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## Computational interpretations of classical linear logic (2007)

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Venue: | LNCS |

Citations: | 4 - 3 self |

### Citations

803 | Linear logic
- Girard
- 1987
(Show Context)
Citation Context ...alizability see chapter III of [20] or the book chapter [21]. Background on Gödel’s Dialectica interpretation can be obtained in [1]. For an introduction to linear logic see Girard’s original papers =-=[9,10]-=-. 1.1 Classical Linear Logic LLω We work with an extension of classical linear logic to the language of all finite types. The set of finite types T is inductively defined as follows: – o ∈ T ; – if ρ,... |

185 |
Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Dialectica 12
- Gödel
- 1958
(Show Context)
Citation Context ...near logic based on one-move two-player (Eloise and Abelard) games. As we will see, these are related to functional interpretations of intuitionistic logic such as Gödel’s Dialectica interpretations =-=[11]-=- and Kreisel’s modified realizability [14]. The intuition behind the interpretation is that each formula A defines an adjudication relation between arguments pro (Eloise’s move) and against (Abelard’s... |

153 |
Metamathematical Investigation of Intuitionistic Arithmetic and Analysis
- Troelstra
- 1973
(Show Context)
Citation Context .... Leivant and R. de Queiroz. (Eds.): WoLLIC 2007, LNCS 4576, pp. 285–296, 2007. c© Springer-Verlag Berlin Heidelberg 2007 286 P. Oliva For an introduction to modified realizability see chapter III of =-=[20]-=- or the book chapter [21]. Background on Gödel’s Dialectica interpretation can be obtained in [1]. For an introduction to linear logic see Girard’s original papers [9,10]. 1.1 Classical Linear Logic ... |

105 |
Interpretation of analysis by means of constructive functionals of finite types
- Kreisel
- 1959
(Show Context)
Citation Context ...loise and Abelard) games. As we will see, these are related to functional interpretations of intuitionistic logic such as Gödel’s Dialectica interpretations [11] and Kreisel’s modified realizability =-=[14]-=-. The intuition behind the interpretation is that each formula A defines an adjudication relation between arguments pro (Eloise’s move) and against (Abelard’s move) the truth of A. If the formula is i... |

95 | Gödel’s functional (‘Dialectica’) interpretation
- Avigad, Feferman
- 1998
(Show Context)
Citation Context ...Berlin Heidelberg 2007 286 P. Oliva For an introduction to modified realizability see chapter III of [20] or the book chapter [21]. Background on Gödel’s Dialectica interpretation can be obtained in =-=[1]-=-. For an introduction to linear logic see Girard’s original papers [9,10]. 1.1 Classical Linear Logic LLω We work with an extension of classical linear logic to the language of all finite types. The s... |

48 |
Strong majorizable functionals of finite type: a model for bar recursion containing discontinuous functionals
- Bezem
- 1985
(Show Context)
Citation Context ...s developed in order to deal with strong analytical principles in classical feasible analysis. The interpretation makes use of Howard-Bezem’s strong majorizabilty relation ≤∗ between functionals (cf. =-=[2]-=-). Using the majorizability relation, we can define well-behaved bounded sets which can be used as moves in the treatment of the exponential games. More precisely, the interpretation of the exponentia... |

32 | A dialectica-like model of linear logic
- Paiva
- 1989
(Show Context)
Citation Context ...nown functional interpretations of intuitionistic logic, including Gödel’s Dialectica interpretation and Kreisel’s modified realizability. 1 Introduction This article surveys several interpretations =-=[3,16,17,18]-=- of classical linear logic based on one-move two-player (Eloise and Abelard) games. As we will see, these are related to functional interpretations of intuitionistic logic such as Gödel’s Dialectica ... |

31 | Henkin quantifiers and complete problems
- Blass, Gurevich
- 1986
(Show Context)
Citation Context ...ple tensor as a Henkin quantifier is a common feature of a number of interpretations of linear logic”. The simultaneous quantifier can be viewed as a simplification of Henkin’s (branching) quantifier =-=[4,12]-=-, in which no alternation of quantifiers is allowed on the two branches. See Bradfield [5] as well, where this simple form of branching quantifier is also used. 290 P. Oliva Theorem 2. Extend the inte... |

22 |
Some remarks on infinitely long formulas, in: Infinitistic Methods
- Henkin
- 1959
(Show Context)
Citation Context ...ple tensor as a Henkin quantifier is a common feature of a number of interpretations of linear logic”. The simultaneous quantifier can be viewed as a simplification of Henkin’s (branching) quantifier =-=[4,12]-=-, in which no alternation of quantifiers is allowed on the two branches. See Bradfield [5] as well, where this simple form of branching quantifier is also used. 290 P. Oliva Theorem 2. Extend the inte... |

21 |
Eine Variante zur Dialectica Interpretation der Heyting Arithmetik endlicher Typen. Archiv für Mathematische Logik und Grundlagenforschung
- Diller, Nahm
- 1974
(Show Context)
Citation Context ...ing system in order to deal with finite sets of arbitrary type. This choice for the treatment of the exponentials corresponds to a variant of Gödel’s Dialectica interpretation due to Diller and Nahm =-=[6]-=-. 4.3 Interpretation 3: Stein’s Interpretation A hybrid interpretation between options 1 and 2 can also be given for each parameter n ∈ N. The natural number n dictates from which type level we should... |

20 | Bounded functional interpretation
- Ferreira, Oliva
(Show Context)
Citation Context ...is choice corresponds to Stein’s interpretation [19], and again leads to a sound interpretation of full classical linear logic. 294 P. Oliva 4.4 Interpretation 4: Bounded Dialectica Interpretation In =-=[7,8]-=-, a “bounded” variant of Gödel’s Dialectica interpretation was developed in order to deal with strong analytical principles in classical feasible analysis. The interpretation makes use of Howard-Beze... |

13 | Independence: logics and concurrency
- Bradfield
- 2000
(Show Context)
Citation Context ...r logic”. The simultaneous quantifier can be viewed as a simplification of Henkin’s (branching) quantifier [4,12], in which no alternation of quantifiers is allowed on the two branches. See Bradfield =-=[5]-=- as well, where this simple form of branching quantifier is also used. 290 P. Oliva Theorem 2. Extend the interpretation (Definition 1) to the system LLωq by defining | ÆvwA(v,w)|f ,vg,w :≡ |A(v,w)|fw... |

13 | Proof theory in the abstract
- Hyland
(Show Context)
Citation Context ...s ( Æx y A)⊥ ≡ Æy x A ⊥ and corresponds precisely to the switch of roles between the players. Let us refer to the extension of LLω with the new simultaneous quantifier by LLωq . 2 According to Hyland =-=[13]-=- (footnote 18) “the identification of a sufficiently simple tensor as a Henkin quantifier is a common feature of a number of interpretations of linear logic”. The simultaneous quantifier can be viewed... |

11 | Questions and answers – a category arising in linear logic, complexity theory, and set theory
- Blass
(Show Context)
Citation Context ...nown functional interpretations of intuitionistic logic, including Gödel’s Dialectica interpretation and Kreisel’s modified realizability. 1 Introduction This article surveys several interpretations =-=[3,16,17,18]-=- of classical linear logic based on one-move two-player (Eloise and Abelard) games. As we will see, these are related to functional interpretations of intuitionistic logic such as Gödel’s Dialectica ... |

7 |
Unifying functional interpretations, Notre Dame Journal of Formal Logic (2006), to appear, downloadable from the author’s Web
- Oliva
(Show Context)
Citation Context ...be used as moves in the treatment of the exponential games. More precisely, the interpretation of the exponentials can also be given as: |!A|xf :≡ !∀y≤∗fx |A|xy |?A|fy :≡ ?∃x≤∗fy |A|xy . As argued in =-=[15]-=-, in this case we must first perform a relativisation of the quantifiers to Bezem’s modelM of strongly majorizable functionals. After that, all candidate witnesses and challenges are monotone, and the... |

7 |
Interpretation der Heyting-Arithmetik endlicher Typen
- Stein
- 1976
(Show Context)
Citation Context ...ρ)A[y] and ∃y∈ rng(b(n−1)→ρ)A[y] are used as an abbreviation for ∀in−1A[bi] and ∃in−1A[bi], respectively (n − 1 is the pure type of type level n− 1). This choice corresponds to Stein’s interpretation =-=[19]-=-, and again leads to a sound interpretation of full classical linear logic. 294 P. Oliva 4.4 Interpretation 4: Bounded Dialectica Interpretation In [7,8], a “bounded” variant of Gödel’s Dialectica in... |

6 | Bounded functional interpretation and feasible analysis
- Ferreira, Oliva
(Show Context)
Citation Context ...is choice corresponds to Stein’s interpretation [19], and again leads to a sound interpretation of full classical linear logic. 294 P. Oliva 4.4 Interpretation 4: Bounded Dialectica Interpretation In =-=[7,8]-=-, a “bounded” variant of Gödel’s Dialectica interpretation was developed in order to deal with strong analytical principles in classical feasible analysis. The interpretation makes use of Howard-Beze... |

5 | Modified realizability interpretation of classical linear logic
- Oliva
- 2007
(Show Context)
Citation Context ...nown functional interpretations of intuitionistic logic, including Gödel’s Dialectica interpretation and Kreisel’s modified realizability. 1 Introduction This article surveys several interpretations =-=[3,16,17,18]-=- of classical linear logic based on one-move two-player (Eloise and Abelard) games. As we will see, these are related to functional interpretations of intuitionistic logic such as Gödel’s Dialectica ... |

4 | The Dialectica interpretation of first-order classical affine logic
- Shirahata
(Show Context)
Citation Context |

2 |
Towards a geomery of interaction
- Girard
- 1989
(Show Context)
Citation Context ...alizability see chapter III of [20] or the book chapter [21]. Background on Gödel’s Dialectica interpretation can be obtained in [1]. For an introduction to linear logic see Girard’s original papers =-=[9,10]-=-. 1.1 Classical Linear Logic LLω We work with an extension of classical linear logic to the language of all finite types. The set of finite types T is inductively defined as follows: – o ∈ T ; – if ρ,... |

1 |
Handbook of proof theory 137
- Troelstra
- 1998
(Show Context)
Citation Context ...oz. (Eds.): WoLLIC 2007, LNCS 4576, pp. 285–296, 2007. c© Springer-Verlag Berlin Heidelberg 2007 286 P. Oliva For an introduction to modified realizability see chapter III of [20] or the book chapter =-=[21]-=-. Background on Gödel’s Dialectica interpretation can be obtained in [1]. For an introduction to linear logic see Girard’s original papers [9,10]. 1.1 Classical Linear Logic LLω We work with an exten... |