of the modal logics $\mathsf{K}$ and $\mathsf{KD}$ with the axiom for shift reflexivity. The latter of these extensions is also known as $\mathsf{SDL}^+$ in the context of deontic logic. All our calculi enjoy syntactic cut elimination and can be used in backwards proof search procedures of optimal complexity. The tableaufication of the calculi for $\mathsf{K5}$ and $\mathsf{KD5}$ yields simplified prefixed tableau calculi for these logic reminiscent of the simplified tableau system for $\mathsf{S5}$, which might be of independent interest.

[1]  Balder ten Cate,et al.  Hybrid logics , 2007, Handbook of Modal Logic.

[2]  Grigori Mints,et al.  Indexed systems of sequents and cut-elimination , 1997, J. Philos. Log..

[3]  Paul McNamara,et al.  Deontic logic , 2006, Logic and the Modalities in the Twentieth Century.

[4]  M. Fitting.  Proof Methods for Modal and Intuitionistic Logics , 1983 .

[5]  Linh Anh Nguyen.  Analytic Tableau Systems and Interpolation for the Modal Logics KB, KDB, K5, KD5 , 2001, Stud Logica.

[6]  Patrick Blackburn,et al.  Terminating Tableau Calculi for Hybrid Logics Extending K , 2009, M4M.

[7]  M. Takano.  A MODIFIED SUBFORMULA PROPERTY FOR THE MODAL LOGICS K5 AND K5D , 2007 .

[8]  Fabio Massacci,et al.  Single Step Tableaux for Modal Logics , 2000, Journal of Automated Reasoning.

[9]  Francesca Poggiolesi,et al.  Gentzen Calculi for Modal Propositional Logic , 2010 .

[10]  Rajeev Goré,et al.  Tableau Methods for Modal and Temporal Logics , 1999 .

[11]  Nuel Belnap,et al.  Display logic , 1982, J. Philos. Log..

[12]  Fabio Massacci,et al.  Strongly Analytic Tableaux for Normal Modal Logics , 1994, CADE.

[13]  Sara Negri,et al.  Proof Analysis in Modal Logic , 2005, J. Philos. Log..

[14]  M. de Rijke,et al.  Modal Logic , 2001, Cambridge Tracts in Theoretical Computer Science.

[15]  Björn Lellmann,et al.  Axioms vs Hypersequent Rules with Context Restrictions: Theory and Applications , 2014, IJCAR.

[16]  G. Mints.  A Short Introduction to Modal Logic , 1992 .

[17]  Francesca Poggiolesi,et al.  The Method of Tree-Hypersequents for Modal Propositional Logic , 2009, Towards Mathematical Philosophy.

[18]  Franco Montagna,et al.  Algebraic and proof-theoretic characterizations of truth stressers for MTL and its extensions , 2010, Fuzzy Sets Syst..

[19]  Sara Negri,et al.  Structural proof theory , 2001 .

[20]  Greg Restall,et al.  Proofnets for S5: sequents and circuits for modal logic , 2007 .

[21]  II. Mathematisches.  Power and Weakness of the Modal Display Calculus , 1996 .

[22]  Leandro Chaves Rêgo,et al.  Characterizing the NP-PSPACE Gap in the Satisfiability Problem for Modal Logic , 2006, IJCAI.

[23]  Heinrich Wansing,et al.  Sequent Systems for Modal Logics , 2002 .

[24]  Kazuo Matsumoto,et al.  Gentzen method in modal calculi. II , 1957 .

[25]  Melvin Fitting,et al.  Prefixed tableaus and nested sequents , 2012, Ann. Pure Appl. Log..

[26]  Dirk Pattinson,et al.  Correspondence between Modal Hilbert Axioms and Sequent Rules with an Application to S5 , 2013, TABLEAUX.

[27]  Heinrich Wansing,et al.  Sequent Calculi for Normal Modal Proposisional Logics , 1994, J. Log. Comput..

[28]  Melvin Fitting,et al.  Tableau methods of proof for modal logics , 1972, Notre Dame J. Formal Log..

[29]  G. Gentzen.  Untersuchungen über das logische Schließen. I , 1935 .

[30]  Kai Br.  Deep Sequent Systems for Modal Logic , 2006 .

[31]  T. Braüner.  Hybrid Logic and its Proof-Theory , 2010 .

[32]  Richard E. Ladner,et al.  The Computational Complexity of Provability in Systems of Modal Propositional Logic , 1977, SIAM J. Comput..

[33]  A. Avron.  The method of hypersequents in the proof theory of propositional non-classical logics , 1996 .

[34]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[35]  Torben Braüner,et al.  First-order modal logic , 2007, Handbook of Modal Logic.