#### DMCA

## Cut-elimination and proof-search for bi-intuitionistic logic using nested sequents (2008)

Citations: | 15 - 4 self |

### Citations

211 | A modal analysis of staged computation
- Davies, Pfenning
(Show Context)
Citation Context ...lly uses Hilbert calculi with algebraic, topological or relational semantics. We omit details since our interest is primarily proof-theoretic. Sequent and natural deduction calculi for IMLs are rarer =-=[13,1,16,3,5,11,7]-=-. Extending them with “converse” modalities like and causes cut-elimination to fail as it does for classical modal logic S5 where ♦ is a self-converse. Labels [14,23,15] can help but are not purel... |

203 | Basic Proof Theory - Troelstra, Schwichtenberg - 2000 |

182 | An Introduction to Substructural Logics - Restall - 2000 |

133 | Display logic - Belnap - 1982 |

128 | The Proof Theory and Semantics of Intuitionistic Modal Logic
- SIMPSON
- 1994
(Show Context)
Citation Context ...quent calculus with cut-elimination [9] and a labelled sequent calculus [17] with cut-free-completeness have been found for BiInt. The literature on Intuitionistic Modal/Tense Logics (IM/TLs) is vast =-=[6,23]-=- and typically uses Hilbert calculi with algebraic, topological or relational semantics. We omit details since our interest is primarily proof-theoretic. Sequent and natural deduction calculi for IMLs... |

96 | Eine Interpretation des intuitionistischen Aussagenkalkuls - Gödel - 1933 |

80 | Intuitionistic propositional logic is polynomial-space complete - Statman - 1979 |

65 | The method of hypersequents in the proof theory of propositional nonclassical logics - Avron - 1996 |

56 | A symmetric modal lambda calculus for distributed computing (extended technical report
- Murphy, Crary, et al.
- 2004
(Show Context)
Citation Context ...for IMLs are rarer [13,1,16,3,5,11,7]. Extending them with “converse” modalities like and causes cut-elimination to fail as it does for classical modal logic S5 where ♦ is a self-converse. Labels =-=[14,23,15]-=- can help but are not purely proof-theoretic since they encode the Kripke semantics. The closest to our work is that of Sadrzadeh and Dyckhoff [22] who give a cut-free sequent calculus using deep infe... |

49 | Substructural logics on display - Goré - 1997 |

41 | Deep sequent systems for modal logic - Brünnler - 2006 |

39 | Efficient loopcheck for backward proof search in some non-classical propositional logics - Heuerding, Seyfried, et al. - 1996 |

34 | Atomic cut elimination for classical logic - Brünnler - 2003 |

33 | A formulae-as-types interpretation of subtractive logic
- Crolard
(Show Context)
Citation Context ...Rauszer [21] obtained BiInt by extending Int with a binary connective −< called “exclusion” which is adjoint to ∨ in that A→ (B ∨C) is valid iff (A −< B)→ C is valid iff (A −< C)→ B is valid. Crolard =-=[4]-=- showed that BiInt has a computational interpretation in terms of continuation passing style semantics. Uustalu and Pinto recently showed that Rauszer’s sequent calculus [20] and Crolard’s extensions ... |

33 | Sequent calculi for normal modal propositional logics - Wansing - 1994 |

26 | Subtractive logic - Crolard - 2001 |

26 |
Cut-free sequent calculi for some tense logics
- Kashima
- 1994
(Show Context)
Citation Context ...,” which binds tighter than “.”. Thus, we write •X,Y . Z to mean (•(X), Y ) . Z. A nested sequent is a structure of the form X.Y . This notion of nested sequents generalises Kashima’s nested sequents =-=[12]-=- for classical tense logics, Brünnler’s nested sequents [2] and Poggiolesi’s tree-hypersequents [18] for classical modal logics. Figure 1 shows the formula-translation of nested sequents. On both sid... |

26 |
Intuitionistic tense and modal logic
- EWALD
- 1986
(Show Context)
Citation Context ...quent calculus with cut-elimination [9] and a labelled sequent calculus [17] with cut-free-completeness have been found for BiInt. The literature on Intuitionistic Modal/Tense Logics (IM/TLs) is vast =-=[6,23]-=- and typically uses Hilbert calculi with algebraic, topological or relational semantics. We omit details since our interest is primarily proof-theoretic. Sequent and natural deduction calculi for IMLs... |

24 | Power and weakness of the modal display calculus - Kracht - 1996 |

19 | Hypersequent Calculi for Gödel Logics - a Survey - Baaz, Ciabattoni, et al. - 2003 |

19 |
An algebraic and Kripke-style approach to a certain extension of intuitionistic logic
- Rauszer
- 1980
(Show Context)
Citation Context ...ng derivations automatically. In traditional intuitionistic logic, the connectives → and ∧ form an adjoint pair in that (A ∧ B) → C is valid iff A → (B → C) is valid iff B → (A → C) is valid. Rauszer =-=[21]-=- obtained BiInt by extending Int with a binary connective −< called “exclusion” which is adjoint to ∨ in that A→ (B ∨C) is valid iff (A −< B)→ C is valid iff (A −< C)→ B is valid. Crolard [4] showed t... |

18 | The complexity of propositional tense logics - Spaan - 1993 |

14 | Symmetries in natural language syntax and semantics: the Lambek-Grishin calculus - Moortgat - 2007 |

13 | Dual Intuitionistic Logic Revisited - Goré |

12 | A formalization of the propositional calculus of H-B logic - Rauszer - 1974 |

12 |
On some calculi of modal logic
- Mints
- 1971
(Show Context)
Citation Context ...for IMLs are rarer [13,1,16,3,5,11,7]. Extending them with “converse” modalities like and causes cut-elimination to fail as it does for classical modal logic S5 where ♦ is a self-converse. Labels =-=[14,23,15]-=- can help but are not purely proof-theoretic since they encode the Kripke semantics. The closest to our work is that of Sadrzadeh and Dyckhoff [22] who give a cut-free sequent calculus using deep infe... |

11 | Hypersequent Calculi for some Intermediate Logics with Bounded Kripke Models - Ciabattoni, Ferrari - 2001 |

10 | A uniform tableau method for intuitionistic modal logics i
- Amati, Pirri
- 1994
(Show Context)
Citation Context ...lly uses Hilbert calculi with algebraic, topological or relational semantics. We omit details since our interest is primarily proof-theoretic. Sequent and natural deduction calculi for IMLs are rarer =-=[13,1,16,3,5,11,7]-=-. Extending them with “converse” modalities like and causes cut-elimination to fail as it does for classical modal logic S5 where ♦ is a self-converse. Labels [14,23,15] can help but are not purel... |

10 | Taming displayed tense logics using nested sequents with deep inference
- Goré, Postniece, et al.
(Show Context)
Citation Context ...t Σ which itself contains a formula A and a •-structure, such that the •-structure contains A. DBiKt achieves the goal of merging the DBiInt calculus [19] and a two-sided version of the DKt calculus =-=[10]-=-. While in the shallow inference case, a calculus for BiKt could be obtained relatively easily by merging shallow inference calculi for BiInt and tense logics, the combination of calculi is not so obv... |

10 |
On a modal lambda-calculus for S4
- Pfenning, Wong
- 1995
(Show Context)
Citation Context ...lly uses Hilbert calculi with algebraic, topological or relational semantics. We omit details since our interest is primarily proof-theoretic. Sequent and natural deduction calculi for IMLs are rarer =-=[13,1,16,3,5,11,7]-=-. Extending them with “converse” modalities like and causes cut-elimination to fail as it does for classical modal logic S5 where ♦ is a self-converse. Labels [14,23,15] can help but are not purel... |

9 |
2-sequent calculus: Intuitionism and natural deduction
- Masini
- 1993
(Show Context)
Citation Context |

9 | Positive logic with adjoint modalities: Proof theory, semantics and reasoning about information
- Sadrzadeh, Dyckhoff
(Show Context)
Citation Context ...al logic S5 where ♦ is a self-converse. Labels [14,23,15] can help but are not purely proof-theoretic since they encode the Kripke semantics. The closest to our work is that of Sadrzadeh and Dyckhoff =-=[22]-=- who give a cut-free sequent calculus using deep inference for a logic with an adjoint pair of modalities (,) plus only ∧, ∨, > and ⊥. As all their connectives are “monotonic”, cutelimination presen... |

8 | Sequent-Systems for modal logic - Doˇsen - 1985 |

8 | Gödel’s Collected Works - Feferman, ed - 1980 |

8 | Modular sequent systems for modal logic
- Brünnler, Straßburger
(Show Context)
Citation Context ...ean (•(X), Y ) . Z. A nested sequent is a structure of the form X.Y . This notion of nested sequents generalises Kashima’s nested sequents [12] for classical tense logics, Brünnler’s nested sequents =-=[2]-=- and Poggiolesi’s tree-hypersequents [18] for classical modal logics. Figure 1 shows the formula-translation of nested sequents. On both sides of the sequent, ◦ is interpreted as a white (modal) opera... |

8 | Proof search and counter-model construction for bi-intuitionistic propositional logic with labelled sequents
- Pinto, Uustalu
- 2009
(Show Context)
Citation Context ...at Rauszer’s sequent calculus [20] and Crolard’s extensions of it Goré, Postniece and Tiu fail cut-elimination, but a nested sequent calculus with cut-elimination [9] and a labelled sequent calculus =-=[17]-=- with cut-free-completeness have been found for BiInt. The literature on Intuitionistic Modal/Tense Logics (IM/TLs) is vast [6,23] and typically uses Hilbert calculi with algebraic, topological or rel... |

7 | L.: Combining derivations and refutations for cut-free completeness in biintuitionistic logic - Goré, Postniece - 2010 |

6 | A cut-free sequent calculus for bi-intuitionistic logic - Buisman, Goré - 2007 |

6 |
The tree-hypersequent method for modal propositional logic
- Poggiolesi
- 2009
(Show Context)
Citation Context ...a structure of the form X.Y . This notion of nested sequents generalises Kashima’s nested sequents [12] for classical tense logics, Brünnler’s nested sequents [2] and Poggiolesi’s tree-hypersequents =-=[18]-=- for classical modal logics. Figure 1 shows the formula-translation of nested sequents. On both sides of the sequent, ◦ is interpreted as a white (modal) operator and • as a black (tense) operator. No... |

5 | Deep inference in bi-intuitionistic logic
- Postniece
- 2009
(Show Context)
Citation Context ...iKt is the structural contraction rules, which allow contraction on arbitrary structures, not just formulae as in traditional sequent calculi. LBiKt is as a merger of two calculi: the LBiInt calculus =-=[9,19]-=- for the intuitionistic connectives, and the display calculus [8] for the tense connectives. Note that we use ◦ and • as structural proxies for the non-residuated pairs (♦,) and (,) respectively, w... |

4 | Multiple sequent calculus for tense logics - Indrzejczak - 2000 |

4 | Cut-free double sequent calculus for S5 - Indrzejczak - 1998 |

3 | Gentzen-style axiomatization of tense logic - Trzesicki - 1984 |

3 | Days in logic ’06 conference abstract. Online at http://www.mat.uc.pt/ ∼ kahle/dl06/tarmo-uustalu.pdf, accessed on 27th - Uustalu, Pinto - 2006 |

2 | Modal tableaux based on residuation - Wansing - 1997 |

2 |
Semantics and proof theory of an intuitionistic modal sequent calculus
- Collinson, Hilken, et al.
- 1999
(Show Context)
Citation Context |

2 | Calculi for intuitionistic normal modal logic - Kakutani - 2007 |

1 | Constructive negation, implication and co-implication. Manuscript, March:to appear, 2007. Rajeev Goré - Wansing |

1 | Calculi for an intuitionistic hybrid modal logic - Galmiche, Salhi - 2008 |

1 |
Substructural logics on display. Log
- Goré
- 1998
(Show Context)
Citation Context ...arbitrary structures, not just formulae as in traditional sequent calculi. LBiKt is as a merger of two calculi: the LBiInt calculus [9,19] for the intuitionistic connectives, and the display calculus =-=[8]-=- for the tense connectives. Note that we use ◦ and • as structural proxies for the non-residuated pairs (♦,) and (,) respectively, whereas Wansing [24] uses only one • as a structural proxy for the... |